Regression and Modeling AI Prompts

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.

Diagnose and validate a fitted linear regression model. Model: {{model_description}} (outcome, predictors, n) Fitted model output: {{model_output}} 1. The four core OLS assumptions (LINE): L — Linearity: - Residuals vs fitted values plot: should show random scatter around zero - Pattern in residuals = non-linearity → add polynomial terms, interaction terms, or transform predictors - Partial regression plots (added variable plots): check linearity of each predictor separately I — Independence of errors: - By design: is this a cross-sectional dataset (no natural ordering)? - For time series or clustered data: Durbin-Watson test for serial autocorrelation (target: DW near 2) - If observations are clustered: use clustered standard errors or mixed effects model N — Normality of residuals: - Q-Q plot of standardized residuals: points should fall on the diagonal - Shapiro-Wilk test for normality (reliable for n < 2000) - Note: normality of residuals is the LEAST critical assumption for large samples (CLT) - Skewed residuals suggest: log-transform the outcome, or consider a GLM with appropriate family E — Equal variance (homoscedasticity): - Scale-location plot (sqrt(|standardized residuals|) vs fitted): should be flat - Breusch-Pagan test: p < 0.05 indicates heteroscedasticity - Fix: use heteroscedasticity-consistent (HC) standard errors (HC3 is robust in small samples) - Or: weighted least squares if variance structure is known 2. Influential observations: - Leverage (h_ii): measures how far an observation's predictor values are from the mean High leverage: h_ii > 2(k+1)/n - Cook's Distance: measures overall influence of each observation on all fitted values Influential if D_i > 4/n (rule of thumb) - DFFITS and DFBETAS: influence on fitted values and specific coefficients - Action: investigate (not automatically remove) flagged observations 3. Multicollinearity: - Variance Inflation Factor (VIF) per predictor - VIF > 5: concerning, VIF > 10: severe multicollinearity - Fix: remove redundant predictors, combine correlated predictors via PCA, or use ridge regression 4. Model fit assessment: - R-squared: proportion of variance explained (note: always increases with more predictors) - Adjusted R-squared: penalizes for adding unhelpful predictors - AIC/BIC: for model comparison (lower is better) - RMSE on a holdout set: most honest measure of predictive accuracy Return: assumption check results per criterion, influential observation list, multicollinearity report, and model fit summary.

Compare candidate statistical models and select the most appropriate one. Outcome variable: {{outcome}} Candidate models: {{models}} (list of model specifications) Data: {{data_description}} Goal: {{goal}} (inference / prediction / both) 1. Information criteria: AIC = 2k - 2 ln(L) BIC = k ln(n) - 2 ln(L) where k = number of parameters, L = maximized likelihood, n = sample size - Lower AIC/BIC = better model - AIC minimizes prediction error; BIC penalizes complexity more (prefers parsimonious models) - Delta AIC: difference from the best model Delta < 2: substantial support for this model Delta 4-7: considerably less support Delta > 10: essentially no support - For purely predictive goals: use AIC or cross-validation - For inference with parsimony: use BIC 2. Likelihood ratio test (LRT) for nested models: LRT statistic = -2(ln L_restricted - ln L_full) Follows chi-square distribution with df = difference in number of parameters Reject the restricted model if p < 0.05 Use LRT when: comparing a simpler model to a more complex one that contains it as a special case 3. Cross-validation: For predictive model selection, k-fold cross-validation gives the most honest estimate: - Split data into k folds (k=10 is standard) - Train on k-1 folds, test on held-out fold - Average test metric (RMSE for continuous, AUC for binary) across folds - Select model with best mean CV metric, accounting for standard error - One-standard-error rule: prefer the simpler model within 1 SE of the best 4. Goodness-of-fit tests: - For linear regression: overall F-test (are any predictors useful?) - For logistic regression: Hosmer-Lemeshow test (is the calibration good?) - For count models: overdispersion test (is Poisson appropriate, or do we need negative binomial?) 5. Parsimony principle: - Between models with similar fit: prefer the simpler one - A model that is too complex will overfit: good in-sample fit, poor out-of-sample prediction - Report confidence/credible intervals for all selected model parameters Return: AIC/BIC comparison table, LRT results (if applicable), cross-validation scores, and model selection recommendation with rationale.