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Method Article
Revised

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

[version 2; peer review: 1 approved, 2 not approved]
PUBLISHED 05 Sep 2024
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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 points 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 compared 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 Auto regressive of order 1 type of correlation structure in longitudinal data. The graphical presentation is given for better visualization of the scenarios.

Results: Sample size required for highly correlated longitudinal data using multi timepoint sample size derivation method led to much smaller sample size requirement as compared to 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 carrying out the experiment. Also, we must carefully assess which method to go with when the correlation is weak. 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

Revised Amendments from Version 1

The introduction section has been updated with respect to motivation and description of the problem.
Justification has been added for obtaining the estimates of variance and correlation.
Information have been added on the attenuation with too many time points.
More clarification has been added to the parameters used.
Better explanation has been provided for the rationale of the contrast.
Plots are placed in the right order and the correlation range has been changed to 0.1 - 0.95.

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

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 testing 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. The power of a statistical test is the probability that it will correctly reject the null hypothesis when it is false, thus avoiding a Type II error. 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.

In clinical trials, carefully calculating the sample size is essential. It 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, ultimately 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 100 professional athletes for our study at random; in this case, 100 would be the sample size. Based on the data gathered from a sample of 100 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 is the term used to describe this level of uncertainty or inaccuracy. It affects the estimator’s precision, which is a metric that is important for the chosen target population of all professional athletes.

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 addresses the non-sampling errors.

Background

Longitudinal studies take longer but help determine causality and monitor the trend over time. To see how sample size calculation was addressed in published longitudinal studies we searched the databases such as 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”. The ensuing literature review did not reveal much information on details of how the sample size was calculated for these published longitudinal studies.3,4

Formulae for deriving sample size in longitudinal studies is available from several papers.5,6 Basagana, Liao and Spiegelma7 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.57

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.

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

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

Let Yij (Xij) be the outcome of interest at jth (j = 1, 2, 3, …, t) time point for the ith (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 μ1&μ2 be the population mean for the test drug and comparator drug respectively.

Let Y¯&X¯ be the sample mean outcome for the test drug and comparator drug respectively, where Y¯Nμ1σ2n and X¯Nμ2σ2n.

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:

H0:μ1μ2=δ=0, No difference between the effects of test drug and comparator drug.

Vs

Ha:μ1μ2=δ, Test drug effect (where δ > 0) is greater than comparator.

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=Y¯X¯VarY¯X¯=Y¯X¯σ2n+σ2n=Y¯X¯2σ2n

Now If H0 is true (and μ1=μ2),then TN01, else if Ha is true (i.e., δ0), 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 1β and power is the probability to reject H0 when Ha is true. In probability equation it could be written as

(2)
PrRejectH0Hais true=PrT>z1αδ]=1β,
where z1α is the threshold or the critical value which is 1α quantile from Gaussian distribution and α is the type I error or the level of significance.
(3)
TNδ2σ2n1,UnderHaTδ2σ2n=ZN01
(4)
Power function=PrT>z1αδ]=PrY¯X¯δ2σ2n>z1αδ2σ2n=1Φz1α2nδ2σ,whereΦz=PrZz

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

(5)
1Φz1αn2δσ1ββΦz1αn2δσorzβz1αn2δσ

After solving equation 5, we get

(6)
n2z1α+z1β2σ2μ1μ22=2zα+zβ2μ1μ2/σ2

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

Method for Sample size with multiple time point 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 multiple time points let the hypothesis be set as:

Ha:ψc=0, there is no difference between the effects of test and comparator drug.

Ha:ψc>0, test drug is having larger effect as compared to comparator.

Where ψc=,

Let C=c1c2cict be the contrast to be tested for the hypothesis and let Λ=μ11μ21μ12μ22μ1iμ2iμ1tμ2t

μ1i&μ2i are the mean effect in arm one and arm two at time point “i” respectively in a study with t time points. ci can take any value depending on the hypothesis we want to test.

For example, if we want to see the difference between two drugs when t = 2, then c1=1andc2=1 and the resulting ψc will be

(7)
ψc==11.μ11μ21μ12μ22=μ11μ21+μ12μ22=μ12μ11μ22μ21=effect in Drug1effect in Drug2.
Common Variance covariance matrix=σ12σ12σ1tσ12σ22σ2tσi1σi2σitσ1tσ2tσt2

Where σi2 is the variance at time point i and σij represents the covariance between time point i and j.

The test statistic assuming similar variance-covariance matrix for both arms will be given by

(8)
T=ψcVarψc=ψcVar=ψcC.VarΛ.C

Consider the VarΛ,

VarΛ=Varμ11μ21μ12μ22μ1iμ2iμ1tμ2t=Varδ1δ2δiδt=2n.σ124n.σ124n.σ1t4n.σ122n.σ224n.σ2t4n.σi14n.σi24n.σit4n.σ1t4n.σ2t2n.σt2
(9)
VarΛ=2n.σ122σ122σ1tσ12σ22σ2tσi12σi22σit2σ1t2σ2tσt2

Solving for C.VarΛ.C, we get

(10)
C.VarΛ.C=c1c2cict.2n.σ122σ122σ1tσi12σi22σit2σ1t2σ2tσt2c1c2cict=2n.i=1nci2σi2+2i<jncicjσij=2n.σc2,whereσc2=i=1nci2σi2+2i<jncicjσij

Solving equations Equation 8 and Equation 9 for T we get

(11)
T=ψcC.VarΛ.C=ψc2n.i=1nci2σi2+2i<jncicjσij=n2ψcσc2

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

(12)
βΦz1αn2ψcσc2orzβz1αn2ψcσc2

And solving Equation 12, we get12

(13)
n2zα+zβ2σc2ψc2withψc=i=1tciμ1iμ2iandσc2=i=1tci2σi2+2i<jtcicjσij

σi2=common variance in thetwogroupsattimepointi.

σij=common covariance in thetwogroups between timepointiandj.

ci=contrast appliedattimepointiandtrepresents 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

Appropriate sample size was calculated for multi-time and single time cases with different scenarios to achieve a difference of 0.9 points at the last time point between two treatment groups with an increasing trend from baseline. The common standard deviation (SD) used was 3.6 points allowing 5% two-sided type I error and 85% power. The effect size and standard deviation used here are based on a real study.13 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 lots of missing data, the characteristics till the second year's assessment were used as it had equal numbers of patients in both arms and stabilized assessments.

Sample size (single time point case)

We considered a two-arm parallel group scenario with one baseline and one post baseline timepoints to assess the change from baseline in absolute scale. Using the formulae in Equation 6 above for single timepoint analysis the sample size required per arm was 287 cases 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 10 timepoints. Each of these cases correspond to several 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 and so on.

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

94f43805-e5b8-4a8b-9536-e73146f90e5b_figure1.gif

Figure 1. Simulation results with compound symmetry correlation structure.

94f43805-e5b8-4a8b-9536-e73146f90e5b_figure2.gif

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.

σ2σ2ρσ2ρσ2ρσ2ρσ2σ2ρσ2ρσ2ρσ2ρσ2σ2ρσ2ρσ2ρσ2ρσ2

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.

σ2σ2ρσ2ρ2σ2ρt1σ2ρσ2σ2ρσ2ρt2σ2ρ2σ2ρσ2σ2ρt3σ2ρt1σ2ρt2σ2ρt3σ2

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.

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

Contrast for 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 the interest lies in observing the mean treatment effect on the subjects 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).

  • 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 subjects 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).

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. Sample Size was derived using the formulae mentioned 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, 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, 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 assessment with more timepoints requires 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 10 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 10 timepoints requires 192 per arm. This trend shows that the sample size required reduces when the correlation increases.

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 3 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 10 time points the sample size required does not changes 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 hence may help in retaining the power.

Another thing to notice was that under CS type of covariance structure for the mean over time contrast the sample size required with fewer visit is more as compared to the sample size required with higher number of visits. Whereas for the mean diff contrast the sample size remained same irrespective of the number of visits. On the other hand, under AR(1) type of covariance structure for mean over time contrast the sample size increases as the number of visit increases for ρ > 0.35 but for ρ ≤ 0.35 the sample size decreases as the number of visits increase. 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. This change with AR(1) may be due to the fact that the correlation weakens as number of visit increases.

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. 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.

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/547747570.14

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

License: MIT License.

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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 2; peer review: 1 approved, 2 not approved]. F1000Research 2024, 11:1550 (https://doi.org/10.12688/f1000research.124917.2)
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ApprovedThe 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 approvedFundamental flaws in the paper seriously undermine the findings and conclusions
Version 2
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PUBLISHED 05 Sep 2024
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Reviewer Report 21 Nov 2024
Simon Vandekar, Vanderbilt University Medical Center, Tennessee, USA 
Not Approved
VIEWS 3
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, ... Continue reading
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Vandekar S. Reviewer Report For: Sample size variation in single-time post-dose assessment vs multi-time post-dose assessment [version 2; peer review: 1 approved, 2 not approved]. F1000Research 2024, 11:1550 (https://doi.org/10.5256/f1000research.170509.r333804)
NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article.
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Reviewer Report 23 Oct 2024
Kiranmoy Das, Indian Statistical Institute, Kolkata, West Bengal, India 
Approved
VIEWS 4
I have seen the revised version of the paper and the author's comments.
I ... Continue reading
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Das K. Reviewer Report For: Sample size variation in single-time post-dose assessment vs multi-time post-dose assessment [version 2; peer review: 1 approved, 2 not approved]. F1000Research 2024, 11:1550 (https://doi.org/10.5256/f1000research.170509.r320982)
NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article.
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Reviewer Report 28 Sep 2024
Ronald Geskus, Oxford University, Oxford, England, UK 
Not Approved
VIEWS 10
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 ... Continue reading
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Geskus R. Reviewer Report For: Sample size variation in single-time post-dose assessment vs multi-time post-dose assessment [version 2; peer review: 1 approved, 2 not approved]. F1000Research 2024, 11:1550 (https://doi.org/10.5256/f1000research.170509.r320983)
NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article.
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Reviewer Report 10 Jul 2024
Ronald Geskus, Oxford University, Oxford, England, UK 
Not Approved
VIEWS 15
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.
... Continue reading
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Geskus R. Reviewer Report For: Sample size variation in single-time post-dose assessment vs multi-time post-dose assessment [version 2; peer review: 1 approved, 2 not approved]. F1000Research 2024, 11:1550 (https://doi.org/10.5256/f1000research.137160.r294678)
NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article.
  • 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
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
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Reviewer Report 30 May 2024
Kiranmoy Das, Indian Statistical Institute, Kolkata, West Bengal, India 
Not Approved
VIEWS 18
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 ... Continue reading
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Das K. Reviewer Report For: Sample size variation in single-time post-dose assessment vs multi-time post-dose assessment [version 2; peer review: 1 approved, 2 not approved]. F1000Research 2024, 11:1550 (https://doi.org/10.5256/f1000research.137160.r273771)
NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article.
  • 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
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. 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

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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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