Multiple Regression Analysis

Relationships With More Than Two Variables

• Multiple regression is a statistical method used to study multivariate relationships.
• It can assess the relationship between two specific variables.
• Multiple regression controls the influence of potential confounding variables.
• Predictor variables only predict outcomes.
• Relationships are noted, but causation is not explained.

Multivariate Regression

• Multivariate designs involve more than two variables.
• Multiple regression analysis identifies primary predictor(s), criterion, and third variables in a multiple regression.
• Multiple regression is a statistical analysis computing the relationship between the predictor variable and the criterion variable.
• It has control for other predictor variables.

Ruling Out Possible Third Variables

• Controlling for a third variable means holding it constant statistically or experimentally.
• This is done when investigating the relationship between two other variables.
• Multiple regression helps address internal validity questions by ruling out some third variables.

Ruling out Third Variables Example

• Exposure to sexual TV content correlates with teen pregnancy rate.
• The predictor is exposure to sexual TV content.
• The criterion is teen pregnancy rate.
• A possible third variable is age.

Describing Multiple Regression Example

• Sexual TV content and age predict teen pregnancy rate in a multiple regression analysis.
• Teen pregnancy rate was regressed on sexual TV content and age.
• Teen pregnancy was regressed on sexual TV content, controlling for age.

Venn Diagram as a Metaphor

• A circle represents the variability in a variable.
• The variability includes differences of each score from the mean and differences among the scores.
• Overlapping areas in circles represent the degree of relationship between two variables.
• The relationship can be described as positive or negative.

Variable Prediction

• To start, pick the variable intended to be predicted; this is the criterion.
• The remaining two variables are predictors.
• All three variables are typically intercorrelated.
• Multiple regression determines if one variable predicts the criterion, controlling for the others.

Regression Outcome #1 and Visual Representation

• Sexual TV content does not have an independent effect when using both sexual TV content and age to predict teen pregnancy, controlling for age.

Regression Outcome #2

• When both sexual TV content and age predict teen pregnancy, both variables have independent effects.
• Age predicts teen pregnancy, controlling for sexual TV content.
• Sexual TV content predicts teen pregnancy, controlling for age.

Unlikely Regression Outcome #3

• Age does not have an independent effect, while sexual TV content independently predicts teen pregnancy, controlling for age.

Controlling for a Third Variable

• Controlling for a third variable involves finding the unique overlap between the predictor and the criterion of interest.
• Analyze the proportion of variability attributable to the third variable.
• Determines the unique proportion of variability attributable to the primary predictor.

Beta in Regression Analysis

• β (beta) is a standardized regression coefficient.
• Metaphorically, beta represents the unique overlap between predictor and criterion.
• It does not include overlaps with potential third variables.
• Beta shows both the strength and direction of relationship.

Regression

• Regression is a statistical process that finds the Finds the linear equation.
• It produces the most accurate predicted values for Y using one predictor variable (X).

Multiple Regression

• Researchers often use several predictor variables to make more accurate predictions.
• The variables are used in a multiple regression equation for calculations.

Beta vs. b

• Both beta and b are often seen in publications.
• Beta (standardized regression coefficient) can compare relative strength.
• It can compare relative strength from the same multiple regression analysis (i.e., same result table).
• Beta cannot compare across different regression analyses.
• The letter b (unstandardized regression coefficient) cannot compare within the same regression analysis but is easy to interpret.

Interpreting b

• Template: For every 1 unit change in X, there is a “b” unit(s) change in Y, holding third variables (C, D, and E) constant.
• Example: X: proneness to hunger (b = -1.26) Y: self-control; C: gender; D: age; E: neuroticism.
• For every 1 unit change in proneness to hunger, there is a -1.26 unit change in self-control.

Interpreting Beta

• Template with X as predictor of interest, Y as the criterion, and C, D, and E as possible third variables.
• This beta indicates whether X is associated with Y, such that [high/low] scores on X align with [high/low] scores on Y.
• Requires controlling for predictors C, D, and E.

Beta in the Example

• Multiple-Regression Results from a Study Predicting Pregnancy from Sexual Content on TV and Age
• The predictor (independent) variables assessed were Exposure to sex on TV, and Age
• Each has a 95% CI for BETA value, and Statistical significance.

Finding the Strongest Predictors

• Larger beta values indicate a stronger predictor.
• For CI and p-value, 95% CI does not include zero
• p < .05 is considered significant and can use * to designate significance.
• When 95% CI includes zero, p = or > .05 (nonsignificant)
• It can be designated using the letters n.s.

Features of Multiple Regression

• Regression analysis can help answer two questions.
• It helps control for several third variables at once.
• It examines the betas for the other predictor variables.
• Used to assess which factors most strongly predict the criterion.

Predicting Stress Levels

• Predict stress levels (measured on a scale of 1-10) from weekly exercise hours and weekly screen time (in hours) among university students.
• The result showed a negative correlation, where stress decreased as weekly exercise increased (Beta = -0.35)
• Also, a high p-value suggests time spent on screens doesn't affect stress

Predicting Environmental Concern

• Environmental concern (scale 1-10) can be predicted, based on political ideology and education level among adults in a survey.

Predicting Creativity Scores

• Creativity scores can be predicted based on time spent on social media and time spent engaging in outdoor activities among high school students.

Mediation

• Mediation addresses why there is an association between the predictor and criterion.
• It explains how the predictor transmits its effect on the criterion.
• A mediator is a variable that helps explain the relationship between the predictor and criterion.
• It identifies an outcome of the predictor that is responsible for the criterion.

Workplace Autonomy Example

• High levels of workplace autonomy tend to report higher job satisfaction.
• It might occur because workplace autonomy leads to a sense of control over one’s tasks, which in turn increases job satisfaction.
• Predictor: workplace autonomy, criterion: job satisfaction, and mediator: locus of control.

Deep Talk and Well-Being Example

• Kenny's (2008) Five Regression Steps are:

  • Is predictor associated with criterion? (c)
  • Is predictor associated with M? (a)
  • Is M associated with criterion? (b)
  • Multivariate Regression: use predictor and M to predict criterion (c')
  • Measure predictor and/or M before measuring criterion

• Total Effect (c) = Direct Effect (c') + Indirect Effect (ab)

• Mediation effect is, if the indirect effect (ab) is significant, c' is smaller than c.

Identifying Simple Mediation

• In correlational research, the amount of deep talk is positively associated with the quality of social ties, which is positively associated with well-being.
• Quality of social ties mediated the association between the amount of deep talk and well-being.
• In experimental research, a high amount of deep talk led to higher quality of social ties, which in turn led to higher well-being.
• Quality of social ties mediated the effect of amount of deep talk on well-being.

Causal Interpretation

• Mediation may imply causal orders, with experimental studies as best cases.
• Mediation can explore correlational research, but must measure P and M before C, test reverse models, and be specific while avoiding causal language.

Moderators

• Assess complex questions and inform external validity through moderation.
• Moderation occurs in an association between A and B.
• The strength and/or direction of the association differs depending on the levels of C.
• C is a moderator of the relationship between A and B.

Moderation Example: Sleep Quality and Productivity

• A study investigates the relationship between sleep quality and productivity at work.
• Researchers found a positive correlation between sleep quality and productivity (r = 0.40).
• People who work in high-stress jobs have positive correlation (r = 0.60).
• Those in low-stress jobs had weaker correlation (r = 0.20).
• Job stress moderates the relationship between sleep quality and productivity.

Another Moderation Example

• A is percentage of Wins, B is attendance, and C is city Mobility.
• A and B for Arizona: positive correlation (r = .29*).
• A and B for Pittsburgh: there is a non-significant correlation (r = -.16).

Differences Among Variables

• Requires some theoretical judgment.
• A Third Variable Problem (The Villain!) is external to the relationship between P and C.
• It needs to rule out lurking, distraction, accidental.
• A Mediator relates internally to P and is the study's focal interest to the researchers.
• A mediator shows Why is the P associated/influencing the C?
• A Moderator questions For Whom or When.
• It also analyzes if the relationship becomes more or less intense, depending on different levels of the moderator?

Exercises Overview

• Indicate whether each statement below is describing a mediation hypothesis, a third-variable problem, or a moderator result.
• First, identify the key bivariate relationship.
• Next, decide whether the extra variable is a mediator, moderator, or third variable.

Exercise 1

• A mentally demanding job associates with cognitive benefits in later years
• Highly educated people take mentally demanding jobs.
• Highly educated people have better cognitive skills.

Exercise 2

• A mentally demanding job is associated with cognitive benefits in later years, in men only.

Exercise 3

• A mentally demanding job is associated with greater cognitive skills.
• Mentally demanding job tasks build lasting connections in the brain.
• Resulting in greater cognitive skills.

Exercise 4

• Regular physical exercise reduces stress levels.
• Exercise helps reduce stress more for younger adults compared with older adults.

Exercise 5

• People with strong social networks tend to have lower levels of depression.
• Social support helps individuals develop better coping strategies.
• This directly reduces depressive symptoms.

The Longitudinal Developmental Research Design

• Explores growth by watching or measuring a group/cohorts with time.
• For example, researchers measured IQ in a group of 40-year-olds, then again at 60 and 80.

Example Longitudinal Design

• Burmmelman et al., 2015 studied parental overpraising and child narcissism.
• Measurements occurred four times, once in every 6 months.
• They used eight variables.

Cross-Sectional Longitudinal Correlations

• Whether two variables measured at the same point are related.
• It cannot establish temporal precedence.

Autocorrelations

• Correlation of each variable across different time points.
• It cannot establish temporal precedence.
• It assesses if a variable is consistent over time.

Cross-Lag Correlations

• Correlation between an earlier measure of one variable with a later measure of the other variable.
• It measures correlation between X at time 1 and Y at time 2.
• Significant cross-lag correlations establish temporal precedence.

Study Finding Outcome 1

• Overvaluation early leads to more narcissism later (Figure 9.3)

Study Finding Outcome 2

• Narcissism at earlier periods with significantly correlated with overvaluation at later periods.
• Solid path = significant correlation, dashed path = nonsignificant correlation.

Study Finding Outcome 3

• One and Two are significant.
• Overvaluation and narcissism are mutually reinforcing.
• Sold path equals a significant correlation, and a dashed path equals a nonsignificant one.

Longitudinal Studies

• Covariance indicates significant relationships, and temporal precedence uses cross-lag correlations for evidence on causal direction.
• Internal Validity should assess third variables and rule them out, as they cannot automatically be ruled out.

Strengths and Weaknesses in Longitudinal Research Designs

• Strengths: no cohort or generation effects, and assesses individual behavioral changes.
• Weaknesses: time-consuming, participant attrition may create bias, and potential for practice effects.