PSCH 443 Multiple Regression Partial and Semi Partial Correlation

Introduction to Multiple Regression

• Multiple regression is a statistical technique used to model the effects of several predictor variables on an outcome variable.

• In this approach, there is typically one dependent variable (Y) and multiple independent variables (X).

• These predictor variables are usually measured rather than manipulated, making multiple regression appropriate for correlational research designs.

• It is commonly applied when the outcome variable is continuously scaled, allowing for a graded variation in the measurements.

Nature of Predictor Variables

• Most predictor variables are continuous.

• Categorical data can be included after appropriate transformation to approximate continuous data for analysis.

• Researchers must ensure that certain assumptions about the data are met.

Analyzing Effects of Predictor Variables

• As predictor variables are included in the analysis, it is essential to evaluate their individual contributions and their collective impact on the outcome variable.

• Overlapping relationships between predictors and outcomes introduce complexity in analysis:

  • Individual predictors may correlate with the outcome.

  • Predictors may also correlate with one another, necessitating careful consideration in analysis strategies.

Complex Research Questions

• Multiple regression facilitates the exploration of complex research questions beyond simple bivariate relationships.

• Example: The relationship between GRE scores and graduate school performance, considering underlying factors such as undergraduate GPA.

  • The GRE may predict performance; however, undergraduate GPA could influence both GRE scores and graduate success.

Venn Diagram Example

• In exploring the example, the Venn diagram illustrates:

  • GRE and Undergraduate GPA predict the number of years to complete a master's degree.

  • Unique overlap exists among these variables, which may affect the analysis of the outcome variable.

Partial Correlation

• A partial correlation examines the relationship between two variables while controlling for a third variable.

• In this context:

  • X = GRE

  • Y = Years to completion

  • Z = Undergraduate GPA

  • This technique allows for isolating the relationship between GRE and completion time by controlling for GPA.

Formula and Interpretation of Partial Correlation

• The formula for partial correlation helps isolate X and Y relationships by factoring out the influence of Z.

• It's not mandatory to compute this by hand; software (like SPSS) handles the computation.

• Understanding the concept is vital: the goal is to assess the unique relationship between X and Y after excluding the influence of Z.

Semi-Partial Correlation

• A semi-partial correlation examines the unique influence of X on Y while controlling for Z's influence on X.

• This focuses on the variability in Y that can be uniquely attributed to X after accounting for Z, maintaining the integrity of Y.

• The semi-partial correlation is crucial in multiple regression, highlighting the total influence of independent predictors.

Residual Variance and Multiple Variables

• Residual variance (E) represents the unexplained variability in the outcome variable that predictors do not account for.

• Overlap among predictors can lead to multicollinearity, complicating the analysis and interpretation of results.

• It’s essential to identify and manage this overlap in predictors to maintain clarity in regression models.

Multicollinearity

• Defined as the overlap among predictor variables, which can dilute their individual explanatory power in a model.

• Excessive correlation among predictors can hinder the ability to accurately assess their unique contributions to the outcome variable.

• The ideal scenario is when predictors are independent of each other while correlating with the outcome, allowing distinct contributions to the explained variance.

• Future discussions will delve deeper into recognizing and addressing multicollinearity issues in multiple regression models.