PSCH 443 Multiple Regression 3 Evaluating Assumptions Part 1

Assumptions of Multiple Regression

• Discussion of multiple regression involves addressing several key assumptions that need to be met to ensure the reliability of the regression analysis.

Sample Size Considerations

• Importance of Sample Size: A larger sample size is preferred for stable estimates and evaluating statistical significance.

  • Encourages stability in estimates of coefficients and the overall model.

  • Essential for assessing significance levels like R-squared, individual parameters, and t-tests.

• Statistical Power: Refers to the ability to detect effects when they exist.

  • Guidelines suggest 10-15 cases per predictor as a minimum, but more is better.

• Minimum Acceptable Sample Size:

  • Field's suggestion: Minimum of 50 + 8 times the number of predictors for overall models.

  • For individual predictors, 104 participants plus the number of predictors are recommended.

• Realistic Expectations: Collect as many cases as possible, ideally around 200 if feasible.

  • Ensure participants are a representative sample, avoiding biased convenience samples.

• Research Area Norms: Check what peers are collecting to gauge reasonable sample sizes in your field.

Situations with Limited Participants

• Constraints on Participant Availability: Sometimes bound by pre-existing datasets or specialized populations (e.g., rare conditions).

  • It remains valuable to conduct studies even with smaller sample sizes if they provide insights—especially in fields like mental health where participants may be limited.

Role of Power Analysis

• Power Analysis: Crucial for determining required sample size based on expected relationships between predictors and outcomes.

  • Aim for 0.8 power—80% probability of correctly rejecting a false null hypothesis.

• Strength of Relationships: Assess through bivariate correlations or existing research to gauge effect sizes and impact.

Multicollinearity

• Definition: Occurs when predictors in the model are highly correlated, causing issues in model estimation.

  • Strong correlations obscure individual predictor effects, complicating interpretations.

• Assessing Severity of Multicollinearity:

  • Raised standard errors of estimates and reduced R-squared values.

• Ideal vs. Problematic Cases:

  • An ideal regression would show no multicollinearity; predictors explain unique variance in outcomes.

  • Perfect multicollinearity leads to inability to differentiate predictors.

Evaluating Multicollinearity

• Correlation Matrix: Use simple correlations to check for high correlations (e.g., >0.8 to 0.9).

  • Moderate correlations may also lead to compounded issues in regression.

• Beta Weights and Diagnostics: Utilize diagnostics such as tolerances and variance inflation factors (VIF) to assess multicollinearity.

  • VIF: A VIF greater than 10 signals potential multicollinearity.

  • Tolerances: Values lower than 0.2 indicate concern.

Outliers in Data

• Identification of Outliers: Outliers can skew regression results and violate normal distribution assumptions.

  • Univariate outliers affect single variable distributions, while multivariate outliers disrupt prediction accuracy in combination analyses.

• Residuals and Diagnosing Outliers:

  • Analyzing residuals can uncover extreme prediction errors.

  • Generally, only 5% of residuals should exceed two standard deviations from the mean in a normal distribution.

Handling Outliers

• Investigation of Outliers: Understand why outliers occurred before deciding to retain or remove them.

  • Possible reasoning includes coding errors or respondent biases.

  • Can run analyses with and without outliers to gauge impact on results.

• Best Practices: Always be transparent about how outliers were treated in analysis.

  • Report findings and methodologies clearly to maintain integrity.