Why GLM Interpretation Traps Matter
A fitted GLM can be numerically correct and still be communicated incorrectly. In logistic regression, for example, a coefficient describes a change in log odds, not a change in probability. A factor coefficient compares its level with a reference level, and an interaction can make a coefficient’s interpretation depend on another predictor. Diagnostics have similar traps: a large residual signals an unusual outcome relative to its fitted value, but does not by itself show that the observation is influential or erroneous.
These distinctions matter when translating a model into an actuarial conclusion. A model association is not automatically a causal effect, and a striking diagnostic value is a prompt for investigation, not a reason to delete a record automatically. The examples below use hypothetical data and focus on accurate interpretation rather than model selection.
Odds, Probabilities, and Logistic Coefficients
For a binary response \(Y\), let \(p=P(Y=1\mid x)\). A logistic GLM with the logit link specifies that the conditional log odds are a linear predictor:
The odds corresponding to probability \(p\) are \(p/(1-p)\). Conversely, odds \(o\) correspond to probability \(o/(1+o)\). These are different scales: multiplying odds by a factor does not multiply probability by that same factor.
The coefficient interpretation follows directly from the model. If \(x\) increases by one unit and all other terms are unchanged, then \(\eta\) increases by \(\beta\). Exponentiating both sides of the logit equation shows that odds equal \(e^\eta\). Therefore, the new odds are \(e^{\eta+\beta}=e^\beta e^\eta\): the odds are multiplied by \(e^\beta\). Probability does not have this constant-multiplication property.
This result is conditional on the model’s specified linear predictor. It does not say that the probability changes by \(\beta\), or by \(e^\beta\), and it does not establish that changing the predictor would cause an outcome to change.
Worked Example: Converting an Odds Ratio to a Probability
A hypothetical insurer fits a logistic model for whether a submitted service receives an additional review. The model has a baseline probability of \(0.2000\) for category A. The estimated coefficient for category B, relative to A, is \(\log(1.5)=0.4055\). Find the fitted probability for B and state the correct interpretation.
Method. Convert the baseline probability to odds, apply the odds ratio, then convert the resulting odds back to probability. The factor coding makes A the reference category.
For A, the odds are \(0.2000/(1-0.2000)=0.2000/0.8000=0.2500\). Category B has odds \(1.5(0.2500)=0.3750\). The corresponding probability is
Thus the fitted probability is \(0.2727\), or 27.27% to two decimal places as a percentage. The model estimates that B has 1.5 times the odds of A, or 50% higher odds. The fitted probability is 7.27 percentage points higher, not 50% higher.
Common mistake and correction. The incorrect calculation \(0.2000\times1.5=0.3000\) treats an odds ratio as a probability multiplier. Instead, apply the ratio to odds and then convert back, giving \(0.2727\). This interpretation is an association in the fitted model, not evidence that category membership causes review.
Reference Categories and Interaction Effects
A categorical predictor is commonly represented by indicator variables. With two levels, A and B, if A is the reference, the indicator \(I(B)\) equals 1 for B and 0 for A. In a model without interactions, the coefficient of \(I(B)\) compares B with A, holding the other model terms fixed. Reversing the reference category reverses the comparison: the coefficient changes sign and the odds ratio becomes its reciprocal.
For a predictor involved in an interaction, a coefficient is not generally an overall effect. In a logistic model containing \(xz\), the coefficient of \(x\) describes the change associated with a one-unit increase in \(x\) when \(z=0\). At another value of \(z\), the interaction contributes to that change. For binary \(z\), the conditional odds ratio for a one-unit increase in \(x\) is \(e^{\beta_x+\beta_{xz}z}\).
Worked Example: Reading a Logistic Interaction
A hypothetical study of policy renewals uses two binary predictors: \(x=1\) indicates that a policy is in a specified risk class, and \(z=1\) indicates enrollment in an optional monitoring program. The fitted model is
Calculate the fitted probabilities for all four combinations, then interpret the risk-class coefficient and interaction.
Method. Substitute each combination into the linear predictor, convert \(\eta\) to probability using \(p=e^\eta/(1+e^\eta)\), and compare conditional odds ratios.
- For \(x=0,z=0\), \(\eta=-2\), so \(p=1/(1+e^2)=0.1192\).
- For \(x=1,z=0\), \(\eta=-2+0.5=-1.5\), so \(p=1/(1+e^{1.5})=0.1824\).
- For \(x=0,z=1\), \(\eta=-2+0.8=-1.2\), so \(p=1/(1+e^{1.2})=0.2315\).
- For \(x=1,z=1\), \(\eta=-2+0.5+0.8+0.4=-0.3\), so \(p=1/(1+e^{0.3})=0.4256\).
The risk-class odds ratio when \(z=0\) is \(e^{0.5}=1.6487\). When \(z=1\), it is \(e^{0.5+0.4}=e^{0.9}=2.4596\). The interaction coefficient of 0.4 means the risk-class log-odds contrast is 0.4 larger when \(z=1\) than when \(z=0\); equivalently, the ratio of those two conditional odds ratios is \(e^{0.4}=1.4918\).
The probability difference for risk class is \(0.1824-0.1192=0.0632\) when \(z=0\), but \(0.4256-0.2315=0.1941\) when \(z=1\). These differences are not the coefficients and are not constant across \(z\). Rounding probabilities to four decimals, the four fitted values are 0.1192, 0.1824, 0.2315, and 0.4256.
Common mistake and correction. Saying “the risk-class odds ratio is \(e^{0.5}\) for everyone” ignores the interaction. That is correct only at \(z=0\). At \(z=1\), include the interaction term, giving \(e^{0.9}\). Also, neither odds ratio is a probability ratio.
A changed reference category changes coefficient labels and signs, but not the fitted probabilities or the model’s underlying comparisons. When explaining a factor effect, name both the level being compared and the reference level. When explaining an interaction, state the condition or value of the other predictor.
Residual Size Is Not the Same as Influence
A residual compares an observed outcome with what the model fitted. For an individual Bernoulli observation \(y\in\{0,1\}\) with fitted probability \(\hat p\), the Pearson residual is
A large absolute residual indicates an outcome that is surprising relative to its fitted probability, subject to the model and the observation’s scale. It does not, by itself, show that the data are wrong, that the model is globally poor, or that the observation materially changes the fitted coefficients.
Influence concerns how much a fitted result changes when an observation is removed or perturbed. Leverage concerns how unusual an observation’s predictor values are relative to the fitted model’s predictor pattern. These ideas are related but distinct: a case can have a large residual and little influence, or modest residual size and considerable influence if its predictor values give it substantial leverage. Residual plots and diagnostics discussed in the earlier material on GLM residuals, leverage, and case deletion should be interpreted together and in context.
Worked Example: A Large Residual, an Influential Case, and a Causal Claim
An analyst fits a hypothetical logistic model for whether a policy has a claim during a year. For one policy, the fitted probability is \(\hat p=0.0500\), but a claim occurred (\(y=1\)). A second policy has \(\hat p=0.8000\), but no claim occurred (\(y=0\)). Separately, one policy has an unusual predictor profile; a case-deletion refit changes the estimated coefficient for a continuous coverage measure from 0.60 to 0.31. The data are observational. Explain what each result supports.
Method. Calculate the individual Pearson residuals, distinguish their role from the stated case-deletion change, and limit conclusions to what an observational model can establish.
For the first policy,
This is a large positive residual: the observed claim is unexpected under the fitted probability of 0.0500. It warrants checking the record and model fit, but it is not proof of an error.
For the second policy,
This is a negative residual: no claim occurred although the fitted probability was high. Its absolute value is smaller than the first residual’s. These calculations describe outcome discrepancies; they do not establish which case most affects the fitted coefficients.
The case-deletion change is \(0.60-0.31=0.29\) in the coverage coefficient. This is direct evidence that the fitted coefficient is sensitive to deleting that particular case, so its predictor profile and data quality merit investigation. It is not a residual calculation. Without additional information, such as the refit’s uncertainty and the model specification, the change alone does not establish that the case should be removed or that the model is unusable.
Finally, even if the coverage coefficient is positive and stable, the observational association does not prove that increasing coverage causes claim probability to change. Policyholders may differ in other ways related to both coverage and claims. A causal conclusion requires a defensible causal design and assumptions addressing confounding; a fitted GLM coefficient alone does not supply them.
Common mistake and correction. It would be wrong to declare the first policy “influential” solely because its residual is about 4.36, or to delete the unusual-profile case solely because its coefficient changes. Use residuals to flag outcome discrepancies, leverage and case-deletion diagnostics to assess sensitivity, and substantive review to decide what action is justified. Do not convert a fitted association into a causal claim.
A Practical Interpretation Checklist
Before reporting a GLM result, check the scale of the coefficient, its coding, and the diagnostic evidence separately. This prevents a numerically accurate output from becoming a misleading business statement.
For logistic regression, state that the linear predictor is on the log-odds scale and that the response is binary.
Name the reference category and the comparison level. For interactions, specify the value of the interacting predictor when interpreting a coefficient.
Exponentiate a logistic coefficient for an odds ratio. To report a probability, calculate it from the model’s full linear predictor at stated predictor values.
Use residuals to assess observed-versus-fitted discrepancies, and leverage or case-deletion evidence to assess unusual predictor configurations and influence.
Describe a fitted relationship as an association unless the study design and assumptions justify a causal interpretation.
This disciplined interpretation is aligned with the SOA Exam PA (Predictive Analytics) syllabus. A concise report can still be precise: name the comparison, report the odds ratio or fitted probabilities on the proper scale, qualify the diagnostic evidence, and distinguish prediction or association from causation.
Check Your Understanding
For each hypothetical situation, identify the interpretation or calculation that avoids the relevant GLM trap.
- A logistic model estimates a coefficient of \(\log(2)\) for level B relative to reference level A. If the fitted probability for A is 0.1000, calculate the fitted probability for B. (Answer: A odds \(=0.1000/0.9000=0.1111\); B odds \(=0.2222\); B probability \(=0.1818\).)
- A logistic model includes \(0.3x+0.7z+0.5xz\), where \(x\) and \(z\) are binary. What is the odds ratio for changing \(x\) from 0 to 1 when \(z=1\)? (Answer: \(e^{0.3+0.5}=e^{0.8}=2.2255\).)
- An individual binary outcome is \(y=0\) with fitted probability \(\hat p=0.2500\). Calculate its Pearson residual and state its sign. (Answer: \((0-0.2500)/\sqrt{0.2500(0.7500)}=-0.5774\), negative.)
- A case has a small residual but an unusual predictor profile, and deleting it materially changes a coefficient. Which conclusion is supported? (Answer: The fit is sensitive to that case; a small residual does not rule out influence. Investigate the case and model rather than deleting it automatically.)
- An observational logistic model estimates lower claim odds for policies with a certain optional feature after adjusting for recorded variables. Does the coefficient alone show that adopting the feature reduces claims? (Answer: No. It describes a conditional association; causal interpretation requires a design and assumptions that address confounding.)