# Sample size of logistic regression with interaction

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

The result:

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


My calculations:

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."