Could you please help me calculate the minimal sample size in order to detect an interaction effect? I have estimated effect sizes (% of successes) of binary variables:
A - 0.055 (5,5%) B - 0.065 (6,5%) AB - 0.075 (7,5%) When all variables are at zero - 0.05 (5%)
And the factorial design is (used for simulation):
A B C Y 0 0 0 0,05 0 0 1 0,05 1 0 1 0,055 1 0 0 0,055 0 1 1 0,065 0 1 0 0,065 1 1 0 0,075 1 1 1 0,075
For calculations I used NCSS PASS calculator. It uses "Tests for the Interaction Odds Ratio in Logistic Regression with Two Binary X's (Wald Test)"
So my input is:
Solve For: Sample Size Alternative Hypothesis: Two-Sided Power: 0,80 Alpha: 0,05 P0 [Pr(Y = 1 | X = 0, Z = 0)]: 0,05 ORint (X,Z Interaction Odds Ratio): 1,0778 ORyx (Y,X Odds Ratio): 1,056 ORyz (Y,Z Odds Ratio): 1,067 ORxz (X,Z Odds Ratio): 1 Percent with X = 1: 50 Percent with Z = 1: 50 Logistic regression equation: Log(P/(1-P)) = β0 + β1×X + β2×Z + β3×X×Z, where P = Pr(Y = 1|X, Z) and X and Z are binary. Power is the probability of rejecting a false null hypothesis. N is the sample size. P0 is the response probability at X = 0, Z = 0. That is, P0 = Pr(Y = 1|X = 0, Z = 0). Percent X=1 is the percent of the population in which the exposure is 1. Percent Z=1 is the percent of the population in which the confounder is 1. ORint = Exp(β3) is the odds ratio of the interaction. This is the effect size. ORyx = Exp(β1) is the odds ratio of Y versus X. ORyz = Exp(β2) is the odds ratio of Y versus Z. ORxz is the odds ratio of X versus Z in a logistic regression of X on Z. Alpha is the probability of rejecting a true null hypothesis. Beta is the probability of accepting a false null hypothesis.
Here is the similar calculator: http://www.dartmouth.edu/~eugened/power-samplesize.php
Numeric Results for Two-Sided Wald Test Alternative Hypothesis: ORint ≠ 1 Percent Percent Power N X=1 Z=1 P0 ORint ORyx ORyz ORxz Alpha Beta 0,8000 440023 50,0 50,0 0,050 1,078 1,056 1,067 1,000 0,050 0,2000
Could you confirm please whether I made calculations correctly?
ORint (X,Z Interaction Odds Ratio): exponent of 0.075 (AB interaction (7,5%)) = 1,0778 ORyx (Y,X Odds Ratio): exponent of 0.055 ( A 5,5% ) = 1,056 ORyz (Y,Z Odds Ratio): exponent of 0.065 ( B 6,5% ) = 1,067
But I don't understand what does "ORxz" mean. Software says: "one or more values of the Odds Ratio of X and Z, a measure of the relationship between the exposure X and the confounder Z. Note that this measure does NOT involve the outcome variable, Y."