Regression diagnostics and analysis workflow

The video introduces a detailed workflow for conducting regression analysis, emphasizing the empirical testing of assumptions inherent in the process. The presenter opts for R as the primary tool, acknowledging that Stata and SPSS are viable alternatives. The workflow begins with hypothesis formulation, followed by data collection and exploration to understand variable relationships. The initial regression model involves identifying dependent and independent variables, then briefly reviewing the results. The focus shifts to diagnostics, favoring plots over statistical tests for a more nuanced understanding of issues like heteroscedasticity, rather than merely confirming their presence. The presenter iteratively refines the regression model by addressing identified problems such as non-linear relationships, outliers, or heteroscedasticity, until a satisfactory model is achieved. This model is then subjected to nested model tests against alternatives. Finally, the presenter emphasizes the importance of interpreting regression coefficients within the research context, using the Prestige dataset as an example to explore education, income, and gender distribution as factors influencing occupational prestige.

This segment delves into the diagnostics phase of regression analysis, highlighting the critical role of residuals—differences between observed and predicted values—as estimates of error terms. The presenter begins with the normal Q-Q plot to assess the normality of residuals, indicating a well-fitting model if residuals are normally distributed. This step also helps identify outliers and potential issues with the model. The discussion then moves to the residual vs. fitted plot, which is used to detect non-linear relationships and heteroscedasticity. The presenter describes various patterns that might emerge in this plot, such as a 'butterfly shape' indicating heteroscedasticity or a 'megaphone' shape suggesting both nonlinearity and heteroscedasticity. The aim is to observe a pattern-free distribution of residuals against fitted values, signaling a well-fitting model.

This part continues the exploration of regression diagnostics, focusing on further scrutiny of the variance of estimates and the presence of outliers. The presenter uses residual vs. leverage plots to identify influential observations, leveraging Cook's distance as a measure of an observation's influence. Specific professions are examined for their prestigiousness relative to the model's predictions, revealing instances where the model may significantly over or under-predict. This section suggests potential responses to identified issues, such as considering heteroscedasticity robust standard errors or modifying the data (e.g., removing outliers) to refine the model. The emphasis is on applying judgment and domain knowledge to interpret and act upon diagnostic findings.

The final segment discusses the steps following diagnostics, focusing on adjusting the regression model to address identified issues. The presenter suggests practical interventions, like using heteroscedasticity robust standard errors or considering the removal of influential observations (e.g., 'general managers') to check the robustness of the model. Additionally, the importance of log transformation for variables like income is highlighted to account for the relative significance of changes in such variables. The video underscores that these adjustments, informed by diagnostic plots and tests, are crucial for refining the regression model to better reflect the underlying data and research questions.