Link Functions for Non-Continuous Mixed-Effects Models

Why Link Functions Are Needed

A mixed-effects model for a non-continuous response must connect an individual’s covariates and random effects to a response probability or mean. A straight-line predictor can take any real value, but a probability must lie between zero and one, and a positive mean must be greater than zero. A link function makes this connection by transforming the response mean onto a scale where the predictor can take any real value.

Let \(Y_{ij}\) denote the response for individual \(i\) at observation \(j\), and let \(\mu_{ij}=E(Y_{ij}\mid b_i)\) be its conditional mean given that individual’s random effects \(b_i\). Write \(\eta_{ij}\) for the individual-specific predictor. A common form is \(\eta_{ij}=x_{ij}^{T}\beta+z_{ij}^{T}b_i\), where \(x_{ij}\) and \(z_{ij}\) are covariate vectors, \(\beta\) is a fixed-effects parameter vector, and \(b_i\) is an individual random-effects vector. More generally, the predictor may be a nonlinear function of covariates and parameters. The link specifies how \(\mu_{ij}\) and \(\eta_{ij}\) are related; it does not, by itself, specify the response distribution or the random-effects distribution.

The key requirement is that the inverse link maps every permitted predictor value to a value allowed by the response mean. This is why the choice of link is tied to the response scale. The four links in this tutorial are widely used, but they are not interchangeable: their inverse functions have different shapes and, for the log link, a different range.

Four Links and Their Response Scales

For a binary response, write \(p_{ij}=P(Y_{ij}=1\mid b_i)\). The logit, probit, and complementary log-log links each map \(0<p_{ij}<1\) to an unrestricted real-valued predictor. Their inverse functions therefore produce valid probabilities for any finite \(\eta_{ij}\).

All three inverse links are increasing, but they do not assign the same probability to a given predictor. The logit curve is symmetric around \((0,0.5)\). The probit curve is also symmetric around that point, but its shape differs from the logistic curve. The complementary log-log curve is asymmetric: it approaches zero and one at different rates. Consequently, choosing a link affects the model’s assumed response curve, not merely the way its equation is written.

The complementary log-log link also has a useful event-probability interpretation. If an event occurs according to a constant hazard \(\lambda_{ij}\) over an interval of length \(\Delta\), then the probability of at least one event in that interval is \(p_{ij}=1-\exp(-\lambda_{ij}\Delta)\). Setting \(\eta_{ij}=\log(\lambda_{ij}\Delta)\) gives \(p_{ij}=1-\exp\{-\exp(\eta_{ij})\}\), the complementary log-log inverse. This interpretation depends on the constant-hazard-over-the-interval assumption; the link alone does not establish that assumption.

For a positive conditional mean, a common choice is the log link. Unlike the three probability links, it is not a probability model: its inverse produces positive values with no upper bound. That range is appropriate for a positive expected count or rate, but not generally for a probability.

How the Inverse Links Preserve the Response Scale

The probability links are valid because their inverse functions stay in the probability range. For the logit inverse, \(\exp(-\eta)>0\), so its denominator \(1+\exp(-\eta)\) is greater than one. Thus \(0<1/(1+\exp(-\eta))<1\). For the probit inverse, a cumulative distribution function takes values strictly between zero and one at every finite real argument. For the complementary log-log inverse, \(\exp(\eta)>0\), so \(0<\exp\{-\exp(\eta)\}<1\), and subtracting this quantity from one also gives a value strictly between zero and one.

The log link preserves positivity because \(\exp(\eta)>0\) for every real \(\eta\). These range properties matter in an NLME model because the random effects change the individual predictor. No matter what finite value the random-effects contribution gives that predictor, the inverse link still returns a value on its required scale. This guarantee concerns the range, not whether the resulting predictions fit the data or whether the chosen response distribution is appropriate.

This sequence makes explicit a useful distinction: the predictor is on the link scale, while the mean is on the response scale. A value of \(\eta_{ij}=0.5\), for example, is not itself a probability or an expected count. It must be transformed using the selected inverse link.

Interpreting Shape and Choosing Among Links

The links differ in how quickly probabilities change as the predictor changes. For the logit inverse, the derivative with respect to \(\eta\) is \(p(1-p)\); for probit it is the standard normal density evaluated at \(\eta\); for complementary log-log it is \(\exp(\eta)\exp\{-\exp(\eta)\}\). Each derivative is positive, so each probability increases with the predictor. Their different derivative patterns explain why the curves can yield different predictions, especially away from the center.

For the log link, \(\mu=\exp(\eta)\), and the derivative is \(\mu\). A fixed increase \(c\) in the predictor multiplies the mean by \(\exp(c)\), because \(\exp(\eta+c)=\exp(c)\exp(\eta)\). This multiplicative structure is often useful for positive means and rates. It also means that an additive predictor change has a larger absolute effect when the starting mean is larger.

The link is only one part of the model specification. A link that produces valid probabilities does not guarantee that the assumed binary response distribution, random-effects structure, or predictor is suitable. Similarly, a positive mean from a log link does not establish that a particular count distribution is appropriate. Those assumptions must be specified and assessed separately. In repeated-measures models, the predictor and mean here are conditional on the individual random effects; averaging over individuals is a different operation and should not be confused with this mapping.

Diagnostics, Limitations, and Common Mistakes

A first check is mathematical: calculate predictions on the response scale and verify that they are valid. A probability outside \([0,1]\) signals an implementation error or an unsuitable mapping; the log link cannot be used as an unrestricted probability link because \(\exp(\eta)>1\) whenever \(\eta>0\). For a positive mean, a nonpositive prediction would likewise contradict the log-link mapping and indicate a calculation or implementation problem.

A valid range does not establish good fit. Predictions that cluster extremely close to zero or one may be sensitive to small changes in the predictor, and predicted event frequencies should be checked against observed frequencies across relevant covariate or predictor ranges. For count outcomes, compare predicted means with observed counts in context and check whether the response distribution represents the data’s variability. Such checks address model adequacy; they do not change the inverse-link formulas.

A common mistake is to treat the link-scale predictor as if it were already on the response scale. For example, saying “the probability is \(-0.70\)” from a logit predictor of \(-0.70\) is incorrect. The correction is to apply the logit inverse, which gives a probability of approximately \(0.3318\). Another mistake is to choose a link solely because it returns a valid range. Several links can return valid probabilities, but their curve shapes and scientific interpretations differ. The response scale, the expected relationship, and the assumptions supporting an interpretation should all guide the choice.