Tutorial on Biostatistics: Sample Size and Power Calculation for Ophthalmic Studies With Correlated Binary Eye Outcomes

Abstract

Purpose: To describe and demonstrate sample size and power calculation for ophthalmic studies with a binary outcome from one or both eyes.

Methods: We describe sample size and power calculation for four commonly used eye designs: (1) one-eye design or person-design: one eye per subject or outcome is at person-level; (2) paired design: two eyes per subject and two eyes are in different treatment groups; (3) two-eye design: two eyes per subject and both eyes are in the same treatment group; and (4) mixture design: mixture of one eye and two eyes per subject. For each design, we demonstrate sample size and power calculations in real ophthalmic studies.

Results: Using formulas and commercial or free statistical packages including SAS, STATA, R, and PS, we calculated sample size and power. We demonstrated that different statistical packages require different parameters and provide similar, yet not identical, results. We emphasize that studies using data from two eyes of a subject need to account for the intereye correlation for appropriate sample size and power calculations. We demonstrate the gain in efficiency in designs that include two eyes of a subject compared to one-eye designs.

Conclusions: Ophthalmic studies use different eye designs that include one or both eyes in the same or different treatment groups. Appropriate sample size and power calculations depend on the eye design and should account for intereye correlation when two eyes from some or all subjects are included in a study. Calculations can be executed using formulas and commercial or free statistical packages.

In ophthalmic research, the primary outcome measure is often binary (e.g., presence or absence of an eye disease), which can be obtained from one or two eyes of a subject to evaluate the effect of a systemic or ocular treatment. When the treatment is systemic, two eyes of a subject receive the same treatment and thus are in the same comparison group. When the treatment is ocular, two eyes can be in the same comparison group or different comparison groups depending on whether two eyes receive the same treatment or different treatments. Because the outcome measures from two eyes of a subject are usually positively correlated, the appropriate sample size and power calculation require accounting for the intereye correlation when binary outcome measures are obtained from both eyes of a subject. Furthermore, in some studies, some subjects only have one eligible eye whereas others have both eyes eligible, leading to a mixture of one eye and two eyes per subject for the study, further complicating the sample size/power calculation. 

Although determination of sample size and power is a critical component of the study design, researchers in ophthalmology and vision science are often uncertain how to perform the proper calculation. A recent review of 96 clinical trials published in four top ophthalmology journals (Ophthalmology, JAMA Ophthalmology, American Journal of Ophthalmology, and British Journal of Ophthalmology) found that 22% of trials did not provide sample size or power calculations.¹ Although previous articles^2–9 have discussed sample size and power calculations for ophthalmology and vision research, these articles did not provide hands-on instructions with examples from real ophthalmic studies with various designs. In this tutorial article, we demonstrate the sample size and power calculation for real ophthalmic studies using commercial or free statistical packages for each common ophthalmic design with the provision of sample size calculation codes and outputs. 

Methods

We describe methods for sample size and power calculation for each of the following four designs that are commonly used in ophthalmology and vision research (Fig.). 

Figure.

 

The types of eye design that are determined based on whether the treatment is at person level or eye level, whether the outcome measure is taken at person level or eye level, how many study eyes per person provide the outcome measure, and whether two eyes of a person are in the same or different treatment groups.

One-Eye Design or Person-Level Design

In a one-eye design, only one eye of each person is in the study, whereas in a person-level design, the outcome measure is taken at the person level (e.g., quality of life). In both one-eye design and person-level design, one outcome measure is obtained from each subject; thus intervention comparisons are made between persons using standard statistical methods without the need for adjustment of intereye correlation. 

Paired Design (or Contralateral-Eye Design)

In this design, two eyes per person are included, and each eye is assigned to a different treatment. Having one treatment given to one eye and another treatment (or control) to the contralateral fellow eye, the person acts as their own control; thus it has the advantage of balancing confounding effects between interventions. Under this design, it is assumed that two eyes of a person are very similar (although they may not be identical) in their ocular measures before treatment. Taking the average of ocular measures across all persons in one eye should be the same as taking the average across all persons of the fellow eye; thus the differences in outcome measure between two eyes of a person after treatment are likely due to different treatment the two eyes receive. 

Two-Eye Design

In this design, two eyes of a person are in the same treatment group. It is different from the one-eye design or person-level design; although the interventions can be systemic or ocular, the outcome measures are obtained from both eyes of a person, and comparisons of ocular outcome measures are made between persons in one treatment group and persons in another treatment group. 

Mixture Design

In this design, some people have two eyes in the study and receive the same treatment, whereas other people only have one eye in the study because their fellow eyes are not eligible for the study. Thus the data are from a mixture of one or two eyes per person. 

For each design, we will first describe an overview of the method for sample size and power calculation. We will then introduce a previously published ophthalmic study as an example, followed by the demonstration of the sample size calculation for the example study using the formula, free software PS (download at https://biostat.app.vumc.org/wiki/Main/PowerSampleSize) or R (https://cran.r-project.org), and commercial statistical software including SAS and STATA. 

Factors Affecting Sample Size and Statistical Power

Sample size calculations are dependent on the following factors or assumptions: 

Eye Design

Before conducting an ophthalmic study, we first have to decide whether the study will include one or two eyes per subject and whether the study will use two eyes per subject, regardless of whether both eyes are in the same or different treatment or comparison group. 

Intereye Correlation (r)

If the study includes both eyes from some or all study subjects, we need to consider the intereye correlation in the outcome measure because the sample size and statistical power depend on its magnitude. The intereye correlation can often be estimated from previous studies. 

Proportion of Occurrence of Binary Outcome in the Control Group

This is usually estimated based on previous studies. 

Proportion of Occurrence of Binary Outcome in the Treatment Group or Difference in the Proportion of Binary Outcome Between Treatment Group and Control Group

The difference in the event proportion between treated and control groups or proportion in the treatment group are usually unknown and thus need to be determined based on the clinically meaningful difference that the study is targeted to detect in comparison to the control group. 

Ratio of Sample Size Between Groups

In clinical trials, study subjects are usually 1:1 randomized to two treatment groups as this typically maximizes power for a fixed total sample size. 

Type 1 Error (α)

This is usually set as two-sided 0.05. In some studies with multiple primary outcomes, the type I error rate can be set smaller than 0.05 to account for multiple comparisons for primary outcomes. For example, if there are two co-primary outcomes, the type I error rate can be set as 0.025 each to account for the two comparisons. 

Power (1-β)

We usually set the power to be at least 0.80. For a large clinical trial, we may set power equal to 0.90 or even higher to ensure the study has adequate power to detect the targeted difference because missing the detection of such a difference can keep an effective treatment from being adopted. 

Attrition Rate

In clinical studies, it is common that some subjects cannot complete the study because of death, loss to follow-up, dropout, or the primary outcome cannot be obtained because of poor image quality or lack of patient cooperation. The sample size for the number of subjects for enrollment needs to account for the attrition rate, which usually can be estimated from previous studies. This sample size adjustment for attrition assumes missing data from attrition are at random (e.g., noninformative).¹⁰ 

Sample Size for Each Eye Design

Design 1: One-Eye Design or Person-level Design

This design is common when the study only includes one eye per subject, or when the outcome is determined at person-level, such as quality of life. This eye design often leads to the comparison of two independent proportions using a χ² test or Fisher's exact test. 

For example, in the myo-inositol trial for the Type 1 retinopathy of prematurity (ROP),¹¹ preterm infants <28 weeks’ gestational age were 1:1 randomized to intravenous treatment with 40 mg/kg myo-inositol or placebo (a solution of 5% glucose) every 12 hours for 10 weeks. The primary outcome is person-level, defined as the incidence of type 1 ROP by 55 weeks’ postmenstrual age or death before the ROP outcome could be determined. Thus the sample size calculation for this trial is based on the comparison of two independent proportions. 

Sample Size Calculation for Comparison of Two-independent Groups

We assume the proportion of the occurrence of an event in the experimental group and the control group (with equal sample size n) are P_E and P_C, respectively, and their event proportion difference d = P_E − P_C. The null hypothesis for the comparison of two proportions at significance level α and power 1 − β is: 

\begin{eqnarray*}{{{{\bf H}}}_{{\bf 0}}}{\rm{: }}\, {{{\rm{P}}}_{\rm{E}}} - {{{\rm{P}}}_{\rm{C}}} = {\rm{0,}}\,{\rm{and}}\,{\rm{alternative}}\,{\rm{hypothesis}}\,{{{{\bf H}}}_{{\bf 1}}}\,{\rm{is}}\! :\,{{{\rm{P}}}_{\rm{E}}} - {{{\rm{P}}}_{\rm{C}}} \ne {\rm{0}}{\rm{.}}\end{eqnarray*}

 

The test statistic is: 

\begin{eqnarray*} Z = \frac{{{{P}_E} - {{P}_C}}}{{\sqrt {\frac{{2PQ}}{n}} }}\end{eqnarray*}

where P = (P_E + P_C)/2, and Q = 1 − P. 

The formula for sample size per group is: 

\begin{eqnarray}n = {{\left[ {{{z}_{1 - \alpha /2}}\sqrt {2PQ} + {{z}_{1 - \beta }}\sqrt {{{P}_E}{{Q}_E} + {{P}_C}{{Q}_C}} } \right]}^2}/{{d}^2}\end{eqnarray}

(1)

where Q_E = 1 − P_E, Q_C = 1 − P_C, z_{1 − α/2} and z_{1  −  β} are the (1 – α/2)^th and the (1-β)^th quantiles of the standard normal distribution. 

If the experimental group and the control group do not have equal sample size, such as the sample size in the experimental group (n_E) is k times (k > 0) as many as that in the control group (n_C) (i.e., n_E = k * n_C), the sample size formula in Equation 1 becomes:  

\begin{eqnarray*}{n}_C = \frac{\left[ \sqrt {PQ\left( {1 + \frac{1}{k}} \right)}{z}_{1 - \alpha/2} + \sqrt {P_C Q_C + \frac{P_EQ_E} {k}} z_{1 - \beta } \right]^2}{d^2}\end{eqnarray*}

where \(P = \frac{{{{P}_C}\, +\, k{{P}_E}}}{{1\, +\, k}}\) and Q = 1 − P. 

The above sample size formulas are based on the z-test that works well for a large sample study. For a small sample study, especially when any of the cells of a contingency table have an expected frequency <5,¹² Fisher's exact test is commonly used to compare two independent proportions. There is no explicit formula for sample size calculation based on the Fisher's exact test, but some statistical packages (e.g., PS, SAS, STATA) can provide sample size calculation based on Fisher's exact test¹³ as demonstrated in the following example. 

Example: Sample size Calculation for the Myo-inositol Trial

The Myo-inositol Trial¹¹ was a double-blind placebo-controlled randomized clinical trial approved by the US Food and Drug Administration, to evaluate the safety and efficacy of myo-inositol among infants younger than 28 weeks gestational age to reduce the rates of type 1 ROP. The primary outcome of the trial, defined at the person level, is the occurrence of the death or incidence of type 1 ROP in either eye up to a maximum of 55 weeks postmenstrual age. The trial was designed to enroll a total of 1760 preterm infants to provide 90% power for detecting an absolute reduction of 7% or greater in the incidence of type 1 ROP in either eye or death (i.e., 30% in the placebo group¹⁴ vs. 23% in the Myo-inositol group) at a type I error rate of 0.05 anticipating a loss to follow-up rate of 5%. We demonstrated sample size calculation for the Myo-inositol Trial using different approaches as described below and also summarized in Table 1. 

Table 1.

 

Summary for the Sample Size Calculation Using Various Approaches for the Design 1 in the Myo-inositol Trial

 

Sample Size Calculation Using Formula

Using the sample size formula for the comparison of two independent proportions, and the assumed proportion of events in each treatment group as described above, we calculated the sample size as follows: 

\begin{eqnarray*}\begin{array}{@{}r@{\;}c@{\;}l@{}} {{{\rm{P}}}_{\rm{E}}} &=& 0.23,\,{{{\rm{Q}}}_{\rm{E}}} = 1 - {{{\rm{P}}}_{\rm{E}}} = 0.77,\\ {{{\rm{P}}}_{\rm{C}}} &=& 0.30,\,{{{\rm{Q}}}_{\rm{c}}} = 1 - {{{\rm{P}}}_{\rm{C}}} = 0.70,\\ {\rm{d}} &=& {{{\rm{P}}}_{\rm{E}}} - {{{\rm{P}}}_{\rm{C}}} = 0.23-0.30 = - 0.07\\ {\rm{P}} &=& \left( {{{{\rm{P}}}_{\rm{E}}} + {{{\rm{P}}}_{\rm{C}}}} \right)/2 = \left( {0.23 + 0.30} \right)/2 = 0.265,\, \\ {\rm{Q}} &=& 1 - 0.265 = 0.735 \\ {{{\boldsymbol{z}}}_{1 - {\boldsymbol{\alpha }}/2}} &=& 1.96,\,{{{\boldsymbol{z}}}_{1 - {\boldsymbol{\beta \ }}}} = 1.282 \end{array}\end{eqnarray*}

 

\begin{eqnarray*}\begin{array}{@{}rcl@{}} n &=& \displaystyle \frac{{{{{[{{Z}_{1 - \alpha /2}}\sqrt {2PQ^{\vphantom{|}}} + {{Z}_{1 - \beta }}\sqrt {{{P}_E}{{Q}_E} + {{P}_C}{{Q}_C}^{\vphantom{|}}} ]}}^2}}}{{{{d}^2}}} \\[6pt] &=& \displaystyle \frac{{{{{\left[ {1.96 \times \sqrt {2 \times 0.265 \times 0.735^{\vphantom{|}}} + 1.282 \times \sqrt {0.23 \times 0.77 + 0.30 \times 0.70^{\vphantom{|}}} } \right]}}^2}}}{{{{{( - 0.07)}}^2}}}\\[6pt] &=& 833.5 \approx 834 \end{array}\end{eqnarray*}

 

Thus the study needs a sample size of 834 infants per group. This calculated sample size is the number of infants in each group with primary outcome available for statistical comparison of primary outcome between two treatment groups. To determine the sample size for enrollment, we need to account for a 5% attrition rate that the trial anticipated. Thus, the sample size for enrollment is 834/(1 − 0.05) = 878 infants per group (i.e., total sample size of 1756 infants). 

Sample Size Calculation Using Free Software PS

For sample size calculation using free software PS as shown in Supplementary Appendix 1A, we provided alpha (0.05), power (0.90), the event proportion in the Myo-inositol treatment group (0.23) and in the placebo control group (0.30), and the randomization ratio (m = 1). In addition, we also provided the study design information that is needed for PS. Because the sample size is for the comparison of two independent proportions using a χ² test for a prospective cohort trial involving two independent groups, we selected “Independent” for the question about matched or independent, and “Prospective” for the question about “case-control design or prospective cohort design,” and selected “Two proportions” and “uncorrected χ² test” for the questions about the statistical test. As shown in Supplementary Appendix 1A, PS provided the same sample size of 834 infants per treatment group as obtained using Equation 1. 

Although sample size calculation based on the Fisher's exact test is not recommended because the expected frequency in each cell is >5 in the Myo-inositol trial, we calculated the sample size based on the Fisher's exact test as 862 infants per treatment group (Supplementary Appendix 1B), which is larger than the sample size (834 infants per group) based on the χ² test. However, if the expected frequency in any cell is 5 or less, the sample size calculation should be based on Fisher's exact test. 

Sample Size Calculation in SAS

In SAS, the procedure PROC POWER is designed to perform sample size and power calculations. Similar to the sample size calculation in PS, we provided the information on the statistical test used for comparing two independent proportions, which is the χ² test, the event proportion in the two treatment groups (0.30 in the placebo group and 0.23 in the Myo-inositol group), the alpha (0.05), and the power (0.90). In addition, SAS requires a specification of the null hypothesis for the difference in proportions (0). Because we want to calculate sample size, we leave sample size blank (indicated by .). With these values provided for the required parameters, SAS calculated a sample size of 834 infants per treatment group for final statistical analysis (Supplementary Appendix 1C). If we want to calculate the power for a given sample size, we can provide a value for sample size (e.g., 834 per group), and leave power blank as shown in Supplementary Appendix 1D. 

Sample Size Calculation in STATA

In STATA (Supplementary Appendix 1E), we provided the event proportion of 0.23 for the Experimental Group (e.g. the Myo-inositol group), and 0.30 for the Control Group (in fact, which proportion is assigned to which group does not matter), type I error rate (0.05), and power (0.9). STATA output also showed a sample size of 834 infants per group for analysis. 

Sample Size Calculation in R

There are several R packages available for sample size and power calculations. One commonly used package is “pwr” which was developed based on Cohen's effect sizes.¹⁵ For sample size calculation using this package, we need to first calculate effect size based on the difference between two proportions, which can be calculated using function ES.h( ) available in “pwr”. Based on the determined effect size, the R function pwr.2p.test ( ) was used to calculate the required sample size. As shown in Supplementary Appendix 1F, for the Myo-inositol Trial, we provided the two proportions (0.30, 0.23), and the effect size of 0.159 which was calculated by using function ES.h( ) to calculate arcsin(0.30)-arcsin(0.23). With this effect size as the input, along with the specification of type I error (0.05), power (0.90), and two-sided test, the sample size from function pwr.2p.test ( ) is 832 patients per group, which is somewhat different from that of SAS, STATA and PS, because the alternative hypothesis was specified as a difference based on the arcsin of p1 and arcsin of p2 (instead of the difference between p1 and p2),¹⁵ which is not the standard two-sample test for binomial proportions. 

Design 2: Paired Design

Many ocular diseases (e.g., myopia, glaucoma, age-related macular degeneration [AMD]) are often symmetric, affecting both eyes of a subject simultaneously.¹⁶ When the treatment of interest is localized to an eye, many ophthalmic trials use a paired design by randomizing one eye to study treatment, and the other eye serves as a control. For example, in the US Diabetic Retinopathy Study,¹⁷ one eye of each eligible subject was randomly assigned to immediate photocoagulation and the other eye to follow-up without treatment. When the treatment is locally restricted to an eye (such as laser treatment for diabetic retinopathy), the paired design is efficient in that it enables the comparison between two interventions within a subject, eliminates between-subject variation and, hence, improves the efficiency in estimating the treatment effect. However, the paired design is not valid if treatments are systemic or a treatment to one eye affects the other eye. 

In ophthalmology, Lee et al.¹⁸ found that a paired-eye design was used in nine (13%) of 69 ophthalmic trials published in four general clinical ophthalmology journals (American Journal of Ophthalmology, Archives of Ophthalmology, the British Journal of Ophthalmology, and Ophthalmology) between January and December of 2009. Similarly, among 96 papers from randomized clinical trials published in these same four ophthalmology journals from 2020 to 2021, 10 (10%) studies used a paired-eye design.¹ 

Sample Size for Paired Designs with a Binary Outcome

For a paired-design ophthalmic study with a binary outcome (e.g., treatment success or failure), the following 2 × 2 table (Table 2) can be constructed to estimate the parameters needed for the sample size calculation. The within-subject difference for the proportion of positive outcomes between the two treatment groups is:  

\begin{eqnarray*}d = {{\pi }_B} - {{\pi }_A}\end{eqnarray*}

 

Table 2.

 

The 2 × 2 Table for the Comparison of a Binary Outcome From Ophthalmic Studies Using a Paired Design With N Subjects

 

The proportion of subjects with a discordant binary outcome between the eye assigned to treatment A and the contralateral eye assigned to treatment B is calculated as:  

\begin{eqnarray*}f = \left( {{\rm{b}} + {\rm{c}}} \right)/{\rm{N}}\end{eqnarray*}

f is needed because only discordant pairs contribute information to the comparison between two treatment groups. 

For the paired-eye design, the number of subjects needed (n) with a two-sided type I error rate (α), and power (1 − β) can be calculated using the following formula:  

\begin{eqnarray}n = {{\left[ {{{z}_{\alpha /2}}\sqrt f + {{z}_\beta }\sqrt {f - {{d}^2}} } \right]}^2}/{{d}^2}\end{eqnarray}

(2)

 

To calculate the sample size n using Equation 2, we need to estimate the discordant percentage f. If the information on f is not available, then f can be estimated based on the anticipated proportion with the event π_A and π_B as follows:  

\begin{eqnarray*}f = {{\pi }_{\rm{A}}}(1 - {{\pi }_{_{\rm{B}}}}) + {{\pi }_{\rm{B}}}(1 - {{\pi }_{\rm{A}}}),\end{eqnarray*}

assuming independence within pairs, which provides a conservative (i.e., overestimate) estimate of sample size. 

Example: Sample Size Calculation for a Paired Design

The Complications of Age-Related Macular Degeneration Prevention Trial (CAPT) is a randomized clinical trial to evaluate whether prophylactic laser treatment to patients with large drusen (e.g., intermediate AMD) can prevent progression to AMD. The trial was designed with one eye randomized to receive laser treatment of drusen and the other eye to observation (e.g., as control). The primary outcome of the trial was the loss of three or more lines in visual acuity (VA) in an eye in five years. The required sample size was calculated to compare the proportion of VA loss of three or more lines in five years between treated and observed eyes. The proportion of eyes with VA loss in five years in the observed eye group was assumed to be 14% ([1 − (1 − 0.04)⁵] × 0.75 = 0.14) based on the estimated proportion of choroidal neovascularization (4%/year) through five years and 75% of patients who develop choroidal neovascularization will have the VA loss of three or more lines based on data from the Macular Photocoagulation Study.¹⁹ The trial assumed 4.4% of subjects would develop loss of VA in both eyes based on previous data from the Macular Photocoagulation Study. The trial was designed to detect the minimal treatment effect of 30% relative reduction, or a five-year incidence of three or more lines loss of VA of 9.8% (14% * 70%), with 90% statistical power at a two-sided type I error rate of 0.05. The trial also assumed an attrition rate of 16% because of death and loss to follow-up. 

Based on the above information about the proportion with VA loss of three lines or more in the observed eye group (14%), in the laser-treated group (9.8%), and the intereye agreement (4.4% of subjects would develop loss of VA in both eyes), we calculated the two discordant proportions as: 

p₀₁ = 9.8% − 4.4% = 5.4%, and p₁₀ = 14% − 4.4% = 9.6%) (see Table 3). 

We can then derive the following 2 × 2 table (Table 3). From this 2 × 2 table, we can get the discordant proportion f = 0.054 + 0.096 = 0.150. We demonstrated sample size calculation for the CAPT study using different approaches as described below and also summarized in Table 4. 

Table 3.

 

The 2 × 2 Table for the Probability Combination of Outcome in the Eye in the Laser Treated Group and Observed Eye in the No Treatment Group

 

Table 4.

 

Summary for the Sample Size Calculation Using Various Approaches for Paired Design in the CAPT Study

 

Sample Size Calculation Using Formula

By using the sample size formula for paired design,  

\begin{eqnarray*}n = {{\left[ {{{z}_{\alpha /2}}\sqrt f + {{z}_\beta }\sqrt {f - {{d}^2}} } \right]}^2}/{{d}^2}\end{eqnarray*}

 

\begin{eqnarray*} && {{{\boldsymbol{z}}}_{1 - {\boldsymbol{\alpha }}/2}} = 1.96,\,{{{\boldsymbol{z}}}_{1 - {\boldsymbol{\beta \ }}}} = 1.282,\, \\ && {\rm{d}} = 0.140 - 0.098 = 0.042,\,f = 0.150\end{eqnarray*}

we calculated the sample size of n = 889 subjects for final analysis. After we accounted for a 16% attrition rate, the total number subjects for enrollment is: 890/(1 − 0.16) = 1060 subjects. 

Sample Size Calculation Using SAS

We also calculated the sample size using PROC POWER of SAS by providing the discordant proportions (0.096, 0.054), alpha (0.05), and power (0.90), the SAS program calculated the sample size of 890 subjects for final analysis (Supplementary Appendix 2A), the same as the sample size from the formula calculation. 

Sample Size Calculation Using STATA

STATA can calculate the sample size based on McNemar's test for comparing two paired proportions by either specifying discordant proportions, or by specifying marginal proportions (e.g., proportion with binary outcome in each treatment group) and intereye correlation ɸ. The intereye correlation can be calculated using the formula:  

\begin{eqnarray*}\phi = \frac{{{{p}_{11}}{{p}_{00}} - {{p}_{10}}{{p}_{01}}}}{{\sqrt {{{p}_1}{{q}_1}{{p}_0}{{q}_0}} }}\end{eqnarray*}

 

Based on the values in Table 3, the intereye correlation was calculated to be 0.2935. 

As shown in Supplementary Appendix 2B, by specifying two discordant proportions (0.096, 0.054), STATA calculated a sample size of 890 patients. As shown in Supplementary Appendix 2C, by specifying the two marginal proportions (0.140, 0.098) and the intereye correlation of 0.2935, STATA calculated the same sample size of 890 patients. 

Sample Size Calculation in R

The R package “TrialSize” can perform the sample size calculation based on McNemar's test for paired proportions. As shown in Supplementary Appendix 2D, we only provided the type I error (0.05), beta (1 – power = 0.10), the ratio “psai” of two discordant proportions (0.054/0.096), and sum “paid” of these two proportions (0.150). The function McNemar.Test( ) calculated that 889 subjects are needed for the final analysis. 

Design 3: Two-Eye Design

In some studies using a two-eye design, two eyes of a subject receive the same treatment, such as when the treatment is systemic. Thus two eyes of a subject are in the same comparison group. An example is the Age-Related Eye Disease Study 2, which was designed to evaluate the efficacy and safety of lutein plus zeaxanthin (L + Z) or w-3 long-chain polyunsaturated fatty acid (LCPUFA) supplementation in reducing the risk of developing advanced AMD. The subjects of the Age-Related Eye Disease Study 2 were randomly assigned to placebo, L + Z, w-3 LCPUFAs, or the combination of L + Z and w-3 LCPUFAs. 

The two-eye design is very common in ophthalmic clinical trials. Among 96 randomized clinical trials published in the top four ophthalmology journals in 2020 or 2021, 21 (22%) trials used a two-eye design.¹ 

Sample size Calculation for Two-Eye Design

The general approach for sample size calculation for a two-eye design usually follows the following steps: 

• (1) Use the same sample size formula for a one-eye design to calculate the number of independent eyes per group = n_ind;
• (2) Calculate the number of correlated eyes per group as n_ind * (1 + r), where r is the intereye correlation in the outcome measure.⁴ This adjustment of sample size is similar to adjustment for design effect in cluster randomized trials.⁸
• (3) Calculate the number of subjects per group as n_ind * (1 + r)/2.

Example: Myopia Prevention Trial

A randomized clinical trial was conducted from October 1, 2018, to December 31, 2020, among grade 1 students in Guangzhou, China, to evaluate the effect of a school-based family health education intervention via WeChat in raising parents’ awareness of myopia prevention for controlling the development of myopia in children.²⁰ 

The trial was designed by assuming the myopia incidence rate in the control group was 10% per year between grade 1 to grade 3 (i.e., incidence rate equal to 20% in the control group); the intervention can reduce the annual myopia incidence rate by 8% during the two-year follow-up (i.e., an incidence rate of 12% in the intervention group). The trial was designed to provide 95% power using a two-sided type I error of 0.05. The trial also assumed a two-year loss to follow-up of 15% and a baseline participation rate of 90%. The trial was designed with only using the right eye data for statistical analysis with the justification that the refractive error of the right eye and the left eye are highly correlated. Thus the trial was designed with a total sample size of 1420 students for enrollment. We demonstrated sample size calculation for this trial using SAS and STATA as described below and also summarized in Table 5. 

Table 5.

 

Summary for the sample size calculation using various approaches for design 3 in the Myopia Prevention Trial

 

Sample Size Calculation Using SAS

We first calculate the number of independent eyes needed per group using a one-eye design. In SAS (Supplementary Appendix 3A), we specified a myopia rate of 20% in the control group and 12% in the intervention group, power 95% and two-sided type I error of 0.05. It was determined that 543 independent eyes per group are needed. After accounting for the 15% loss to follow-up and 90% participation rate, the number of subjects for enrollment per group (assuming only using the right eye for analysis) is 710 (543/(0.90 * 0.85) = 710). 

Because the two eyes of a student receive the same treatment and provide the refractive error measures in both eyes, a better analysis would be to include both eyes using a per-eye analysis while accounting for the intereye correlation.²¹ Thus we also calculated the sample size based on a two-eye design. 

The sample size for a two-eye design requires the specification of the intereye correlation. From a previous myopia study,²² the intereye correlation for the incidence of myopia is equal to 0.80. The sample size for the number of correlated eyes per group is 543 * (1 + 0.8) = 978 eyes. The sample size for the number of subjects per group is 978/2 = 489 subjects. 

After accounting for a 15% loss to follow-up and a 90% participation rate, the number of subjects for enrollment per group = 489/(0.90 * 0.85) = 639 subjects, a reduction of 71 subjects per group (total reduction of 142 patients for enrollment, or a 10% reduction in the sample size) when compared to a one-eye design (which only use one eye per subject for statistical analysis). 

Sample Size Calculation Using STATA

The sample size for a two-eye design can also be performed in STATA by using methods for a clustered clinical trial²³ as shown in Supplementary Appendix 3B. In STATA, we specified the myopia incidence of 0.20 in the control group and 0.12 in the intervention group, power (0.95), type I error rate (0.05), and the intereye correlation coefficient (0.80). STATA calculated that 978 eyes from 489 subjects per group are needed. After accounting for 15% loss to follow-up and a 90% participation rate, the number of subjects for enrollment in each group is 639 subjects (1278 eyes), the same sample size as that calculated from SAS. 

Instead of calculating sample size, we can also calculate the power for a given sample size in STATA using the methods for a clustered clinical trial as shown in Supplementary Appendix 3C. Using the following command in STATA:  

\begin{eqnarray*} && { {power \, twoproportions}}\,0.20\,0.12,\,m1\left( 2 \right)\,m2\left( 2 \right)\, \\ && k1\left( {489} \right)\,k2\left( {489} \right)\,rho\left( {0.8} \right),\end{eqnarray*}

where m1, m2 = number of eyes per subject and k1, k2 = number of subjects per group in groups 1 and 2, respectively, we obtained a power of 95%, consistent with the previous sample size. 

Design 4: Mixture of One-Eye and Two-Eye Design

In clinical studies, some subjects (proportion p₁) only have one eye eligible for the study while other subjects (proportion p₂) have both eyes eligible for the study. For those with both eyes eligible for the study, two eyes of a subject can be in the same treatment group or in different treatment groups. 

An example of a study that used a mixture design is the FLuorometholone as Adjuntive MEdical Therapy for Trachomatous Trichiasis (TT) Surgery (FLAME) Trial. The FLAME trial was a prospective 1:1 randomized, parallel design, double-masked, placebo-controlled clinical trial of fluorometholone 0.1% eye drops vs. placebo in eyes with trachomatous trichiasis (TT) undergoing lid rotation surgery. In this trial, the unit of randomization was the person, and one or both eligible eyes of a subject were enrolled into the study. The primary outcome is the eye-specific incidence of postoperative TT by one year as determined by the trained field team members at four weeks, six months, and one year. Postoperative TT was defined as the presence of one of the following: (1) one or more lashes touching the globe in an eye operated for TT; (2) clinical evidence of epilation; (3) history of repeat TT surgery. 

Sample Size Calculation for a Mixture Design

For a study with a mixture design, we assume the proportion of subjects contributing one eye is p₁ and the proportion of subjects contributing both eyes is p₂. We also assumed that among those contributing both eyes to the study, the two eyes are in the same treatment group with intereye correlation r in the binary outcome measure. 

The sample size calculation for the mixture design involves the following steps: 

• (1) Use a one-eye design to calculate the number of independent eyes per group n_ind
• (2) Calculate the number of correlated eyes per group n_eyes = n_ind * [1 + 2rp₂/(p₁ + 2p₂)]
• (3) Calculate number of subjects per group n_subject = n_eyes/(p₁ + 2p₂).

We demonstrated sample size calculation for this trial using SAS and STATA as described below and also summarized in Table 6. 

Table 6.

 

Summary for the sample size calculation using various approaches for design 4 in the FLAME Study

 

Sample Size Calculation in SAS

The FLAME trial assumed an incidence rate of postoperative TT of 20% in the placebo group by one year, a 25% reduction in the rate of postoperative TT in the fluorometholone group (e.g., incidence rate of postoperative TT 15%), 35% subjects eligible in both eyes, and an intereye correlation in postoperative TT between 2 eligible eyes is equal to 0.46 based on previous studies. The trial was designed with 90% power, two-sided type I error rate 5%, and a rate of loss to follow-up 5%. 

As shown in Supplementary Appendix 4A, we calculated that the number of independent eyes per group using a one-eye design is 1212. By applying the above formula to accommodate the mixture of one or two eyes per subject, we calculated the number of correlated eyes per group as following:  

\begin{eqnarray*}\begin{array} {@{}rcl@{}} {{{\rm{n}}}_{{\rm{eyes}}}} &=& {{{\rm{n}}}_{{\rm{ind}}}}\,{\rm{* }}\left[ {{\rm{1}} + {\rm{2r}}{{{\rm{p}}}_{\rm{2}}}{\rm{/}}\!\left( {{{{\rm{p}}}_{\rm{1}}} + {\rm{2}}{{{\rm{p}}}_{\rm{2}}}} \right)} \right] \\[3pt] &=& 1212*{\rm{ }}\left[ {1 + 2*0.46*0.35/\!\left( {0.65 + 2*0.35} \right)} \right]\ = 1501 \end{array}\end{eqnarray*}

 

We calculated the number of subjects n_subjects per group  

\begin{eqnarray*} {{{\rm{n}}}_{{\rm{subjects}}}} &=& {{{\rm{n}}}_{{\rm{eyes}}}}{\rm{/}}\!\left( {{{{\rm{p}}}_{\rm{1}}} + {\rm{2}}{{{\rm{p}}}_{\rm{2}}}} \right) \\ &=& {\rm{1501/}}\!\left( {{\rm{0}}{\rm{.65}} + {\rm{2*0}}{\rm{.35}}} \right)\ \\ &=& 1501/\!1.35 = 1112.\end{eqnarray*}

 

After adjustment by a 5% attrition rate, the number of subjects for enrollment per group is 1170 (1112/0.95). 

Sample Size Calculation in STATA

We can also perform the sample size calculation for the mixture design in STATA for comparing two independent proportions using a cluster randomized design.²⁴ For this design, we need to use a mean cluster size of 1.35 in each group (because 35% of subjects are bilateral), and a coefficient of variation of cluster size of 0.353 (calculated by\(\frac{{\sqrt {{{p}_2}(1 - {{p}_2})} }}{{1 + {{p}_2}}}\), where p₂ is the proportion of subjects contributing two eyes). The other parameters needed for sample size calculation included postoperative TT of 20% in the placebo and 15% in the fluorometholone group, 90% power, two-sided type I error rate 5%. With these parameters as shown in Supplementary Appendix 4B, the number of subjects per group is 1076, which is somewhat smaller than the sample size of 1112 subjects per group from the calculation in SAS. 

The Reduction in the Sample Size From Two-Eye Design and Mixture Design Compared to One-Eye Design

In the mixture design with a sample size of n subjects and p₂ of subjects contributing two eyes (p2 = 1 for two-eye design) into the study with intereye correlation r in the outcome measure, the number of independent eyes from these n subjects can be calculated as: \(n + \frac{{(1\, -\, r){{p}_2}n}}{{1\, +\, r}}\), and the percentage of reduction in the sample size compared to one-eye design can be calculated as \(\frac{{(1\, -\, r){{p}_2}}}{{(1\, +\, r)\, +\, (1\, -\, r){{p}_2}}} \times 100\% \). Table 7 shows the efficiency (e.g., the percent reduction in the sample size) from using both eyes from some or all subjects under varying degree of intereye correlation (1, 0.75, 0.50, 0.25 and 0.0) and varying proportion (1.00, 0.75, 0.50, 0.25) of subjects contributing both eyes (in the same treatment group) for the study. For example, for a study with half of subjects contributing both eyes to the study, sample size (in terms of number of subjects) can be reduced by 23% if the intereye correlation is low (0.25), and by 14% if the intereye correlation is moderate (0.50). 

Table 7.

 

The Reduction in the Sample Size in the Two-Eye Design or Mixture Design Compared to One-Eye Design Under Various Combinations of Percent of Bilateral Subjects And Intereye Correlation

 

Discussion

In this tutorial article, we described how to perform sample size and power calculations based on a comparison of two proportions for each of four types of eye designs commonly used in ophthalmology and vision research including a one-eye design, paired-eye design, two-eye design, and mixture design. Using examples of real clinical studies, we demonstrated the calculation of sample size and power using a formula, free or commercial statistical software including PS, R, SAS, and STATA. From our review of ophthalmic clinical trials published in top ophthalmology journals, we previously reported that many ophthalmic clinical trials did not perform appropriate sample size or power calculations,¹ likely because of the lack of knowledge of sample size and power calculation for ophthalmic studies involving one or two eyes per subject. This tutorial article was developed to provide step-by-step guidance to researchers in ophthalmology and vision research on how to perform the proper sample size and power calculations for their studies. 

One unique feature of this tutorial article is to demonstrate the sample size and power calculation using both free and commercial software because researchers have variable access to resources and skills in using different statistical software for sample size and power calculation. Different software packages can use different parametrizations of the same statistical test or use different statistical tests and thus require different inputs for sample size calculation. Although we found that sample sizes from different software packages are similar most of the time, they are not always the same, especially for the mixture design. Because the sample size calculation is mainly to guide study planning and small differences in sample size from different estimation approaches are expected, the investigators only need to perform the sample size calculation using one approach/software that they are most comfortable with. 

Because many eye diseases are bilateral, clinical trials often include both eyes in the trial. When the treatment is systemic, both eyes of a subject receive the same treatment. 

However, several previous review articles found that when the intervention is systemic and thus has an effect on both eyes of a subject, many ophthalmic studies only used one eye (e.g., right eye, random eye, or worse eye) for statistical analysis,^{1,16,18,25–27} and the sample size/power calculation was also based on one eye per subject for the statistical analysis. This practice is not optimal because the data from the fellow eye were not used, which results in a larger sample size than for a two-eye design. As we demonstrated in the Myopia Prevention Trial, if data from both eyes were included in the analysis for the same power, the sample size would be reduced by 10% compared to an analysis of one eye per subject, when the intereye correlation in refractive error is 0.80. The percentage of reduction in the sample size can be calculated as 100 * (1 − r)/2. When the intereye correlation is low or moderate, the reduction in sample size is substantial. For example, when the intereye correlation is 0.50, the sample size can be reduced by 25% without any loss of statistical power. Therefore, for efficiency, when both eyes are included in a trial, the statistical analysis should include data from all available eyes of study subjects. 

This article demonstrates the sample size calculation for a mixture design that includes a mixture of one eye and two eyes per subject into the study. Although the sample size and power calculation are less straightforward than for the one-eye and two-eye designs, this design capitalizes on the availability of two eligible eyes from some subjects, which can reduce the number of subjects required when compared with the one-eye design. Table 7 shows the efficiency (e.g., the percent of reduction in the sample size) gained from using both eyes from some or all subjects is dependent on the degree of intereye correlation and the proportion of subjects contributing both eyes to the study. For example, for a study with half of the subjects contributing both eyes to the study, sample size (in terms of the number of subjects) can be reduced by 23% if the intereye correlation is low (0.25), and by 14% if the intereye correlation is moderate (0.50). This mixture design can be particularly useful when (a) intereye correlation is low; or (b) the disease is not common, and enrollment is difficult. The statistical analysis of data from this mixture design follows the same approach as for a two-eye design. 

This tutorial article only focuses on the sample size and power calculation that is based on the difference of two proportions which is usually the most straightforward. The sample size for studies with binary outcomes can also be calculated based on the risk ratio or odds ratio of two proportions.^28,29 Although the examples used to demonstrate approaches for sample size and power calculations in this tutorial article are from clinical trials, these approaches can be applied to observational studies. Furthermore, our sample size calculation did not account for the covariates, which is appropriate in randomized clinical trials but may not be appropriate in the epidemiological studies that usually require the adjustment of confounders in the comparisons of binary outcomes between groups using the multivariable regression models.^30,31 

In conclusion, ophthalmic studies use various types of eye designs depending on the use of one eye or two eyes per subject, and whether two eyes of a subject receive the same or different treatments. These different eye designs lead to different ways to calculate the required sample size and power. This tutorial paper provides step-by-step guidance on how to perform these calculations for ophthalmic studies of various eye designs, using free or commercial statistical software. Proper sample size and power calculations following this tutorial article can lead to the efficient study design of ophthalmic studies. 

Acknowledgments

We thank Maureen G. Maguire for her helpful guidance on this article. 

Supported by grants R01EY022445 and P30 EY01583-26 from the National Eye Institute, National Institutes of Health, Department of Health and Human Services, and an unrestricted grant from Research to Prevent Blindness to the University of Pennsylvania. 

Disclosure: G.-S. Ying, None; R.J. Glynn, None; B. Rosner, None