Interpreting Coefficients on Link and Response Scales

Two Scales, Two Kinds of Interpretation

In a generalized linear model, a coefficient first describes a change in the linear predictor, not necessarily a fixed change in the mean response. The link function connects that linear predictor to the response mean. Consequently, interpreting a coefficient requires two steps: identify its change on the link scale, then apply the inverse link to understand the corresponding response-scale comparison.

This distinction matters especially for logit and log links. A logit coefficient gives a change in log odds and, after exponentiation, an odds ratio. A log-link coefficient gives a change in log mean and, after exponentiation, a ratio of means. Neither coefficient generally gives a constant probability or mean difference. Those differences depend on the starting probability or mean.

These interpretations concern the model’s specified mean structure. For a logit model with a Bernoulli or binomial response, the mean is a probability \(p\). For a log-link model, the mean \(\mu\) is positive, as in common Poisson and Gamma mean models. The algebraic coefficient interpretation requires the stated link and linear predictor. Statistical inference about the coefficient additionally depends on the model’s distributional and sampling assumptions; a fitted association alone does not establish a causal effect.

Logit Coefficients: Log Odds and Odds Ratios

For a binary response with probability \(p\), the logit link is \(g(p)=\log\{p/(1-p)\}\). Thus the linear predictor is the log odds. If \(x_j\) increases by one unit, with the other predictors unchanged, the log odds change by \(\beta_j\). Exponentiating that contrast gives the ratio of the two odds.

Here \(p_0\) and \(p_1\) are the model probabilities at predictor values differing by one unit, with all other predictors fixed. The quantity \(e^{\beta_j}\) is the odds ratio for that one-unit increase. An odds ratio above 1 means the modeled odds increase; an odds ratio below 1 means they decrease. It does not say that probability increases or decreases by that same ratio.

Deriving the Probability Comparison

An odds ratio can be translated into a probability comparison when a starting probability is specified. Write \(R=e^{\beta_j}\). The odds at the second predictor value are \(R\) times the starting odds. Solving this relation for the second probability gives the response-scale formula.

The probability difference is \(p_1-p_0\). Since the formula contains \(p_0\), the same odds ratio can correspond to different probability differences at different starting probabilities. A probability difference is often expressed in percentage points: a change from 0.60 to 0.77 is about 17 percentage points, not a 17% relative increase.

Log-Link Coefficients: Log Means and Mean Ratios

For a positive conditional mean \(\mu\), the log link is \(g(\mu)=\log(\mu)\). If \(x_j\) increases by one unit while other predictors remain fixed, the change in log mean is \(\beta_j\). Exponentiating gives a ratio of conditional means:

When \(\beta_j>0\), the mean is multiplied by a factor greater than 1; when \(\beta_j<0\), it is multiplied by a factor between 0 and 1. The corresponding mean difference is \(\mu_1-\mu_0=\mu_0(e^{\beta_j}-1)\). It depends on the starting mean \(\mu_0\), even though the mean ratio is constant across starting means under this model. The means retain the response’s units; the ratio does not have units.

Comparing the Interpretations

Both links turn a coefficient into a multiplicative comparison on a transformed scale. The response-scale quantities are different: the logit link concerns odds for a probability response, whereas the log link concerns the conditional mean itself. Keeping these interpretations separate prevents a common error: treating an exponentiated coefficient as if it were a probability ratio in a logit model, or as an absolute mean difference in a log-link model.

A useful reporting practice is to state both a scale-specific effect and its context. For a logit model, report the odds ratio and, when a relevant baseline is available, the modeled probabilities or probability difference. For a log-link model, report the mean ratio and, when useful, the comparison in outcome units at a specified starting mean. In each case, identify the one-unit predictor contrast and make clear that other predictors are held fixed.

Assumptions, Diagnostics, and Common Pitfalls

These interpretations follow from a particular model specification: the chosen link is used for the response mean, the predictor enters the linear predictor as stated, and the comparison changes only the predictor of interest. For example, a single \(\beta_j\) describes a constant link-scale change per unit only when the model’s linear predictor contains the corresponding linear term and does not include an interaction that changes the contrast with another predictor. If an interaction is present, the contrast must be calculated at the specified value of the interacting predictor.

Before interpreting a coefficient, check how the predictor is coded and scaled. A one-unit increase might mean one year, one kilogram, or one category change; if the predictor was recorded in tens, the unit in the model is ten original units. For a binary indicator coded 0 and 1, its coefficient compares the modeled groups, conditional on the other predictors, rather than describing a continuous change. Also check that the response family and link are those actually specified and that the requested starting probability or mean corresponds to the covariate profile being discussed.

A common mistake is to say, “A logit coefficient of 0.8 raises the probability by 0.8.” That is incorrect: 0.8 is a change in log odds, and \(e^{0.8}\) is the odds ratio. The probability difference must be calculated from a starting probability. Likewise, a log-link coefficient of 0.2 is not an increase of 0.2 outcome units; it implies a mean ratio \(e^{0.2}\), and the absolute difference depends on the starting mean.

These are model-based comparisons, not automatic causal effects. A coefficient may describe an adjusted association under the model, but causal interpretation requires additional design and identification assumptions. Further, an exponentiated point estimate alone does not express its uncertainty: inferential conclusions require an appropriate standard error or interval for the coefficient and the relevant modeling assumptions.