The SF-36 is the most commonly used health status measure in the world today 8. It originated in the USA 9, but has been validated for use in the UK 10. It contains 36 questions measuring health across eight different dimensions – physical functioning (PF), role limitation because of physical health (RP), social functioning (SF), vitality (VT), bodily pain (BP), mental health (MH), role limitation because of emotional problems (RE) and general health (GH). Responses to each question within a dimension are combined to generate a score from 0 to 100, where 100 indicates "good health". Thus, the SF-36 generates a profile of HRQoL outcomes, (see Figure 1), on eight dimensions.

In principle, there are no major differences in planning a study using HRQoL outcomes, such as the SF-36, to those using conventional clinical outcomes. Pocock 11 outlines five key questions regarding sample size:

1. What is the main purpose of the trial?

2. What is the principal measure of patient outcome?

3. How will the data be analysed to detect a treatment difference?

4. What type of results does one anticipate with standard treatment?

5. How small a treatment difference is it important to detect and with what degree of certainty?

Given answers to all of the five questions above, we can then calculate a sample size.

The choice of the sample size formulae strictly depends on the way data will be analysed, which in turn depends on specific characteristics of the data analysed. For this reason this paper is not only a comparison of four methods of sample size calculation, but also the comparison of the power of four different methods of analysis. We describe four methods of sample-size estimation when using the SF-36 in the comparative clinical trials of two treatments (Table 1). The first method (Method 1) assumes the various individual dimensions of the SF-36 are continuous and Normally distributed. The second method (Method 2) assumes the SF-36 dimensions are continuous. The third method (Method 3) assumes the SF-36 is an ordered categorical outcome. The fourth method uses a bootstrap approach.

In this paper the bootstrap has two roles. It is one of the four methods of sample size calculation and consequently analysis but it is also the method used to estimate the power curves presented in the figures. The bootstrap, in the way used in this paper, is a procedure for evaluating the performance of the statistical procedures, including tests. The bootstrap is non parametric in the sense that it evaluates the performance of any test statistic without making assumptions about the form of the distribution. For the methods of sample size estimation, we consider three test statistics, Methods 1, 2 and 3 and evaluate two of them in two ways, one using the usual assumptions (Normality or continuity), and the other by generating bootstrap distributions from the data.

Suppose we have two independent random samples x₁, x₂,...,x_m and y₁, y₂,....,y_n, of HRQoL data of size m and n respectively. The x's are y's are random samples from continuous HRQoL distributions having cumulative distribution functions (cdfs), F_x and F_y respectively. We will consider situations where the distributions have the same shape, but the locations may differ. Thus if δ denotes the location difference (i.e. mean (y) - mean (x) = δ), then F_Y(y) = F_X(y - δ), for every y. We shall focus on the null hypothesis H₀: δ = 0 against the alternative H_A: δ ≠ 0. We can test these hypotheses using an appropriate significance test (e.g. t-test). With a Normal distribution under the location shift assumption and with n = m, the necessary sample size to achieve a power of 1-β is given in Table 1.

If the HRQoL outcome data (i.e. the GH dimension of the SF-36) is assumed continuous and plausibly not sampled from a Normal distribution then the most popular (not necessarily the most efficient), non-parametric test for comparing two independent samples is the two-sample Mann-Whitney U (also known as the Wilcoxon rank sum test) 12.

Suppose (as before) we have two independent random samples of x's and y's and we want to test the hypothesis that the two samples have come from the same population against the alternative that the Y observations tend to be larger than the X observations. As a test statistic we can use the Mann-Whitney (MW) statistic U, i.e., U = #(y_j > x_i), i = 1,...,m; j = 1,...,n, which is a count of the number of times the y_js are greater than the x_is. The magnitude of U has a meaning, because U/nm is an estimate of the probability that an observation drawn at random from population Y would exceed an observation drawn at random from population X, i.e. Pr(Y > X).

Noether 13 derived a sample size formula for the MW test (see Table 1), using an effect size p_Noether, (i.e. Pr(Y > X)), that makes no assumptions about the distribution of the data (except that it is continuous), and can be used whenever the sampling distribution of the test statistic U can be closely approximated by the Normal distribution, an approximation that is usually quite good except for very small n 14.

Hence to determine the sample size, we have to find the 'effect size' p_Noether or the equivalent statistic Pr(Y > X). There are several ways of estimating p_Noether, under various assumptions, one non-parametric possibility is p_Noether = U/nm. Unfortunately, this can only be estimated after we have collected the data and calculated the U statistic or by computer simulation (as we shall see later). If we assume that X ~ N(μ_X, σ²_X) and Y ~ N(μ_Y, σ²_Y) then a parametric estimate of Pr(Y > X) using the sample estimates of the mean and variance () is given by 15:

where Φ is the Normal cumulative distribution function.

If we assume the SF-36 is Normally distributed then equation 1 allows the calculation of two comparable 'effect sizes' p_Noether and Δ_Normal thus enabling the two methods of sample size estimation to be directly contrasted. If the SF-36 is not Normally distributed then we cannot use equation 1 to calculate comparable effect sizes and must rely on the empirical estimates of p_Noether = U/nm calculated post hoc from the data. Alternatively, under the location shift assumption, we can use bootstrap methods to estimate p_Noether.

Whitehead 16 has derived a method for estimating sample sizes for ordinal data and suggested the odds ratio (OR_Ordinal), which is the odds of a subject being in a given category or lower in one group compared with the odds in the other group, as an effect size. To use Whitehead's formulae the proportion of subjects in each scale category for one of the groups must also be specified.

Suppose there are two groups T and C and the HRQoL outcome measure of interest Y has k ordered categories y_i denoted by i = 1,2,...,k. Let π_iT be the probability of a randomly chosen subject being in category i in Group T and γ_iT be the expected cumulative probability of being in category i or less in Group T (i.e. γ_iT = Pr(Y ≤ y_i)). For category i, where i takes values from 1 to k-1, the OR_i is given in Table 1.

The assumption of proportional odds specifies that the OR_i will be the same for all categories from i = 1 to k-1. As the derivation of the sample size formulae and analysis of data is based on the Mann-Whitney U test, Whitehead's method can be regarded as a 'non-parametric' approach, although it still relies on the assumption of a constant OR for the data. Whitehead's method also assumes a relatively small log odds ratio and a large sample size, which will often be the case in HRQoL studies where dramatic effects are unlikely 4. Table 1 gives the number of subjects per group n_Ordinal for a two-sided significance level α and power 1-β.

Whitehead's 16 method for sample determination is derived from the proportional odds model. The proportional odds model is equivalent to the MW test when there is only a 0/1 (or group) variable in the regression 17. The advantage of the proportional odds model, over the MW test is that it allows the estimation of confidence intervals for the treatment group effect and for the adjustment of the HRQoL outcome for other covariates.

If the number of categories is large, it is difficult to postulate the proportion of subjects who would fall in a given category. Whitehead 16 points out that there is little increase in power (and hence saving in the number of subjects recruited) to be gained by increasing the number of categories beyond five. An even distribution of subjects within categories leads to the greatest efficiency.

Shepstone 18 demonstrates how the three seemingly different effect size measures Δ_Normal, OR_Ordinal and p_Noether, which are all numerical expressions of treatment efficacy can be combined into a common scale. If Y and X are the values of an outcome (higher values more preferable) for randomly selected individuals from the Treatment and Control groups respectively, then A_YX = Pr(Y > X), i.e. the probability that the Treatment patient has an outcome preferable to that of the Control patient, is equivalent to the effect size statistic p_Noether. If we let A_XY = Pr(X > Y), i.e. the probability that a random individual from group 2 (Control) has a better outcome than a random individual from group 1 (Treatment), then

λ = A_YX - A_XY = Pr(Y >X) - Pr(X >Y)     (2)

and

Shepstone 18 shows that for ordinal and continuous outcomes A_YX - A_XY = λ and A_YX / A_XY = θ are equivalent to the Absolute Risk Reduction (ARR) and OR for binary outcomes. A_XY and A_YX, or their equivalent statistics Pr(X > Y) and Pr(Y > X) can be calculated by either a parametric approach for continuous outcomes (Equation 1) via a theoretical distribution (e.g. Normal) or a non-parametric approach without any distributional assumptions via the Mann-Whitney U statistic. (Since A_XY and A_YX can be estimated by and where U_XY and U_YX are the values of the Mann-Whitney U statistics).

If the outcomes are continuous and/or can be fully ranked and there are no ties in the data then Pr(X = Y) = 0 and λ = A_YX - A_XY = Pr(Y > X) - Pr(X > Y) and θ = A_YX/A_XY = Pr(Y > X)/Pr(X > Y) can be estimated exactly. Conversely, if there are a large number of ties in the data, i.e. x_i = y_i, (which is likely for HRQoL outcomes, with their discrete response categories) then Pr(X = Y) > 0. In this case any pairs for which x_i = y_i, contribute 1/2 a unit to both U_YX and U_XY. Hence the two A statistics A_YX and A_XY can only be estimated approximately and thus the approximate estimates of θ and λ in the case of ties will be denoted by θ' and λ' respectively.

Methods 1 and 2 assume the HRQoL outcome is continuous and the simple location shift model is appropriate. Here this would imply that, on a certain scale, the difference in effect of the intervention compared to the control is constant or, at least that the intervention shifts the distribution of the HRQoL scores under the control to the right (or to the left if the intervention is harmful) but keeping its shape. However, the boundedness of the SF-36 outcomes renders this location shift assumption questionable, especially if the proportion of cases the upper limit is high. Therefore, we used bootstrap methods to compare the power of the t-test and MW test with allowance for ties for detecting a shift in location using three dimensions of the SF-36 (GH, RP and V) as outcomes 141920. These three dimensions illustrate the different distributions of HRQoL outcomes that are likely to occur in practice.

Suppose (as before) we have two independent random samples of x's and y's from continuous distributions having cdfs, F_x and F_y respectively. Again we will consider situations where the distributions have the same shape, but the locations may differ; i.e. mean (y) - mean (x) = δ. If we focus on the null hypothesis H₀: δ = 0 against the alternative H_A: δ ≠ 0, then we can test this hypothesis using an appropriate significance test (i.e. t-test, Mann-Whitney or proportional odds model). However, we did not evaluate the proportional odds model as part of the bootstrap. This was because the proportional odds model is equivalent to the MW test when there is only a 0/1 variable in the regression, and the p-values from the MW test and the significance of the regression coefficient for the group variable are indentical 17.

The bootstrap strategy is to use pilot data to a provide a non-parametric estimate of F and to use a simulation method for finding the power of the test associated with any specified sample size n if the data follow the estimated distribution functions under the null and alternative hypotheses. If we denote the distribution function estimate by , under the alternative hypothesis δ, we can estimate the approximate power, (G, δ, α, n) by the following computer simulation procedure 141920.

When planning the CPSW study we went through Pocock's 11 five key questions regarding sample size.

Suppose we believe the GH dimension to be continuous, but not Normally distributed and are intending to compare GH scores in the two groups with a Mann-Whitney U test (with allowances for ties). Therefore, Noether's method will be appropriate. As before if we assume a mean difference of 5 and a standard deviation of 20 for the GH dimension of the SF-36, then using equation 1 leads to an parametric estimate of the effect size p_Noether = Pr(Y > X) of 0.57 and consequently Pr(X > Y) of 0.43. Substituting p_Noether = 0.57 in the formula for Method 2 (in Table 1) with a two-sided 5% significance level and 80% power gives the estimated number of subjects per group as 267.

Method 1 has given us a slightly smaller sample size estimate than Method 2. The two methods can be regarded as equivalent when the two populations are Normally distributed, with equal variances. In this case, the MW test will require about 5% more observations than the two-sample t-test to provide the same power against the same alternative. For non-Normal populations, especially those with long tails, the MW test may not require as many observations as the two-sample t-test 23.

Empirically, calculating a parametric estimate of Pr(Y > X) from the observed effect size data (using the observed sample means and standard deviations), leads to values very similar to the non-parametric estimate. For example, for the GH dimension in the CPSW data in Table 2, the observed non-parametric estimate of Pr(Y > X) was 0.542 compared to a parametric estimate of 0.537.

Assuming a mean difference of 5 (i.e. ) and a common standard deviation of 20 (i.e. ) for the GH dimension of the SF-36, then equation (1) leads to a parametric estimate of the effect size p_Noether = Pr(Y > X) of 0.57. This in turn leads to a parametric estimate of the ARR (from equation 2), λ' = 0.57 - 0.43 = 0.14 and an estimated (from 3) OR θ' = 0.57/0.43 = 1.33.

If we assume OR_Ordinal = OR = 1.33 then the assumption of proportional odds specifies that the OR_Ordinali will be the same for all 34 categories of the GH dimension. If we also assume the proportion of subjects in each category in the control group is the same as in Figure 1g. Then under the assumption of proportional odds OR_Ordinal = 1.33, the anticipated cumulative proportions (γ_iT) for each category of treatment T are given by:

After calculating the cumulative proportions (γ_iT), the anticipated proportions falling into each treatment category, π_iT can be determined from the difference in successive γ_iT. Finally, the combined mean () of the proportions of treatments C and T for each category is calculated.

Substituting OR_Ordinal = 1.33 and = 0.0067 with a two-sided 5% significance level and 80% power gives the estimated number of subjects per group as 584. Given this sample size, and with the distribution shown in Figure 1g and an OR of 1.33, we can work out what the corresponding mean values are. The estimated mean GH score was 77.6 in the treatment group and 75.0 in the control group. This is an estimated mean difference of 2.6 points, which is smaller than the five-point mean difference used earlier.

It is difficult to translate a shift in means into the shift in the probabilities on an ordinal scale, without several assumptions. If we assume proportions in each category in the control group as shown in Figure 1g and proportional odds shift, then an OR_Ordinal of 1.63 is approximately equal to a mean shift of 5.0. This leads to = 0.007 and a sample size estimate of 199 subjects per group with two-sided 5% significance and 80% power. Given this sample size then the corresponding estimated mean GH scores are 74.8 and 79.8 in the control and treatment groups respectively.

Figure 1g shows the skewed distribution of the GH dimension and that the underlying assumption of Normality of the distribution required for Method 1 may not be appropriate. Furthermore the, GH dimension is bounded by 0 and 100. Thus, if a new mother already has a GH score of 100 in the control group, then under the intervention no extra improvement can be seen, at least not by the GH dimension of the SF-36. Seven percent of women (35/487) in the Sheffield data had a GH score of 100 and 14.2% (70/487) had a score of 95 or more.

Figure 2 shows the estimated power curves for Methods 1, 2 and 3 and the two bootstrap methods (t and MW tests) at the 5% two-sided significance level for detecting a location shift (mean difference) δ = 5 in the SF-36 GH dimension using the data from the general population as our pilot sample, for sample sizes per group varying from 50 to 600. For these general population data a location shift of δ = 5 is equivalent to a standardised effect size Δ_Normal = 0.25 and p_Noether = Pr(Y > X) = 0.57. The bootstrap methods taking into account the bounded and non-Normal distribution of the data suggest a mean difference d of 4.5 and p = Pr(Y > X) = 0.58.

The GH dimension (Figure 1g) of the SF-36 has a large number (> 30) of discrete values or categories, most of which are occupied, and the proportion scoring 0 or 100 is low. Figure 2 suggests that the MW test is more powerful than the t-test for the GH dimension based on the bootstrap results for the bounded shift. The power curves shown in Figure 2 do not diverge too greatly and thus, the location shift hypothesis is a useful working model.

In contrast, Figure 3 shows the estimated power curves for another dimension of the SF-36: RP, which can only take one of five discrete values (as shown in Figure 1c), for detecting a simple location shift (mean difference) δ = 5. For these data a simple location shift of δ = 5 is equivalent to a standardised effect size Δ_Normal = 0.17 and p_Noether = Pr(Y > X) = 0.55. Since three-quarters of the pilot sample scored 100, the bootstrap methods under the location shift model, taking into account the bounded and non-Normal distribution of the data suggest a mean difference d of 1.2 and p = Pr(Y > X) = 0.51. The power curves shown in Figure 3 diverge greatly and the simple location shift hypothesis may not be appropriate for this outcome. Figure 3 clearly shows the value of the bootstrap in investigating the impact of the bounded HRQoL distributions on the power of the hypothesis test.

Finally, Figure 4 shows the estimated power curves for the Vitality dimension of the SF-36. This computer simulation suggests that if the distribution of the HRQoL dimension are reasonably symmetric (Figure 1b), and the proportion of patients at each bound is low, then under the location shift alternative hypothesis, the t-test appears to be slightly more powerful than the MW test at detecting differences in means.

The bootstrap methodology provides an ideal opportunity to consider Type I errors. Resampling Algorithm 1 can easily be adapted for this. It simply involves modification of step 1 and not adding δ to the second simulated sample of patients. Under the true null hypothesis of no difference in distributions, the actual Type I error rate can be computed by determining the proportion of simulated cases which had significance levels at or below its nominal value. For a nominal Type I error rate of α = 0.05, we would expect 5% (or 0.05) of the bootstrap samples to give a (false-positive) significant result under the true null hypothesis of no difference in distributions. The robustness of each test can then be determining by comparing the actual Type I error rates to the nominal Type I error rates.

Statistical tests are said to be robust if the observed Type I error rates are close to the pre-selected or nominal, Type I error rates in the presence of violations of assumptions 2425. Sullivan and D'Agostino 26 describe a test as 'robust' if the actual significance level does not exceed 10% of the nominal significance level (e.g. less than or equal to 0.055 when the nominal significance level is 0.05). They describe a test as 'liberal' if the observed significance exceeds the nominal level by more than 10%. Finally, they describe a test as 'conservative' if the actual significance level is less than the nominal level. A 'conservative' test is of less concern, as the probability of making a Type I error is controlled.

The overall actual significance levels relative to a nominal level of 0.05 under the null hypothesis of no treatment differences for the GH and RP dimensions are displayed in Table 3 for a variety of sample sizes. Both tests (t-test and MW) are 'robust' tests of the equality of means (and distributions) for both the GH and RP outcomes.

When using the proportional odds model to estimate sample size, Whitehead 16 and Julious et al 27 have pointed out that there is little increase in power (and hence saving in the number of subjects recruited) to be gained by increasing the number of categories in a proportional odds model beyond five. Although the model is robust to mild departures from the assumption of proportional odds, with increasing numbers of categories it is less likely that the proportional odds assumption remains true. Therefore, to illustrate this point, we shall use the five discrete category outcome of the RP dimension of the SF-36 to show the effect of the bootstrap sample size estimator when the alternative to the null hypothesis is an odds ratio transformation rather than a simple location shift.

Figure 5 shows the power curves for t-test and MW test for the RP dimension of the SF-36 assuming the alternative hypothesis is a proportional odds shift in HRQoL of OR_Ordinal = 1.50. As one would expect, the bootstrap power curves in Figure 5 show that the MW test or the equivalent proportional odds model is more powerful than the t-test when the alternative hypothesis is an odds ratio shift, although the differences in power for a given sample size are small.

Sample sizes of over 450 patients per group are required to have an 80% chance of detecting this 'small to moderate' odds ratio (OR = 1.5) effect as statistically significant at the 5% two-sided level. With such 'large' sample sizes statistical theory, via Central Limit Theorem (CLT), guarantees that the sample means will be approximately Normally distributed, which ensures the relatively good performance of the t-test in detecting an OR location shift. The robustness of the two independent samples t-test when applied to three-, four- and five-point ordinal scaled data has previously been demonstrated by Heeren and D'Agostino 25 for far smaller sample sizes than this (as small as 20 per group).

It is important to make maximum use of the information available from other related studies or extrapolation from other unrelated studies. The more precise the information the better we can design the trial. We would recommend that researchers planning a study with HRQoL measures as the primary outcome pay careful attention to any evidence on the validity and frequency distribution of the HRQoL measures and its dimensions.

The frequency distribution of HRQoL dimension scores from previous studies should be assessed to see what methods should be used for sample size calculations and analysis. If the HRQoL outcome has a limited number of discrete values (say less than seven categories e.g. RP and RE dimensions, in the case of the SF-36, Figures 1c and 1f) and/or the proportion of cases at the upper bound (i.e. scoring 100) is high (e.g. PF, SF and BP dimensions in our general population sample example dataset, Figures 1a,1e,1f), then we would recommend using Method 3 to estimate the required sample size (Figure 6) 16. In this case, the alternative hypothesis of a location shift model is questionable and the proportional odds model will provide a suitable alternative with such bounded discrete outcomes.

If the HRQoL outcome has a larger number of discrete values (greater than or equal to seven categories say), most of which are occupied and the proportion of cases at the upper or bounds (i.e. scoring 0 or 100, in the case of the SF-36) is low (e.g. VT, GH and MH dimensions in our general population sample example dataset, Figures 1b,1g and 1h), then the simple location shift model is a useful working hypothesis. We would therefore recommend using Methods (1) or (2) to estimate the required sample size (Figure 6).

Computer simulation has suggested that if the distributions of the HRQoL dimensions are reasonably symmetric (Figure 1b), and the proportion of patients at each bound is low, then under the location shift alternative hypothesis, the t-test appears to be slightly more powerful than the MW test at detecting differences in means (Figure 4). Therefore, if the distribution of the HRQoL outcomes is symmetric or expected to be reasonably symmetric and the proportion of patients at the upper or lower bounds is low then Method 1 could be used for sample size calculations and analysis (Figure 6). The use of parametric methods for analysis (i.e. t-test) also enables the relatively easy estimation of confidence intervals, which is regarded as good statistical practice 1.

If the distribution of the HRQoL outcome is expected to be skewed then the MW test appears to be more powerful at detecting a location shift (difference in means) than the t-test (Figures 2 and 3). Therefore, in these circumstances, the MW test is preferable to the t-test and Method 2 could be used for sample size calculations and analysis. However, using Method 2 for sample size estimation requires the effect size to be defined in terms of Pr(Y > X), which is difficult to quantify and interpret. Pragmatically we would recommend Method 1 as the effect size Δ_Normal is rather easier to quantify and interpret than the effect size p_Noether required for sample size estimation using Method 2 (Figure 6).

If the HRQoL data have a symmetric distribution, the mean and median will tend to coincide so either measure is a suitable summary measure of location. If the HRQoL data have an asymmetric distribution, then conventional statistical advice would suggest that the median is the preferred summary statistic 1. However, a case when the mean and mean difference might be preferred (even for skewed outcome data) as a summary measure is when heath care providers are deciding whether to offer a new treatment or not to its population. The mean (along with the sample size) provides information about the total benefit (and total cost) from treating all patients, which is needed as the basis for health care policy decisions 28. We cannot estimate the total benefit (or cost) from the sample median.

If the sample size is "sufficiently large" then statistical theory, using the CLT, guarantees that the sample means will be approximately Normally distributed. Thus, if the investigator is planning a large study and the sample mean is an appropriate summary measure of the HRQoL outcome, then pragmatically there is no need to worry about the distribution of the HRQoL outcome and we can use conventional methods to calculate sample sizes. Although the Normal distribution is strictly only the limiting form of the sampling distribution of the sample mean as the sample size n increases to infinity, it provides a remarkably good approximation to the sampling distribution even when n is small and the distribution of the data is far from Normal. Generally, if n is greater than 25, these approximations will be good. However, if the underlying distribution is symmetric, unimodal and continuous a value of n as small as 4 can yield a very adequate approximation 29. If a reliable pilot or historical dataset of HRQoL data is readily available (to estimate the shape of the distribution) then bootstrap simulation (Method 4) will provide a more accurate and reliable sample size estimate than Methods 1 to 3 (Figure 6) 30.

We had a reliable historical data set of over 400 subjects so we had a large sample to estimate the distributions (cdfs) F_x and F_y under the null and alternative hypotheses using Method 4. Lesaffre et al 31 show that bootstrap can give fairly unbiased estimates of power, though for small pilot samples with large variability. In the absence of a reliable pilot set, bootstrapping is not appropriate and we will need to use conventional methods of sample size estimation or simulation models 32. Fortunately, with the increasing use of HRQoL outcomes in research, historical datasets are becoming more readily available. White and Thompson 33 suggest the estimation of (and hence ) should be derived from a pilot dataset, and that the use of baseline data or related data sets (which we have used) is somewhat less satisfactory. They suggest a third possibility for estimating is to use follow-up data viewed in a blinded manner, although only when the blinding can demonstrably be preserved.

Strictly speaking, our results and conclusions only apply to the SF-36 outcome measure. Further empirical work is required to see whether or not these results hold true for other HRQoL outcomes, populations and interventions. However, the SF-36 has many features in common with other HRQoL outcomes, such as the NHP and QLQ-C30, i.e. multi-dimensional, ordinal or discrete response categories with upper and lower bounds, and skewed distributions; therefore, we see no theoretical reason why these results and conclusions with the SF-36 may not be appropriate for other HRQoL measures.

Throughout this paper, we only considered the situation where a single dimension of HRQoL is used at a single endpoint. We have assumed a rather simple form of the alternative hypothesis that the new treatment/intervention would improve HRQoL compared to the control/standard therapy. This form of hypothesis (superiority vs. equivalence) may be more complicated than actually presented. However, the assumption of a simple form of the alternative hypothesis that new treatment/intervention would improve HRQoL compared to the control/standard therapy, is not unrealistic for most superiority trials and is frequently used for other clinical outcomes.

We have based the calculations above on the assumption that there is a single identifiable endpoint, or HRQoL outcome, upon which treatment comparisons are based (in our case the GH dimension of the SF-36). Sometimes there is more than one endpoint of interest; HRQoL outcomes are multi-dimensional (e.g. the SF-36 has eight dimensions including GH). If one of these dimensions is regarded as more important than the others, it can be named as the primary endpoint and the sample size estimates calculated accordingly. The remainder should be consigned to exploratory analyses or descriptions only 3. Fairclough gives a more comprehensive discussion of multiple endpoints and suggests several methods for analysing HRQoL outcomes 34.

More work is required on what is a clinically meaningful effect sizes for the SF-36 and other HRQoL outcomes. There is an extensive literature on the important issue of clinically meaningful change and the minimum important difference (MID) for HRQoL outcomes. As the subject of this paper is the use of computer intensive methods such as the bootstrap we have played down the issue of the MID. Again for brevity and practical purposes of sample size estimation this paper has assumed the MID for the SF-36 outcome is around five points for each dimension. This is an important issue in sample size estimation. The interested reader is referred to a series of papers from the Clinical Significance Consensus Meeting Group for more detailed discussion 35.