# Why is a variable with a comparatively smaller odds ratio statistically significant (when the larger OR is not)? [closed]

I have two predictors in a binary logistic regression model, where both variables are on the same scale (standardized). I find it puzzling that the variable with a smaller OR is significant at p <. 05 (OR=1.3) when the one on the same scale and has a larger OR=2 is NOT significant in the same model.

More importantly, when the two variables are tested in separate models, the variable with the larger OR also shows a higher probability estimate for the outcome variable (75%) (vs. smaller: 60%)

Could someone please explain how/why this might be or if this could be diagnostic of a problem (multicollinearity?)

Results

s01=summary(pool(f1))

s01

                     est        se         t       df   Pr(>|t|)

(Intercept) -1.7805826 0.1857663 -9.585070 391.0135 0.00000000
X1           0.2662796 0.1308970  2.034268 390.4602 0.04259997
X2           0.6757952 0.3869652  1.746398 395.6098 0.08151794


cbind(exp(s01[, c("est", "lo 95", "hi 95")]), pval=s01[, "Pr(>|t|)"])

                             est     lo 95     hi 95       pval
(Intercept) 0.1685399 0.1169734 0.2428389 0.00000000
X1          1.3051000 1.0089684 1.6881459 0.04259997
X2          1.9655955 0.9185398 4.2062035 0.08151794

• As you can see, s01 estimates are log odds ratios.

logOR=log(1.9655955)

logOR [1] 0.6757953

Despite X2 having larger OR and log odds ratio, it is not significant in the model.

Update

One possibility I have not tried is testing the mean difference between the two OR (or log OR?) using the following method: Statistical test for difference between two odds ratios?

I am curious if testing the difference between the two log OR might change the conclusions and would appreciate suggestions on how I might do so using the output above (getting the standard error of logOR for the two coefficients).

## closed as unclear what you're asking by gung♦Jun 11 '18 at 23:43

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• Significance is not the same as effect size--of which the OR is an example. It is determined in logistic regression by comparing the OR to its standard error. Doesn't your computation produce and report standard errors of the estimates? What are they equal to? – whuber Jun 3 '18 at 20:34
• SE for the non-significant predictor is in fact larger (0.4) vs. (0.13 for the significant predictor)! Given this, is it fair to conclude that the significant predictor (with smaller OR) predicts the binary outcome ABOVE and Beyond the other? Does the interpretation work the same with logistic regression? – ksroogl Jun 3 '18 at 20:41
• As for viewing OR as a form of an effect size, is it your suggestion that I divide the OR by its SE? If not, would you have any particular suggestion? – ksroogl Jun 3 '18 at 20:56
• Usually it's the log OR that is estimated and the SE refers to that. In that case, dividing the former by the latter gives a t-statistic that is either referred to a Student t distribution or a Normal distribution to determine its significance. That detail aside, what you want to compare is the sizes of those ratios, not the ORs themselves. – whuber Jun 3 '18 at 22:04
• No, because p-values and effect sizes tell you two completely different things. – whuber Jun 4 '18 at 13:31