Regression Model Diagnostics with Julia

Carrying out model diagnostics for regression models in Julia can help you evaluate the performance and validity of your models. Here are some steps you can follow:

1. Fit the Regression Model: First, fit your regression model using the GLM.jl package.

   using GLM
   model = lm(@formula(y ~ x1 + x2), data)

2. Check Residuals: Examine the residuals to check for any patterns. Plotting residuals vs. fitted values is a good start.

   using Plots
   residuals = residuals(model)
   fitted_values = predict(model)
   scatter(fitted_values, residuals, xlabel="Fitted Values", ylabel="Residuals")

3. Normality of Residuals: Assess if the residuals are normally distributed using a histogram and Q-Q plot.

   histogram(residuals, bins=30, xlabel="Residuals", ylabel="Frequency")
   qqplot(residuals)

4. Heteroscedasticity: Check for heteroscedasticity using methods like the Breusch-Pagan test.

   using HypothesisTests
   bp_test = BreuschPaganTest(model)
   println(bp_test)

5. Influence and Leverage: Identify influential observations using Cook’s distance and leverage plots.

   using StatsPlots
   cooks_distance = cooks_distance(model)
   scatter(1:length(cooks_distance), cooks_distance, xlabel="Observation", ylabel="Cook's Distance")
   
   leverage = leverage(model)
   scatter(1:length(leverage), leverage, xlabel="Observation", ylabel="Leverage")

6. Multicollinearity: Check for multicollinearity using the Variance Inflation Factor (VIF).

   using RegressionDiagnostics
   vif_values = vif(model)
   println(vif_values)

7. Autocorrelation: Examine the residuals for autocorrelation using the Durbin-Watson test.

   using HypothesisTests
   dw_test = DurbinWatsonTest(residuals)
   println(dw_test)

8. Summary Statistics: Look at the summary statistics of your model to understand its overall performance.

   println(coef(model))
   println(r2(model))
   println(adjr2(model))

These steps will help you diagnose your regression model and ensure that it meets the necessary assumptions for a valid analysis.