PSCH 443 Multiple Regression Overview and Example

Overview of Multiple Regression

Definition: Multiple regression aims to explain the variation in an outcome variable (Y) using multiple predictor variables (X).
Goal: Similar to bivariate regression, it seeks to explain why some individuals score high or low on the Y variable.
Overlap Consideration: It is essential to account for the unique contributions of each predictor as well as the overlap between predictors themselves concerning their influence on the outcome variable.

Estimate Model Parameters:
- Use least squares techniques for estimating model parameters.
- Include an intercept, multiple B parameters, and regression weights for each predictor.
Evaluate Model Fit:
- Assess overall model fit using R-squared.
- Examine individual predictors' contributions to ascertain if they add meaningful value to the model.

Generic Structure: Similar to a simple regression equation, including:
- A parameter: Intercept (value of Y when all X's are zero).
- B weights: Slope parameters for each predictor variable.
- Error term: Accounts for variability not explained by the predictors.

Scenario: Assess whether GRE scores can predict graduate school performance.
Dataset Details:
- 36 participants with variables including undergraduate GPA, GRE scores, and years to complete a master's degree.
Predictors:
- GRE scores (verbal sub-score)
- Undergraduate GPA
Outcome: Years taken to complete a master's program.

Examine Bivariate Relationships:
- Look at the correlation matrix between variables to identify expected associations (e.g., higher GPA correlates with higher GRE scores).
- Check for potential issues such as unexpected correlation directions or strong correlations between predictor variables (multicollinearity).

Multiple Correlation (R): Correlation of all predictor variables with the outcome Y variable.
R-squared Value:
- Represents the proportion of variance explained by predictors; in this example, about 26% of the variance in completion years is explained.
Adjusted R-squared:
- Adjusts R-squared based on sample size and number of predictors; relevant when considering generalizability of the results.
Standard Error of Estimate: Reflects average prediction error, relevant for model accuracy.

ANOVA Overview: Ratio of effect sums of squares to error sums of squares to test the overall significance of the model.
Degrees of Freedom: Determines total degrees of freedom based on the number of participants.
- Total DF: Total participants - 1 (e.g., 35 for 36 participants).
- Regression DF: Coefficients estimated - 1 (e.g., 2 predictors + intercept = 3 - 1 = 2).

Constant (Intercept): Found in the table as unstandardized coefficients.
B Weights:
- Indicate the change in Y for a one-unit change in X.
- Example: Holding GPA constant, each GRE point results in a 0.002 years reduction in completion time; a full GPA grade point increase results in a 0.634 years reduction.
Standardized Coefficients (Beta Weights): Reflect changes in Y in standard deviation units, allowing comparison of predictor effects.
- GRE: A standard deviation increase predicts a 0.201 standard deviation reduction in years to completion.
- GPA: A standard deviation increase predicts a 0.393 standard deviation reduction in years to completion.

T Statistics: Evaluate significance of B parameters (unstandardized coefficients).
- GRE scores significance: P = 0.233 (not significant).
- Undergraduate GPA significance: P = 0.023 (significant).
Practical Importance of Effects: Consider the size of coefficients in relation to real-world applications beyond p-values; significant testing alone does not confirm practical importance.

Importance of Assumptions: Results interpretation depends on meeting regression assumptions, types of regression used, and the overall purpose of the analysis.
Critical Evaluation: Emphasize understanding beyond statistical significance for practical application and insights.