Week 9

Lesson 9.1: Goodness of Fit

• Introduction to Goodness of Fit
    - Importance in assessing regression models and their predictive capability for datasets.
    - Objectives of the lesson include:
      - Defining goodness of fit.
      - Explaining model fit in the context of aggression.
      - Interpreting R-squared as a key measure of model fit.
      - Identifying and describing factors that influence model fit.
      - Conducting visual assessments of model fit through residual plots.

• Definition of Goodness of Fit
    - Goodness of fit assesses how well a fitted regression line represents a dataset.
    - Evaluated primarily through the R² statistic:
      - R² is the proportion of variance in Y explained by X.
      - Categories of variance (small, medium, large) based on Cohen's suggested cut-offs.
    - Alternative assessments include:
      - Residual plots
      - Statistics like Cook's distances

• Factors Influencing Goodness of Fit
    - Actual Relationship:
      - Some variables predict outcomes more effectively than others. Example: IQ as a predictor of academic performance vs. shoe size.
    - Measurement Error:
      - Inaccurate measurements introduce bias that affects model fit.
    - Individual Differences:
      - Variability among participants hampers perfect model accuracy.
    - Range Restrictions:
      - Sampling only a small portion of a population can skew model accuracy.
    - Nonlinearity:
      - Linear regression assumes straight-line relationships; curvilinear data need nonlinear modeling.
    - Heterogeneous Subgroups:
      - Distinct groups within the dataset may mislead when aggregated without using subgroup analysis.
    - Outliers and Influential Cases:
      - Outliers are Y values significantly distant from their predicted values (high residuals).
      - Influential observations alter regression coefficients (slope and intercept).

• Residuals
    - Residuals are differences between observed values and predicted values of Y:
      - A large residual indicates a poorly fitted model.
    - Residual plots:
      - X-axis: predicted values of Y.
      - Y-axis: residual values.
      - Line at zero shows the expected average distance of the points from the regression line.
    - Interpretation of residual values:
      - Positive residual = model's predicted value is too low.
      - Negative residual = predicted value is too high.
      - Zero residual = perfect prediction.

• Visual Assessment Using Residual Plots
    - Assessing assumptions of regression:
      - Normality: dispersed residuals should appear random around zero.
      - Linearity: absence of curvature suggests a good fit.
      - Homoscedasticity: constant variance across all X values.
    - Violations lead to observable patterns:
      - Clustering suggests non-normality.
      - Curvy patterns imply nonlinearity.
      - Fanning suggests violations in homoscedasticity.

• Identifying Outliers
    - Outliers have high residual values.
    - Standardizing residuals (Z-scores) helps formalize outlier detection:
      - A Z-score greater than +/- 3 indicates candidates for removal.
    - Cook’s Distances:
      - A Cook's distance greater than 1 signals influential cases.
    - Leverage:
      - Measures how far an observation is from the mean value of X, with high leverage indicating extreme scores.
      - High leverage combined with significant residuals marks influential cases.

• Influential Cases
    - Influence is defined by changes in regression coefficients upon including a data point.
    - Examples of outlier characteristics:
      - Outlier with low leverage: Minimal impact on the regression line.
      - High leverage without being an outlier: Potentially low impact if close to the fitted line.
      - High leverage and outlier: Significant impact on model fit and coefficients.

• Summary of Goodness of Fit
    - Good fit characterized by minimal errors, consistent predictive power, and absence of influential cases.
    - Factors affecting goodness of fit include:
      - Model relationships
      - Measurement accuracy
      - Participant variability
      - Sampling scope
      - Data distributions
      - Outliers
      

Lesson 9.2: Multiple Regression

• Introduction to Multiple Regression
    - Aim to perform multiple linear regression, understanding both predictor and outcome variables.
    - Key learning outcomes include:
      - Identifying when multiple regression is appropriate.
      - Interpreting unstandardized and standardized coefficients.
      - Understanding assumptions of multiple regressions.

• Distinction Between Simple and Multiple Regression
    - Simple Linear Regression:
      - One continuous predictor and one continuous outcome.
      - Predicts Y based on a single X.
    - Multiple Linear Regression:
      - Use of multiple predictors (e.g., 2 to 10) to estimate Y.

• Example Application
    - Variables affecting psychological distress:
      - Predictors may include:
          - Negative life events.
          - Degree of social support.
          - Physical health.
  
  

• Unstandardized and Standardized Regression Equations
    - Unstandardized Equation:
      - Involves multiple B’s and X’s (e.g., Y = B1X1 + B2X2 + B3X3 + A).
      - Each B indicates the change in Y for each unit change in the predictor, holding others constant.
    - Standardized Equation:
      - Same format but allows comparison across predictors by using beta coefficients.
      - Beta shows standard deviation change in Y per standard deviation change in X.
    - Importance of standardization:
      - Provides context for contrasting different predictors with varying measurement scales.
      

• Requirements for Multiple Regression
    - Continuous outcome variable.
    - Two or more predictors (independent variables); combination of continuous and categorical is possible.

• Hypotheses for Multiple Regression
    - Overall Model Hypothesis:
      - Null: No linear relationships between predictors and Y.
      - Alternative: At least one predictor has a significant relationship with Y.
    - Individual Predictors Hypotheses:
      - Null: No unique relationship of each predictor with Y.
      - Alternative: Each predictor contributes uniquely.

• Assumptions for Multiple Regression
    - Independence of observations.
    - Normality: Residuals are normally distributed.
    - Linearity: Straight line relationship between predictors and outcome.
    - Homoscedasticity: Constant variance of errors.
    - Absence of multicollinearity among predictors.
      - Check using tolerance statistics (ideally greater than 0.10).

• Outliers and Influential Points
    - Cook’s distance < 1 indicates no influential points present.
    - Detection of univariate and multivariate outliers through plots and distance measures.
  
  

• Running Multiple Regression Analysis
    - Generate essential statistics:
      - Descriptive stats for predictors and outcomes (means, standard deviations, correlations).
      - Overall model fit tested using an F-test.
      - Reporting on null hypothesis significance tests for both overall and individual predictors.
  
  

• Reporting in APA Format
    - Presentation of means, standard deviations, and correlation coefficients in tables for clarity.
    - Summary of the model’s significance (B2 < 0.05, R² indicating variance explained).
    - Clear interpretation of coefficients:
      - Positive vs. negative relationships between predictors and outcome.

• Conclusion
    - Multiple regression assists in understanding relationships between multiple predictors and a continuous outcome.
    - Models allow predictions of Y based on different combinations of X’s, facilitating nuanced analysis.