Reading the Log-Mean Scale

From a Count Model to Its Log-Mean Scale

A Poisson model describes a count through its conditional mean. The mean is the expected count at a specified covariate profile; it is not itself the observed count. In a log-mean model, the logarithm of that conditional mean is written as an additive predictor. Reading that predictor correctly is essential: its terms add on the log scale, while the corresponding expected count is obtained by exponentiating the total.

This tutorial focuses on interpreting a change in that total when a predictor or model term changes and all other terms are held fixed. The resulting change in expected counts depends both on the size of the log-scale shift and on the starting profile.

For example, with an intercept and one numeric predictor \(x_i\), the predictor is \(\eta_i=\beta_0+\beta_1x_i\). With several predictors, it can be written \(\eta_i=\beta_0+\beta_1x_{i1}+\cdots+\beta_px_{ip}\). These expressions are additive: each term contributes to the value of \(\eta_i\). The expected count is then \(\mu_i=\exp(\eta_i)\).

The distinction between these scales matters. A coefficient describes a contribution to the log mean, not a fixed number of count units. To find the expected count at a profile, first add the predictor terms at that profile and then exponentiate the result.

Reading Additive Contributions

The intercept is the predictor value when all included covariates are zero, provided that this profile is meaningful for the model. Other terms contribute according to their coefficients and covariate values. For a numeric predictor with coefficient \(\beta_j\), its contribution at value \(x_j\) is \(\beta_jx_j\). To compare two profiles, calculate the predictor at each one; if only one term changes, all the other contributions cancel when finding the difference between the predictors.

Translating a Log-Scale Shift into Counts

The general calculation can be stated directly. Suppose a change to a profile shifts the predictor from \(\eta\) to \(\eta+\delta\), where \(\delta\) is the net change in the additive terms. The starting expected count is \(\exp(\eta)\), and the new expected count is \(\exp(\eta+\delta)\). Their difference is

Proof. By the log-mean specification, the starting conditional mean is \(\exp(\eta)\). After the change, the predictor is \(\eta+\delta\), so the new conditional mean is \(\exp(\eta+\delta)\). Subtract the starting mean and use \(\exp(\eta+\delta)=\exp(\eta)\exp(\delta)\): $$ \exp(\eta+\delta)-\exp(\eta) =\exp(\eta)\exp(\delta)-\exp(\eta) =\exp(\eta)\{\exp(\delta)-1\}. $$ This is the difference in the two conditional means, hence the change in expected counts. \(\square\)

This formula makes two features explicit. First, the direction of the change is determined by \(\delta\): since the exponential function is increasing, a positive shift raises the expected count, and a negative shift lowers it. Second, the size of the count-scale change depends on the starting mean \(\exp(\eta)\) as well as on \(\delta\). Thus, the same shift on the log scale need not correspond to the same number of expected counts at different starting profiles.

Why the Starting Mean Matters

For a fixed positive shift \(\delta\), the count difference is \(\exp(\eta)\{\exp(\delta)-1\}\). The factor in braces is fixed, but \(\exp(\eta)\), the starting mean, can vary across profiles. Consequently, a given log-scale shift produces a larger absolute increase when the starting mean is larger.

Proof. Denote the increase by \(D(\eta)=\exp(\eta+\delta)-\exp(\eta)\). Differentiating with respect to \(\eta\) gives $$ D'(\eta)=\exp(\eta+\delta)-\exp(\eta) =\exp(\eta)\{\exp(\delta)-1\}. $$ For every real \(\eta\), \(\exp(\eta)>0\). Since \(\delta>0\), \(\exp(\delta)-1>0\). Thus \(D'(\eta)>0\) for every \(\eta\), which proves that the increase is strictly larger at a larger starting predictor, and therefore at a larger starting mean. \(\square\)

This result concerns an absolute difference in expected counts. It does not say that the observed count in a particular case must increase by that amount: the response \(Y\) remains random, and an individual observed count can fall even when the model’s conditional mean rises. The comparison is between model-implied conditional means at two covariate profiles.

Common Mistake: Treating a Coefficient as Count Units

A common error is to say, “The coefficient is \(0.5\), so increasing the predictor by one adds \(0.5\) expected counts.” That reasoning treats the coefficient as though it were on the count scale. In a log-mean model, the coefficient contributes to the change in \(\eta=\log(\mu)\), not directly to the change in \(\mu\).

The correction is to compute the predictor at each profile and exponentiate each total. For a one-unit increase in \(x\) with coefficient \(\beta\), the predictor changes by \(\beta\), but the expected-count difference is \(\exp(\eta+\beta)-\exp(\eta)\), where \(\eta\) is the starting predictor. Since this expression includes the starting predictor, a coefficient alone does not determine a fixed count-unit change.