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1 Chapter 12 Sample Size and Power Calculations Chapter Table of Contents Introduction Hypothesis Testing Confidence Intervals Equivalence Tests One-Way ANOVA Power Computation Details Hypothesis Tests One-WayANOVA ConfidenceIntervals EquivalenceTests References...279

2 252 Chapter 12. Sample Size and Power Calculations

3 Chapter 12 Sample Size and Power Calculations Introduction Figure Sample Size Menu When you are planning a study or experiment, you often need to know how many units to sample to obtain a certain power, or you may want to know the power you would obtain with a specific sample

4 254 Chapter 12. Sample Size and Power Calculations size. The power of a hypothesis test is the probability of rejecting the null hypothesis when the alternative hypothesis is true. With an inadequate sample size, you may not reach valid conclusions with your work; with an excessive sample size, you may waste valuable resources. Thus, performing sample size and power computations is often quite important. The power and sample size calculations depend on the planned data analysis strategy. That is, if the primary hypothesis test is a twosample t-test, then the power calculations must be based on that test. Otherwise, if the sample size calculations and data analyses are not aligned, the results may not be correct. Determining sample size requirements ahead of the experiment is a prospective exercise. Then, you proceed to select the appropriate number of sampling units and perform data collection and analysis. However, power and sample size calculations are also useful retrospectively. For a given analysis, you may want to calculate what level of power you achieved or what sample size would have been needed for a given power. Power and sample size calculations are a function of the specific alternative hypothesis of interest, in addition to other parameters. That is, the power results will vary depending on which value of the alternative hypothesis you specify, so sometimes it is useful to do these analyses for a range of values to see how sensitive the power analysis is to changes in the alternative hypothesis value. Often, you produce plots of power versus sample size, called power curves, to see how sample size and power affect each other. The Sample Size tasks provide prospective sample size and power calculations for several types of analyses: t-tests, confidence intervals, and tests of equivalence. Each of these calculations is available for one-sample, paired-sample, and two-sample study designs. Power and sample size calculations are also available for the oneway ANOVA design. Multiple parameter values can be input, and results and power curves are produced for each combination of values. Note that retrospective power computations are also available in a number of the statistical tasks in the Analyst Application such as the Hypothesis Test, Regression, and ANOVA tasks.

5 Hypothesis Testing 255 Hypothesis Testing Sample size and power calculations are available for one-sample and two-sample paired and independent designs where the proposed analysis is hypothesis testing of a mean or means via a t-test. These computations assume equally sized groups. Suppose you want to compute the power for a one-sample t-test. The alternative hypothesis mean and the standard deviation have the values and , respectively. You are interested in testing whether the null mean has the value 8, at an alpha level of 0.05, and you are interested in looking at a range of sample sizes from 11 to 211. The study for which these statistics were computed had a sample size of 51. Requesting Power Computations for the One-Sample t-test To access this task, select Statistics! Sample Size! One-Sample t-test ::: Figure 12.2 displays the resulting dialog. Note that, unlike the other statistical tasks that require a data set for analysis, performing one of the Sample Size tasks requires only entering information in the appropriate dialog. The data table is not involved.

6 256 Chapter 12. Sample Size and Power Calculations Figure Sample Size Dialog for One-Sample t-test In this task, you specify whether you want to compute sample size or power, enter values for the test hypothesis and parameters, specify the alpha level (0.05 is the default), specify whether you want a power curve produced, and specify a range of power values or sample sizes depending on whether you are computing sample size or power. To enter the information for this example, follow these steps: 1. Select Power. 2. Enter 8 as the Null mean: value. 3. Enter as the Alternate mean: 4. Enter as the Standard deviation: 5. Make sure that the Alpha: valueis Enter 11 as the value for the From: field in the line for N: 7. Enter 211 and 20 as the values under To: and By:, respectively, in the line for N: 8. Select Power vs. N to produce a plot.

7 Hypothesis Testing Enter 51 as the value for N ref line: 10. Select 2-sided for Tails if it is not already selected. Note that you can enter multiple values in fields such as for Alpha: and Null mean:, separated by commas or blanks, and the analysis will be performed for all combinations of the entered values. Here, power will be computed for sample sizes ranging from 11 to 211 in multiples of 20. Figure 12.3 contains the completed dialog. Figure Sample Size Dialog for One-Sample t-test Figure 12.4 contains the power computations for the sample sizes ranging from 11 to 211.

8 258 Chapter 12. Sample Size and Power Calculations Figure Sample Size Results for One-Sample t-test The interpretation of a power of for n =51is as follows: suppose the true mean and standard deviation are and , and suppose a random sample of 51 observations is taken. Then the probability that the hypothesis test will reject the null hypothesis (H 0 : =8:0) and conclude (correctly) that the alternative hypothesis (H A : =8:6137) is true is The requested plot is shown in Figure 12.5 with a reference line at n =51.

9 Hypothesis Testing 259 Figure Plot of Power versus Sample Size More on Hypothesis Tests In the two-sample cases, you must enter the null means of each group and the standard deviation. In the paired case, the standard deviation entered is the standard deviation of the differences between the two groups. In the independent case, the standard deviation is the pooled standard deviation, which is calculated as follows: s (n 1, 1)S 2 S +(n 1 2, 1)S 2 2 p = (n 1 + n 2, 2)

10 260 Chapter 12. Sample Size and Power Calculations Confidence Intervals Sample size and power calculations are available for one-sample and two-sample paired and independent designs, when the proposed analysis is construction of confidence intervals of a mean (onesample) or difference of two means (two-sample), via the t-test. To understand the power of a confidence interval, first define the precision to be half the length of a two-sided confidence interval (or the distance between the endpoint and the parameter estimate in a onesided interval). The power can then be considered to be the probability that the desired precision is achieved, that is, the probability that the length of the two-sided interval is no more than twice the desired precision. Here, a slight modification of this concept is used. The power is considered to be the conditional probability that the desired precision is achieved, given that the interval includes the true value of the parameter of interest. The reason for the modification is that there is no reason for the interval to be particularly small if it does not contain the true value of the parameter. These computations assume equally sized groups. Requesting Power Computations for a Confidence Interval in a Paired t-test To perform this task, select Statistics! Sample Size! Paired Confidence Interval ::: Figure 12.6 displays the resulting dialog.

11 Confidence Intervals 261 Figure Sample Size Dialog for Paired Confidence Interval You specify whether you want to compute sample sizes or power, enter values for desired precision and standard deviation, enter the alpha levels, enter the sample sizes or power, and select if you want a power curve. To request power for a paired confidence interval where the desired precision is 0.5 and the standard deviation is 2.462, follow these steps: 1. Select Power. 2. Enter 0.5 as the Desired precision: 3. Enter as the Std dev of diff: 4. Enter 0.01, 0.025, 0.05, and 0.1 as values in the field for Alpha: 5. Enter 11 as the value for the From: field in the line for N: 6. Enter 211 and 5 as the values under To: and By:, respectively, in the line for N: 7. Select Power vs. N. 8. Select 2-sided for Interval if it is not already selected.

12 262 Chapter 12. Sample Size and Power Calculations Note that you can enter multiple values in these fields, for example, for Alpha: and Desired precision:, separated by commas or blanks, and the analysis will be performed for all combinations of the input values. Here, power will be computed for sample sizes ranging from 11 to 211 in multiples of 5. Figure 12.7 contains the completed dialog. Note that, because multiple alpha values were entered, sets of results will be created for each one. Figure Completed Sample Size Dialog for Paired Confidence Interval Figure 12.8 contains the power computations for the sample sizes ranging from 36 to 171.

13 Confidence Intervals 263 Figure Sample Size Results for Paired Confidence Interval The power analysis results in Figure 12.8 show that, for a two-sided paired confidence interval where the standard deviation of the differences is 2.462, the significance level is = 0.025, the sample size is 121, and the power is That is, represents the probability that a confidence interval containing the true parameter value has a length of no more than twice the desired precision of 0.5. The requested plot is displayed in Figure 12.9 and includes one power curve for each specified alpha value.

14 264 Chapter 12. Sample Size and Power Calculations Figure Plot for Paired Confidence Interval Equivalence Tests In a test of equivalence, a treatment mean and a reference mean are compared to each other. Equivalence is taken to be the alternative hypothesis, and the null hypothesis is nonequivalence. The model assumed may be additive or multiplicative. In the additive model (Phillips 1990), the focus is on the difference between the treatment mean and the reference mean, while in the multiplicative model (Diletti, Hauschke, and Steinijans 1991), the focus is on the ratio of the treatment mean to the reference mean. In the additive model, the null hypothesis is that the difference between the treatment mean and the reference mean is not near zero. That is, the difference is less than the lower equivalence bound or greater than the upper equivalence bound and thus nonequivalent.

15 Equivalence Tests 265 The alternative is that the difference is between the equivalence bounds; therefore, the two means are considered to be equivalent. In the multiplicative model, the null hypothesis is that the ratio of the treatment mean to the reference mean is not near one. That is, the ratio is below the lower equivalence bound or above the upper equivalence bound, and thus the two means are not equivalent. The alternative is that the ratio is between the bounds; thus, the two means are considered to be equivalent. The power of a test is the probability of rejecting the null hypothesis when the alternative is true. In this case, the power is the probability of accepting equivalence when the treatments are in fact equivalent, that is, the treatment difference or ratio is within the prespecified boundaries. Often, the null difference is specified to be 0; the null hypothesis is that the treatment difference is less than the lower bound or greater than the upper bound, and the alternative is that the difference is not outside the bounds specified. However, in a case where you suspect that the treatments differ slightly (for example, 1 =6, 2 =5, 1, 2 = 1), but you want to rule out a larger difference (for example, j 1, 2 j > 2) with probability equal to the power you select, you would specify the null difference to be 1 and the lower and upper bounds to be,2 and 2, respectively. Note that the null difference must lie within the bounds you specify. Requesting Sample Sizes for Paired Samples In Equivalence As an example of computing sample sizes for an equivalence test, consider determining sample sizes for an additive model. The coefficient of variation is 0.2, and the null differences of interest are 0, 0.05, 0.10, and The significance level under investigation is 0.05, and the power of interest in The lower and upper equivalence bounds are,0:2 and 0.2, respectively. To perform this computation, select Statistics! Sample Size! Paired Equivalence ::: Figure displays the resulting dialog.

16 266 Chapter 12. Sample Size and Power Calculations Figure Sample Size Dialog for Paired Equivalence For this analysis, you need to input the model type, null difference, coefficient of variation, and the usual alpha level. In addition, you need to specify the equivalence bounds. These bounds should be chosen to be the minimum difference so that, if the treatments differed by at least this amount, you would consider them to be different. For the multiplicative model, enter the bioequivalence lower and upper limits. For the additive model, enter the bioequivalence lower and upper limits as percentages of the reference mean lowerbound R and upperbound R. For the null difference or ratio, specify one or more values for the null hypothesis difference between the treatment and reference means (additive model) or the ratio of means (multiplicative model). The null difference/ratio value must lie within the equivalence bounds you specify. For the additive model, specify the null difference as a percentage of the reference mean j T, R j R,where T is the hypothesized treatment mean, and R is the hypothesized reference mean. For the multiplicative model, calculate the null ratio as T R.

17 Equivalence Tests 267 You must also input one or more values for the coefficient of variation (c.v.). For the additive model, enter this as a percentage of the reference mean p R, which can be estimated by MSE R. For pthe multiplicative model, the coefficient of variation is defined as e ( 2 ), 1. You can estimate by ^, where^ 2 is the residual variance of the logarithmically transformed observations. That is, can be estimated by p MSE from the ANOVA of the transformed observations. To produce sample size computations for the preceding problem, follow these steps: 1. Select N. 2. Select Additive. 3. Enter 0, 0.05, 0.10, and 0.15 as values for Null difference: 4. Enter 0.20 for Coeff of variation: 5. Enter 0.05 as the Alpha: 6. Enter 0.80 as the Power: 7. Enter,0:2 and 0.2 as the values for Lower: and Upper:, respectively, for the Equivalence bounds. 8. Click OK to perform the analysis. Figure displays the completed dialog.

18 268 Chapter 12. Sample Size and Power Calculations Figure Sample Size Dialog for Paired Equivalence The results are displayed in Figure Figure Results for Paired Equivalence

19 One-Way ANOVA 269 The results consist of the sample sizes for a power of 0.80 for the values of the null difference, as displayed in Figure These results are for the alpha level of For a null difference of 0.10, the sample size is 51. For a null difference of 0.15, the sample size jumps to 200. One-Way ANOVA When you are planning to analyze data from more than two groups with a one-way ANOVA, you need to calculate your sample size and power accordingly. These computations are available, prospectively, for use in the planning stages of the study, using the Sample Size task. Retrospective calculations are available, for use in the analysis stage, from the One-Way ANOVA task. This section discusses the prospective computations available in the Analyst Application, which assume equally sized groups. You must supply two quantities in order to produce these computations: the corrected sum of squares of means (CSS) and the standard deviation. CSS is calculated as where CSS = GX g=1 ( g, : ) 2 g = mean of the gth group : = overall mean You must enter one or more values for the standard deviation, which in this case is the square root of the Mean Squared Error (MSE). Requesting Power Computations for ANOVA The following is an example of calculating the power for a one-way ANOVA with specified values of sample size. Suppose that you are comparing three groups, the overall mean is 5.5, and the group means

20 270 Chapter 12. Sample Size and Power Calculations are 4.5, 5.5, and 6.5. Therefore, the corrected sum of squares of means (CSS) is (4:5, 5:5) 2 +(5:5, 5:5) 2 +(6:5, 5:5) 2 =2 The standard deviation is the square root of the MSE, which is You are interested in studying sample sizes that range from 6 to 20. To perform these computations, select Statistics! Sample Size! One-Way ANOVA ::: Figure displays the resulting dialog. For this analysis, you need to enter the number of treatments, or factor levels, the CSS of means, the standard deviation, and the alpha level. Figure Sample Size Dialog for One-Way ANOVA

21 One-Way ANOVA 271 To produce power computations for the preceding problem, follow these steps: 1. Select Power. 2. Enter 3 for #oftreatments: 3. Enter 2 for CSS of means: 4. Enter for Standard deviation: 5. Enter 0.05 for Alpha: 6. Enter 6, 20, and 1 for the fields N per group:,forfrom:, To:, and By:, respectively. 7. Click OK to perform the analysis. Figure displays the completed dialog. Figure Sample Size Dialog for One-Way ANOVA Requested are power computations for sample sizes ranging from 6 to 20. The results are displayed in Figure

22 272 Chapter 12. Sample Size and Power Calculations Figure Results for Power Computations for One-Way ANOVA Note that, to achieve a minimum of 80% power, 11 units per group would be needed. Power Computation Details This section provides information on how the power is computed in the Analyst Application. When you request that sample size be computed, the computations produce the smallest sample size that provides the specified power. Hypothesis Tests The power for the one-sample t-test, the paired t-test, and the twosample t-test is computed in the usual fashion. That is, power is the probability of correctly rejecting the null hypothesis when the alternative is true. The sample size is the number per group; these calculations assume equally sized groups. To compute the power of a t-test, you make use of the noncentral t distribution. The formula

23 Hypothesis Tests 273 (O Brien and Lohr 1984) is given by Power = Prob(t <t crit ;;NC) for a one-sided alternative hypothesis and Power = Prob(t <t critu ;;NC)+1, Prob(t <t critl ;;NC) for a two-sided alternative hypothesis where t is distributed as noncentral t(nc;v). t crit = t (1,;) is the (1, ) quantile of the t distribution with df t critu = t (1,=2;) is the (1-=2) quantile of the t distribution with df t critl = t (=2;) is the (=2) quantile of the t distribution with df For one sample and paired samples, = n, 1 NC = p n are the df is the noncentrality parameter For two samples, =2(n, 1) NC = p 2=n df the noncentrality parameter Note that n equals the sample size (number per group). The other parameters are ( ja, 0 j = s ( 1, 2 ) s for one-sample for two-sample and paired samples

24 274 Chapter 12. Sample Size and Power Calculations s = 8 < : standard deviation for one-sample standard deviation of the differences for paired samples pooled standard deviation for two samples One-Way ANOVA The power for the one-way ANOVA is computed in a similar manner as for the hypothesis tests. That is, power is the probability of correctly rejecting the null (all group means are equal) in favor of the alternative hypothesis (at least one group mean is not equal), when the alternative is true. The sample size is the number per group; these calculations assume equally sized groups. To compute the power, you make use of the noncentral F distribution. The formula (O Brien and Lohr 1984) is given by Power = Prob(F <F crit ; 1 ; 2 ;NC) where F is distributed as the noncentral F (NC;, 1 ;, 2 ) and F crit = F (1,;1 ; 2 ) is the (1, ) quantile of the F distribution with 1 and 2 degrees of freedom. 1 = r, 1 2 = r(n, 1) n r NC = ncss 2 is the numerator df is the denominator df is the number per group is the number of groups is the noncentrality parameter CSS = P G g=1 ( g, : ) 2 is the corrected sum of squares g : 2 is the mean of the gth group is the overall mean is estimated by the mean squared error (MSE)

25 Confidence Intervals 275 Confidence Intervals Power calculations are available when the proposed analysis is construction of confidence intervals of a mean (one-sample) or difference of two means (two-samples or paired-samples). To understand the power of a confidence interval, first define the precision to be half the length of a two-sided confidence interval (or the distance between the endpoint and the parameter estimate in a one-sided interval). The power can then be considered to be the probability that the desired precision is achieved, that is, the probability that the length of the two-sided interval is no more than twice the desired precision. Here, a slight modification of this concept is used. The power is considered to be the conditional probability that the desired precision is achieved, given that the interval includes the true value of the parameter of interest. The reason for the modification is that there is no reason to want the interval to be particularly small if it does not contain the true value of the parameter. To compute the power of a confidence interval or an equivalence test, you make use of Owen s Q formula (Owen 1965). The formula is given by where and Q (t; ; a; b) = = Z x (x) =,1 (t)dt p 2 1 p e (,x2 =2) (2),( 2 )2(,2)=2 Z b a ( tx p, )x,1 (x)dx The power of a confidence interval (Beal 1989) is given by Power = 2[Q v(t c ; 0; 0;B), Q v (0; 0; 0;B)] 1, s

26 276 Chapter 12. Sample Size and Power Calculations where t c = t (1,=2;v) t c = t (1,;v) is the (1, =2) quantile of a t distribution with df for a two-sided confidence interval is the (1, ) quantile of a t distribution with degrees of freedom for a one-sided confidence interval is the confidence level s = 2 for a two-sided confidence interval for a one-sided confidence interval B = p t c = n, 1 1=n = p p = 2(n, 1) = 2=n for the one-sample and paired confidence intervals for the two-sample confidence interval = desired precision is the upper bound of the interval half-length standard deviation Equivalence Tests In a test of equivalence, a treatment mean and a reference mean are compared to each other. Equivalence is taken to be the alternative hypothesis, and the null hypothesis is nonequivalence. The power of a test is the probability of rejecting the null hypothesis when the alternative is true, so in this case, the power is the probability of failing to reject equivalence when the treatments are in fact equivalent, that is, the treatment difference or ratio is within the prespecified boundaries.

27 Equivalence Tests 277 The computational details for the power of an equivalence test (refer to Phillips 1990 for the additive model; Diletti, Hauschke, and Steinijans 1991 for the multiplicative) are as follows: Power = Prob(t 1 t (1,;n,2) and t 2,t (1,;n,2) jbioequivalence) Owen (1965) showed that (t 1 ;t 2 ) has a bivariate noncentral t distribution that can be calculated as the difference of two definite integrals (Owen s Q function): Power = Q v (,t (1,;) ; 2 ;0; R), Q v (t (1,;) ; 1 ;0; R) where t (1,;) is the (1, ) quantile of a t distribution with df. = 8 < : n, 1 for the one-sample test n, 2 for the paired test 2(n, 1) for the two-sample test and 1 =,b l V 2 =,bu V = null difference 1 = log(),log(b l) p log(1+v 2 ) 2 = log(),log(bu) p log(1+v 2 ) = null ratio 9 = for the additive model ; 9 >= >; for the multiplicative model V b l b u = is the coefficient of variation is the lower equivalence bound is the upper equivalence bound p p 1=n for the one-sample test 2=n for the two-sample and paired tests

28 278 Chapter 12. Sample Size and Power Calculations R = p (1, 2 ) 2 t (1,;) For equivalence tests, alpha is usually set to 0.05, and power ranges from 0.70 to 0.90 (often set to 0.80). For the additive model of equivalence, the values you must enter for the null difference, the coefficient of variation (c.v.), and the lower and upper bioequivalence limits must be expressed as percentages of the reference mean. More information on specifications follow: Calculate the null difference as j T, R j R,where T is the hypothesized treatment mean and R is the hypothesized reference mean. The null difference is often in the range of 0 to For the coefficient of variation value, can be estimated by ^, where ^ 2 is the residual variance of the observations (MSE). Enter the c.v. as a percentage of the reference mean, so for the p ^ c.v., enter R,or MSE R. This value is often in the range of 0.05 to Enter the bioequivalence lower and upper limits as percentages of the reference mean as well. That is, for the bounds, enter lowerbound R and upperbound R. These values are often,0:2 and 0.2, respectively. For the multiplicative model of equivalence, calculate the null ratio as T R,where T is the hypothesized treatment mean and R is the hypothesized reference mean. This value is often in the range of 0.80 to More information on specifications follow: The coefficient of variation (c.v.) is defined as p e (2 ), 1. You can estimate by ^, where^ 2 is the residual variance of the logarithmically transformed observations. That is, can be estimated by p MSE from the ANOVA of the transformed observations. The c.v. value is often in the range of 0.05 to 0.30.

29 References 279 The bioequivalence lower and upper limits are often set to 0.80 and 1.25, respectively. References Beal, S.L. (1989), Sample Size Determination for Confidence Intervals on the Population Mean and on the Differences Between Two Population Means, Biometrics, 45, Diletti, E., Hauschke, D., and Steinijans, V.W. (1991), Sample Size Determination for Bioequivalence Assessment by Means of Confidence Intervals, International Journal of Clinical Pharmacology, Therapy and Toxicology, Vol. 29, 1 8. O Brien, R., and Lohr, V. (1984), Power Analysis For Linear Models: The Time Has Come, Proceedings of the Ninth Annual SAS User s Group International Conference, Owen, D.B. (1965), A Special Case of a Bivariate Non-central t-distribution, Biometrika, 52, Phillips, K.F. (1990), Power of the Two One-Sided Tests Procedure in Bioequivalence, Journal of Pharmacokinetics and Biopharmaceutics, Vol. 18, No. 2,

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