Basic Concepts
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
ncp = n ⋅ s ⋅ η2
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
1 – β = 1 – NF_DIST(fcrit, df1, df2, ncp)
where
df1 = k(g – 1) df2 = s(n – g – k + s)
fcrit = F_INV_RT(α, df1, df2)
and g = number of groups and k = number of dependent variables. This is the same approach used by G*Power.
Example
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
Worksheet Functions
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 (default), 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
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.
Data Analysis Tool
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.
Examples Workbook
Click here to download the Excel workbook with the examples described on this webpage.
References
Faul, F., Erdfelder, E., Buchner, A., & Lang, A. G. (2009) Statistical power analyses using G*Power 3.1: Tests for correlation and regression analyses. Behavior Research Methods, 41, 1149-1160.
http://link.springer.com/article/10.3758/BRM.41.4.1149
Hintze, J. L. (2011) Multivariate analysis of variance (MANOVA). NCSS PASS
https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/PASS/Multivariate_Analysis_of_Variance-MANOVA.pdf
Castelloe, J. (2014) Power and sample size for MANOVA and repeated measures with the GLMPOWER procedure
https://support.sas.com/resources/papers/proceedings14/SAS030-2014.pdf