Sample size variation in single-time post-dose assessment vs multi-time post-dose assessment [version 4; peer review: 2 approved, 2 not approved]

Дата публикации: 30-09-2026 06:30:56

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.

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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 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. The distribution of Y¯ & X¯ can be approximated as N(μ1,σ2n) and N(μ2,σ2n) 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

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:μ1=μ2 , no difference between the effects of test drug and comparator drug.

Ha:μ1>μ2 , test drug effect is greater than comparator and μ1−μ2= 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=Y¯−X¯Var(Y¯−X¯)=Y¯−X¯σ2n+σ2n=Y¯−X¯2σ2n

Now if H0 is true ( μ1=μ2 ), then T∼N(0,1) , else if Ha is true (i.e., μ1>μ2 ) , 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 H0 when Ha is true. In probability equation it could be written as

(2)Pr[RejectH0|Hais true]=Pr[T>z1−α|δ]=1−β,

where z1−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∼Nδ2σ2n1,underHa→T−δ2σ2n=Z∼N01

(4)Power function=PrT>z1−αδ]=PrY¯−X¯−δ2σ2n>z1−α−δ2σ2n=1−Φz1−α−2nδ2σ,whereΦz=PrZ≤z

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

(5)1−Φz1−α−n2δσ≥1−β→β≥Φz1−α−n2δσorzβ≥z1−α−n2δσ

Upon solving Equation 5, we get¹⁵

(6)n≥2(z1−α+z1−β)2σ2(Y¯−X¯)2=2(zα+zβ)2[(Y¯−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=c1c2⋮ci⋮ct be the contrast to be tested for the hypothesis and let Λ=μ11−μ21μ12−μ22⋮μ1i−μ2i⋮μ1t−μ2t

With the hypothesis to be tested as:

H0: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=C′Λand C′ is the transpose of C.

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 evaluate.

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=C′Λ=−11.μ11−μ21μ12−μ22=−μ11−μ21+μ12−μ22=μ12–μ11−μ22−μ21=effect in Drug1−effect in Drug2.

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

[σ12σ12⋯σ1tσ21σ22⋱σ2t⋮⋮⋱⋮σi1σi2⋱σit⋮⋮⋱⋮σt1σt2⋯σt2]

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

The test statistic is given by

(8)T=ψ̂cVar(ψ̂c)=ψ̂cVar(C′Λ̂)=ψ̂cC′.Var(Λ̂).C

Where, ψ̂c and Λ̂ denote the sample-based estimators of the corresponding population quantities, such that the resulting expression constitutes a valid test statistic such that

ψ̂c=C′Λ̂andΛ̂=[Y¯1−X¯1Y¯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(Λ̂) ,

(9)Var(Λ̂)=Var[Y¯1−X¯1Y¯2−X¯2⋮Y¯i−X¯i⋮Y¯t−X¯t]=[2n.σ122n.σ12⋯2n.σ1t2n.σ212n.σ22⋱2n.σ2t⋮⋮⋱⋮2n.σi12n.σi2⋱2n.σit⋮⋮⋱⋮2n.σt12n.σt2⋯2n.σt2]=2n.[σ12σ12⋯σ1tσ21σ22⋱σ2t⋮⋮⋱⋮σi1σi2⋱σit⋮⋮⋱⋮σt1σt2⋯σt2]

Solving for C′.Var(Λ̂).C using Equation 9, we get

(10)C′.Var(Λ̂).C=[c1c2…ci…ct].2n.[σ12σ12⋯σ1tσ21σ22⋱σ2t⋮⋮⋱⋮σi1σi2⋱σit⋮⋮⋱⋮σt1σt2⋯σt2].[c1c2⋮ci⋮ct]=2n.[∑i=1tci2σi2+2∑i<jtcicjσij]=2n.σc2,whereσc2=[∑i=1tci2σi2+2∑i<jtcicjσij]

Using Equation 10 in Equation 8 for T we get

(11)T=ψ̂cC′.Var(Λ̂).C=ψ̂c2n.[∑i=1nci2σi2+2∑i<jncicjσij]=n2(ψ̂cσc2)

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

(12)β≥Φz1−α−n2ψ̂cσc2orzβ≥z1−α−n2ψ̂cσc2

And solving Equation 12, we get¹²

(13)n≥2(zα+zβ)2σc2ψ̂c2withψ̂c=∑i=1tci(Y¯i−X¯i)andσc2=[∑i=1tci2σi2+2∑i<jtcicjσij]

σi2 = common variance(assumed known)in the two groups at timepoint ‘i’

σij = common covariance(assumed known) in the two groups between timepoint ‘i’ and ‘j’.

ci = 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, H0:μ1=μ2 vs Ha:μ1>μ2(whereμ1−μ2=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.

Keeping the SD as 3.6 we tried to vary over two different correlation structures Compound symmetry and Discrete Auto regressive of order 1.

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.

[σ2σ2ρσ2ρ⋯σ2ρσ2ρσ2σ2ρ⋯σ2ρσ2ρσ2ρσ2⋯σ2ρ⋮⋱⋮⋮⋮σ2ρσ2ρσ2ρ⋯σ2]

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.

[σ2σ2ρσ2ρ2⋯σ2ρt−1σ2ρσ2σ2ρ⋯σ2ρt−2σ2ρ2σ2ρσ2⋯σ2ρt−3⋮⋱⋮⋮⋮σ2ρt−1σ2ρt−2σ2ρt−3⋯σ2]

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].

CS_diff( i), CS_mean( i), 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.

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.

Figure 1. Simulation results with compound symmetry correlation structure.

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

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 and the mathematically reason behind it is added under Appendix 1 in the Appendices document.

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 affected 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.

An important distinction should be noted when interpreting these findings. The single-time-point approach considered in this study represents analyses based on a single post-baseline assessment. In many clinical trials, however, treatment effects are estimated using baseline-adjusted methods such as ANCOVA or change-from-baseline analyses, which already account for the correlation between baseline and follow-up measurements. Consequently, the present results should not be interpreted as demonstrating that repeated-measures approaches are universally superior to all single-endpoint methods. Rather, they illustrate the potential efficiency gained when multiple post-baseline assessments are incorporated into sample size determination instead of relying solely on a single endpoint assessment.

The findings for the mean difference contrast should be interpreted in the context of the assumed AR(1) covariance structure. The apparent increase in sample size with a greater number of assessment occasions arises because the correlation between baseline and the final assessment decreases as the temporal separation between them increases. While this affects the variance used in sample size calculations, it should not be viewed as a justification for reducing the number of follow-up assessments. The choice and timing of assessments should primarily be driven by scientific and clinical objectives, with covariance assumptions evaluated accordingly during study planning.

Conclusion

Sample size requirements depend on both the planned analysis and the structure of the collected data. The results of this study demonstrate that when information from multiple post-baseline assessments is incorporated into sample size determination, substantial reductions in required sample size may be achieved compared with approaches based on a single endpoint assessment, particularly when within-subject correlations are high. These findings should not be interpreted as implying that repeated-measures methods are universally superior to baseline-adjusted single-endpoint approaches such as ANCOVA or change-from-baseline analyses, which already account for the correlation between baseline and follow-up measurements. Rather, the primary implication is that explicit incorporation of multiple post-baseline observations may provide additional efficiency when longitudinal data are available and the covariance structure is appropriately characterised. This has the potential to reduce study cost, duration, and resource requirements while maintaining statistical 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 difference contrast, the observed increase in sample size with a larger number of assessment time points was primarily driven by the AR(1) correlation structure assumed in the illustrative examples. Under this structure, the correlation between baseline and the final assessment decreases as additional intervals are introduced, resulting in larger sample size requirements. This mathematical consequence should not be interpreted as evidence that collecting fewer measurements is beneficial. In practice, additional assessments may provide valuable information regarding disease progression, treatment response, and longitudinal treatment effects. The results therefore emphasize the importance of carefully specifying and evaluating covariance assumptions during study planning rather than advocating for fewer measurement occasions (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.

These findings extend the commonly used practice, described in the Introduction, of calculating sample size based on change from baseline to a single end-of-study assessment. The results suggest that additional efficiency may be achieved when multiple post-baseline measurements are explicitly incorporated into the sample size derivation, particularly in studies with moderate to high within-subject correlations.

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

1. Biemer PP, Lyberg LE: Introduction to Survey quality. Hoboken, New Jersey: John Wiley and Sons; 2003.
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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The author(s) declared that no grants were involved in supporting this work.

Article Versions (4)

Published: 30 Sep 2026, 11:1550

Published: 05 Feb 2026, 11:1550

Published: 05 Sep 2024, 11:1550

Published: 21 Dec 2022, 11:1550

© 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 4; peer review: 2 approved, 2 not approved]. F1000Research 2026, 11:1550 (https://doi.org/10.12688/f1000research.124917.4)

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Open Peer Review

Current Reviewer Status: ?

Key to Reviewer Statuses VIEW HIDE

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 4

VERSION 4

PUBLISHED 30 Sep 2026

Revised

Reviewer Report 14 Oct 2026

Lucas Damiani, Brazilian Clinical Research Institute, São Paulo, Brazil

Approved

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Biostatistics; Sample Size; Generalized Linear Equations; Bayesian Statistics

Version 3

VERSION 3

PUBLISHED 05 Feb 2026

Revised

Reviewer Report 31 Aug 2026

Lucas Damiani, Brazilian Clinical Research Institute, São Paulo, Brazil

Not Approved

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

Yes
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?

Partly
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; sample size

Version 2

VERSION 2

PUBLISHED 05 Sep 2024

Revised

Reviewer Report 21 Nov 2024

Simon Vandekar, Vanderbilt University Medical Center, Tennessee, USA

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

Reviewer Report 23 Oct 2024

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

Approved

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Biostatistics, Longitudinal data analysis, Bayesian modeling

Reviewer Report 28 Sep 2024

Ronald Geskus, Oxford University, Oxford, England, UK

Not Approved

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: biostatistics

Version 1

VERSION 1

PUBLISHED 21 Dec 2022

Reviewer Report 10 Jul 2024

Ronald Geskus, Oxford University, Oxford, England, UK

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

Reviewer Report 30 May 2024

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

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

Comments on this article Comments (0)

Version 4

VERSION 4 PUBLISHED 21 Dec 2022

Comment

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Sample size variation in single-time post-dose assessment vs multi-time post-dose assessment [version 4; peer review: 2 approved, 2 not approved]

Дата публикации: 30-09-2026 06:30:56

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