# Example of interpretation of logistic regression

I was looking at a paper by Pell JS 2009, regarding smoking and survival following acute coronary syndrome. Part of the analysis carried out in the paper was a logistic regression with results as shown below: I have 3 questions about these results:

1. As seen in the univariate analysis, is it right to say that only serum cotinine (>0.9) is statistically significant, despite the p-value for serum cotinine being 0.005?

2. Has the strength of association between serum cotinine and all-cause death after adjustment has increased or decreased after multivariate analysis?

Given that the p-value has decreased but the point estimate and SE of all categories of serum cotinine has increased (except <=0.1 which has decreased) after multivariate analysis, I am more inclined to say it has increased but I am not sure.

3. Given the fact that of the 3 factors tested in multivariate analysis only age is associated with all-cause death, I was wondering if we could determine the nature of the association between serum cotinine and age given the changes of ORs for serum cotinine?

I understand that if serum cotinine and age are collinear, the p-value for serum cotinine would become less significant. Therefore, since this did not happen, are Age and serum cotinine independent from one another?

References:

• Pell JS, Haw S, Cobbe S, Newby DE, et al. Secondhand smoke exposure and survival following acute coronary syndrome: prospective cohort study of 1261 consecutive admissions among never-smokers. Heart 2009;95:1415-1418

I think part of what's happening here is that nominal values with ORs of 1 are actually chosen as baselines for the regression. That means, serum cotinine <= 0.1, sex Male, and deprivation quintile 1 are all baselines for evaluating the remaining levels of their respective categorical variables. So, to answer your questions:

1.) My interpretation of the results is that serum cotinine > 0.9 is indeed the only concentration at which the difference from the baseline (i.e. serum cotinine <= 0.1) is indeed statistically significant, with a mean OR of 3.79 and 95% C.I. of 1.77 to 8.13.

2.) The only instance where you could really say that use of multivariate analysis has led to an increase in strength of association is, again at serum cotinine > 0.9 due to the slightly higher OR value (and associated CIs) than in the univariate analysis. The other serum cotinine level ORs' 95% CI still "straddle" 1.0, so their effect is uncertain at the 5% level of significance.

3.) You really couldn't make that pronouncement from these results. You should, instead, test specifically for that using correlation matrices, VIF scores, etc.

@habu has given a good answer here, let me elaborate on a couple points.

1. To determine if a multi-category factor is related to a response, the entire factor needs to be dropped and the nested model tested for a decrease in fit. That is why you get a single p-value.

When a factor is purely nominal (i.e., unrelated categories), it needs to be represented in a way that the analysis can use. The most common way to represent factors is to use reference level coding. That was done here. However, this variable is clearly ordinal in nature. It could have been coded as an ordinal factor, but many people find the output confusing and so reference level coding is still often preferred (I typically use it myself). With a nominal factor, people will sometimes perform post-hoc tests to determine which levels differ, but since this variable is ordinal, that is less appropriate here. Despite the fact that only the first and last categories differ 'significantly', it is better to think of the levels as measuring an underlying / latent continuous variable, serum cotinine, and that the odds of all cause death increase with higher levels of serum cotinine (not just when it's >.9).

2. Since the point estimates of the odds ratios for serum cotinine have increased, I would say that the conditional association is stronger than the marginal association.

3. You largely cannot say from the results displayed here. If the variables were perfectly multicollinear, the model would not have been fitable, so we can conclude they weren't perfectly collinear. The 95% CIs for serum cotinine tended to get slightly wider in the multivariate model, which would not happen if the variables were perfectly independent, so we can conclude that wasn't the case either (plus data are never perfectly independent except in designed experiments where all variables are manipulated so as to be perfectly independent).