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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

Could you confirm please whether I made calculations correctly?

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."

kjetil b halvorsen
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Gustavjung
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  • relevant link: http://andrewgelman.com/2018/03/15/need-16-times-sample-size-estimate-interaction-estimate-main-effect/ – R J Mar 23 '18 at 10:48
  • Title of that Gelman link: *You need 16 times the sample size to estimate an interaction than to estimate a main effect* – kjetil b halvorsen Mar 12 '19 at 11:39
  • Possible dups: https://stats.stackexchange.com/questions/162294/multiple-logistic-regression-power-analysis/396681#396681 and https://stats.stackexchange.com/questions/35940/simulation-of-logistic-regression-power-analysis-designed-experiments and https://stats.stackexchange.com/questions/26016/sample-size-for-logistic-regression – kjetil b halvorsen Mar 12 '19 at 11:40

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