Introduction
Power analysis is the name given to the process for determining the sample size for a research study. The technical definition of power is that it is the probability of detecting a “true” effect when it exists. Many students think that there is a simple formula for determining sample size for every research situation. However, the reality it that there are many research situations that are so complex that they almost defy rational power analysis. In most cases, power analysis involves a number of simplifying assumptions, in order to make the problem tractable, and running the analyses numerous times with different variations to cover all of the contingencies.
In this unit we will try to illustrate how to do a power analysis for multiple regression model that has two control variables, one continuous research variable and one categorical research variable (three levels).
Description of the Experiment
A school district is designing a multiple regression study looking at the effect of gender, family income, mother’s education and language spoken in the home on the English language proficiency scores of Latino high school students. The variables gender and family income are control variables and not of primary research interest. Mother’s education is a continuous research variable that measures the number of years that the mother attended school. The range of this variable is expected to be from 4 to 20. The variable language spoken in the home is a categorical research variable with three levels: 1) Spanish only, 2) both Spanish and English, and 3) English only. Since there are three levels, it will take two dummy variables to code language spoken in the home.
The full regression model will look something like this,
 engprof = b_{0} + b_{1}(gender) + b_{2}(income) + b_{3}(momeduc) + b_{4}(homelang1) + b_{5}(homelang2)
Thus, the primary research hypotheses are the test of b_{3} and the joint test of b_{4} and b_{5}. These tests are equivalent the testing the change in R^{2} when momeduc (or homelang1 & homelang2) are added last to the regression equation.
The Power Analysis
We will make use of the SAS proc power to do the power analysis. To begin with, we believe, from previous research, that the R^{2} for the fullmodel (r2f) with five predictor variables (2 control, 1 continuous research, and 2 dummy variables for the categorical variable) will be will be about 0.48.
Let’s start with the continuous predictor (momeduc). We think that it will add about 0.03 to the R^{2} when it is added last to the model. This means that the R^{2} for the model without the variable (the reduced model) would be about 0.45, which leads to the difference in R^{2} (rsquarediff) of .03. The total number of variables (nfullpredictors) is 5 and the number being tested (ntestpredictors) is one. We will run proc power for powers equal to .7, .8 and .9.
proc power; multreg model = fixed nfullpredictors = 5 ntestpredictors = 1 rsquarefull = 0.48 rsquarediff = 0.03 ntotal = . power = 0.7 to .9 by .1; run;
The POWER Procedure Type III F Test in Multiple Regression Fixed Scenario Elements Method Exact Model Fixed X Number of Predictors in Full Model 5 Number of Test Predictors 1 Rsquare of Full Model 0.48 Difference in Rsquare 0.03 Alpha 0.05 Computed N Total Nominal Actual N Index Power Power Total 1 0.7 0.704 110 2 0.8 0.803 139 3 0.9 0.901 185
This gives us a range of sample sizes ranging from 110 to 185 depending on power.
Let’s see how this compares with the categorical predictor (homelang1 & homelang2) which uses two dummy variables in the model. We believe that the change in R^{2} attributed to the two dummy variables will be about 0.025. This would give an R^{2 } of 0.455. The nfullpredictors stays at 5 while the ntestpredictors is now 2.
proc power; multreg model = fixed nfullpredictors = 5 ntestpredictors = 2 rsquarefull = 0.48 rsquarediff = 0.025 ntotal = . power = 0.7 to .9 by .1; run;
Type III F Test in Multiple Regression Fixed Scenario Elements Method Exact Model Fixed X Number of Predictors in Full Model 5 Number of Test Predictors 2 Rsquare of Full Model 0.48 Difference in Rsquare 0.025 Alpha 0.05 Computed N Total Nominal Actual N Index Power Power Total 1 0.7 0.702 164 2 0.8 0.801 204 3 0.9 0.901 267
This series of power analyses yielded sample sizes ranging from 164 to 267. These sample sizes are larger than those for the continuous research variable.
If it is the case that both of these research variables are important, we might want to take into that we are testing two separate hypotheses (one for the continuous and one for the categorical) by adjusting the alpha level. The simplest but most draconian method would be to use a bonferroni adjustment by dividing the nominal alpha level, 0.05, by the number of hypotheses, 2, yielding an alpha of 0.025. We will rerun the categorical variable power analysis using the new adjusted alpha level.
proc power; multreg model = fixed nfullpredictors = 5 ntestpredictors = 2 rsquarefull = 0.48 rsquarediff = 0.025 ntotal = . alpha = .025 power = 0.7 to .9 by .1; run;
Type III F Test in Multiple Regression Fixed Scenario Elements Method Exact Model Fixed X Number of Predictors in Full Model 5 Number of Test Predictors 2 Alpha 0.025 Rsquare of Full Model 0.48 Difference in Rsquare 0.025 Computed N Total Nominal Actual N Index Power Power Total 1 0.7 0.700 199 2 0.8 0.800 243 3 0.9 0.900 311
The bonferroni adjustment assumes that the tests of the two hypotheses are independent which is, in fact, not the case. The squared correlation between the two sets of predictors is about .2 which is equivalent to a correlation of approximately .45. Using an internet applet to compute a bonferroni adjusted alpha taking into account the correlation gives us an adjusted alpha value of 0.034 to use in the power analysis.
proc power; multreg model = fixed nfullpredictors = 5 ntestpredictors = 2 rsquarefull = 0.48 rsquarediff = 0.025 ntotal = . alpha = .034 power = 0.7 to .9 by .1; run;
Type III F Test in Multiple Regression Fixed Scenario Elements Method Exact Model Fixed X Number of Predictors in Full Model 5 Number of Test Predictors 2 Alpha 0.034 Rsquare of Full Model 0.48 Difference in Rsquare 0.025 Computed N Total Nominal Actual N Index Power Power Total 1 0.7 0.702 184 2 0.8 0.801 226 3 0.9 0.901 292
Based on the series of power analyses the school district has decided to collect data on a sample of about 226 students. This sample size should yield a power of around 0.8 in testing hypotheses concerning both the continuous research (momeduc) variable and the categorical research variable language spoken in the home (homelang1 & homelang2). The nominal alpha level is 0.05 but has been adjusted to .034 to take into account the number of hypotheses tested and the correlation between the predictors.
See Also
 Data Analysis Examples
 References

Cohen, J. 1988. Statistical Power Analysis for the Behavioral Sciences, Second Edition.
Mahwah, NJ: Lawrence Erlbaum Associates.