Why an Adjusted Coefficient Is Not Automatically Causal
A generalized linear model can compare outcomes for individuals who differ in one predictor while holding the model’s other predictors fixed. That is an adjusted conditional association. It does not, by itself, show what would happen if the same individuals were made to change that predictor. The distinction matters whenever a report moves from “the model associates X with Y” to “changing X will change Y.”
For prediction, an association can be useful even when it is not causal: a predictor may improve forecasts because it tracks other risk factors. A causal question is different. It asks about the outcome under an intervention or change in exposure, and requires assumptions beyond fitting a GLM. The interpretation of GLM coefficients covered in the earlier material on links and contrasts describes model-based comparisons; this tutorial focuses on what those comparisons can and cannot establish.
To make the distinction precise, let \(X\) be a binary exposure and let \(Y(1)\) and \(Y(0)\) denote the potential outcomes for an individual under exposure values 1 and 0. A population average causal effect on the mean scale is \(E[Y(1)]-E[Y(0)]\). Only one of these potential outcomes is observed for each individual, so estimating this contrast requires a design or assumptions that let the unobserved comparison be identified.
Common requirements include a well-defined exposure and outcome, consistency between the observed outcome and the potential outcome under the exposure actually received, and conditional exchangeability: after conditioning on sufficient pre-exposure variables \(C\), exposure assignment is unrelated to the potential outcomes. There must also be positivity: within each relevant covariate profile, both exposure levels must be possible. A correctly specified GLM is not a substitute for these requirements. Nor does adjusting for every available variable necessarily help: a variable may be a mediator, a collider, measured with substantial error, or unrelated to the confounding problem.
Why Confounding Can Change the Comparison
A confounder is a variable that is associated with exposure and is also relevant to the outcome, such that failing to account for it can distort the exposure-outcome comparison. The basic arithmetic explains why. For a binary exposure \(X\) and a covariate \(C\), the observed mean in exposure group \(x\) averages over that group’s own covariate distribution:
If the covariate mix differs between exposure groups, these two observed means combine the within-covariate comparisons using different weights. By contrast, a standardized comparison uses the same covariate weights for both exposure levels. This is why crude and adjusted comparisons can differ, even when the adjusted model’s within-group comparisons are simple.
Worked Example: A Reversal in a Claim-Frequency Comparison
A hypothetical insurer compares annual claim counts for policyholders who completed a safety course with counts for those who did not. Policies are classified as low-risk or high-risk using information available before course participation. The table gives claim counts and policy-year exposures. For each risk group, calculate the rate ratio for course completion; then compare those results with the crude rate ratio. Explain what an adjusted log-link model would say, and what it would not establish.
| Risk group | Completed course: claims / policy-years | Did not complete: claims / policy-years |
|---|---|---|
| Low-risk | 8 / 100 | 18 / 180 |
| High-risk | 24 / 100 | 6 / 20 |
Within the low-risk group. The completion rate is \(8/100=0.08\) claims per policy-year. The non-completion rate is \(18/180=0.10\). Thus, the rate ratio is \(0.08/0.10=0.80\).
Within the high-risk group. The completion rate is \(24/100=0.24\); the non-completion rate is \(6/20=0.30\). The rate ratio is \(0.24/0.30=0.80\). In both groups, the observed rate is 20% lower among course completers.
Crude comparison. Course completers have \(8+24=32\) claims over \(100+100=200\) policy-years, giving a rate of \(32/200=0.16\). Non-completers have \(18+6=24\) claims over \(180+20=200\) policy-years, giving a rate of \(24/200=0.12\). The crude rate ratio is \(0.16/0.12=4/3\), or approximately 1.333. In this unadjusted comparison, completers appear to have a higher rate.
Model interpretation. Consider a Poisson log-link model for the claim count with the logarithm of policy-years as an offset, a completion indicator, and a high-risk indicator. In this hypothetical table, the completion-to-non-completion rate ratio is 0.80 in each risk group, so the model without an interaction represents the observed rates with a completion coefficient \(\log(0.80)\approx-0.2231\). Exponentiating that coefficient gives the conditional rate ratio \(0.80\), holding risk group fixed.
What follows—and what does not. The crude comparison is distorted by the different risk-group mix: 100 of 200 completion policy-years are high-risk, compared with 20 of 200 non-completion policy-years. The adjusted coefficient summarizes the within-risk-group association. It does not prove that completing the course caused a 20% reduction. For example, unmeasured motivation or prior driving behavior could affect both course completion and claims. A causal statement would need a defensible account of such factors and the other assumptions for causal identification.
Common mistake and correction. A mistaken conclusion is: “The crude rate ratio is \(4/3\), so the course increases claim frequency by 33.3%.” That calculation compares groups with different risk mixes. The correction is to report the crude contrast as an unadjusted association, then give the risk-adjusted contrast of 0.80 as a conditional association. Neither number alone establishes a causal effect.
Selection Can Create an Association
Even when two variables are independent in the source population, restricting analysis to a selected subset can make them associated. This can occur when inclusion depends on both the predictor of interest and a factor related to the outcome. The selection process changes the combinations of variables represented in the analyzed data. A model fitted to that selected data describes the selected sample; adding predictors does not automatically undo the selection mechanism.
Worked Example: An Association Created by Enrollment
In a hypothetical source population, a binary feature \(X\) and a binary outcome-risk factor \(U\) are independent, each with probability 0.5 of being 1. The outcome is \(Y=U\), so \(X\) has no effect on \(Y\) in the source population. A study enrolls everyone with \(X=1\) or \(U=1\), but also enrolls 10% of people with \(X=0,U=0\). Calculate the selected-sample outcome risks and the odds ratio comparing \(X=1\) with \(X=0\).
Source cell probabilities. Independence and the two 0.5 probabilities give probability \(0.25\) for each of the four \((X,U)\) combinations. Since \(Y=U\), the source-population outcome risk is \(0.5\) for both \(X=0\) and \(X=1\); the source odds ratio is therefore 1.
Selected cell weights. The three cells with \(X=1\) or \(U=1\) are selected with probability 1, so each contributes \(0.25\) to the selected mass. The \((0,0)\) cell contributes \(0.25(0.10)=0.025\). The total selected mass is \(0.25+0.25+0.25+0.025=0.775\).
Risk among selected people with \(X=0\). The selected mass with \(X=0\) is \(0.025+0.25=0.275\). The outcome-positive mass is \(0.25\), so the risk is \(0.25/0.275=10/11\approx0.9091\). Its odds are \((10/11)/(1/11)=10\).
Risk among selected people with \(X=1\). The selected mass is \(0.25+0.25=0.50\), of which \(0.25\) is outcome-positive. The risk is \(0.25/0.50=0.50\), and the odds are \(0.50/0.50=1\).
The selected-sample odds ratio for \(X=1\) versus \(X=0\) is \(1/10=0.10\), despite a source-population odds ratio of 1 and no effect of \(X\) on \(Y\). Enrollment has induced the association. A logistic regression fitted only to enrolled people could report this selected-sample association; its coefficient would not show that changing \(X\) changes the outcome.
Common mistake and correction. A tempting error is to regard the enrolled sample as if it had the source population’s distribution because the original variables were independent. But conditioning on enrollment changes that distribution. The correction is to examine how selection depends on the variables and to define the population to which any estimate applies.
Prediction-Focused Interpretation
A predictive GLM uses measured features to estimate an outcome for cases with those features. Its coefficient can be useful for forecasts, ranking, or resource planning without representing an intervention effect. For example, an observed participation indicator might predict claim frequency because it captures differences in behavior, access, or underlying risk. A model can exploit that information even if intervening to change the indicator would not change claim frequency.
Worked Example: Interpreting a Predictive Frequency Coefficient
A hypothetical predictive model for annual claim counts uses a Poisson distribution with mean \(\mu\), a log link, and a policy-year exposure offset. At a fixed covariate profile, the predicted rate for a nonparticipant is 0.12 claims per policy-year. The fitted coefficient for participation is \(-0.30\). Calculate the participant rate and the predicted mean claims for 2.5 policy-years at each participation value. State the proper interpretation.
Model comparison. With a log link, holding the other predictors fixed, changing the participation indicator from 0 to 1 multiplies the fitted rate by \(\exp(-0.30)\). This factor is approximately \(0.7408\).
Participant rate. The fitted rate is \(0.12(0.7408)=0.088896\), or about 0.0889 claims per policy-year when rounded to four decimal places. The rate difference between the two model predictions is \(0.088896-0.12=-0.031104\), or about \(-0.0311\) claims per policy-year.
Predicted counts for 2.5 policy-years. The nonparticipant mean is \(2.5(0.12)=0.3000\) claims. The participant mean is \(2.5(0.088896)=0.22224\), or 0.2222 claims when rounded to four decimal places. These are fitted expected counts, not predictions that an individual policy must have a fractional number of claims.
Interpretation. At the specified covariate profile, the model predicts a rate about 25.9% lower for participants, since \(1-0.7408=0.2592\). This is a model-based conditional prediction contrast. It is not evidence that making a nonparticipant participate would reduce that person’s rate by 25.9%. That causal reading would require additional design and identification assumptions.
Common mistake and correction. The coefficient is \(-0.30\), not a 30% reduction on the response scale. The log-link rate ratio is \(\exp(-0.30)\approx0.7408\), corresponding to an approximately 25.9% lower fitted rate. Even that correctly transformed result remains an association unless the causal assumptions are justified.
A Practical Test for Causal Language
Before describing a GLM result as an effect of changing a predictor, ask what population and intervention the claim concerns, and whether the data support the required comparison. In particular, consider whether important pre-exposure common causes were measured and adjusted for, whether the analysis conditions on a selection process, and whether adjustment includes variables that occur after exposure. A regression table alone cannot answer these questions.
State whether the goal is prediction or a causal contrast, and identify the population, exposure, outcome, and scale.
Specify the link and the conditioning variables. For example, a log-link coefficient describes a conditional rate ratio after exponentiation.
Look for different covariate mixes, unmeasured common causes, selection into the analysis, and adjustment for variables that may be consequences of exposure.
If causal identification is not supported, describe an adjusted association or model prediction rather than an intervention effect.
A precise report can still be useful without causal language: “At the same values of the included covariates, the fitted model estimates a rate ratio of 0.80 for course completers versus non-completers.” This states the model comparison. It does not imply that the model has controlled every source of bias. This distinction is aligned with the SOA Exam PA (Predictive Analytics) syllabus, where interpreting a model appropriately includes recognizing what its coefficients do not establish.
Check Your Understanding
For each problem, distinguish the model-based comparison from any causal conclusion.
- A log-link GLM estimates a coefficient of \(-0.20\) for an indicator, holding the other predictors fixed. Calculate the rate ratio and approximate percentage difference in fitted rate. What can be concluded if the analysis is observational? (Answer: Rate ratio \(e^{-0.20}\approx0.8187\), or an approximately 18.1% lower fitted rate; this is a conditional association, not automatically a causal effect.)
- A binary outcome is modeled with logistic regression. The adjusted coefficient for a feature is \(0.40\). Calculate the adjusted odds ratio and state what it compares. (Answer: \(e^{0.40}\approx1.4918\); the fitted conditional odds are about 1.49 times as high for feature value 1 versus 0, at the same included covariate values, without a causal conclusion by itself.)
- In a source population, \(X\) and \(U\) are independent and each has probability 0.5 of being 1; \(Y=U\). A study selects everyone with \(X=1\) or \(U=1\), and no one else. Find the selected-sample risk of \(Y=1\) for each \(X\) value. (Answer: For \(X=0\), only \(U=1\) is selected, so risk \(=1\); for \(X=1\), \(U\) remains equally likely to be 0 or 1, so risk \(=0.5\). Selection creates an association although the source-population risk is 0.5 for both \(X\) values.)
- A model predicts 0.20 claims per policy-year for a nonparticipant and estimates a participation rate ratio of 0.75. What is the participant rate and the difference for 3 policy-years? Does the difference establish a causal reduction? (Answer: Participant rate \(=0.20(0.75)=0.15\); the three-year predicted means are 0.60 and 0.45, a difference of \(-0.15\) claims. These are predictive contrasts and do not establish a causal reduction.)
- A report adjusts for a variable measured after exposure and calls the exposure coefficient the total causal effect. Give one reason this conclusion may fail and the safer interpretation if causal assumptions are not established. (Answer: The post-exposure variable may be a mediator, so adjusting for it can remove part of the effect being targeted; it may also introduce bias in some settings. Without justified causal assumptions, describe the coefficient as an adjusted conditional association under the specified model.)