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 is 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 Stata program powerreg (search powerreg) (see How can I use the search command to search for programs and get additional help? for more information about using search) 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, r2r) would be about 0.45. The total number of variables (nvar) is 5 and the number being tested (ntest) is one. We will run powerreg three times with power equal to .7, .8 and .9.
powerreg, r2f(.48) r2r(.45) nvar(5) ntest(1) power(.7) Linear regression power analysis alpha=.05 nvar=5 ntest=1 R2full=.48 R2reduced=.45 R2change=0.0300 nominal actual power power n 0.7000 0.6959 108 powerreg, r2f(.48) r2r(.45) nvar(5) ntest(1) power(.8) Linear regression power analysis alpha=.05 nvar=5 ntest=1 R2full=.48 R2reduced=.45 R2change=0.0300 nominal actual power power n 0.8000 0.7998 138 powerreg, r2f(.48) r2r(.45) nvar(5) ntest(1) power(.9) Linear regression power analysis alpha=.05 nvar=5 ntest=1 R2full=.48 R2reduced=.45 R2change=0.0300 nominal actual power power n 0.9000 0.8966 182
This gives us a range of sample sizes ranging from 108 to 182 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 r2r of 0.455. The nvar stays at 5 while the ntest is now 2.
powerreg, r2f(.48) r2r(.455) nvar(5) ntest(2) power(.7) Linear regression power analysis alpha=.05 nvar=5 ntest=2 R2full=.48 R2reduced=.455 R2change=0.0250 nominal actual power power n 0.7000 0.7021 164 powerreg, r2f(.48) r2r(.455) nvar(5) ntest(2) power(.8) Linear regression power analysis alpha=.05 nvar=5 ntest=2 R2full=.48 R2reduced=.455 R2change=0.0250 nominal actual power power n 0.8000 0.7990 203 powerreg, r2f(.48) r2r(.455) nvar(5) ntest(2) power(.9) Linear regression power analysis alpha=.05 nvar=5 ntest=2 R2full=.48 R2reduced=.455 R2change=0.0250 nominal actual power power n 0.9000 0.8997 266
This series of power analyses yielded sample sizes ranging from 164 to 266. 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.
powerreg, r2f(.48) r2r(.455) nvar(5) ntest(2) alpha(.025) power(.7) Linear regression power analysis alpha=.025 nvar=5 ntest=2 R2full=.48 R2reduced=.455 R2change=0.0250 nominal actual power power n 0.7000 0.7000 199 powerreg, r2f(.48) r2r(.455) nvar(5) ntest(2) alpha(.025) power(.8) Linear regression power analysis alpha=.025 nvar=5 ntest=2 R2full=.48 R2reduced=.455 R2change=0.0250 nominal actual power power n 0.8000 0.8042 245 powerreg, r2f(.48) r2r(.455) nvar(5) ntest(2) alpha(.025) power(.9) Linear regression power analysis alpha=.025 nvar=5 ntest=2 R2full=.48 R2reduced=.455 R2change=0.0250 nominal actual power power n 0.9000 0.8971 308
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.
powerreg, r2f(.48) r2r(.455) nvar(5) ntest(2) alpha(.034) power(.7) Linear regression power analysis alpha=.034 nvar=5 ntest=2 R2full=.48 R2reduced=.455 R2change=0.0250 nominal actual power power n 0.7000 0.6966 182 powerreg, r2f(.48) r2r(.455) nvar(5) ntest(2) alpha(.034) power(.8) Linear regression power analysis alpha=.034 nvar=5 ntest=2 R2full=.48 R2reduced=.455 R2change=0.0250 nominal actual power power n 0.8000 0.8030 227 powerreg, r2f(.48) r2r(.455) nvar(5) ntest(2) alpha(.034) power(.9) Linear regression power analysis alpha=.034 nvar=5 ntest=2 R2full=.48 R2reduced=.455 R2change=0.0250 nominal actual power power n 0.9000 0.9031 294
Based on the series of power analyses the school district has decided to collect data on a sample of about 225 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 and 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.