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I run a multinomial logistic regression model on a dependent variable Y with five levels (with one of them set as a reference). Though the outputs of the model were in line with my expectations, I have HUGE relative risk estimates on the intercept values of some variables (in the thousands).

I believe this can be due to the low number of cases in some of the levels; for example, one level has only 42 cases while the reference has 858. Could this be a factor in getting overinflated risk estimates? And is it wrong to report such bizarre estimates?

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I am assuming you are using the odds ratio as an approximation of the relative risk or the relative risk ratio, and that you are using the latter two terms equivalently.

The multinomial logistic regression model does not estimate relative risk for intercepts unless it is parametrized to do so. The usual output shows an intercept which is a "baseline odds" (using Stata terminology) or an expected prevalence/risk if all $X$ = 0. You can, however, perform comparisons between the various intercept terms.

If these comparisons are statistically or practically significant, then all it means that the number of $Y$ categorical levels have a non-uniform frequency when all the $X$ predictors are equal to 0.

This can be an artifact of having an inappropriately scaled $X$. But it can also simply be the truth. As far as reporting, I fail to see why it is ever useful to report intercepts from models. It is not wrong to report these values, provided you can appropriately describe the output to your target audience.

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  • $\begingroup$ Thank you! After scaling (centering) one of the explanatory variables, the estimates for the coefficient has gotten much lower and reasonable (albeit not in the direction I expected, it is now negative -- which I believe means that for my outcomes of interest, people have lower intercepts than the reference categories, making it a bit contrary to my expectations). The coefficients and SE for the scaled variable remain the same. $\endgroup$
    – guaguncher
    Commented Feb 27, 2018 at 14:07
  • $\begingroup$ Regarding the interpretation, I found read here (stats.idre.ucla.edu/r/dae/multinomial-logistic-regression) that the model 'exp(coef)' produces relative risk ratios. Is that correct? $\endgroup$
    – guaguncher
    Commented Feb 27, 2018 at 16:20
  • $\begingroup$ @guaguncher only when "coef" is not an intercept term or a product term. The intercept in a logistic or multinomial model is a log-odds of the outcome with all covariates equal to 0. Think of the linear model as an analogue: the intercept is in units of "y" whereas the slope is in units of "y/x". $\endgroup$
    – AdamO
    Commented Feb 27, 2018 at 17:17
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There is also the possibility that your variable is not centered or scaled well or even that you have a data point with a typo.

For example, say you were addressing the effect of a binary variable on an outcome. All it would take is a mistake with one 1 becoming a 10 or .1 to drastically inflate your relative risk.

My suggestions are as follows-

  1. go back to your data and check for any data entry errors (informally check for gross outliers as well).
  2. assess if your problem is due to inappropriately scaling a variable
  3. assess if your problem is due to a variable being poorly centered
  4. formally assess outliers
  5. formally assess assumptions

That should hopefully point you in the right direction.

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  • $\begingroup$ But what of the prevalence of the outcome? It's possible these results are just due to a large imbalance in two or more levels of the categorical outcome. $\endgroup$
    – AdamO
    Commented Feb 26, 2018 at 23:26
  • $\begingroup$ That is a distinct possibility. In my opinion, though, if there is a large large RR that is associated with a category, and it does not make sense in the literature. Its more likely that it is an error (i.e. in the case of the op and scaling), than that a new finding has been stumbled upon. $\endgroup$
    – JWH2006
    Commented Feb 28, 2018 at 20:28

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