Generalized Linear Models AI Prompt

Specify and interpret the appropriate Generalized Linear Model (GLM) for this outcome.

Outcome variable: {{outcome}} and its distribution: {{distribution}}
Predictors: {{predictors}}
Data structure: {{data_structure}} (cross-sectional, panel, clustered)

1. GLM family and link function selection:

   Gaussian family, identity link:
   - Outcome: continuous, approximately normal
   - Equivalent to OLS linear regression

   Binomial family, logit link (logistic regression):
   - Outcome: binary (0/1) or proportion
   - Coefficient interpretation: log-odds. exp(beta) = odds ratio
   - Alternative links: probit (normal CDF), complementary log-log (for rare events)

   Poisson family, log link:
   - Outcome: count data (non-negative integers)
   - Assumption: mean = variance (equidispersion)
   - exp(beta) = incidence rate ratio
   - Add offset term (log of exposure) for rate models: log(mu) = offset + X'beta

   Negative binomial family, log link:
   - Outcome: overdispersed count data (variance > mean)
   - Adds a dispersion parameter: variance = mu + mu^2/theta
   - Check: if Poisson residual deviance >> df, use negative binomial

   Gamma family, log or inverse link:
   - Outcome: positive continuous, right-skewed (cost, duration, concentration)
   - Log link preferred for interpretability

   Inverse Gaussian family, log link:
   - Outcome: positive continuous, strongly right-skewed

2. Model fitting and interpretation:
   - Fit the GLM using maximum likelihood
   - Coefficients are on the scale of the link function
   - Back-transform for interpretation: exponentiate log-link coefficients for multiplicative effects
   - Confidence intervals: profile likelihood CI preferred over Wald CI for small samples

3. Overdispersion check (for count models):
   - Residual deviance / df: should be close to 1.0 for Poisson
   - If >> 1: overdispersion → switch to negative binomial or quasi-Poisson
   - If << 1: underdispersion (rare) → investigate data generation process

4. Zero inflation:
   - If there are more zeros than the Poisson/NB distribution predicts: zero-inflated model
   - ZIP (Zero-inflated Poisson): mixture of a point mass at zero and a Poisson distribution
   - ZINB: Zero-inflated negative binomial
   - Test: Vuong test comparing Poisson to ZIP

5. Goodness of fit:
   - Pearson chi-square statistic / df
   - Deviance / df
   - Rootogram (for count data): visual comparison of observed vs fitted count distributions

Return: GLM family and link function selection with rationale, coefficient interpretation, overdispersion check, zero-inflation assessment, and goodness-of-fit evaluation.