1. Power and Multiple Regression

2. Relationship between Power and hypothesis testing
Accept Null Hypothesis
Reject Null Hypothesis
Null Hypothesis is true
Correct decision
Type I error( alpha typically set to 5%)
Null Hypothesis is False
Type II error (aka Beta)
Correct decision: Probability of making this decision correctly is defined as Power

3. Why is Power important?
Insure sample is large enough to detect effect of interest

4. Requirements to estimate Power
Alpha
Effect size of interest
Sample size

5. Requirements to estimate Power
Effect size of interest
Determined by theory or intuition
Are men heavier than women? What is an “important” difference?
Two kilograms?
Twenty kilograms?

6. Requirements to estimate Power
Alpha
Risk of committing Type I error (rejecting null hypothesis when it is true) vs Type II error (accepting null hypothesis when false)

7. Power in Stata
Powerreg command: gives power estimates for changes in R2
Example: Predicting student’s weight based on height
Requirements:
Alpha: We will use conventional .05
Effect size: What impact do we expect the use height to have on R2?
This is based on prior research, experience, intuition or theory
Let’s use .20 or a 20% increase in r-square
Sample size: 22 students in class

8. Power in Stata

9. Power in Stata
When choosing effect size of interest you are choosing the change in R- square of interest
In most instances you will not have strong a priori reasons for a specific effect size or change in R-square
For the assignment 5% is reasonable

10. Power for Logistic Regression
Use powerlog in Stata
Gives power estimates for changes in predicted probabilities
Requirements:
Alpha:
Effect size:
Sample size:

11. Power for Logistic Regression
Use powerlog in Stata
Gives power estimates for changes in predicted probabilities
Requirements:
Alpha: Conventional .05
Effect size:
Predicted probability at mean of independent variable (P1)
Predicted probability at mean + 1 SD of independent variable (P2)
Sample size: To be determined by program

12. Power for Logistic Regression
Example: Power for baseball team making playoffs based on ERA
Gives power estimates for changes in predicted probabilities
Requirements:
Alpha: Conventional .05
Effect size:
Predicted probability at mean of ERA = . 25 (P1)
Predicted probability at mean + 1 SD of independent variable = .13 (P2)
Sample size: To be determined by program

13. Obtaining predicted probabilities for independent variables
If you have the data
Use summarize command to obtain standard deviation
Use margins command to produce predicted probabilities
E.g. margins, at independent_variable=(1, 2,3…))
If you don’t have data
Use published research
Make educated guess

14. Power for Logistic Regression

15. Power for Logistic Regression using .025 to assume two tailed test

16. Power for Logistic Regression using different predicted probabilities

17. For Assignment If using OLS regression If using Logistic Regression
Estimate power for bivariate model (you can add additional variables for the rest of assignment)
Estimate power for a r-square and an increase in r-square
Use an r-square consistent with prior research, or
Use a change of five percentage points
If using Logistic Regression
Use power analysis for bivariate model (you can add additional variables for the rest of assignment)
Use two predicted probabilities to estimate power
Determine if sample is sufficient for desired power (e.g. .8)