# Is a logistic regression appropriate for calculating odds ratio with low event rate?

I have what seems to be a simple question: is risk factor x associated with outcome y or z?

I have two samples that were 1:1 propensity score matched with good results (all 19 measurable covariates are within 0.1 SMD of each other).

My resulting contingency table looks like:

        yFALSE yTRUE
xFALSE   908    3
xTRUE    905    6


and

        zFALSE zTRUE
xFALSE   890   21
xTRUE    887   24


My outcomes are pretty rare events, especially y.

In order to quantify the association between x and y/z, I figure I have two options:

A simple Chi-square test of proportions:

                           x
FALSE        TRUE       p
y = TRUE (%)     3 ( 0.3)    6 ( 0.7)   0.504
z = TRUE (%)    21 ( 2.3)   24 ( 2.6)   0.763


Or a univariate logistic regression and a Wald's test (I am not adjusting for the variables I previously matched for due to the oft-cited 10:1 EPV rule of thumb):

For y:

Deviance Residuals:
Min       1Q   Median       3Q      Max
-0.1150  -0.1150  -0.0812  -0.0812   3.3811

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  -5.7126     0.5786  -9.873   <2e-16 ***
xTRUE       0.6965     0.7091   0.982    0.326
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for quasibinomial family taken to be 1.001122)

Null deviance: 113.54  on 1821  degrees of freedom
Residual deviance: 112.52  on 1820  degrees of freedom
AIC: NA


For z:

Deviance Residuals:
Min       1Q   Median       3Q      Max
-0.2311  -0.2311  -0.2160  -0.2160   2.7459

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  -3.7467     0.2209 -16.961   <2e-16 ***
xTRUE         0.1369     0.3027   0.452    0.651
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for quasibinomial family taken to be 1.001126)

Null deviance: 421.97  on 1821  degrees of freedom
Residual deviance: 421.77  on 1820  degrees of freedom
AIC: NA


What other reasonable options do I have for quantifying and reporting the relationship between the IV x and the DVs y and z, especially when y and z are such rare events?

And a slightly unrelated question:

If I were to exponentiate the coefficient from my logistic regression, would this be the marginal OR since it is unadjusted?