Sample size variation in single-time post-dose assessment <i>vs</i> multi-time post-dose assessment

Sarfaraz Sayyed; Ashwini Mathur; Asha Kamath

doi:10.12688/f1000research.124917.3

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Method Article

Revised

Sample size variation in single-time post-dose assessment vs multi-time post-dose assessment

[version 3; peer review: 1 approved, 2 not approved]

Sarfaraz Sayyed ¹, Ashwini Mathur², Asha Kamath³

PUBLISHED 05 Feb 2026

Author details Author details

¹ Analytics, Novartis Healthcare Pvt. Ltd., Hyderabad, Telangana, 500080, India
² Clinical Technology & Innovation, Novartis Ireland Ltd., Dublin, Ireland
³ Department of Data Science, Manipal Academy of Higher Studies, Manipal, Karnataka, 576104, India

Sarfaraz Sayyed
Roles: Formal Analysis, Investigation, Visualization, Writing – Original Draft Preparation

Ashwini Mathur
Roles: Conceptualization, Supervision, Writing – Review & Editing

Asha Kamath
Roles: Conceptualization, Supervision, Writing – Review & Editing

OPEN PEER REVIEW

REVIEWER STATUS

This article is included in the Manipal Academy of Higher Education gateway.

Abstract

Background

Many randomized trials measure a continuous outcome simultaneously at baseline and after taking the drug. For a single continuous post-treatment outcome, the sample size calculation is simple, but if there are assessments at multiple time point post-treatment then this longitudinal data may give more insights by analyzing the data using the repeated measures method. Also, if the sample size is calculated using the single time-point method for longitudinal data, it may lead to a larger than required sample size, increasing the cost and time.

Methods

In this research, an effort is made to determine the size of the sample for repeated measures case and then compare with the single post-baseline case. The sample sizes were examined under different scenarios for the continuous type of response variable. Under ‘Mean contrast’ and ‘Diff contrast’ the sample sizes were calculated with different correlations. These two scenarios were again examined under compound symmetry as well as discrete Auto regressive of order 1 type of correlation structure in longitudinal data. The graphical presentation is given for better visualization of the scenarios.

Results

The sample size required for highly correlated longitudinal data using the multi-timepoint sample size derivation method led to a smaller sample size requirement compared to the single timepoint sample size calculation method.

Conclusions

This study will help researchers to make better decisions in choosing the right method for sample size determination which may reduce the time and cost of conducting the experiment. Additionally, it is crucial to carefully evaluate and choose the appropriate method when the correlation is weak, as this can significantly impact the accuracy of the results. More complex correlation structures are not studied in this article but can be studied in the same fashion.

Keywords

sample size estimation; longitudinal study; repeated measure analysis; univariate analysis

Corresponding author: Sarfaraz Sayyed

Competing interests: The author (Sarfaraz Sayyed) is employed by Novartis Healthcare Pvt. Ltd.

Grant information: The author(s) declared that no grants were involved in supporting this work.

Copyright: © 2026 Sayyed S et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: Sayyed S, Mathur A and Kamath A. Sample size variation in single-time post-dose assessment vs multi-time post-dose assessment [version 3; peer review: 1 approved, 2 not approved]. F1000Research 2026, 11:1550 (https://doi.org/10.12688/f1000research.124917.3) First published: 21 Dec 2022, 11:1550 (https://doi.org/10.12688/f1000research.124917.1) Latest published: 05 Feb 2026, 11:1550 (https://doi.org/10.12688/f1000research.124917.3)

Revised Amendments from Version 2

The introduction and background section has been updated with respect to adding the importance of precision and some spell correction. The ‘Notation and Framework’ section has been updated with the normal approximation assumption. The notation for null and alternative hypothesis as well as notation for sample estimates has been updated under the methods sections. More clarification has been added to the ‘Calculation of sample size section’ regarding timepoints, effect used and the derived sample size. Few appendices have been added to explain the theoretically phenomenon that was observed in the comparison of sample sizes with different correlation and different contrast types. We have also added references to a book and R packages for further exploration on power analysis in longitudinal studies.

See the authors' detailed response to the review by Kiranmoy Das
See the authors' detailed response to the review by Ronald Geskus
See the authors' detailed response to the review by Simon Vandekar

Introduction

Understanding the concept of “sample size” is crucial for anyone involved in scientific research or clinical trials. The sample size refers to the number of subjects selected or observed in an experiment. This sample is a subset of the entire target population, which includes all individuals relevant to the study. For instance, in a study evaluating a new drug for type II diabetes, the target population would consist of all individuals suffering from this condition.

The sample size significantly influences the precision of our estimates and the study’s power, which is probability of correctly rejecting the null hypothesis when it is false. Essentially, higher power means a greater chance of detecting a true effect.

Two primary factors impact the power of a study: the sample size and the effect size. A larger sample size generally increases the study’s power, enhancing our ability to draw accurate conclusions. Effect size, on the other hand, measures the magnitude of the difference or relationship being studied between the groups.

In clinical trials, carefully calculating the sample size is essential. Correctly calculated sample size ensures that the study is adequately powered to meet its objectives, providing reliable and meaningful results. This meticulous planning is fundamental to advancing medical knowledge and improving patient care.

By paying close attention to sample size, researchers can design robust experiments that yield trustworthy insights, contributing to scientific progress and better health outcomes.

As an illustration, consider a study to compare the performance of a professional athlete taking a particular protein shake versus athletes who do not consume any special protein shakes. Narrowing down attention to a portion of the wider group is essential to enable tracking of the eating habits of every elite athlete in the world. Suppose this entails choosing one hundred professional athletes for our study at random; in this case, one hundred would be the sample size. Based on the data gathered from a sample of one hundred elite athletes, the study’s findings potentially characterize the population of all athletes in the sports industry. Lack of full coverage of the target population would result in the study’s outcome having a margin of error. Sampling error¹ refers to the random variation between a sample estimate and the true population parameter due to observing only a subset of the population. It reduces the precision of an estimator (e.g., widening confidence intervals), which is crucial for generalizing results to the target population (e.g., all professional athletes). Greater precision enhances confidence in results, enabling more accurate analysis and informed decision-making.

Although sampling error cannot be completely eliminated, it can be reduced.² A larger sample typically has a narrower margin of error. We require an appropriate sample size to examine and provide an accurate picture of the effects of protein shake consumption on performance. Note that increasing the sample size will help in reducing the sampling error but it does not address the non-sampling errors.

Background

Longitudinal studies typically require longer follow-up periods but offer important advantages, including the assessment of temporal trends and stronger support for causal inference. While numerous longitudinal studies are available in literature, the purpose of this section is not to provide a comprehensive review of such research. Rather, it aims to illustrate that even in well-established and prestigious journals, longitudinal studies often provide limited detail regarding sample size calculation.

To examine how sample size calculations are reported in applied longitudinal studies, we conducted a targeted search of major bibliographic databases, including ‘Scopus’, ‘Web of Science’, ‘PubMed’, ‘ScienceDirect’, and ‘Google Scholar’ using a range of key terms: “designing clinical trials”, “sample size calculation”, “longitudinal studies”, “randomized trials” and “repeated measures”. This targeted review indicated that detailed descriptions of sample size calculation were frequently limited or absent in the reported methods of published longitudinal studies, as illustrated by selected examples.³^,⁴ In contrast, methodological papers providing statistical formulae and frameworks for sample size determination in longitudinal designs are well documented in the literature.⁵^,⁶ Additional examples from applied studies,⁷^–¹¹ discussed in the following paragraph, further highlight this recurring shortfall in reporting practice.¹^–³

Basagana, Liao and Spiegelman⁷ published a study in which the power as well as the sample size are discussed for time-varying exposures, but how this is practically applied to a longitudinal study design and its outcome is undocumented in published papers. Pourhoseingholi et al.,⁸ and Karimollah⁹ both published about the importance of various components for calculating the sample size in medical studies or clinical trials where often there would be more than one post baseline assessment, but sample size calculation is shown assuming single post baseline assessment. Manja and Lakshminrusimha published a two-part study¹⁰^,¹¹ which does give a good explanation on clinical research design, but sample size is not discussed in detail.

Most of the published studies which have assessments at multiple time points calculate the sample size based on the change from study end time point to baseline whereas a smaller number of papers emphasize on the use of multiple time points into consideration for calculating the sample size.⁵^–⁷

The need for this research was prompted by this lack of proper usage of sample size calculation for longitudinal studies and to further explore which method for sample size calculation should be used in a longitudinal study resulting in correlated outcome data.

A wide range of methodologies and software tools for sample size and power calculation in longitudinal studies already exist¹²^,¹³ (e.g., Diggle et al.; Ahn et al.; [R package references]). The present work does not seek to replace these approaches, but rather to complement them by explicitly contrasting single-time-point and multiple-time-point designs under simple and commonly assumed correlation structures.

Objective

To explore the variation in sample size by considering multiple time point assessment versus the change from baseline to a single endpoint.

Notation and framework

Consider an experiment for testing certain hypothesis with parallel group design, having two independent groups to be treated under different scenarios to compare the outcome between the two groups. In our study we would consider the objective of comparison of two drugs.

Let $Y_{ij}$ ( $X_{ij}$ ) be the outcome of interest at $j^{th}$ (j = 1, 2, 3, … , t) time point for the $i^{th}$ (i = 1, 2, 3, … , n) patient in the two groups.

For the parallel group design, these 2n patients will be divided into two groups with 1: 1 ratio where one arm is assigned to receive the test drug, and the other arm is assigned to receive the comparator drug.

Let $μ_{1}$ & $μ_{2}$ be the population mean for the test drug and comparator drug, respectively.

Let $\bar{Y}$ & $\bar{X}$ be the sample mean outcome for the test drug and comparator drug respectively. The distribution of $\bar{Y}$ & $\bar{X}$ can be approximated as $N (μ_{1}, \frac{σ^{2}}{n})$ and $N (μ_{2}, \frac{σ^{2}}{n})$ respectively under common standard deviation assumption. Under standard regularity conditions, the sampling distribution of the sample mean $\bar{X}$ is approximately normal, with the accuracy of this approximation improving as the sample size increases, in accordance with the central limit theorem.¹⁴

Methods

Method for sample size with single time point assessment analysis

Change at single post-baseline assessment (single time assessment analysis)

In a parallel group design study with two arms of equal size let the hypothesis be set as:

$H_{0} : μ_{1} = μ_{2}$ , no difference between the effects of test drug and comparator drug.

$H_{a} : μ_{1} > μ_{2}$ , test drug effect is greater than comparator and $μ_{1} - μ_{2} =$ δ, δ > 0 is the expected difference.

The test statistic assuming known common standard deviation ‘ $σ$ ’ (estimated from previous clinical trial data with same molecule which could be phase1, phase2 trials for same indication or pivotal trials with same molecule for different indication) for both arms will be given by

(1)

T = \frac{\bar{Y} - \bar{X}}{\sqrt{Var (\bar{Y} - \bar{X})}} = \frac{\bar{Y} - \bar{X}}{\sqrt{\frac{σ^{2}}{n} + \frac{σ^{2}}{n}}} = \frac{\bar{Y} - \bar{X}}{\sqrt{\frac{2 σ^{2}}{n}}}

Now if $H_{0}$ is true ( $μ_{1} = μ_{2}$ ), then $T \sim N (0, 1)$ , else if $H_{a}$ is true (i.e., $μ_{1} > μ_{2}$ ) , then T will still follow Gaussian distribution but with a mean greater than zero.

If Type II error is denoted by $β$ then power will be simply $1 - β$ and power is the probability to reject $H_{0}$ when $H_{a}$ is true. In probability equation it could be written as

(2)

Pr [Reject H_{0} | H_{a} is true] = Pr [T > z_{1 - α} | δ] = 1 - β,

where

z_{1 - α}

is the threshold or the critical value which is

(1 - α)

quantile from Gaussian distribution and

α

is the type I error or the level of significance.

(3)

T \sim N (\frac{δ}{\sqrt{\frac{2 σ^{2}}{n}}}, 1), under H_{a} \to T - \frac{δ}{\sqrt{\frac{2 σ^{2}}{n}}} = Z \sim N (0, 1)

(4)

\begin{matrix} Power function = Pr [T > z_{1 - α}| δ] = Pr \{\frac{\bar{Y} - \bar{X} - δ}{\sqrt{\frac{2 σ^{2}}{n}}} > z_{1 - α} - \frac{δ}{\sqrt{\frac{2 σ^{2}}{n}}}\} \\ = 1 - Φ (z_{1 - α} - \sqrt{2 n} (\frac{δ}{2 σ})), where Φ (z) = Pr (Z \leq z) \end{matrix}

Now in any study we would be looking for an inequality as below.

(5)

\begin{matrix} 1 - Φ (z_{1 - α} - \sqrt{\frac{n}{2}} (\frac{δ}{σ})) \geq 1 - β \\ \to β \geq Φ (z_{1 - α} - \sqrt{\frac{n}{2}} (\frac{δ}{σ})) or z_{β} \geq z_{1 - α} - \sqrt{\frac{n}{2}} (\frac{δ}{σ}) \end{matrix}

Upon solving Equation 5, we get¹⁵

(6)

n \geq \frac{2 {(z_{1 - α} + z_{1 - β})}^{2} σ^{2}}{{(\bar{Y} - \bar{X})}^{2}} = \frac{2 {(z_{α} + z_{β})}^{2}}{{[(\bar{Y} - \bar{X}) / σ]}^{2}}

Here, n is the sample size required per arm. We will use these formula in calculating the sample size for single post baseline time point analysis.

Method for sample size with multiple time points assessment analysis

Post baseline assessment at multiple timepoints (multiple time points analysis)

In a parallel group design study with two arms of equal size and assessments taken at ‘t’ time points,

Let $C = [\begin{matrix} c_{1} \\ c_{2} \\ ⋮ \\ c_{i} \\ ⋮ \\ c_{t} \end{matrix}]$ be the contrast to be tested for the hypothesis and let $Λ = [\begin{matrix} μ_{11} - μ_{21} \\ μ_{12} - μ_{22} \\ ⋮ \\ μ_{1 i} - μ_{2 i} \\ ⋮ \\ μ_{1 t} - μ_{2 t} \end{matrix}]$

With the hypothesis to be tested as:

$H_{0} : ψ_{c} = 0$ , there is no difference between the effects of test and comparator drug.

$H_{a} : ψ_{c} > 0$ , test drug is having larger effect as compared to comparator.

Where $ψ_{c} = C^{'} Λ and$ $C^{'}$ is the transpose of $C .$

$μ_{1 i}$ & $μ_{2 i}$ are the mean effect in arm one and arm two at time point ‘i’ respectively in a study with ‘t’ time points. $c_{i}$ can take any value depending on the hypothesis we want to evaluate.

For example, if we want to see the difference between two drugs when t = 2, then $c_{1} = - 1 and c_{2} = 1$ and the resulting $ψ_{c}$ will be

(7)

\begin{matrix} ψ_{c} = C^{'} Λ = [- 1 1] . [\begin{matrix} μ_{11} - μ_{21} \\ μ_{12} - μ_{22} \end{matrix}] \\ = - (μ_{11} - μ_{21}) + μ_{12} - μ_{22} \\ = (μ_{12} - μ_{11}) - (μ_{22} - μ_{21}) \\ = effect in Drug 1 - effect in Drug 2 . \end{matrix}

The variance-covariance matrix (estimated from previous clinical trial data with same molecule which could be phase1, phase2 trials for same indication or pivotal trials with same molecule for different indication), assumed to be known and similar between the two arms is given by

[\begin{matrix} σ_{1}^{2} & σ_{12} & \dots & σ_{1 t} \\ σ_{21} & σ_{2}^{2} & ⋱ & σ_{2 t} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ σ_{i 1} & σ_{i 2} & ⋱ & σ_{it} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ σ_{t 1} & σ_{t 2} & \dots & σ_{t}^{2} \end{matrix}]

Where $σ_{i}^{2}$ is the variance at time point ‘i’ and $σ_{ij}$ represents the covariance between time point ‘i’ and ‘j’.

The test statistic is given by

(8)

T = \frac{{\hat{ψ}}_{c}}{\sqrt{Var ({\hat{ψ}}_{c})}} = \frac{{\hat{ψ}}_{c}}{\sqrt{Var (C^{'} \hat{Λ})}} = \frac{{\hat{ψ}}_{c}}{\sqrt{C^{'} . Var (\hat{Λ}) . C}}

Where, ${\hat{ψ}}_{c}$ and $\hat{Λ}$ denote the sample-based estimators of the corresponding population quantities, such that the resulting expression constitutes a valid test statistic such that

{\hat{ψ}}_{c} = C^{'} \hat{Λ} and \hat{Λ} = [\begin{matrix} {\bar{Y}}_{1} - {\bar{X}}_{1} \\ {\bar{Y}}_{2} - {\bar{X}}_{2} \\ ⋮ \\ {\bar{Y}}_{i} - {\bar{X}}_{i} \\ ⋮ \\ {\bar{Y}}_{t} - {\bar{X}}_{t} \end{matrix}]

with ${\bar{Y}}_{i}$ & ${\bar{X}}_{i}$ being the sample mean at time ‘i’ in both the arms respectively.

Consider the $Var (\hat{Λ})$ ,

(9)

Var (\hat{Λ}) = Var [\begin{matrix} {\bar{Y}}_{1} - {\bar{X}}_{1} \\ {\bar{Y}}_{2} - {\bar{X}}_{2} \\ ⋮ \\ {\bar{Y}}_{i} - {\bar{X}}_{i} \\ ⋮ \\ {\bar{Y}}_{t} - {\bar{X}}_{t} \end{matrix}] = [\begin{matrix} \frac{2}{n} . σ_{1}^{2} & \frac{2}{n} . σ_{12} & \dots & \frac{2}{n} . σ_{1 t} \\ \frac{2}{n} . σ_{21} & \frac{2}{n} . σ_{2}^{2} & ⋱ & \frac{2}{n} . σ_{2 t} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ \frac{2}{n} . σ_{i 1} & \frac{2}{n} . σ_{i 2} & ⋱ & \frac{2}{n} . σ_{it} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ \frac{2}{n} . σ_{t 1} & \frac{2}{n} . σ_{t 2} & \dots & \frac{2}{n} . σ_{t}^{2} \end{matrix}] = \frac{2}{n} . [\begin{matrix} σ_{1}^{2} & σ_{12} & \dots & σ_{1 t} \\ σ_{21} & σ_{2}^{2} & ⋱ & σ_{2 t} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ σ_{i 1} & σ_{i 2} & ⋱ & σ_{it} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ σ_{t 1} & σ_{t 2} & \dots & σ_{t}^{2} \end{matrix}]

Solving for $C^{'} . Var (\hat{Λ}) . C$ using Equation 9, we get

(10)

C^{'} . Var (\hat{Λ}) . C = [\begin{matrix} c_{1} c_{2} \dots c_{i} \dots c_{t} \end{matrix}] . \frac{2}{n} . [\begin{matrix} σ_{1}^{2} & σ_{12} & \dots & σ_{1 t} \\ σ_{21} & σ_{2}^{2} & ⋱ & σ_{2 t} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ σ_{i 1} & σ_{i 2} & ⋱ & σ_{it} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ σ_{t 1} & σ_{t 2} & \dots & σ_{t}^{2} \end{matrix}] . [\begin{matrix} c_{1} \\ c_{2} \\ ⋮ \\ c_{i} \\ ⋮ \\ c_{t} \end{matrix}] = \frac{2}{n} . [\sum_{i = 1}^{t} c_{i}^{2} σ_{i}^{2} + 2 \sum_{i < j}^{t} c_{i} c_{j} σ_{ij}] = \frac{2}{n} . σ_{c}^{2}, where σ_{c}^{2} = [\sum_{i = 1}^{t} c_{i}^{2} σ_{i}^{2} + 2 \sum_{i < j}^{t} c_{i} c_{j} σ_{ij}]

Using Equation 10 in Equation 8 for T we get

(11)

T = \frac{{\hat{ψ}}_{c}}{\sqrt{C^{'} . Var (\hat{Λ}) . C}} = \frac{{\hat{ψ}}_{c}}{\sqrt{\frac{2}{n} . [\sum_{i = 1}^{n} c_{i}^{2} σ_{i}^{2} + 2 \sum_{i < j}^{n} c_{i} c_{j} σ_{ij}]}} = \sqrt{\frac{n}{2}} (\frac{{\hat{ψ}}_{c}}{\sqrt{σ_{c}^{2}}})

Now, if we follow similar steps as we did in single time point analysis above, we get the following inequality.

(12)

β \geq Φ (z_{1 - α} - \sqrt{\frac{n}{2}} (\frac{{\hat{ψ}}_{c}}{\sqrt{σ_{c}^{2}}})) or z_{β} \geq z_{1 - α} - \sqrt{\frac{n}{2}} (\frac{{\hat{ψ}}_{c}}{\sqrt{σ_{c}^{2}}})

And solving Equation 12, we get¹²

(13)

\begin{matrix} n \geq \frac{2 {(z_{α} + z_{β})}^{2} σ_{c}^{2}}{{\hat{ψ}}_{c}^{2}} \\ with \\ {\hat{ψ}}_{c} = \sum_{i = 1}^{t} c_{i} ({\bar{Y}}_{i} - {\bar{X}}_{i}) and σ_{c}^{2} = [\sum_{i = 1}^{t} c_{i}^{2} σ_{i}^{2} + 2 \sum_{i < j}^{t} c_{i} c_{j} σ_{ij}] \end{matrix}

$σ_{i}^{2}$ = common variance(assumed known)in the two groups at timepoint ‘i’

$σ_{i j}$ = common covariance(assumed known) in the two groups between timepoint ‘i’ and ‘j’.

$c_{i}$ = contrast applied at timepoint ‘i’ and ‘t’ represents the number of time points.

We will use the formulae specified in Equation 13 to calculate the sample size for multiple time point analysis.

Calculation of sample size

Consider a 3-year study with assessments at baseline, 3 months, 6 months, 9 months, 12 months, 15 months, 18 months, 24 months, 30 months, 36 months. Appropriate sample size was calculated for multi-timepoint and single timepoint cases with different scenarios to achieve a difference of 0.9 points at 36 months between two treatment groups with an increasing trend from baseline. The different scenarios considered were with respect to equally spaced time points. Below table shows the timepoints selected for each scenario under longitudinal data sample size calculation. For single timepoint assessment only the baseline and 36 months were used.

Table 1. Scenarios with timepoints selected.

Scenarios	Visits considered
3 timepoints	Baseline, 18m, 36m
4 timepoints	Baseline, 12m, 24m, 36m
5 timepoints	Baseline, 9m, 18m, 24m, 36m
6 timepoints	Baseline, 6m, 15m, 24m, 30m, 36m
8 timepoints	Baseline, 6m, 9m, 12m, 15m, 18m, 24m, 36m
10 timepoints	Baseline, 3m, 6m, 9m, 12m, 15m, 18m, 24m, 30m, 36m

A common standard deviation (SD) of 3.6 points was used; with type I error rate chosen as 5% and the power set at 85%. The effect size and standard deviation used here are observed in a real study.¹⁶ This was a three-year study with primary endpoint assessment at the end of year 3, but the sample size calculation in this study was done based on single time point. Since this study failed to recruit the expected number of patients and had a lot of missing data, the characteristics up until the second year’s evaluation were utilized because they featured a balanced number of patients in both treatment groups and had consistent measurements.

Sample size (single time point case)

We considered a two-arm parallel group scenario with assessment at baseline and 36 months to assess the change from baseline in absolute scale. Using the formulae in Equation 6 above for single timepoint analysis to test the hypothesis, $H_{0} : μ_{1} = μ_{2}$ vs $H_{a} : μ_{1} > μ_{2} (where μ_{1} - μ_{2} = 0.9,$ which is the expected difference), the sample size required per arm was 287 cases to achieve a difference of 0.9 points at the end of 36 months between two treatment groups with a common standard deviation (SD) of 3.6 points choosing type I error rate as 5% and the power set at 85% to show statistical significance.

Sample size (longitudinal case)

Here again we considered two arm parallel groups with multiple timepoints and for studying we investigated six cases i.e., three, four, five, six, eight, and ten timepoints. Each of these cases correspond to multiple assessments including baseline. Three timepoints corresponded to the case with one baseline and two post baseline assessments, four timepoints corresponded to the case with one baseline and three post baseline assessments, five timepoints corresponded to the case with one baseline and four post baseline assessments, similarly other cases corresponded to one visit of baseline and remaining visits from pot baseline assessments.

Figure 1 and Figure 2 represents each of these cases as a line in the plot under different contrast types and correlation structures.

Figure 1. Simulation results with compound symmetry correlation structure.

Figure 2. Simulation results with discrete auto regressive of order 1 correlation structure.

Keeping the SD as 3.6 we tried to vary over two different correlation structures:

Compound symmetry (CS)

Compound symmetry just means that all the variances are equal and all the covariances are equal. So, the same variance and covariance are used for all subjects. In compound symmetry the covariances across the subjects and the variances (pooled within the group) of the different repeated measures are homogeneous.

[\begin{matrix} σ^{2} & σ^{2} ρ & σ^{2} ρ & \dots & σ^{2} ρ \\ σ^{2} ρ & σ^{2} & σ^{2} ρ & \dots & σ^{2} ρ \\ σ^{2} ρ & σ^{2} ρ & σ^{2} & \dots & σ^{2} ρ \\ ⋮ & ⋱ & ⋮ & ⋮ & ⋮ \\ σ^{2} ρ & σ^{2} ρ & σ^{2} ρ & \dots & σ^{2} \end{matrix}]

Where σ² is the common variance assumed to be similar over time and ρ is the assumed correlation. The order of variance covariance matrix will be $t$ × $t$ , where ‘t’ is the number of time points. Generally, σ² and ρ are estimated from previous clinical trial data with same molecule which could be phase1, phase2 trials for same indication or pivotal trials with same molecule for different indication.

Discrete Auto regressive of order 1(AR1)

This is the homogeneous variance first-order autoregressive structure. Any two elements that are adjacent have a correlation that is equal to rho (ρ), those separated by a third will have correlation ρ², and so on. rho is restricted such that –1< ρ <1.

[\begin{matrix} σ^{2} & σ^{2} ρ & σ^{2} ρ^{2} & \dots & σ^{2} ρ^{t - 1} \\ σ^{2} ρ & σ^{2} & σ^{2} ρ & \dots & σ^{2} ρ^{t - 2} \\ σ^{2} ρ^{2} & σ^{2} ρ & σ^{2} & \dots & σ^{2} ρ^{t - 3} \\ ⋮ & ⋱ & ⋮ & ⋮ & ⋮ \\ σ^{2} ρ^{t - 1} & σ^{2} ρ^{t - 2} & σ^{2} ρ^{t - 3} & \dots & σ^{2} \end{matrix}]

Also, we considered different scenarios of how we want to analyze the results at the end as different contrasts as described below.

Contrast of repeated measures

We investigated two types of contrasts.

1. Time-related contrasts i.e., mean over time (mean contrast).
Rationale: This contrast is more applicable in the situation where interest lies in observing the mean treatment effect on the subject’s disease condition over the period of time the subject is exposed to a prescribed dosing regimen compared to baseline.
This will be labelled in the legend of Figure 1 as CS_mean(i) and in the legend of Figure 2 as AR1_mean(i). For example, for five timepoints the contrast would look like $C^{'} =$ [-1, ¼, ¼, ¼, ¼].
The mean-over-time contrast with equal weights was chosen to estimate the average treatment effect across follow-up, which is clinically meaningful in chronic disease settings and does not rely on the treatment effect following a strictly linear pattern over time.
2. Mean difference (change at last time point from baseline) (diff contrast).
Rationale: This contrast is more applicable in the situation where the interest lies in observing the treatment effect once the subject is exposed to a prescribed dosing regimen and what is the change at the end of the treatment exposure period in the subject’s disease condition. Here the total effect at the end of the treatment course as compared to the baseline is of interest.
This will be labelled in the legend of Figure 1 as CS_diff(i) and in the legend of Figure 2 as AR1_diff(i). For example, for five timepoints the contrast would look like $C^{'} =$ [-1, 0, 0, 0, 1].

The sample size was calculated for correlation ranging from 0.1 – 0.95 with intervals of 0.05 for both the plots Figure 1 and Figure 2, using the formulas outlined in Equation 13.

CS variance structure with ‘mean over time’ and ‘mean difference’ contrasts

The red horizontal line represents the sample size from the single time point assessment approach.

CS_diff(i), CS_mean(i) – ‘i’ represents the no. of visits used for sample size calculation, and i = 3,4,5,6,8,10.

AR(1) covariance structure with ‘mean over time’ and ‘mean difference’ contrasts

The red horizontal line represents the sample size from the single time point assessment approach.

AR1_diff(i), AR1_mean(i) – ‘i’ represents the no. of visits used for sample size calculation, and i = 3,4,5,6,8,10.

Results

Under CS type of variance covariance structure ( Figure 1)

All the trend lines for mean difference type of contrast overlaps each other. For mean difference type of contrast, the sample size does not change for an increase/decrease in the number of visits. It changes with the correlation i.e., highly correlated (rho > 0.5) timepoints would need less sample size as compared to low correlated timepoints. Also, for correlation = 0.5 the multiple assessment sample size coincides with that of single time point assessment.

However, the sample size does vary when the contrast is set to mean over time. Multiple time point assessments with more timepoints require less sample size as compared to that of multiple time point assessment with less time points for example, the multiple time point assessment with three timepoint requires 86 per arm with correlation 0.8 and the multiple time point assessment with ten timepoints requires 64 per arm. On the same lines the multiple time point assessment with three timepoints requires 258 per arm with correlation 0.4 and the multiple time point assessment with ten timepoints requires 192 per arm. This trend shows that under compound symmetry, the efficiency gain from repeated measurement depends on the within subject correlation. The required sample size decreases as the correlation between timepoints increases. However, under CS, at moderate correlation levels (ρ = 0.5), this gain may be neutral, whereas lower correlations reduce efficiency and increase the required sample size.

Under AR(1) type of variance covariance structure ( Figure 2)

Under mean over time contrast the multiple time point assessment requires lower sample size as compared to single time point assessment (287 per arm) for correlation greater than 0.35 and the sample size increases as correlation goes below 0.35. For correlation 0.35 the sample size coincides with that of single time point assessment for all the cases except the case of three timepoints which requires slightly higher sample number.

Whereas for mean difference contrast the multiple time point assessment requires lower sample size as compared to single time point assessment (287 per arm) for correlation greater than 0.7 but requires higher sample size for correlation less than 0.7.

The trend changes shape for mean difference contrast vs mean over time contrast. Also, at certain point the increase in sample size attenuates for example, in case of mean difference type contrast with ten time points the sample size required does not change when correlation drops below 0.55.

Discussion

One of the hurdles in considering the longitudinal methodology for sample size calculation is the assumption on the covariance matrix. It is often easy to estimate the variance of single timepoint as compared to estimating the variance-covariance matrix for multiple time points.

The above derivations were done for trial design with parallel group, 1:1 ratio and two arms. If the ratio changes or if we have more than two arms or if the design is crossover, then the effective overall sample size would get effected in both the cases i.e., sample size with single time point as well as sample size with multiple timepoints, but the trend would remain the same as shown above in the figures and the results will still hold good. Similar trends should hold for other variance – covariance structures though they have not been simulated here.

Conclusion

Sample size changes depending on the analysis type and the data collected. Both the graphs in Figure 1 and Figure 2 in this study reveal that if response is assessed at multiple timepoints and the correlation between the paired observations is high (> 0.6) then one should consider using repeated measures analysis and consequently determine the size of the sample that is based on the multiple time points scenario which results in lower sample size requirement as compared to the sample size derived assuming single timepoint response assessment. This would reduce the cost, resources, and time in conducting the experiment fastening the new drug development. Also, repeated measures analyses will not drop the patients in which they have certain missing data as compared to single point analysis where the patient will be dropped if the response is missing and hence may help in retaining the power.

Under a compound symmetry (CS) covariance structure, the required sample size for the mean over time contrast decreases when information from a greater number of time points is available, reflecting the additional information gained from repeated measurements. In contrast, for the mean diff contrast, which is based on a comparison between two time points only, the required sample size remains unchanged regardless of the total number of additional time points observed.

Under an AR(1) covariance structure, different patterns are observed. For the mean over time contrast, the required sample size increases as more time points are included when the within-subject correlation exceeds ρ = 0.35, whereas for ρ ≤ 0.35 the required sample size decreases with the inclusion of additional time points. For the mean diff contrast, the required sample size with information on fewer timepoints is lower as compared to the sample size required when we have information available at more timepoints (Figure 2).

These patterns are not universal but depend on the assumed correlation structure and the magnitude of the mean difference at the chosen time point. Further investigation is warranted to better understand how these relationships vary across alternative covariance structures and effect trajectories.

Sample size derivation using longitudinal design method for studies with multiple assessments can be considered of substantial benefit in cost and time although the challenge of estimating the variance-covariance matrix remains. When within subject correlations are low, averaging treatment effects across multiple time points may increase estimator variance, thereby reducing efficiency relative to a single and well-chosen time point comparison. Also, to be noted that the distance between the timepoints is not taken into consideration by the sample size with multiple timepoints derivation method which could be a topic of further research.

The sample size derivation for multiple timepoint assessment Equation 13 can directly be used by the readers using their treatment effect, variance covariance matrix and the contrast of their choice. To ease this, we have also provided the link to R code under software availability section where inputs for treatment effect, variance covariance matrix and contrast can be given to the function to obtain the sample size required. To explore more on power analysis in longitudinal studies one may refer to the book “Sample Size Calculations for Clustered and Longitudinal Outcomes in Clinical Research”.¹³ One could also make use of the R packages for different cases of longitudinal power analyses provided under the ‘The R Journal’ article “Power and Sample Size for Longitudinal Models in R”.¹⁷

Software availability

Software available from: The Comprehensive R Archive Network (https://cran.r-project.org/)

Source code available from: https://github.com/Sarfaraz-Sayyed/Sample-Size-Variation.

Archived source code at time of publication: https://zenodo.org/badge/latestdoi/547747570.¹⁸

Archived source code at time of revision: Sample Size Variation (zenodo.org)

License: MIT License.

Data availability

Extended data

Supplementary appendices offering detailed clarification of selected outcomes to support deeper methodological understanding are available on Zenodo: https://doi.org/10.5281/zenodo.18453058.¹⁹

Acknowledgement

We would like to thank Novartis Healthcare Pvt. Ltd. and Manipal Academy of Higher Education for their support in conducting this research and all the reviewers for providing their valuable feedback and suggestions.

References

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2. Nisbet R, Miner G, Yale K: Model Evaluation and Enhancement. Handbook of Statistical Analysis and Data Mining Applications. Elsevier; 2018. Publisher Full Text
3. Khelif A, Saleh MN, Salama A, et al.: Changes in Health-Related Quality of Life with Long-term Eltrombopag Treatment in Adults with Persistent/Chronic Immune Thrombocytopenia: Findings from the EXTEND Study. Am. J. Hematol. 2019; 94: 200–208. PubMed Abstract | Publisher Full Text
4. Palileo-Villanueva LM, Dans AL: Composite endpoints. J. Clin. Epidemiol. 2020; 128: 157–158. Publisher Full Text
5. Bloch DA: Sample size requirements and the cost of a randomized clinical trial with repeated measurements. Stat. Med. 1986; 5(6): 663–667. PubMed Abstract | Publisher Full Text
6. Hedeker D, Gibbons RD, Waternaux C: Sample size estimation for longitudinal designs with attrition: Comparing time-related contrasts between two groups. J. Educ. Behav. Stat. 1999; 24(1): 70–93. Publisher Full Text
7. Basagana X, Liao X, Spiegelman D: Power and sample size calculations for longitudinal studies estimating a main effect of a time-varying exposure. Stat. Methods Med. Res. 2011; 20(5): 471–487. PubMed Abstract | Publisher Full Text
8. Pourhoseingholi MA, Vahedi M, Rahimzadeh M: Gastroenterol Hepatol Bed Bench. Winter. Sample size calculation in medical studies. 2013; 6(1): 14–17.
9. Hajian-Tilaki K: Sample size estimation in epidemiologic studies. Caspian J. Intern. Med. 2011; 2(4): 289–298. PubMed Abstract
10. Veena Manja MD, Satyan Lakshminrusimha MD: Epidemiology and Clinical Research Design, Part 1: Principles. Neoreviews. 2015; 16(2): e94–e108. PubMed Abstract | Publisher Full Text
11. Manja V MD, Lakshminrusimha S MD: Epidemiology and Clinical Research Design, Part 1: Study Types. Neoreviews. Neoreviews. 2014; 15(12): e558–e569. PubMed Abstract | Publisher Full Text
12. Diggle PJ, Heagerty P, Liang K-Y: Analysis of Longitudinal Data. 2nd ed.New York: Oxford Statistical Science Series; 2002. Publisher Full Text
13. Ahn C, Heo M, Zhang S: Sample Size Calculations for Clustered and Longitudinal Outcomes in Clinical Research. Chapman and Hall/CRC; 2014. Publisher Full Text
14. Rao CR: Linear Statistical Inference and its Applications. 2nd ed.Wiley; 1973. Publisher Full Text
15. Chow S-C, Shao J, Wang H, et al.: Sample Size Calculations in Clinical Research. 3rd ed.Chapman and Hall/CRC; 2017. Publisher Full Text
16. Black DM, Reid IR, Cauley JA, et al.: The Effect of 6 Versus 9 Years of Zoledronic AcidTreatment in Osteoporosis: A Randomized SecondExtension to the HORIZON-Pivotal Fracture Trial (PFT). J. Bone Miner. Res. 2014; 30(5): 934–944. Publisher Full Text
17. Iddi S, Donohue MC: Power and Sample Size for Longitudinal Models in R -- The longpower Package and Shiny App. R J. 2022; 14(1):264–281. PubMed Abstract | Publisher Full Text https://journal.r-project.org/articles/RJ-2022-022/
18. Sarfaraz-Sayyed: Sarfaraz-Sayyed/Sample-Size-Variation: Sample Size variation (V1.0.0). Zenodo. Software.2022. Publisher Full Text
19. Sarfaraz-Sayyed: Sarfaraz-Sayyed/Sample-Size-Variation: Sample Size variation (V1.1.0). Zenodo. Software. 2026. Publisher Full Text

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Version 3

VERSION 3 PUBLISHED 21 Dec 2022

Author details Author details

Sarfaraz Sayyed
Roles: Formal Analysis, Investigation, Visualization, Writing – Original Draft Preparation

Ashwini Mathur
Roles: Conceptualization, Supervision, Writing – Review & Editing

Asha Kamath
Roles: Conceptualization, Supervision, Writing – Review & Editing

Competing interests

The author (Sarfaraz Sayyed) is employed by Novartis Healthcare Pvt. Ltd.

Grant information

The author(s) declared that no grants were involved in supporting this work.

Article Versions (3)

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© 2026 Sayyed S et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Sayyed S, Mathur A and Kamath A. Sample size variation in single-time post-dose assessment vs multi-time post-dose assessment [version 3; peer review: 1 approved, 2 not approved]. F1000Research 2026, 11:1550 (https://doi.org/10.12688/f1000research.124917.3)

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Version 2

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Reviewer Report 21 Nov 2024

Simon Vandekar, Vanderbilt University Medical Center, Tennessee, USA

Not Approved

https://doi.org/10.5256/f1000research.170509.r333804

The paper doesn’t reference the huge body of literature on power analysis in longitudinal studies. There are 14 references and only of them is a classical text on longitudinal data analysis (Diggle et al). For example, the book by Ahn, Heo, Zhang “Sample Size Calculations for Clustered and Longitudinal Outcomes in Clinical Research.” There are numerous R package for longitudinal power analyses, but these were not discussed (e.g. https://journal.r-project.org/articles/RJ-2022-022/). The results and topic are not novel and overlook critical references that have studied the topic. Given the very general topic, I would have expected more references to the existing literature.

There need to be more references for the mathematics in the paper, such as “Method for Sample size with single time point assessment analysis.” Grammar in the paper needs to be reviewed. There are some notation errors, e.g. in equations (8) and (9) Var(\Gamma) doesn’t make sense because Gamma is defined as a parameter (so it has zero variance).

Is the rationale for developing the new method (or application) clearly explained?

Yes
Is the description of the method technically sound?

Partly
Are sufficient details provided to allow replication of the method development and its use by others?

Yes
If any results are presented, are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions about the method and its performance adequately supported by the findings presented in the article?

Partly

References

1. Ahn C, Heo M, Zhang S: Sample Size Calculations for Clustered and Longitudinal Outcomes in Clinical Research. 2014. Publisher Full Text

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Biostatistics

I confirm that I have read this submission and believe that I have an appropriate level of expertise to state that I do not consider it to be of an acceptable scientific standard, for reasons outlined above.

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Author Response 05 Feb 2026

Sarfaraz Sayyed, Analytics, Novartis Healthcare Pvt. Ltd., Hyderabad, 500080, India

05 Feb 2026

Author Response

Response to Comment 1
We thank the reviewer for this important and thorough comment and fully agree that there is a substantial body of literature on power and sample size ... Continue reading Response to Comment 1
We thank the reviewer for this important and thorough comment and fully agree that there is a substantial body of literature on power and sample size calculation for longitudinal studies. Our intent was not to provide a comprehensive review or to introduce new theoretical developments in this well‑established area. Rather, the aim of the present paper is to provide a focused and accessible comparison between sample size calculations based on a single post‑baseline assessment and those incorporating multiple post‑baseline time points, under commonly used correlation structures.
To better position our work within the existing literature, we have expanded the related work section to explicitly acknowledge classical and contemporary references on longitudinal power analysis, including Diggle et al. and the monograph by Ahn, Heo, and Zhang. We have also added references to existing R packages for longitudinal power analysis and clarified how our approach complements these tools by highlighting specific design trade‑offs in simple, interpretable settings.
We hope that these revisions more clearly situate the contribution of this paper as illustrative and comparative rather than novel in a methodological sense, while providing practical insights for applied researchers.

Response to Comment 2
We thank the reviewer for this detailed comment. In the revised manuscript, we have added appropriate references for the single‑time‑point sample size calculation methodology. We have also carefully reviewed the manuscript for grammatical issues and corrected several notation inconsistencies. In particular, we revised Equations (8) and (9) to avoid expressions such as Var(), as represents a fixed parameter, and reformulated these expressions using appropriate sample‑based quantities. We believe these revisions improve both the mathematical clarity and presentation of the manuscript.
Response to Comment 1
We thank the reviewer for this important and thorough comment and fully agree that there is a substantial body of literature on power and sample size calculation for longitudinal studies. Our intent was not to provide a comprehensive review or to introduce new theoretical developments in this well‑established area. Rather, the aim of the present paper is to provide a focused and accessible comparison between sample size calculations based on a single post‑baseline assessment and those incorporating multiple post‑baseline time points, under commonly used correlation structures.
To better position our work within the existing literature, we have expanded the related work section to explicitly acknowledge classical and contemporary references on longitudinal power analysis, including Diggle et al. and the monograph by Ahn, Heo, and Zhang. We have also added references to existing R packages for longitudinal power analysis and clarified how our approach complements these tools by highlighting specific design trade‑offs in simple, interpretable settings.
We hope that these revisions more clearly situate the contribution of this paper as illustrative and comparative rather than novel in a methodological sense, while providing practical insights for applied researchers.

Response to Comment 2
We thank the reviewer for this detailed comment. In the revised manuscript, we have added appropriate references for the single‑time‑point sample size calculation methodology. We have also carefully reviewed the manuscript for grammatical issues and corrected several notation inconsistencies. In particular, we revised Equations (8) and (9) to avoid expressions such as Var(), as represents a fixed parameter, and reformulated these expressions using appropriate sample‑based quantities. We believe these revisions improve both the mathematical clarity and presentation of the manuscript.
Competing Interests: No competing interests were disclosed. Close
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COMMENTS ON THIS REPORT

Author Response 05 Feb 2026

Sarfaraz Sayyed, Analytics, Novartis Healthcare Pvt. Ltd., Hyderabad, 500080, India

05 Feb 2026

Author Response

Response to Comment 1
We thank the reviewer for this important and thorough comment and fully agree that there is a substantial body of literature on power and sample size ... Continue reading Response to Comment 1
We thank the reviewer for this important and thorough comment and fully agree that there is a substantial body of literature on power and sample size calculation for longitudinal studies. Our intent was not to provide a comprehensive review or to introduce new theoretical developments in this well‑established area. Rather, the aim of the present paper is to provide a focused and accessible comparison between sample size calculations based on a single post‑baseline assessment and those incorporating multiple post‑baseline time points, under commonly used correlation structures.
To better position our work within the existing literature, we have expanded the related work section to explicitly acknowledge classical and contemporary references on longitudinal power analysis, including Diggle et al. and the monograph by Ahn, Heo, and Zhang. We have also added references to existing R packages for longitudinal power analysis and clarified how our approach complements these tools by highlighting specific design trade‑offs in simple, interpretable settings.
We hope that these revisions more clearly situate the contribution of this paper as illustrative and comparative rather than novel in a methodological sense, while providing practical insights for applied researchers.

Response to Comment 2
We thank the reviewer for this detailed comment. In the revised manuscript, we have added appropriate references for the single‑time‑point sample size calculation methodology. We have also carefully reviewed the manuscript for grammatical issues and corrected several notation inconsistencies. In particular, we revised Equations (8) and (9) to avoid expressions such as Var(), as represents a fixed parameter, and reformulated these expressions using appropriate sample‑based quantities. We believe these revisions improve both the mathematical clarity and presentation of the manuscript.
Response to Comment 1
We thank the reviewer for this important and thorough comment and fully agree that there is a substantial body of literature on power and sample size calculation for longitudinal studies. Our intent was not to provide a comprehensive review or to introduce new theoretical developments in this well‑established area. Rather, the aim of the present paper is to provide a focused and accessible comparison between sample size calculations based on a single post‑baseline assessment and those incorporating multiple post‑baseline time points, under commonly used correlation structures.
To better position our work within the existing literature, we have expanded the related work section to explicitly acknowledge classical and contemporary references on longitudinal power analysis, including Diggle et al. and the monograph by Ahn, Heo, and Zhang. We have also added references to existing R packages for longitudinal power analysis and clarified how our approach complements these tools by highlighting specific design trade‑offs in simple, interpretable settings.
We hope that these revisions more clearly situate the contribution of this paper as illustrative and comparative rather than novel in a methodological sense, while providing practical insights for applied researchers.

Response to Comment 2
We thank the reviewer for this detailed comment. In the revised manuscript, we have added appropriate references for the single‑time‑point sample size calculation methodology. We have also carefully reviewed the manuscript for grammatical issues and corrected several notation inconsistencies. In particular, we revised Equations (8) and (9) to avoid expressions such as Var(), as represents a fixed parameter, and reformulated these expressions using appropriate sample‑based quantities. We believe these revisions improve both the mathematical clarity and presentation of the manuscript.
Competing Interests: No competing interests were disclosed. Close
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Reviewer Report 23 Oct 2024

Kiranmoy Das, Indian Statistical Institute, Kolkata, West Bengal, India

Approved

https://doi.org/10.5256/f1000research.170509.r320982

I have seen the revised version of the paper and the author's comments.
I ... Continue reading

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Reviewer Report 28 Sep 2024

Ronald Geskus, Oxford University, Oxford, England, UK

Not Approved

https://doi.org/10.5256/f1000research.170509.r320983

The article still contains several typos, sentences that abruptly end and capital letters that are misplaced. I suggest the authors do a careful check of typos. Also, I still found a couple of errors, as well as a few statements that are not clear to me.

1. "It affects the estimator’s precision, which is a metric that is important for the chosen target population of all professional athletes". Precision is a characteristic of the data. Explain why precision is an important metric for the target population.

2. The authors found only two papers (references 3 and 4) in their literature review on longitudinal studies? There must be many more.

3. Spiegelma -> Spiegelman

4. Page 4: Make clear that the normal distribution of the sample mean is an approximation, which becomes more accurate with increasing sample size (based on the central limit theorem)

5. The authors replied: "Comment 4: We fully agree with you, this was a generic equation to show we are looking positive change in response and then further with the examples we have assumed for certain fixed delta how the sample size varies. But have modified it to avoid confusion." I doubt whether "H_a is true" and formula (2) avoid confusion. To my opinion, it is clearer to change "H_a is true" and "\delta > 0" by a specific value for \delta.

6. Page 5: H_a -> H_0 for null hypothesis, and \psi_c=C^t \Lambda, with "t" meaning transpose.

7. Formula (8) is not a test statistic, \psi_c is a property of the population not based on random variables. Also, \Lambda is a population property, not determined by random variables.

8. covar(Y_2-X_2,Y_1-X_1)=covar(Y_2,Y_1)+covar(X_2,X_1). I don't see where the value 4 comes from in (9).

9. I still find the section "Calculation of sample size" not very clearly phrased. Specify what is meant with "the last time point" (later it is written to be 3 years). What time trend does it imply? Replace "allowing" by "choosing"

10. Explain how the sample size of 287 is obtained. How large is the assumed difference under the null hypothesis, i.e. what is the chosen time point?

11. The rationale for the mean over time contrast chosen is still not clear to me. Why would one choose 1/4 at later time points, given that a linear trend is assumed?

12. Can the authors explain i) why the sample size for correlation=0.5 (compound symmetry) coincides with single time point assessment and ii) why the required sample size is larger if correlation<0.5?

13. I do not understand why with low correlation the single time point assessment performs best. Please explain.

14. I do not understand why the sample size for AR(1) mean over time contrast increases with increasing number of time points included if correlation > 0.35. Please explain.

15. Page 11: "Whereas for the mean diff contrast the sample size required with fewer visit is lower as compared to the sample size required with higher number of visits". It's not the number of visits that matters (it is always based on two time points only), but the distance between the visits. The required sample size for AR(1) mean difference contrast increases if the second time point is further away from the baseline. I don't think this is a general property. It will depend on the combination of serial correlation and difference in mean at the chosen time point. This needs further investigation.

16. The authors give a detailed introduction for readers who do not have a statistical background. It would be very useful to provide some guidance how the results can directly be used by them.

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: biostatistics

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Author Response 05 Feb 2026

Sarfaraz Sayyed, Analytics, Novartis Healthcare Pvt. Ltd., Hyderabad, 500080, India

05 Feb 2026

Author Response

Response to General comment
We thank the reviewer for the comment. In the revised version, we have carefully corrected typographical errors, incomplete sentences, and improperly placed capital letters, and clarified ... Continue reading Response to General comment
We thank the reviewer for the comment. In the revised version, we have carefully corrected typographical errors, incomplete sentences, and improperly placed capital letters, and clarified statements that were previously unclear.

Response to Comment 1
We thank the reviewer for this important clarification. In the revised manuscript, we have expanded the Introduction to explicitly explain why estimator precision is a critical metric for the target population of professional athletes. Specifically, we emphasize that professional athletes exhibit high inter- and intra-individual variability, and that precise estimation is essential for accurate performance assessment, monitoring, and interpretation of small but practically meaningful effects. We believe this added explanation clarifies the relevance of precision for the chosen population.

Response to Comment 2
We thank the reviewer for this comment. We agree that there are numerous longitudinal studies in the literature. However, our intention was not to provide a comprehensive review of longitudinal research, but rather to illustrate that even in well‑established and prestigious journals, longitudinal studies often provide limited detail regarding sample size calculation. We have now clarified this scope in the manuscript to avoid any misunderstanding and to emphasize that references 3 and 4 were cited as illustrative examples rather than as an exhaustive review of the literature.

Response to Comment 3
We thank the reviewer for pointing this out. The spelling has been corrected in the revised version of the manuscript

Response to Comment 4
We thank the reviewer for this clarification. We have now explicitly stated that the normal distribution of the sample mean is an approximation that becomes increasingly accurate with larger sample sizes, as justified by the central limit theorem. This clarification has been added to the ‘Notation and framework’ section in the revised manuscript.

Response to Comment 5
We thank the reviewer for this helpful suggestion. To improve clarity, we have revised the hypotheses to explicitly state “” and “ , where , and δ represents the expected difference. This definition clarifies the role of δ in the sample size derivation and avoids ambiguous statements. The corresponding revision has been made in the Methods section and aligns with Equation (2) accordingly.

Response to Comment 6
We thank the reviewer for pointing this out. The null and alternative hypothesis notation and the definition of the transpose operator have been corrected in the revised manuscript.

Response to Comment 7
We thank the reviewer for this important clarification. We agree that ψ_c and Λ, as population parameters, cannot constitute a test statistic. In the revised manuscript, we have therefore replaced these quantities with their corresponding sample estimates and revised the notation accordingly. As a result, Formula (8) is now explicitly defined as a function of sample‑based random variables, ensuring that it represents a valid test statistic.

Response to Comment 8
We thank the reviewer for carefully examining the derivation. We agree that the covariance expansion was incorrectly stated and that the origin of the value 4 in Equation (9) was not justified. In the revised manuscript, we have corrected the covariance expression by explicitly accounting for all relevant covariance terms and have revised Equation (9) accordingly. The incorrect constant has been removed, and the derivation is now consistent with standard covariance identities.

Response to Comment 9
We thank the reviewer for these constructive suggestions to improve clarity. In the revised manuscript, we have explicitly defined and clarified the assumed time trend in the sample size calculation. We have also revised the wording as suggested, replacing “allowing” with “choosing.” In addition, we have added a summary table to clearly indicate the selected time points under each scenario.

Response to Comment 10
We thank the reviewer for this important query. In the revised manuscript, we have explicitly detailed how the sample size of 287 was derived, including the expected difference under the alternative hypothesis, the corresponding null hypothesis, and the selected time point used in the sample size calculation, thereby making the derivation and underlying assumptions transparent and reproducible.

Response to Comment 11
We thank the reviewer for this insightful comment. Our objective in choosing a mean‑over‑time contrast was to target the average treatment effect across the follow‑up period rather than the effect at a single time point. Accordingly, equal weights (1/4 at each post‑baseline time point) were used to reflect this estimand.
Although a linear time trend is assumed for the purpose of modeling and sample size calculation, the choice of an average contrast does not require linearity of the treatment effect over time. This approach is particularly relevant in chronic disease settings, where sustained control of biomarkers over time is often of greater clinical interest than the maximum effect at a specific visit, and it also provides robustness to missing data due to loss to follow‑up.
We have clarified this rationale in the revised manuscript to better explain the choice of the equal‑weight contrast and its relationship to the assumed time trend.

Response to Comment 12
We thank the reviewer for this insightful question. Under a compound symmetry correlation structure, when the within‑subject correlation is 0.5, the variance reduction gained from repeated measurements is exactly offset by their correlation, resulting in a required sample size equivalent to that of a single‑time‑point assessment. When the correlation is lower than 0.5, repeated observations provide less shared information across time points, leading to a reduced efficiency and consequently a larger required sample size.
To provide a detailed mathematical justification of these results, we have added two appendices that explicitly derive the variance expressions and resulting sample size formulas under different correlation assumptions.

Response to Comment 13
We thank the reviewer for this important question. When within‑subject correlation is low, repeated measurements provide relatively little shared information about the underlying treatment effect, as observations across time behave nearly independently. For contrasts that average treatment effects over time, this can lead to increased variability because the additional measurements contribute more noise than signal.
In contrast, a single time‑point assessment, particularly at a clinically relevant follow‑up time point may yield a more efficient estimator under low-correlation settings, as it avoids the accumulation of variability from weakly correlated repeated measures. This explains why, in our framework, the single‑time‑point assessment can require a smaller sample size when the correlation is low. Appendix added can also give a mathematical glimpse.

Response to Comment 14
We thank the reviewer for this insightful question. Under an AR(1) correlation structure, increasing the number of time points does not necessarily lead to increased efficiency when within‑subject correlation is moderate to high. As additional time points are included, the covariance between observations accumulates, and beyond a certain correlation threshold, the variance component associated with the mean‑over‑time contrast () increases rather than decreasing.
This occurs because strongly correlated repeated measurements contribute limited new information while increasing the overall covariance contribution to the contrast variance. As a result, when the correlation exceeds approximately 0.35 in our setting, adding more time points inflates the variance of the mean‑over‑time estimator, leading to a higher required sample size. To clarify this behavior, we have added an appendix that explicitly compares variance components across different correlation levels and side by side layout for 4 timepoints and 10 timepoints case.

Response to Comment 15
We thank the reviewer for this important clarification. We agree that, for the mean difference contrast, the comparison is based on two time points (baseline and a post‑baseline visit), and that it is the temporal distance between these time points, rather than the total number of visits that determines the correlation and variance under an AR(1) structure. We have revised the manuscript to clarify this wording and avoid any implication that the total number of visits alone drives the sample size.
We also agree that the observed increase in required sample size when the post‑baseline assessment is further from baseline is not a general property but depends on the joint behaviour of the serial correlation and the mean difference at the chosen time point. This dependence is now acknowledged more explicitly in the revised text and reiterated in the conclusion.

Response to Comment 16
We thank the reviewer for this helpful suggestion. To facilitate direct use by readers without a statistical background, the manuscript provides explicit formulas for both single and multiple time point sample size calculations, along with openly available R code (see Software Availability). Users can obtain required sample sizes by specifying interpretable inputs such as the anticipated treatment effect, the assumed variance-covariance structure, and the contrast of interest. In addition, we have added a worked example in the Appendices demonstrating how these inputs are selected and applied in a realistic clinical trial setting.
Response to General comment
We thank the reviewer for the comment. In the revised version, we have carefully corrected typographical errors, incomplete sentences, and improperly placed capital letters, and clarified statements that were previously unclear.

Response to Comment 1
We thank the reviewer for this important clarification. In the revised manuscript, we have expanded the Introduction to explicitly explain why estimator precision is a critical metric for the target population of professional athletes. Specifically, we emphasize that professional athletes exhibit high inter- and intra-individual variability, and that precise estimation is essential for accurate performance assessment, monitoring, and interpretation of small but practically meaningful effects. We believe this added explanation clarifies the relevance of precision for the chosen population.

Response to Comment 2
We thank the reviewer for this comment. We agree that there are numerous longitudinal studies in the literature. However, our intention was not to provide a comprehensive review of longitudinal research, but rather to illustrate that even in well‑established and prestigious journals, longitudinal studies often provide limited detail regarding sample size calculation. We have now clarified this scope in the manuscript to avoid any misunderstanding and to emphasize that references 3 and 4 were cited as illustrative examples rather than as an exhaustive review of the literature.

Response to Comment 3
We thank the reviewer for pointing this out. The spelling has been corrected in the revised version of the manuscript

Response to Comment 4
We thank the reviewer for this clarification. We have now explicitly stated that the normal distribution of the sample mean is an approximation that becomes increasingly accurate with larger sample sizes, as justified by the central limit theorem. This clarification has been added to the ‘Notation and framework’ section in the revised manuscript.

Response to Comment 5
We thank the reviewer for this helpful suggestion. To improve clarity, we have revised the hypotheses to explicitly state “” and “ , where , and δ represents the expected difference. This definition clarifies the role of δ in the sample size derivation and avoids ambiguous statements. The corresponding revision has been made in the Methods section and aligns with Equation (2) accordingly.

Response to Comment 6
We thank the reviewer for pointing this out. The null and alternative hypothesis notation and the definition of the transpose operator have been corrected in the revised manuscript.

Response to Comment 7
We thank the reviewer for this important clarification. We agree that ψ_c and Λ, as population parameters, cannot constitute a test statistic. In the revised manuscript, we have therefore replaced these quantities with their corresponding sample estimates and revised the notation accordingly. As a result, Formula (8) is now explicitly defined as a function of sample‑based random variables, ensuring that it represents a valid test statistic.

Response to Comment 8
We thank the reviewer for carefully examining the derivation. We agree that the covariance expansion was incorrectly stated and that the origin of the value 4 in Equation (9) was not justified. In the revised manuscript, we have corrected the covariance expression by explicitly accounting for all relevant covariance terms and have revised Equation (9) accordingly. The incorrect constant has been removed, and the derivation is now consistent with standard covariance identities.

Response to Comment 9
We thank the reviewer for these constructive suggestions to improve clarity. In the revised manuscript, we have explicitly defined and clarified the assumed time trend in the sample size calculation. We have also revised the wording as suggested, replacing “allowing” with “choosing.” In addition, we have added a summary table to clearly indicate the selected time points under each scenario.

Response to Comment 10
We thank the reviewer for this important query. In the revised manuscript, we have explicitly detailed how the sample size of 287 was derived, including the expected difference under the alternative hypothesis, the corresponding null hypothesis, and the selected time point used in the sample size calculation, thereby making the derivation and underlying assumptions transparent and reproducible.

Response to Comment 11
We thank the reviewer for this insightful comment. Our objective in choosing a mean‑over‑time contrast was to target the average treatment effect across the follow‑up period rather than the effect at a single time point. Accordingly, equal weights (1/4 at each post‑baseline time point) were used to reflect this estimand.
Although a linear time trend is assumed for the purpose of modeling and sample size calculation, the choice of an average contrast does not require linearity of the treatment effect over time. This approach is particularly relevant in chronic disease settings, where sustained control of biomarkers over time is often of greater clinical interest than the maximum effect at a specific visit, and it also provides robustness to missing data due to loss to follow‑up.
We have clarified this rationale in the revised manuscript to better explain the choice of the equal‑weight contrast and its relationship to the assumed time trend.

Response to Comment 12
We thank the reviewer for this insightful question. Under a compound symmetry correlation structure, when the within‑subject correlation is 0.5, the variance reduction gained from repeated measurements is exactly offset by their correlation, resulting in a required sample size equivalent to that of a single‑time‑point assessment. When the correlation is lower than 0.5, repeated observations provide less shared information across time points, leading to a reduced efficiency and consequently a larger required sample size.
To provide a detailed mathematical justification of these results, we have added two appendices that explicitly derive the variance expressions and resulting sample size formulas under different correlation assumptions.

Response to Comment 13
We thank the reviewer for this important question. When within‑subject correlation is low, repeated measurements provide relatively little shared information about the underlying treatment effect, as observations across time behave nearly independently. For contrasts that average treatment effects over time, this can lead to increased variability because the additional measurements contribute more noise than signal.
In contrast, a single time‑point assessment, particularly at a clinically relevant follow‑up time point may yield a more efficient estimator under low-correlation settings, as it avoids the accumulation of variability from weakly correlated repeated measures. This explains why, in our framework, the single‑time‑point assessment can require a smaller sample size when the correlation is low. Appendix added can also give a mathematical glimpse.

Response to Comment 14
We thank the reviewer for this insightful question. Under an AR(1) correlation structure, increasing the number of time points does not necessarily lead to increased efficiency when within‑subject correlation is moderate to high. As additional time points are included, the covariance between observations accumulates, and beyond a certain correlation threshold, the variance component associated with the mean‑over‑time contrast () increases rather than decreasing.
This occurs because strongly correlated repeated measurements contribute limited new information while increasing the overall covariance contribution to the contrast variance. As a result, when the correlation exceeds approximately 0.35 in our setting, adding more time points inflates the variance of the mean‑over‑time estimator, leading to a higher required sample size. To clarify this behavior, we have added an appendix that explicitly compares variance components across different correlation levels and side by side layout for 4 timepoints and 10 timepoints case.

Response to Comment 15
We thank the reviewer for this important clarification. We agree that, for the mean difference contrast, the comparison is based on two time points (baseline and a post‑baseline visit), and that it is the temporal distance between these time points, rather than the total number of visits that determines the correlation and variance under an AR(1) structure. We have revised the manuscript to clarify this wording and avoid any implication that the total number of visits alone drives the sample size.
We also agree that the observed increase in required sample size when the post‑baseline assessment is further from baseline is not a general property but depends on the joint behaviour of the serial correlation and the mean difference at the chosen time point. This dependence is now acknowledged more explicitly in the revised text and reiterated in the conclusion.

Response to Comment 16
We thank the reviewer for this helpful suggestion. To facilitate direct use by readers without a statistical background, the manuscript provides explicit formulas for both single and multiple time point sample size calculations, along with openly available R code (see Software Availability). Users can obtain required sample sizes by specifying interpretable inputs such as the anticipated treatment effect, the assumed variance-covariance structure, and the contrast of interest. In addition, we have added a worked example in the Appendices demonstrating how these inputs are selected and applied in a realistic clinical trial setting.
Competing Interests: No competing interests were disclosed. Close
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COMMENTS ON THIS REPORT

Author Response 05 Feb 2026

Sarfaraz Sayyed, Analytics, Novartis Healthcare Pvt. Ltd., Hyderabad, 500080, India

05 Feb 2026

Author Response

Response to General comment
We thank the reviewer for the comment. In the revised version, we have carefully corrected typographical errors, incomplete sentences, and improperly placed capital letters, and clarified ... Continue reading Response to General comment
We thank the reviewer for the comment. In the revised version, we have carefully corrected typographical errors, incomplete sentences, and improperly placed capital letters, and clarified statements that were previously unclear.

Response to Comment 1
We thank the reviewer for this important clarification. In the revised manuscript, we have expanded the Introduction to explicitly explain why estimator precision is a critical metric for the target population of professional athletes. Specifically, we emphasize that professional athletes exhibit high inter- and intra-individual variability, and that precise estimation is essential for accurate performance assessment, monitoring, and interpretation of small but practically meaningful effects. We believe this added explanation clarifies the relevance of precision for the chosen population.

Response to Comment 2
We thank the reviewer for this comment. We agree that there are numerous longitudinal studies in the literature. However, our intention was not to provide a comprehensive review of longitudinal research, but rather to illustrate that even in well‑established and prestigious journals, longitudinal studies often provide limited detail regarding sample size calculation. We have now clarified this scope in the manuscript to avoid any misunderstanding and to emphasize that references 3 and 4 were cited as illustrative examples rather than as an exhaustive review of the literature.

Response to Comment 3
We thank the reviewer for pointing this out. The spelling has been corrected in the revised version of the manuscript

Response to Comment 4
We thank the reviewer for this clarification. We have now explicitly stated that the normal distribution of the sample mean is an approximation that becomes increasingly accurate with larger sample sizes, as justified by the central limit theorem. This clarification has been added to the ‘Notation and framework’ section in the revised manuscript.

Response to Comment 5
We thank the reviewer for this helpful suggestion. To improve clarity, we have revised the hypotheses to explicitly state “” and “ , where , and δ represents the expected difference. This definition clarifies the role of δ in the sample size derivation and avoids ambiguous statements. The corresponding revision has been made in the Methods section and aligns with Equation (2) accordingly.

Response to Comment 6
We thank the reviewer for pointing this out. The null and alternative hypothesis notation and the definition of the transpose operator have been corrected in the revised manuscript.

Response to Comment 7
We thank the reviewer for this important clarification. We agree that ψ_c and Λ, as population parameters, cannot constitute a test statistic. In the revised manuscript, we have therefore replaced these quantities with their corresponding sample estimates and revised the notation accordingly. As a result, Formula (8) is now explicitly defined as a function of sample‑based random variables, ensuring that it represents a valid test statistic.

Response to Comment 8
We thank the reviewer for carefully examining the derivation. We agree that the covariance expansion was incorrectly stated and that the origin of the value 4 in Equation (9) was not justified. In the revised manuscript, we have corrected the covariance expression by explicitly accounting for all relevant covariance terms and have revised Equation (9) accordingly. The incorrect constant has been removed, and the derivation is now consistent with standard covariance identities.

Response to Comment 9
We thank the reviewer for these constructive suggestions to improve clarity. In the revised manuscript, we have explicitly defined and clarified the assumed time trend in the sample size calculation. We have also revised the wording as suggested, replacing “allowing” with “choosing.” In addition, we have added a summary table to clearly indicate the selected time points under each scenario.

Response to Comment 10
We thank the reviewer for this important query. In the revised manuscript, we have explicitly detailed how the sample size of 287 was derived, including the expected difference under the alternative hypothesis, the corresponding null hypothesis, and the selected time point used in the sample size calculation, thereby making the derivation and underlying assumptions transparent and reproducible.

Response to Comment 11
We thank the reviewer for this insightful comment. Our objective in choosing a mean‑over‑time contrast was to target the average treatment effect across the follow‑up period rather than the effect at a single time point. Accordingly, equal weights (1/4 at each post‑baseline time point) were used to reflect this estimand.
Although a linear time trend is assumed for the purpose of modeling and sample size calculation, the choice of an average contrast does not require linearity of the treatment effect over time. This approach is particularly relevant in chronic disease settings, where sustained control of biomarkers over time is often of greater clinical interest than the maximum effect at a specific visit, and it also provides robustness to missing data due to loss to follow‑up.
We have clarified this rationale in the revised manuscript to better explain the choice of the equal‑weight contrast and its relationship to the assumed time trend.

Response to Comment 12
We thank the reviewer for this insightful question. Under a compound symmetry correlation structure, when the within‑subject correlation is 0.5, the variance reduction gained from repeated measurements is exactly offset by their correlation, resulting in a required sample size equivalent to that of a single‑time‑point assessment. When the correlation is lower than 0.5, repeated observations provide less shared information across time points, leading to a reduced efficiency and consequently a larger required sample size.
To provide a detailed mathematical justification of these results, we have added two appendices that explicitly derive the variance expressions and resulting sample size formulas under different correlation assumptions.

Response to Comment 13
We thank the reviewer for this important question. When within‑subject correlation is low, repeated measurements provide relatively little shared information about the underlying treatment effect, as observations across time behave nearly independently. For contrasts that average treatment effects over time, this can lead to increased variability because the additional measurements contribute more noise than signal.
In contrast, a single time‑point assessment, particularly at a clinically relevant follow‑up time point may yield a more efficient estimator under low-correlation settings, as it avoids the accumulation of variability from weakly correlated repeated measures. This explains why, in our framework, the single‑time‑point assessment can require a smaller sample size when the correlation is low. Appendix added can also give a mathematical glimpse.

Response to Comment 14
We thank the reviewer for this insightful question. Under an AR(1) correlation structure, increasing the number of time points does not necessarily lead to increased efficiency when within‑subject correlation is moderate to high. As additional time points are included, the covariance between observations accumulates, and beyond a certain correlation threshold, the variance component associated with the mean‑over‑time contrast () increases rather than decreasing.
This occurs because strongly correlated repeated measurements contribute limited new information while increasing the overall covariance contribution to the contrast variance. As a result, when the correlation exceeds approximately 0.35 in our setting, adding more time points inflates the variance of the mean‑over‑time estimator, leading to a higher required sample size. To clarify this behavior, we have added an appendix that explicitly compares variance components across different correlation levels and side by side layout for 4 timepoints and 10 timepoints case.

Response to Comment 15
We thank the reviewer for this important clarification. We agree that, for the mean difference contrast, the comparison is based on two time points (baseline and a post‑baseline visit), and that it is the temporal distance between these time points, rather than the total number of visits that determines the correlation and variance under an AR(1) structure. We have revised the manuscript to clarify this wording and avoid any implication that the total number of visits alone drives the sample size.
We also agree that the observed increase in required sample size when the post‑baseline assessment is further from baseline is not a general property but depends on the joint behaviour of the serial correlation and the mean difference at the chosen time point. This dependence is now acknowledged more explicitly in the revised text and reiterated in the conclusion.

Response to Comment 16
We thank the reviewer for this helpful suggestion. To facilitate direct use by readers without a statistical background, the manuscript provides explicit formulas for both single and multiple time point sample size calculations, along with openly available R code (see Software Availability). Users can obtain required sample sizes by specifying interpretable inputs such as the anticipated treatment effect, the assumed variance-covariance structure, and the contrast of interest. In addition, we have added a worked example in the Appendices demonstrating how these inputs are selected and applied in a realistic clinical trial setting.
Response to General comment
We thank the reviewer for the comment. In the revised version, we have carefully corrected typographical errors, incomplete sentences, and improperly placed capital letters, and clarified statements that were previously unclear.

Response to Comment 1
We thank the reviewer for this important clarification. In the revised manuscript, we have expanded the Introduction to explicitly explain why estimator precision is a critical metric for the target population of professional athletes. Specifically, we emphasize that professional athletes exhibit high inter- and intra-individual variability, and that precise estimation is essential for accurate performance assessment, monitoring, and interpretation of small but practically meaningful effects. We believe this added explanation clarifies the relevance of precision for the chosen population.

Response to Comment 2
We thank the reviewer for this comment. We agree that there are numerous longitudinal studies in the literature. However, our intention was not to provide a comprehensive review of longitudinal research, but rather to illustrate that even in well‑established and prestigious journals, longitudinal studies often provide limited detail regarding sample size calculation. We have now clarified this scope in the manuscript to avoid any misunderstanding and to emphasize that references 3 and 4 were cited as illustrative examples rather than as an exhaustive review of the literature.

Response to Comment 3
We thank the reviewer for pointing this out. The spelling has been corrected in the revised version of the manuscript

Response to Comment 4
We thank the reviewer for this clarification. We have now explicitly stated that the normal distribution of the sample mean is an approximation that becomes increasingly accurate with larger sample sizes, as justified by the central limit theorem. This clarification has been added to the ‘Notation and framework’ section in the revised manuscript.

Response to Comment 5
We thank the reviewer for this helpful suggestion. To improve clarity, we have revised the hypotheses to explicitly state “” and “ , where , and δ represents the expected difference. This definition clarifies the role of δ in the sample size derivation and avoids ambiguous statements. The corresponding revision has been made in the Methods section and aligns with Equation (2) accordingly.

Response to Comment 6
We thank the reviewer for pointing this out. The null and alternative hypothesis notation and the definition of the transpose operator have been corrected in the revised manuscript.

Response to Comment 7
We thank the reviewer for this important clarification. We agree that ψ_c and Λ, as population parameters, cannot constitute a test statistic. In the revised manuscript, we have therefore replaced these quantities with their corresponding sample estimates and revised the notation accordingly. As a result, Formula (8) is now explicitly defined as a function of sample‑based random variables, ensuring that it represents a valid test statistic.

Response to Comment 8
We thank the reviewer for carefully examining the derivation. We agree that the covariance expansion was incorrectly stated and that the origin of the value 4 in Equation (9) was not justified. In the revised manuscript, we have corrected the covariance expression by explicitly accounting for all relevant covariance terms and have revised Equation (9) accordingly. The incorrect constant has been removed, and the derivation is now consistent with standard covariance identities.

Response to Comment 9
We thank the reviewer for these constructive suggestions to improve clarity. In the revised manuscript, we have explicitly defined and clarified the assumed time trend in the sample size calculation. We have also revised the wording as suggested, replacing “allowing” with “choosing.” In addition, we have added a summary table to clearly indicate the selected time points under each scenario.

Response to Comment 10
We thank the reviewer for this important query. In the revised manuscript, we have explicitly detailed how the sample size of 287 was derived, including the expected difference under the alternative hypothesis, the corresponding null hypothesis, and the selected time point used in the sample size calculation, thereby making the derivation and underlying assumptions transparent and reproducible.

Response to Comment 11
We thank the reviewer for this insightful comment. Our objective in choosing a mean‑over‑time contrast was to target the average treatment effect across the follow‑up period rather than the effect at a single time point. Accordingly, equal weights (1/4 at each post‑baseline time point) were used to reflect this estimand.
Although a linear time trend is assumed for the purpose of modeling and sample size calculation, the choice of an average contrast does not require linearity of the treatment effect over time. This approach is particularly relevant in chronic disease settings, where sustained control of biomarkers over time is often of greater clinical interest than the maximum effect at a specific visit, and it also provides robustness to missing data due to loss to follow‑up.
We have clarified this rationale in the revised manuscript to better explain the choice of the equal‑weight contrast and its relationship to the assumed time trend.

Response to Comment 12
We thank the reviewer for this insightful question. Under a compound symmetry correlation structure, when the within‑subject correlation is 0.5, the variance reduction gained from repeated measurements is exactly offset by their correlation, resulting in a required sample size equivalent to that of a single‑time‑point assessment. When the correlation is lower than 0.5, repeated observations provide less shared information across time points, leading to a reduced efficiency and consequently a larger required sample size.
To provide a detailed mathematical justification of these results, we have added two appendices that explicitly derive the variance expressions and resulting sample size formulas under different correlation assumptions.

Response to Comment 13
We thank the reviewer for this important question. When within‑subject correlation is low, repeated measurements provide relatively little shared information about the underlying treatment effect, as observations across time behave nearly independently. For contrasts that average treatment effects over time, this can lead to increased variability because the additional measurements contribute more noise than signal.
In contrast, a single time‑point assessment, particularly at a clinically relevant follow‑up time point may yield a more efficient estimator under low-correlation settings, as it avoids the accumulation of variability from weakly correlated repeated measures. This explains why, in our framework, the single‑time‑point assessment can require a smaller sample size when the correlation is low. Appendix added can also give a mathematical glimpse.

Response to Comment 14
We thank the reviewer for this insightful question. Under an AR(1) correlation structure, increasing the number of time points does not necessarily lead to increased efficiency when within‑subject correlation is moderate to high. As additional time points are included, the covariance between observations accumulates, and beyond a certain correlation threshold, the variance component associated with the mean‑over‑time contrast () increases rather than decreasing.
This occurs because strongly correlated repeated measurements contribute limited new information while increasing the overall covariance contribution to the contrast variance. As a result, when the correlation exceeds approximately 0.35 in our setting, adding more time points inflates the variance of the mean‑over‑time estimator, leading to a higher required sample size. To clarify this behavior, we have added an appendix that explicitly compares variance components across different correlation levels and side by side layout for 4 timepoints and 10 timepoints case.

Response to Comment 15
We thank the reviewer for this important clarification. We agree that, for the mean difference contrast, the comparison is based on two time points (baseline and a post‑baseline visit), and that it is the temporal distance between these time points, rather than the total number of visits that determines the correlation and variance under an AR(1) structure. We have revised the manuscript to clarify this wording and avoid any implication that the total number of visits alone drives the sample size.
We also agree that the observed increase in required sample size when the post‑baseline assessment is further from baseline is not a general property but depends on the joint behaviour of the serial correlation and the mean difference at the chosen time point. This dependence is now acknowledged more explicitly in the revised text and reiterated in the conclusion.

Response to Comment 16
We thank the reviewer for this helpful suggestion. To facilitate direct use by readers without a statistical background, the manuscript provides explicit formulas for both single and multiple time point sample size calculations, along with openly available R code (see Software Availability). Users can obtain required sample sizes by specifying interpretable inputs such as the anticipated treatment effect, the assumed variance-covariance structure, and the contrast of interest. In addition, we have added a worked example in the Appendices demonstrating how these inputs are selected and applied in a realistic clinical trial setting.
Competing Interests: No competing interests were disclosed. Close
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Version 1

VERSION 1

PUBLISHED 21 Dec 2022

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Reviewer Report 10 Jul 2024

Ronald Geskus, Oxford University, Oxford, England, UK

Not Approved

https://doi.org/10.5256/f1000research.137160.r294678

The authors describe some formulas for sample size calculations with longitudinal data with a continuous outcome. The authors only consider one specific approach, namely generalized least squares and comparing changes from baseline. This latter approach is discouraged, see e.g. https://www.bmj.com/content/323/7321/1123.

More general methods to perform sample size and power calculations with longitudinal data have been described and implemented in software (e.g. https://journal.r-project.org/articles/RJ-2022-022/).

Also, there are some descriptions that need further attention.

1. The introduction describes the basic principles of hypothesis testing. This can be assumed known and therefore I suggest to leave it out of the paper.

2. Page 3: The authors describe sampling error as the uncertainty arising from: "the limitation to examine every situation, the absolute precision of the effect of protein shake on the athlete's performance would be hard to measure." I am not sure whether I understand what is written here, but I think it is not correct.

3. Page 3: Sampling error decreases with increasing sample size, but it is not true that " there comes a point where increasing the sample size has no further effect on the sampling error".

4. Formula (2): power is typically calculated for a specific choice of the effect size, not for all delta>0 at once.

5. Page 7: "Appropriate sample size was calculated for multi-time and single time cases with different scenarios to achieve an overall mean treatment difference (0.9 points)." What is meant with mean difference in the situation with more than 2 time points? The difference could vary per time point but still give a mean difference of 0.9.
The last sentence of that paragraph is irrelevant information.

6. How was \sigma_{ij} chosen?

7. Page 8: make more explicit that a discrete AR(1) is assumed. There also exitsts a continuous AR(1)

8. Page 9: The authors write: "For correlation = 0.1 to 0.35 the sample size coincides with that of single time point assessment." I don't see this in the figure

9. Page 9: I don't understand the rationale "effect remains only for some time and then the disease condition reverses back' and "each dose reduces the disease severity and the over the course it is totally removed from the patient’s body" for the two contrasts.

10. Explain why the sample size goes down to zero if the correlation approaches 1. A correlation of 1 does not make sense. It implies that the second measurement is completely determined by the first, and the intervention cannot have any impact.

11. Legends Fig 1 and 2 have been mixed up.

Is the rationale for developing the new method (or application) clearly explained?

No
Is the description of the method technically sound?

Yes
Are sufficient details provided to allow replication of the method development and its use by others?

Yes
If any results are presented, are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions about the method and its performance adequately supported by the findings presented in the article?

Partly

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Biostatistics

CITE

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Author Response 05 Sep 2024

Sarfaraz Sayyed, Analytics, Novartis Healthcare Pvt. Ltd., Hyderabad, 500080, India

05 Sep 2024

Author Response

Thanks Dr. Ronald Geskus for your review and advice.
Please find below our responses.
- Regarding analyzing change from baseline, we completely agree with you. Our intention was just to ... Continue reading Thanks Dr. Ronald Geskus for your review and advice.
Please find below our responses.
- Regarding analyzing change from baseline, we completely agree with you. Our intention was just to propose a framework, but we have modified it now to make it more precise.

- Regarding methods to perform sample size calculation, we agree with you, but our intention here is to highlight using simple methods to the clinical researchers, the difference in samples required when we have longitudinal information available under 2 different correlation structures.

Comment 1: We kept this as this would be helpful to new researchers not having statistical background.
Comment 2: We have modified it for better understanding.
Comment 3: We have modified and added a note to have more clarity.
Comment 4: We fully agree with you, this was a generic equation to show we are looking positive change in response and then further with the examples we have assumed for certain fixed delta how the sample size varies. But have modified it to avoid confusion.
Comment 5: We have modified the sentence for better clarity and also removed the last sentence of the paragraph.
Comment 6: Sigma was taken from a trial data (reference given) and was assumed to be same over all the time periods.
Comment 7: Thanks for bringing this up, we have updated this in the manuscript now.
Comment 8: This was wrongly written under figure 1, it was supposed to be under figure 2 conclusion. we have made the necessary changes now.
Comment 9: The first rationale refers to chronic diseases where the disease can usually be controlled but not cured. So here we are trying to see that the average effect from the treatment is the one which is expected to control the disease. Have changed the wordings for better understanding. The second rationale is for infectious diseases where with the right treatment regimen the disease will be cured. Have now tried to put in better words.
Comment 10: Agree with your comment on both the extreme cases that a correlation of 1 would imply all the effect is due to intervention not leaving any space for error and also that correlation 0 would mean no effect from the intervention. We have now revised the plots to start from 0.05 till 0.9
Comment 11: Have made the necessary changes.

- Regarding clarifying the rationale, basically in clinical trials the use of all time point data is not generally used. The effect at intermittent time points and the covariance structure between time points would be helpful which otherwise not taken into consideration results in large sample size.

- Regarding conclusion on the methods, we have updated the conclusion to have more insights, how taking into correlation would benefit and also how the effects diminish even if too many timepoints are added.
Thanks Dr. Ronald Geskus for your review and advice.
Please find below our responses.
- Regarding analyzing change from baseline, we completely agree with you. Our intention was just to propose a framework, but we have modified it now to make it more precise.

- Regarding methods to perform sample size calculation, we agree with you, but our intention here is to highlight using simple methods to the clinical researchers, the difference in samples required when we have longitudinal information available under 2 different correlation structures.

Comment 1: We kept this as this would be helpful to new researchers not having statistical background.
Comment 2: We have modified it for better understanding.
Comment 3: We have modified and added a note to have more clarity.
Comment 4: We fully agree with you, this was a generic equation to show we are looking positive change in response and then further with the examples we have assumed for certain fixed delta how the sample size varies. But have modified it to avoid confusion.
Comment 5: We have modified the sentence for better clarity and also removed the last sentence of the paragraph.
Comment 6: Sigma was taken from a trial data (reference given) and was assumed to be same over all the time periods.
Comment 7: Thanks for bringing this up, we have updated this in the manuscript now.
Comment 8: This was wrongly written under figure 1, it was supposed to be under figure 2 conclusion. we have made the necessary changes now.
Comment 9: The first rationale refers to chronic diseases where the disease can usually be controlled but not cured. So here we are trying to see that the average effect from the treatment is the one which is expected to control the disease. Have changed the wordings for better understanding. The second rationale is for infectious diseases where with the right treatment regimen the disease will be cured. Have now tried to put in better words.
Comment 10: Agree with your comment on both the extreme cases that a correlation of 1 would imply all the effect is due to intervention not leaving any space for error and also that correlation 0 would mean no effect from the intervention. We have now revised the plots to start from 0.05 till 0.9
Comment 11: Have made the necessary changes.

- Regarding clarifying the rationale, basically in clinical trials the use of all time point data is not generally used. The effect at intermittent time points and the covariance structure between time points would be helpful which otherwise not taken into consideration results in large sample size.

- Regarding conclusion on the methods, we have updated the conclusion to have more insights, how taking into correlation would benefit and also how the effects diminish even if too many timepoints are added.
Competing Interests: No competing interests were disclosed. Close
Report a concern
Respond or Comment

COMMENTS ON THIS REPORT

Author Response 05 Sep 2024

Sarfaraz Sayyed, Analytics, Novartis Healthcare Pvt. Ltd., Hyderabad, 500080, India

05 Sep 2024

Author Response

Thanks Dr. Ronald Geskus for your review and advice.
Please find below our responses.
- Regarding analyzing change from baseline, we completely agree with you. Our intention was just to ... Continue reading Thanks Dr. Ronald Geskus for your review and advice.
Please find below our responses.
- Regarding analyzing change from baseline, we completely agree with you. Our intention was just to propose a framework, but we have modified it now to make it more precise.

- Regarding methods to perform sample size calculation, we agree with you, but our intention here is to highlight using simple methods to the clinical researchers, the difference in samples required when we have longitudinal information available under 2 different correlation structures.

Comment 1: We kept this as this would be helpful to new researchers not having statistical background.
Comment 2: We have modified it for better understanding.
Comment 3: We have modified and added a note to have more clarity.
Comment 4: We fully agree with you, this was a generic equation to show we are looking positive change in response and then further with the examples we have assumed for certain fixed delta how the sample size varies. But have modified it to avoid confusion.
Comment 5: We have modified the sentence for better clarity and also removed the last sentence of the paragraph.
Comment 6: Sigma was taken from a trial data (reference given) and was assumed to be same over all the time periods.
Comment 7: Thanks for bringing this up, we have updated this in the manuscript now.
Comment 8: This was wrongly written under figure 1, it was supposed to be under figure 2 conclusion. we have made the necessary changes now.
Comment 9: The first rationale refers to chronic diseases where the disease can usually be controlled but not cured. So here we are trying to see that the average effect from the treatment is the one which is expected to control the disease. Have changed the wordings for better understanding. The second rationale is for infectious diseases where with the right treatment regimen the disease will be cured. Have now tried to put in better words.
Comment 10: Agree with your comment on both the extreme cases that a correlation of 1 would imply all the effect is due to intervention not leaving any space for error and also that correlation 0 would mean no effect from the intervention. We have now revised the plots to start from 0.05 till 0.9
Comment 11: Have made the necessary changes.

- Regarding clarifying the rationale, basically in clinical trials the use of all time point data is not generally used. The effect at intermittent time points and the covariance structure between time points would be helpful which otherwise not taken into consideration results in large sample size.

- Regarding conclusion on the methods, we have updated the conclusion to have more insights, how taking into correlation would benefit and also how the effects diminish even if too many timepoints are added.
Thanks Dr. Ronald Geskus for your review and advice.
Please find below our responses.
- Regarding analyzing change from baseline, we completely agree with you. Our intention was just to propose a framework, but we have modified it now to make it more precise.

- Regarding methods to perform sample size calculation, we agree with you, but our intention here is to highlight using simple methods to the clinical researchers, the difference in samples required when we have longitudinal information available under 2 different correlation structures.

Comment 1: We kept this as this would be helpful to new researchers not having statistical background.
Comment 2: We have modified it for better understanding.
Comment 3: We have modified and added a note to have more clarity.
Comment 4: We fully agree with you, this was a generic equation to show we are looking positive change in response and then further with the examples we have assumed for certain fixed delta how the sample size varies. But have modified it to avoid confusion.
Comment 5: We have modified the sentence for better clarity and also removed the last sentence of the paragraph.
Comment 6: Sigma was taken from a trial data (reference given) and was assumed to be same over all the time periods.
Comment 7: Thanks for bringing this up, we have updated this in the manuscript now.
Comment 8: This was wrongly written under figure 1, it was supposed to be under figure 2 conclusion. we have made the necessary changes now.
Comment 9: The first rationale refers to chronic diseases where the disease can usually be controlled but not cured. So here we are trying to see that the average effect from the treatment is the one which is expected to control the disease. Have changed the wordings for better understanding. The second rationale is for infectious diseases where with the right treatment regimen the disease will be cured. Have now tried to put in better words.
Comment 10: Agree with your comment on both the extreme cases that a correlation of 1 would imply all the effect is due to intervention not leaving any space for error and also that correlation 0 would mean no effect from the intervention. We have now revised the plots to start from 0.05 till 0.9
Comment 11: Have made the necessary changes.

- Regarding clarifying the rationale, basically in clinical trials the use of all time point data is not generally used. The effect at intermittent time points and the covariance structure between time points would be helpful which otherwise not taken into consideration results in large sample size.

- Regarding conclusion on the methods, we have updated the conclusion to have more insights, how taking into correlation would benefit and also how the effects diminish even if too many timepoints are added.
Competing Interests: No competing interests were disclosed. Close
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Reviewer Report 30 May 2024

Kiranmoy Das, Indian Statistical Institute, Kolkata, West Bengal, India

Not Approved

https://doi.org/10.5256/f1000research.137160.r273771

Reviewer’s comments on the manuscript “Sample-size variation in single-time post-dose assessment vs multi-time post-dose assessment”

The manuscript attempts to address a very important and interesting problem. However, the presentation is not quite clear, there are lots of typos and grammar issues here and there. The methodology developed in the manuscript can be improved. My comments are given below. After addressing these comments and after a substantial revision, I can make some positive comment on the manuscript.
1. Introduction: Please provide some good motivation and nice description of the problem. There are several typos. Please avoid the phrase For e.g., this should be For example, and also “The size of the sample and the effect size are major factors which affects”, this should be like “The sample size and effect size are major factors that affect”.
2. Methods: Equation (1), you are assuming here that the variance sigma^2 is known. Please specify it, and also how appropriate this assumption is in real application?
3. For the longitudinal studies, your method depends on the auto-correlation \rho. You need to estimate \rho before calculating the sample size. Please clarify that and also discuss how to estimate \rho. Maybe you run a pilot trial and get an estimate?
4. Both the figures 1 and 2 focus on how the sample size varies with correlation \rho. For a longitudinal trial, it maybe more important to know how it covaries with the number of time points. Correlation is something that cannot be adjusted, but the number of time points can be adjusted as per the requirement. I think this issue has been ignored in the manuscript.

Is the rationale for developing the new method (or application) clearly explained?

No
Is the description of the method technically sound?

Yes
Are sufficient details provided to allow replication of the method development and its use by others?

No
If any results are presented, are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions about the method and its performance adequately supported by the findings presented in the article?

Yes

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Biostatistics, Longitudinal data analysis, Bayesian modeling

CITE

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Author Response 05 Sep 2024

Sarfaraz Sayyed, Analytics, Novartis Healthcare Pvt. Ltd., Hyderabad, 500080, India

05 Sep 2024

Author Response

Thanks Dr. Kiranmoy Das for your review and advice.
Please see our response below.
Comment 1: We have updated the introduction section.
Comment 2: Justification now added above equation no. ... Continue reading Thanks Dr. Kiranmoy Das for your review and advice.
Please see our response below.
Comment 1: We have updated the introduction section.
Comment 2: Justification now added above equation no. 1.
Comment 3: An explanation of rho estimation is now added.
Comment 4: We have added a paragraph under the conclusion section to address this.

We have added a few lines in the Introduction section to clarify the rationale.
Regarding details for replication of the method, details on assumptions are specified in the section “Calculation of sample size” in the manuscript. The software availability section provides the link to the related R program residing in Github.
Thanks Dr. Kiranmoy Das for your review and advice.
Please see our response below.
Comment 1: We have updated the introduction section.
Comment 2: Justification now added above equation no. 1.
Comment 3: An explanation of rho estimation is now added.
Comment 4: We have added a paragraph under the conclusion section to address this.

We have added a few lines in the Introduction section to clarify the rationale.
Regarding details for replication of the method, details on assumptions are specified in the section “Calculation of sample size” in the manuscript. The software availability section provides the link to the related R program residing in Github.
Competing Interests: No competing interests were disclosed. Close
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Author Response 05 Sep 2024

Sarfaraz Sayyed, Analytics, Novartis Healthcare Pvt. Ltd., Hyderabad, 500080, India

05 Sep 2024

Author Response

Thanks Dr. Kiranmoy Das for your review and advice.
Please see our response below.
Comment 1: We have updated the introduction section.
Comment 2: Justification now added above equation no. ... Continue reading Thanks Dr. Kiranmoy Das for your review and advice.
Please see our response below.
Comment 1: We have updated the introduction section.
Comment 2: Justification now added above equation no. 1.
Comment 3: An explanation of rho estimation is now added.
Comment 4: We have added a paragraph under the conclusion section to address this.

We have added a few lines in the Introduction section to clarify the rationale.
Regarding details for replication of the method, details on assumptions are specified in the section “Calculation of sample size” in the manuscript. The software availability section provides the link to the related R program residing in Github.
Thanks Dr. Kiranmoy Das for your review and advice.
Please see our response below.
Comment 1: We have updated the introduction section.
Comment 2: Justification now added above equation no. 1.
Comment 3: An explanation of rho estimation is now added.
Comment 4: We have added a paragraph under the conclusion section to address this.

We have added a few lines in the Introduction section to clarify the rationale.
Regarding details for replication of the method, details on assumptions are specified in the section “Calculation of sample size” in the manuscript. The software availability section provides the link to the related R program residing in Github.
Competing Interests: No competing interests were disclosed. Close
Report a concern

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Version 3

VERSION 3 PUBLISHED 21 Dec 2022

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Version 2 (revision) 05 Sep 24	read	read	read
Version 1 21 Dec 22	read	read

Kiranmoy Das, Indian Statistical Institute, Kolkata, India
Ronald Geskus, Oxford University, Oxford, UK
Simon Vandekar, Vanderbilt University Medical Center, Tennessee, USA

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7 Views

21 Nov 2024 | for Version 2

Simon Vandekar, Vanderbilt University Medical Center, Tennessee, USA

7 Views Cite this report Responses(1)

Not Approved

Is the rationale for developing the new method (or application) clearly explained?

Yes
Is the description of the method technically sound?

Partly
Are sufficient details provided to allow replication of the method development and its use by others?

Yes
If any results are presented, are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions about the method and its performance adequately supported by the findings presented in the article?

Partly

References

1. Ahn C, Heo M, Zhang S: Sample Size Calculations for Clustered and Longitudinal Outcomes in Clinical Research. 2014. Publisher Full Text

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Biostatistics

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Author Response

05 Feb 2026

Sarfaraz Sayyed, Analytics, Novartis Healthcare Pvt. Ltd., Hyderabad, 500080, India

Response to Comment 1
We thank the reviewer for this important and thorough comment and fully agree that there is a substantial body of literature on power and sample size calculation for longitudinal studies. Our intent was not to provide a comprehensive review or to introduce new theoretical developments in this well‑established area. Rather, the aim of the present paper is to provide a focused and accessible comparison between sample size calculations based on a single post‑baseline assessment and those incorporating multiple post‑baseline time points, under commonly used correlation structures.
To better position our work within the existing literature, we have expanded the related work section to explicitly acknowledge classical and contemporary references on longitudinal power analysis, including Diggle et al. and the monograph by Ahn, Heo, and Zhang. We have also added references to existing R packages for longitudinal power analysis and clarified how our approach complements these tools by highlighting specific design trade‑offs in simple, interpretable settings.
We hope that these revisions more clearly situate the contribution of this paper as illustrative and comparative rather than novel in a methodological sense, while providing practical insights for applied researchers.

Response to Comment 2
We thank the reviewer for this detailed comment. In the revised manuscript, we have added appropriate references for the single‑time‑point sample size calculation methodology. We have also carefully reviewed the manuscript for grammatical issues and corrected several notation inconsistencies. In particular, we revised Equations (8) and (9) to avoid expressions such as Var(

), as

represents a fixed parameter, and reformulated these expressions using appropriate sample‑based quantities. We believe these revisions improve both the mathematical clarity and presentation of the manuscript.

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6 Views

23 Oct 2024 | for Version 2

Kiranmoy Das, Indian Statistical Institute, Kolkata, West Bengal, India

6 Views Cite this report Responses(0)

Approved

I have seen the revised version of the paper and the author's comments.
I am quite satisfied and I have no additional comments.
I recommend the acceptance of the manuscript

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Biostatistics, Longitudinal data analysis, Bayesian modeling

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

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14 Views

28 Sep 2024 | for Version 2

Ronald Geskus, Oxford University, Oxford, England, UK

14 Views Cite this report Responses(1)

Not Approved

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

biostatistics

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Author Response

05 Feb 2026

Sarfaraz Sayyed, Analytics, Novartis Healthcare Pvt. Ltd., Hyderabad, 500080, India

Response to General comment
We thank the reviewer for the comment. In the revised version, we have carefully corrected typographical errors, incomplete sentences, and improperly placed capital letters, and clarified statements that were previously unclear.

Response to Comment 1
We thank the reviewer for this important clarification. In the revised manuscript, we have expanded the Introduction to explicitly explain why estimator precision is a critical metric for the target population of professional athletes. Specifically, we emphasize that professional athletes exhibit high inter- and intra-individual variability, and that precise estimation is essential for accurate performance assessment, monitoring, and interpretation of small but practically meaningful effects. We believe this added explanation clarifies the relevance of precision for the chosen population.

Response to Comment 2
We thank the reviewer for this comment. We agree that there are numerous longitudinal studies in the literature. However, our intention was not to provide a comprehensive review of longitudinal research, but rather to illustrate that even in well‑established and prestigious journals, longitudinal studies often provide limited detail regarding sample size calculation. We have now clarified this scope in the manuscript to avoid any misunderstanding and to emphasize that references 3 and 4 were cited as illustrative examples rather than as an exhaustive review of the literature.

Response to Comment 3
We thank the reviewer for pointing this out. The spelling has been corrected in the revised version of the manuscript

Response to Comment 4
We thank the reviewer for this clarification. We have now explicitly stated that the normal distribution of the sample mean is an approximation that becomes increasingly accurate with larger sample sizes, as justified by the central limit theorem. This clarification has been added to the ‘Notation and framework’ section in the revised manuscript.

Response to Comment 5
We thank the reviewer for this helpful suggestion. To improve clarity, we have revised the hypotheses to explicitly state “

” and “

, where

, and δ represents the expected difference. This definition clarifies the role of δ in the sample size derivation and avoids ambiguous statements. The corresponding revision has been made in the Methods section and aligns with Equation (2) accordingly.

Response to Comment 6
We thank the reviewer for pointing this out. The null and alternative hypothesis notation and the definition of the transpose operator have been corrected in the revised manuscript.

Response to Comment 7
We thank the reviewer for this important clarification. We agree that ψ_c and Λ, as population parameters, cannot constitute a test statistic. In the revised manuscript, we have therefore replaced these quantities with their corresponding sample estimates and revised the notation accordingly. As a result, Formula (8) is now explicitly defined as a function of sample‑based random variables, ensuring that it represents a valid test statistic.

Response to Comment 8
We thank the reviewer for carefully examining the derivation. We agree that the covariance expansion was incorrectly stated and that the origin of the value 4 in Equation (9) was not justified. In the revised manuscript, we have corrected the covariance expression by explicitly accounting for all relevant covariance terms and have revised Equation (9) accordingly. The incorrect constant has been removed, and the derivation is now consistent with standard covariance identities.

Response to Comment 9
We thank the reviewer for these constructive suggestions to improve clarity. In the revised manuscript, we have explicitly defined and clarified the assumed time trend in the sample size calculation. We have also revised the wording as suggested, replacing “allowing” with “choosing.” In addition, we have added a summary table to clearly indicate the selected time points under each scenario.

Response to Comment 10
We thank the reviewer for this important query. In the revised manuscript, we have explicitly detailed how the sample size of 287 was derived, including the expected difference under the alternative hypothesis, the corresponding null hypothesis, and the selected time point used in the sample size calculation, thereby making the derivation and underlying assumptions transparent and reproducible.

Response to Comment 11
We thank the reviewer for this insightful comment. Our objective in choosing a mean‑over‑time contrast was to target the average treatment effect across the follow‑up period rather than the effect at a single time point. Accordingly, equal weights (1/4 at each post‑baseline time point) were used to reflect this estimand.
Although a linear time trend is assumed for the purpose of modeling and sample size calculation, the choice of an average contrast does not require linearity of the treatment effect over time. This approach is particularly relevant in chronic disease settings, where sustained control of biomarkers over time is often of greater clinical interest than the maximum effect at a specific visit, and it also provides robustness to missing data due to loss to follow‑up.
We have clarified this rationale in the revised manuscript to better explain the choice of the equal‑weight contrast and its relationship to the assumed time trend.

Response to Comment 12
We thank the reviewer for this insightful question. Under a compound symmetry correlation structure, when the within‑subject correlation is 0.5, the variance reduction gained from repeated measurements is exactly offset by their correlation, resulting in a required sample size equivalent to that of a single‑time‑point assessment. When the correlation is lower than 0.5, repeated observations provide less shared information across time points, leading to a reduced efficiency and consequently a larger required sample size.
To provide a detailed mathematical justification of these results, we have added two appendices that explicitly derive the variance expressions and resulting sample size formulas under different correlation assumptions.

Response to Comment 13
We thank the reviewer for this important question. When within‑subject correlation is low, repeated measurements provide relatively little shared information about the underlying treatment effect, as observations across time behave nearly independently. For contrasts that average treatment effects over time, this can lead to increased variability because the additional measurements contribute more noise than signal.
In contrast, a single time‑point assessment, particularly at a clinically relevant follow‑up time point may yield a more efficient estimator under low-correlation settings, as it avoids the accumulation of variability from weakly correlated repeated measures. This explains why, in our framework, the single‑time‑point assessment can require a smaller sample size when the correlation is low. Appendix added can also give a mathematical glimpse.

Response to Comment 14
We thank the reviewer for this insightful question. Under an AR(1) correlation structure, increasing the number of time points does not necessarily lead to increased efficiency when within‑subject correlation is moderate to high. As additional time points are included, the covariance between observations accumulates, and beyond a certain correlation threshold, the variance component associated with the mean‑over‑time contrast (

) increases rather than decreasing.
This occurs because strongly correlated repeated measurements contribute limited new information while increasing the overall covariance contribution to the contrast variance. As a result, when the correlation exceeds approximately 0.35 in our setting, adding more time points inflates the variance of the mean‑over‑time estimator, leading to a higher required sample size. To clarify this behavior, we have added an appendix that explicitly compares variance components across different correlation levels and side by side layout for 4 timepoints and 10 timepoints case.

Response to Comment 15
We thank the reviewer for this important clarification. We agree that, for the mean difference contrast, the comparison is based on two time points (baseline and a post‑baseline visit), and that it is the temporal distance between these time points, rather than the total number of visits that determines the correlation and variance under an AR(1) structure. We have revised the manuscript to clarify this wording and avoid any implication that the total number of visits alone drives the sample size.
We also agree that the observed increase in required sample size when the post‑baseline assessment is further from baseline is not a general property but depends on the joint behaviour of the serial correlation and the mean difference at the chosen time point. This dependence is now acknowledged more explicitly in the revised text and reiterated in the conclusion.

Response to Comment 16
We thank the reviewer for this helpful suggestion. To facilitate direct use by readers without a statistical background, the manuscript provides explicit formulas for both single and multiple time point sample size calculations, along with openly available R code (see Software Availability). Users can obtain required sample sizes by specifying interpretable inputs such as the anticipated treatment effect, the assumed variance-covariance structure, and the contrast of interest. In addition, we have added a worked example in the Appendices demonstrating how these inputs are selected and applied in a realistic clinical trial setting.

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No competing interests were disclosed.

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16 Views

10 Jul 2024 | for Version 1

Ronald Geskus, Oxford University, Oxford, England, UK

16 Views Cite this report Responses(1)

Not Approved

Is the rationale for developing the new method (or application) clearly explained?

No
Is the description of the method technically sound?

Yes
Are sufficient details provided to allow replication of the method development and its use by others?

Yes
If any results are presented, are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions about the method and its performance adequately supported by the findings presented in the article?

Partly

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Biostatistics

Respond to this report

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Author Response

05 Sep 2024

Sarfaraz Sayyed, Analytics, Novartis Healthcare Pvt. Ltd., Hyderabad, 500080, India

Thanks Dr. Ronald Geskus for your review and advice.
Please find below our responses.
- Regarding analyzing change from baseline, we completely agree with you. Our intention was just to propose a framework, but we have modified it now to make it more precise.

- Regarding methods to perform sample size calculation, we agree with you, but our intention here is to highlight using simple methods to the clinical researchers, the difference in samples required when we have longitudinal information available under 2 different correlation structures.

Comment 1: We kept this as this would be helpful to new researchers not having statistical background.
Comment 2: We have modified it for better understanding.
Comment 3: We have modified and added a note to have more clarity.
Comment 4: We fully agree with you, this was a generic equation to show we are looking positive change in response and then further with the examples we have assumed for certain fixed delta how the sample size varies. But have modified it to avoid confusion.
Comment 5: We have modified the sentence for better clarity and also removed the last sentence of the paragraph.
Comment 6: Sigma was taken from a trial data (reference given) and was assumed to be same over all the time periods.
Comment 7: Thanks for bringing this up, we have updated this in the manuscript now.
Comment 8: This was wrongly written under figure 1, it was supposed to be under figure 2 conclusion. we have made the necessary changes now.
Comment 9: The first rationale refers to chronic diseases where the disease can usually be controlled but not cured. So here we are trying to see that the average effect from the treatment is the one which is expected to control the disease. Have changed the wordings for better understanding. The second rationale is for infectious diseases where with the right treatment regimen the disease will be cured. Have now tried to put in better words.
Comment 10: Agree with your comment on both the extreme cases that a correlation of 1 would imply all the effect is due to intervention not leaving any space for error and also that correlation 0 would mean no effect from the intervention. We have now revised the plots to start from 0.05 till 0.9
Comment 11: Have made the necessary changes.

- Regarding clarifying the rationale, basically in clinical trials the use of all time point data is not generally used. The effect at intermittent time points and the covariance structure between time points would be helpful which otherwise not taken into consideration results in large sample size.

- Regarding conclusion on the methods, we have updated the conclusion to have more insights, how taking into correlation would benefit and also how the effects diminish even if too many timepoints are added.

View more View less

Competing Interests

No competing interests were disclosed.

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Reviewer Report

22 Views

30 May 2024 | for Version 1

Kiranmoy Das, Indian Statistical Institute, Kolkata, West Bengal, India

22 Views Cite this report Responses(1)

Not Approved

Is the rationale for developing the new method (or application) clearly explained?

No
Is the description of the method technically sound?

Yes
Are sufficient details provided to allow replication of the method development and its use by others?

No
If any results are presented, are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions about the method and its performance adequately supported by the findings presented in the article?

Yes

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Biostatistics, Longitudinal data analysis, Bayesian modeling

Respond to this report

Responses (1)

Author Response

05 Sep 2024

Sarfaraz Sayyed, Analytics, Novartis Healthcare Pvt. Ltd., Hyderabad, 500080, India

Thanks Dr. Kiranmoy Das for your review and advice.
Please see our response below.
Comment 1: We have updated the introduction section.
Comment 2: Justification now added above equation no. 1.
Comment 3: An explanation of rho estimation is now added.
Comment 4: We have added a paragraph under the conclusion section to address this.

We have added a few lines in the Introduction section to clarify the rationale.
Regarding details for replication of the method, details on assumptions are specified in the section “Calculation of sample size” in the manuscript. The software availability section provides the link to the related R program residing in Github.

View more View less

Competing Interests

No competing interests were disclosed.

Alongside their report, reviewers assign a status to the article:

Approved - the paper is scientifically sound in its current form and only minor, if any, improvements are suggested

Approved with reservations - A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit.

Not approved - fundamental flaws in the paper seriously undermine the findings and conclusions

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Sample size variation in single-time post-dose assessment vs multi-time post-dose assessment

Abstract

Background

Methods

Results

Conclusions

Keywords

Revised Amendments from Version 2

Introduction

Background

Objective

Notation and framework

Methods

Method for sample size with single time point assessment analysis

(1)

(2)

(3)

(4)

(5)

(6)

Method for sample size with multiple time points assessment analysis

(7)

(8)

(9)

(10)

(11)

(12)

(13)

Calculation of sample size

Table 1. Scenarios with timepoints selected.

Sample size (single time point case)

Sample size (longitudinal case)

Figure 1. Simulation results with compound symmetry correlation structure.

Figure 2. Simulation results with discrete auto regressive of order 1 correlation structure.

Contrast of repeated measures

CS variance structure with ‘mean over time’ and ‘mean difference’ contrasts

AR(1) covariance structure with ‘mean over time’ and ‘mean difference’ contrasts

Results

Under CS type of variance covariance structure ( Figure 1)

Under AR(1) type of variance covariance structure ( Figure 2)

Discussion

Conclusion

Software availability

Data availability

Extended data

Acknowledgement

References

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