# Reporting results of a logistic regression

I have the following logistic regression output:

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept)   0.5716     0.1734   3.297 0.000978 ***
R1           -0.4662     0.2183  -2.136 0.032697 *
R2           -0.5270     0.2590  -2.035 0.041898 *


Is it appropriate to report this in the following way:

Beta coefficient, Odds ratio, Zvalue, P value. If yes, how can I obtain the Odds ratio?

Your suggested reporting for a table seems reasonable, although z-values and p-values are redundant. Many journals I am familiar with don't report the z-value/p-value at all and only use asterisks to report statistical significance. I have also seen logistic tables only with the odd's ratios reported, although I personally prefer both the log odds and odds ratios reported if space permits in a table.

But different venues may have different guides as to reporting procedures, so what is expected may vary. If I'm submitting a paper to a journal I will frequently just see how other recent papers have made their tables and just mimic those. If it is your own personal paper, asking whomever may be reviewing it would be a reasonable request. As I mentioned above, space constraints in some venues may prevent you from reporting ultimately redundant information (such as both the log odds and the odds ratios). Some places may force you to report the results entirely in text!

There is also the question of what other model summaries to report. Although many journals I am familiar with frequently report pseudo $R^2$ values, here is a thread on the site that discusses the weaknesses of various measures. I personally prefer classification rates to be reported, but again I suspect this varies by venue (I can imagine some journals would specifically ask for one of the pseudo $R^2$ measures to be reported).

To get the odd's ratio just exponentiate the regression coefficient (i.e. take $e^{\hat{\beta}}$ where $e$ is the base of the natural logarithm and $\hat{\beta}$ is the estimated logistic regression coefficient.) A good guess in any statistical language to calculate this is exp(coefficient).

Also as a note, although this is the current accepted answer, lejohn and Frank Harrell both give very useful advice. While I would typically always want the statistics in the question reported somewhere, the other answers advice about other measures are useful ways to assess effect sizes relative to other estimated effects in the model. Graphical procedures are also useful to examine relative effect sizes, and see these two papers on turning tables into graphs as examples (Kastellec & Leoni, 2007; Gelman et al., 2002)

• The Kastellec & Leoni, 2007 link is broke, but here is an example from the same paper with code. – ACNB Apr 24 '17 at 15:00

Here are some general considerations.

The beta's in logistic regression are quite hard to interpret directly. Thus, reporting them explicitly is only of very limited use. You should stick to odds ratios or even to marginal effects. The marginal effect of variable x is the derivative of the probability that your dependent variables is equal to 1, with respect to x. This way of presenting results is very popular among economists. Personally I believe that marginal effects are more easily understood by laymen (but not only by them ... ) than odds ratios.

Another interesting possibility is to use graphical displays. A place where you will find some illustrations of this approach is the book of Gelman and Hill. I find this even better than reporting marginal effects.

Regarding the question on how to get odds ratios, here is how you can do it in R:

model <- glm(y ~ x1 + x2, family=binomial("logit"))
oddrat <- exp(coef(model))

• Do you have any examples of turning the marginal effect estimates into tables (or specific page references in Gelman and Hill?) – Andy W Aug 22 '11 at 12:51
• Gelman and Hill do not really use the marginal effects, but rather graphs, that are based on predicted probabilities. Have a look at chapter five, starting at page 79. – user5644 Aug 22 '11 at 13:03

It's only in special cases where the coefficients and their anti-logs (odds ratios) are good summaries. This is when the relationships are linear and there is one coefficient associated with a predictor, and when a one-unit change is a good basis for computing the odds ratio (more O.K. for age, no so much for white blood count having a range of 500-100,000). In general, things like inter-quartile-range odds ratios are useful. I have more detail about this at http://biostat.mc.vanderbilt.edu/wiki/pub/Main/RmS/rms.pdf and the R rms package does all this automatically (handling nonlinear terms and interactions, compute quartiles of X, etc.).

It likely depends on your audience and discipline. The answer below is what is normally done for Epidemiology journals, and to a lesser extent medical journals.

To be blunt, we don't care about p-values. Seriously, we don't. Epidemiology won't even let you report them unless you have a really, really pressing need to, and the field has essentially followed suit.

We might not even care about beta estimates, depending on the question. If your report is on something more methodological or simulation oriented, I'd probably report the beta estimate and standard error. If you're trying to report an effect as estimated in the population, I'd stick with the Odds Ratio and 95% Confidence Interval. That's the meat of your estimation, and what readers in that field will be looking for.

Answers have been posted above for how to get the odds ratio, but for the OR & 95% CI:

OR = exp(beta)
95% CI = exp(beta +/- 1.96*std error)

• thanks for the answer! can I ask what the 1.96 stands for in the calculation? – upabove Aug 22 '11 at 23:07
• No problem :) The 95% confidence interval is the span that should cover roughly from the 2.5th percentile to the 97.5th percentile of the normal distribution of your beta estimate. Each of those points is roughly 1.96 standard deviations from the mean (in this case, beta). – Fomite Aug 23 '11 at 5:01
• yes but should I do this for each beta coefficient? also does the 1.96 change? also this is binomial data is it still based on the normal distribution? – upabove Aug 23 '11 at 6:49
• Yes - for each beta coefficient, you can obtain an odds ratio and 95% confidence limit. The 1.96 will not change unless you want to calculate a different percentile for the confidence interval (90%, 99%, etc.), but as 95% is standard, there's no need to do so. And as long as you are working on the log scale, the parameters from a logistic regression model are normally distributed. Once you exponentiate them, this stops being true. – Fomite Aug 23 '11 at 15:05