We can calculate the power and minimum sample size in the same manner as described for one-way ANOVA based on the partial eta-square or eta-square effect size of Pillai’s V statistic and the noncentrality parameter equal to
where η2 = eta-square effect size, n = the sample size and s is as described in MANOVA Basic Concepts. Restricting our attention to the Pillai-Bartlett test, note too that the eta-square effect size can be expressed in terms of the Pillai-Bartlett Trace V or partial eta-square effect size h as follows:
The power can be expressed as
where
and g = number of groups and k = number of dependent variables. This is the same approach used by G*Power.
Example 1: What is the power for the one-way MANOVA in Example 1 of MANOVA Basic Concepts.
The power is 88% as calculated in cell B15 of Figure 1. Note that ‘Manova 1k’ is the name of the worksheet that contains the calculations in Figure 1 and 9 of MANOVA Basic Concepts.
Figure 1 – Power calculation
Real Statistics Functions: The Real Statistics Resource Pack provides the following functions.
MANOVA_POWER(f n, k, g, ttype, alpha, iter, prec) = the statistical power for one-way MANOVA where the sample size is n, the number of dependent variables is k , the number of groups is g and the effect size is f, where f = the partial eta-square effect size if ttype = 1, f = eta-square if ttype = 2 and f = Pillai’s V if ttype = 3.
MANOVA_SIZE(f, k, g, pow, ttype, alpha, iter, prec) = the minimum sample size to obtain statistical power of pow for one-way MANOVA where f, k, g and ttype are as for MANOVA_POWER.
alpha is the significance level (default .05), iter = the maximum number of iterations used in calculating the answer (default 1000) up to a precision of prec (default 0.000000001), the default for pow is .80.
The power for Example 1 can be calculated by any of the following formulas (with reference to Figure 1).
=MANOVA_POWER(B5,B9,B7,B6)
=MANOVA_POWER(B4,B9,B7,B6,2,B13)
=MANOVA_POWER(B3,B9,B7,B6,3,B13)
Example 2: What sample size would be required to detect a partial eta-square effect size of .1 with power 95% if the experiment in Example 1 of MANOVA Basic Concepts is repeated?
The required sample size is calculated as shown in cell G7 of Figure 2.
Figure 2 – Sample size calculation
As we can see, the minimum sample size is 74. Since 74 is not divisible by 4, the number of groups, if we require a balanced model, then the minimum sample is 76, the next highest number larger than 74 that is divisible by 4.
Real Statistics Data Analysis Tool: The Real Statistics Resource Pack also provides a Power and Sample Size data analysis tool that supports One Factor MANOVA. To use this tool press Ctrl-m and select the Power and Sample Size option from the Misc tab. Next, select the One Factor MANOVA option and either the Power or Sample Size option.