GLMM Analysis – How to Back Transform Standard Errors in a GLMM with a Log-Link Function

generalized linear modellink-functionmixed modelreporting

I'm using lme4 and have a GLMM with a log link and gaussian variance structure. I would like to report my fixed effect estimates with their standard errors, as well as the standard deviation of my random effects on the response scale. The output from my model is on the link scale. Can I simply exponentiate the output, or do I need to do more?

Best Answer

Standard errors do not transform. If uncertainty intervals on the link scale are symmetric (as they often approximately are), when you anti-log parameter estimates the uncertainty on the anti-logged scale will be asymmetric. You can anti-log the confidence limits but cannot get a standard error that is useful on the original scale because of the needed asymmetry.

Related Solutions

Solved – Back transform mixed-effects model’s regression coefficients for fixed-effects from log to original scale

The reason those numbers don't make sense is because the whole linear part is exponential, since you log transformed the response.

To get the right values on the original scale, first add the coefficients to the intercept as you would normally do, then backtransform using the exponent. For example:

exp(sum(coef[c(1, 2)]))

As for your second question, you don't do anything with the random effect to do inference on the fixed effects. You already accounted for the structural effect by including it in the model.

Solved – Why does glm() provide estimates and standard errors on the link scale

Hard to know for sure, but there are a few reasons the link scale is useful.

Using standard errors as a summary of uncertainty is generally more reliable on the link scale, where the domain of the parameters is unbounded and where the assumption that the likelihood surface is approximately quadratic ($\leftrightarrow$ sampling distribution of the parameter estimates is approximately Normal) is more likely to be reasonable. For example, suppose you have a log-link model with estimate (on the link scale) 1.0 and standard error 3.0. On the link scale, the confidence interval is approximately $1 \pm 1.96 \times 3$. If you back-transform, exponentiating the parameter and multiplying the standard error by the exponentiated parameter (as in this answer), and then try to construct symmetric CIs, you get $2.718 \pm 1.96 \times 3 \times 2.718$, which includes negative values ... if you do want to back-transform, it makes more sense to back-transform the confidence intervals, i.e. $\exp(1 \pm 1.96 \times 3)$.
Probably more importantly, for the very common logit link, it's basically impossible to sensibly back-transform the parameters all the way to the data scale (i.e., from logit/log-odds-ratios to probability). It is common to exponentiate parameters to move from the log-odds-ratio to the odds-ratio scale, but you can't go back from odds ratios to probabilities without specifying a baseline value. That is, you can say in general "the odds ratio associated with control vs. treatment is XXX", but the change in probability from control to treatment will depend on other covariates (e.g., the odds ratio for females and males may be the same while the change in probability is different because the baseline risk is different for females and males).

Probably the proximal reason is that because of the issues listed above, most people who do a lot of statistical modeling have gotten used to interpreting parameters on the link scale; most epidemiologists and biostatisticians have to spend time learning about odds ratios and log-odds ratios, and there are lots of papers written about their interpretation. For better or worse, R was written by people who are comfortable interpreting parameters on the link scale. Many downstream packages such as broom have options that will exponentiate parameters and CIs for you (putting them on the data (count) scale for the log link; the odds-ratio scale for logit links; and the hazard-ratio scale for cloglog links).

Best Answer

Related Solutions

Solved – Back transform mixed-effects model’s regression coefficients for fixed-effects from log to original scale

Solved – Why does glm() provide estimates and standard errors on the link scale

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