Interpreting Coefficients Under the Log Link

Why a Log-Link Coefficient Is Not a Percentage

In a generalized linear model, a coefficient is estimated on the scale of the link function. With a log link, that means a coefficient describes a change in the logarithm of the conditional mean, not a direct change in the mean itself. Exponentiating the coefficient translates that change into a multiplicative effect on the mean.

This distinction is useful in predictive analytics because it lets an analyst explain how fitted expected counts or positive-valued outcomes differ when a predictor changes. The interpretation depends on the size of the predictor change, the model’s terms, and the other predictors being held fixed. It is not, by itself, a statement about a causal effect.

Definition: A GLM with a log link has conditional mean $\mu(\mathbf{x})=E[Y\mid \mathbf{x}]>0$ and linear predictor $\eta(\mathbf{x})$, related by $$ \log\{\mu(\mathbf{x})\}=\eta(\mathbf{x}). $$ For a model with an intercept and additive predictor terms, $\eta(\mathbf{x})=\beta_0+\beta_1x_1+\cdots+\beta_px_p$. A coefficient $\beta_j$ is the change in the log conditional mean for a one-unit increase in $x_j$, holding the other included predictors fixed, when the model has no interaction involving that change.

From a Coefficient to a Mean Multiplier

Suppose $x_j$ increases by $\Delta$, while the other predictors remain fixed and the model has no interaction involving $x_j$. The linear predictor changes by $\beta_j\Delta$. Because the link is logarithmic, exponentiating that difference gives the ratio of the new conditional mean to the original conditional mean.

\log(\mu_{\text{new}})-\log(\mu_{\text{old}}) =\beta_j\Delta \quad\Longrightarrow\quad \frac{\mu_{\text{new}}}{\mu_{\text{old}}} =\exp(\beta_j\Delta).

Thus the new mean is the old mean multiplied by $\exp(\beta_j\Delta)$. The corresponding relative change, expressed as a percentage, is $100\{\exp(\beta_j\Delta)-1\}\%$. This percentage can be positive or negative. For a one-unit increase, set $\Delta=1$; for a decrease, $\Delta$ is negative. A negative coefficient therefore implies a decrease for an increase in the predictor, but the size of that decrease is determined by exponentiation.

Key result: Under a log-link conditional-mean model, a predictor change of $\Delta$ with coefficient $\beta$ multiplies the fitted mean by $\exp(\beta\Delta)$. The exact percentage change is $100\{\exp(\beta\Delta)-1\}\%$, provided the model’s other terms and predictors are treated consistently with the specified comparison.

The result concerns the conditional mean, not an individual outcome. For a count model, for instance, a fitted mean of 5.1 is an expected count; a realized count must still be a whole number. The particular response distribution determines additional features of the model, such as its variance, but the log-link mean interpretation follows from the link equation itself.

Worked Applications

Worked Example: Expected Annual Claim Counts

A hypothetical insurer models each policy’s annual claim count $Y$ with a Poisson GLM: $Y\mid x\sim\operatorname{Poisson}(\mu(x))$, where $\mu(x)=E[Y\mid x]$ and $\log\{\mu(x)\}=\beta_0+0.08x$. The predictor $x$ is years of vehicle age. For a policy with $x=5$, the fitted mean is 4.00 claims per year. Find the fitted mean at $x=8$, and give the percentage change.

Method. The predictor increases by $\Delta=8-5=3$ years. With no other changing predictors or interaction terms in the stated model, the log mean changes by $0.08(3)=0.24$.

The multiplier is $\exp(0.24)=1.271249$ (to six decimal places). Therefore the new fitted mean is $4.00(1.271249)=5.084996$, or 5.08 claims per year to two decimal places.

The percentage change is $100(1.271249-1)=27.1249\%$, which is a 27.1% increase to one decimal place. This is a change in expected annual claims, not a prediction that this particular policy will have 5.08 realized claims.

Check and common mistake. Since the coefficient is positive, the fitted mean should increase; 5.08 is greater than 4.00. Treating $0.08$ as an exact 8% increase per year would be incorrect: the exact one-year percentage change is $100\{\exp(0.08)-1\}=8.3\%$, and the three-year change compounds through $\exp(0.24)$.

Worked Example: Positive Claim Severity

For a hypothetical portfolio, a Gamma GLM models positive claim severity $Y$ with conditional mean $\mu(x)$ and log link $\log\{\mu(x)\}=\beta_0-0.15x$. The predictor $x$ is measured in increments of 10 units of a repair-cost index. At $x=30$, the fitted mean severity is $240. Assume the conditional Gamma distribution uses shape parameter $\alpha=4$ and scale parameter $\mu(x)/4$, so its conditional mean is $\mu(x)$. Find the fitted mean at $x=50$ and the percentage change.

Method. The increase from 30 to 50 is 20 index units, or $\Delta=2$ of the model’s 10-unit increments. The log mean changes by $(-0.15)(2)=-0.30$.

The multiplier is $\exp(-0.30)=0.740818$ (to six decimal places). The new fitted mean is $$240)(0.740818)=$177.79632$, which rounds to $177.80 to the nearest cent.

The percentage change is $100(0.740818-1)=-25.9182\%$, or a 25.9% decrease to one decimal place. Equivalently, the mean is 74.1% of its original value.

Check and common mistake. The negative coefficient and positive predictor change imply a lower mean, consistent with $177.80 being below $240. The unit matters: the predictor change is two model increments, not 20 increments. Using $-0.15(20)$ would apply the coefficient to the wrong scale and produce an implausibly large decrease.

Worked Example: Two Predictor Changes in an Additive Model

A hypothetical model for annual claim counts is $\log\{\mu(x,z)\}=\beta_0+0.10x-0.20z$, with $Y\mid x,z\sim\operatorname{Poisson}(\mu(x,z))$. There is no interaction term. For a policy’s starting values, the fitted mean is 700 claims per year. Compare it with a policy profile where $x$ increases by 2 units and $z$ decreases by 1 unit, with all other predictors unchanged.

Method. The change in the linear predictor is the sum of the two coefficient contributions: $0.10(2)+(-0.20)(-1)=0.20+0.20=0.40$. The decrease in $z$ contributes a positive change because its coefficient is negative.

The combined multiplier is $\exp(0.40)=1.491825$ (to six decimal places). The new fitted mean is $700(1.491825)=1{,}044.2775$, or 1,044.28 claims per year to two decimal places.

The percentage change is $100(1.491825-1)=49.1825\%$, a 49.2% increase to one decimal place.

Check and common mistake. Both changes raise the linear predictor, so the mean must rise. A common error is to treat a decrease in $z$ as a negative contribution and calculate $0.20-0.20=0$. The signed change is $\Delta z=-1$, so its contribution is $(-0.20)(-1)=+0.20$. This combined interpretation applies to the stated additive model without interaction; a model with interaction terms would require accounting for the relevant interaction contributions as well.

When the Interpretation Applies

The coefficient-to-multiplier calculation is a consequence of the log-link mean equation. It does not require the response to follow one specific distribution. A Poisson GLM is commonly used for count outcomes, and a Gamma GLM can be used for positive outcomes, but each model also relies on its distributional specification and other modeling assumptions. The mean interpretation should not be mistaken for a complete assessment of whether that specification is appropriate.

For a continuous predictor, a coefficient’s interpretation also depends on its recorded units. If age is recorded in years, a one-unit increase means one year; if a cost index is recorded in 10-unit increments, a one-unit increase means 10 index units. Rescaling a predictor changes the numerical coefficient, while the fitted means for equivalent predictor values remain the same.

The model’s comparison must also be stated precisely. For a single coefficient, hold the other predictors fixed and use the predictor’s actual change $\Delta$. For a model with several changing predictors, add their contributions to the linear-predictor change when the model is additive. If the model contains interactions, the coefficient for one predictor alone generally does not describe the full change: the interaction terms can also change. Do not assume an additive interpretation unless it matches the stated model.

Identify the response mean and link.
Confirm that the stated model has $\log\{\mu(\mathbf{x})\}=\eta(\mathbf{x})$, where $\mu(\mathbf{x})=E[Y\mid\mathbf{x}]$.

Specify the comparison.
Write the predictor’s original and new values, calculate $\Delta$, and check the predictor’s units.

Calculate the log-mean change.
For one predictor in an additive model, calculate $\beta\Delta$. Include other changing terms if the model specifies them.

Exponentiate and interpret.
Calculate $\exp(\beta\Delta)$ for the multiplier and $100\{\exp(\beta\Delta)-1\}\%$ for the relative percentage change. Apply the multiplier to a starting mean only if one is supplied.

Common Interpretation Traps

Calling the coefficient an exact percentage. A coefficient of 0.08 does not mean an exact 8% increase for a one-unit change. The exact change is $100\{\exp(0.08)-1\}\%=8.3\%$, rounded to one decimal place.
Ignoring the size or direction of the change. The multiplier is $\exp(\beta\Delta)$, not always $\exp(\beta)$. For a decrease in the predictor, $\Delta$ is negative.
Applying a coefficient to the wrong units. A coefficient estimated per 10 units must be multiplied by the change expressed in 10-unit increments.
Reporting a change in the mean as a change in each outcome. The model describes a conditional mean; individual outcomes can vary around it.
Rounding too early. Keep sufficient precision for the exponential and mean calculation, then round the final dollar amount or percentage as requested.

Key takeaway: A log-link coefficient is a change on the log-mean scale. Multiply it by the predictor change, exponentiate to obtain the conditional-mean multiplier, and convert that multiplier—not the coefficient itself—to a percentage change. This interpretation is aligned with the SOA Exam PA (Predictive Analytics) syllabus.

Check Your Understanding

For each question, identify the predictor change before calculating. Unless stated otherwise, interpret the result as a change in the conditional mean, not as a guaranteed change in an individual outcome.

A hypothetical Poisson model for weekly service requests has $\log\{\mu(x)\}=\beta_0+0.04x$, where $x$ is measured in five-unit increases in a monitoring score. What is the exact percentage change in the fitted mean when the score rises by 25 raw units? (Answer: 22.1% increase.)
A hypothetical Gamma model for positive repair costs has coefficient $-0.12$ for a predictor recorded in single units. What is the percentage change in the fitted mean when the predictor increases by 2 units? (Answer: 21.3% decrease.)
A hypothetical Poisson model has coefficient $0.07$ for $x$. The fitted mean is 3.60 when $x=8$. Find the fitted mean at $x=11$, rounded to two decimals, and the percentage change to one decimal place. (Answer: 4.44; 23.4% increase.)
A hypothetical Gamma model for positive payment amounts has a fitted mean of $520 at a predictor value of 14 and coefficient $0.05$ per unit. Find the fitted mean when the predictor falls to 11, rounded to the nearest cent, and give the percentage change to one decimal place. (Answer: $447.57; 13.9% decrease.)
A hypothetical additive Poisson model with no interaction has coefficients $0.06$ for $x$ and $0.03$ for $z$. The starting fitted mean is 900. If $x$ increases by 2 and $z$ decreases by 1, calculate the new mean to two decimals and the percentage change to one decimal place. (Answer: 984.76; 9.4% increase.)

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