Example:
A researcher conducted a study to predict the marks obtained in final examination by students. His 4 predictor (independent) variables were physical attendance in percentage, score in pre-final examination, gender and score in a separate IQ test. The study revealed R2 = 0.24.
Then he decided to add two more predictors - mother's and father's education. He conducted the study with inclusion of 25 randomly selected students, and he got new effect size, R2 = 0.40, with a total of 6 predictors. What was the power of the study to detect the significant increase in R2, after adding 2 independent variables, at confidence level of 95%?
Solution:
Here
Confidence level = 95%, N = 25, Change in R2=0.16, K1=4, K2=2
After putting these values, we get Power = 41.81%.
How power for simple or multiple regression is calculated (increase in effect size)? (Exclusively for advanced users)
1. Calculation of sample size for Simple / Multiple Linear Regression requires a complex approach.
2. Based on given sample size (N), number of previous predictors (K1), number of additional predictors(K2) : degrees of freedom for numerator (K2) and denominator (N-KN-1) are calculated.
(KN is the total number of predictors = K1 + K2).
Then using given confidence level and these degrees of freedom, F critical value is calculated.
For example, if the number of previous predictors (K1) = 3, additional predictors (K2) = 4, and sample size = 30; degrees of freedom for numerator (d1) will be K2 = 4. Degrees of freedom for denominator (d2) will be N – KN-1 = 22. Now F critical value (x) for 95% confidence level (alpha=0.05), 4 and 22 degrees of freedom is calculated using inverse F distribution function. (It is the F table value for given alpha and degrees of freedom). In this case,critical F value will be 2.8167
3. Non centrality parameter λ (lambda) is calculated using following equation.
λ = f2 * N
. If effect size input is R2 or η2 or f, then f 2 is calculated as follows
f2 = R2/ (1 - R2)
f2 = η2/ (1 - η2)
f2 = f * f
4 Using the values of non-centrality parameter (λ), d1, d2 and x; power of the test is calculated using non central cumulative distribution function formula.
Where, I (q | a, c) is the regularized incomplete beta function.
@ Sachin Mumbare