Understanding Multiple Predictors and Variance in Dependent Variables

Predictors and Variance

• This unit explores how multiple predictors relate to the variance in a dependent variable.
• It covers statistics for understanding and proportioning variance.
• Key statistics: tolerance statistic and semi-partial correlation.

Predictor Usefulness

• Predictors are useful if:
  • Theoretically relevant to the outcome (Y).
  • Correlated with the outcome (Y).
  • Uniquely predictive of the outcome (Y).

Example: Predicting IQ (Y)

• Dependent variable (Y): IQ
• Potential predictors (X):
  • Age
  • Educational level
  • Crossword puzzle ability

Age

• Theoretically relevant to IQ because vocabulary (a component of IQ) typically peaks in mid-adulthood.
• Memory decline with age can affect vocabulary assessment.

Educational Level

• Correlated with IQ, supported by empirical evidence.
• Individuals with more educational experience tend to perform better on IQ tests.

Crossword Puzzle Ability

• Uniquely predictive of IQ because it involves problem-solving and information recall.
• Performance on crossword puzzles may indicate cognitive abilities related to IQ.

Ideal Scenario: Unique Explanations

• Ideal: Predictors (X1, X2, X3) each explain unique portions of the variance in the outcome (Y) without overlap.
• Example: Three predictors explaining 75% of the variance in IQ scores would be highly valuable.

Redundancy

• Overlap among predictors leads to redundancy.
• A redundant variable doesn't add unique explanatory value.
• Example: If X2 is largely overlapped by X1 and X3, it is mostly redundant.

Types of Redundancy

• Partly redundant: Some shared variance explained.
• Wholly redundant: Entirely explained by other predictors.

Correlation

• Independent variables (X's) are typically correlated with the dependent variable (Y).
• Correlation implies shared variance in explaining Y.

R^2:

• R^2 represents the proportion of variance in Y explained by the predictors; includes unique and joint contributions.
• R^2 = (a + c + b) / (a + c + b + d), where:
  • a = unique variance explained by one predictor.
  • c = unique variance explained by another predictor.
  • b = overlapping variance explained by multiple predictors.
  • d = unexplained variance.
• Example: If R^2 = 0.75, the predictors explain 75% of the variance in Y.

Multi-Collinearity

• Multi-collinearity occurs when predictors are highly correlated, which reduces the ability to predict the outcome and violates assumptions.
• Two reasons to avoid high correlation among predictors:
  1. Power is maximized when predictors explain unique components of the outcome.
  2. Violates the assumption of non-multi-collinearity.

Identifying Multi-Collinearity

• Common indicator: correlation (r) > 0.9 between predictors.
• Tolerance is another measure used to assess multi-collinearity.

Assumptions

• Correlation assumptions.
• Simple regression assumptions.
• Multiple regression assumption: no highly redundant information (non-multi-collinearity).

Example: Animal Listing Task

• Three students (Anthony, Louise, Joanna) list animals in 15 seconds without hearing each other.

• Total: 21 animals listed, 13 unique.

• Anthony: two unique animals.

• Louise: three unique animals.

• Joanna: no unique animals.

• Outcome:

  • Joanna lists the most animals overall.
  • Anthony and Louise provide the most unique animals (13).

• Conclusion: Combining Anthony and Louise yields more unique information than Joanna alone.

Example: Employee Skill Sets

• Anthony, Louise, and Joanna have different skill sets (typing, filing, communication, software proficiency).

• If hiring two people, Anthony and Louise would be preferred for their unique skills.

• If hiring one person, Joanna might be chosen due to the total skill set, but Anthony and Louise have unique skill sets that provide a better outcome.

Overlap in Predictors

• As the number of predictors increases, the unique contribution of each predictor tends to decrease.
• More predictors can