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

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.

Vs

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 T = Y ¯ − X ¯ Var ( Y ¯ − X ¯ ) = Y ¯ − X ¯ σ 2 n + σ 2 n = Y ¯ − X ¯ 2 σ 2 n (1)

Now if H 0 is true ( μ 1 = μ 2 ), then T ∼ 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 Pr [ Reject H 0 | H a is true ] = Pr [ T > z 1 − α | δ ] = 1 − β , (2) 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. T ∼ N δ 2 σ 2 n 1 , under H a → T − δ 2 σ 2 n = Z ∼ N 0 1 (3) Power function = Pr T > z 1 − α δ ] = Pr Y ¯ − X ¯ − δ 2 σ 2 n > z 1 − α − δ 2 σ 2 n = 1 − Φ z 1 − α − 2 n δ 2 σ , where Φ z = Pr Z ≤ z (4)

Now in any study we would be looking for an inequality as below. 1 − Φ z 1 − α − n 2 δ σ ≥ 1 − β → β ≥ Φ z 1 − α − n 2 δ σ or z β ≥ z 1 − α − n 2 δ σ (5)

Upon solving Equation 5, we get ¹⁵ n ≥ 2 ( z 1 − α + z 1 − β ) 2 σ 2 ( Y ¯ − X ¯ ) 2 = 2 ( z α + z β ) 2 [ ( Y ¯ − X ¯ ) / σ ] 2 (6)

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.

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 = c 1 c 2 ⋮ c i ⋮ c t be the contrast to be tested for the hypothesis and let Λ = μ 11 − μ 21 μ 12 − μ 22 ⋮ μ 1 i − μ 2 i ⋮ μ 1 t − μ 2 t

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 ψ c = C ′ Λ = − 1 1 . μ 11 − μ 21 μ 12 − μ 22 = − μ 11 − μ 21 + μ 12 − μ 22 = μ 12 – μ 11 − μ 22 − μ 21 = effect in Drug 1 − effect in Drug 2 . (7)

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 [ σ 1 2 σ 12 ⋯ 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 ⋯ σ t 2 ]

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 T = ψ ̂ c Var ( ψ ̂ c ) = ψ ̂ c Var ( C ′ Λ ̂ ) = ψ ̂ c C ′ . Var ( Λ ̂ ) . C (8)

Where, ψ ̂ c and Λ ̂ denote the sample-based estimators of the corresponding population quantities, such that the resulting expression constitutes a valid test statistic such that ψ ̂ c = C ′ Λ ̂ and Λ ̂ = [ Y ¯ 1 − X ¯ 1 Y ¯ 2 − X ¯ 2 ⋮ Y ¯ i − X ¯ i ⋮ Y ¯ t − X ¯ t ]

with Y ¯ i & X ¯ i being the sample mean at time ‘ i’ in both the arms respectively.

Consider the Var ( Λ ̂ ) , Var ( Λ ̂ ) = Var [ Y ¯ 1 − X ¯ 1 Y ¯ 2 − X ¯ 2 ⋮ Y ¯ i − X ¯ i ⋮ Y ¯ t − X ¯ t ] = [ 2 n . σ 1 2 2 n . σ 12 ⋯ 2 n . σ 1 t 2 n . σ 21 2 n . σ 2 2 ⋱ 2 n . σ 2 t ⋮ ⋮ ⋱ ⋮ 2 n . σ i 1 2 n . σ i 2 ⋱ 2 n . σ it ⋮ ⋮ ⋱ ⋮ 2 n . σ t 1 2 n . σ t 2 ⋯ 2 n . σ t 2 ] = 2 n . [ σ 1 2 σ 12 ⋯ 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 ⋯ σ t 2 ] (9)

Solving for C ′ . Var ( Λ ̂ ) . C using Equation 9, we get C ′ . Var ( Λ ̂ ) . C = [ c 1 c 2 … c i … c t ] . 2 n . [ σ 1 2 σ 12 ⋯ 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 ⋯ σ t 2 ] . [ c 1 c 2 ⋮ c i ⋮ c t ] = 2 n . [ ∑ i = 1 t c i 2 σ i 2 + 2 ∑ i < j t c i c j σ ij ] = 2 n . σ c 2 , where σ c 2 = [ ∑ i = 1 t c i 2 σ i 2 + 2 ∑ i < j t c i c j σ ij ] (10)

Using Equation 10 in Equation 8 for T we get T = ψ ̂ c C ′ . Var ( Λ ̂ ) . C = ψ ̂ c 2 n . [ ∑ i = 1 n c i 2 σ i 2 + 2 ∑ i < j n c i c j σ ij ] = n 2 ( ψ ̂ c σ c 2 ) (11)

Now, if we follow similar steps as we did in single time point analysis above, we get the following inequality. β ≥ Φ z 1 − α − n 2 ψ ̂ c σ c 2 or z β ≥ z 1 − α − n 2 ψ ̂ c σ c 2 (12)

And solving Equation 12, we get ¹² n ≥ 2 ( z α + z β ) 2 σ c 2 ψ ̂ c 2 with ψ ̂ c = ∑ i = 1 t c i ( Y ¯ i − X ¯ i ) and σ c 2 = [ ∑ i = 1 t c i 2 σ i 2 + 2 ∑ i < j t c i c j σ ij ] (13)

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.

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.

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.

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.

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.

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. [ σ 2 σ 2 ρ σ 2 ρ ⋯ σ 2 ρ σ 2 ρ σ 2 σ 2 ρ ⋯ σ 2 ρ σ 2 ρ σ 2 ρ σ 2 ⋯ σ 2 ρ ⋮ ⋱ ⋮ ⋮ ⋮ σ 2 ρ σ 2 ρ σ 2 ρ ⋯ σ 2 ]

Where σ ² 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, σ ² 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 ρ ², and so on. rho is restricted such that –1< ρ <1. [ σ 2 σ 2 ρ σ 2 ρ 2 ⋯ σ 2 ρ t − 1 σ 2 ρ σ 2 σ 2 ρ ⋯ σ 2 ρ t − 2 σ 2 ρ 2 σ 2 ρ σ 2 ⋯ σ 2 ρ t − 3 ⋮ ⋱ ⋮ ⋮ ⋮ σ 2 ρ t − 1 σ 2 ρ t − 2 σ 2 ρ t − 3 ⋯ σ 2 ]

Where σ ² 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, σ ² 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.

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.

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.

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.

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.

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 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.