HI5353 - Multiple Regression Summary

When to Consider Use

When you have measures obtained from a sample on one or more independent variables (predictors) and a dependent (criterion) variable having normally distributed errors of prediction around the observed values and you wish to describe the importance of all or individual predictors, or the relationship(s) between predictors and criteria, or make predictions about the criterion variable for the larger population from which the sample was obtained.

Null Hypotheses

Standard: There is no relationship between independent and dependent variables. Sequential or Stepwise: There is no relationship between additional independent variables and the dependent variable.

Assumptions & Some Relevant Tests & Assessment Tools

• Random selection/assignment
• Interval or ratio dependent measures
• Independent errors (residuals) or data observations
• Normal errors (residuals) or data observations for dependent variable - skewness, kurtosis, Kolmogorov-Smirnov, normality plots -
• Linearity - scatterplots, tests for nonlinear trend
• Homoscedasticity, homogeneity of variance of residuals around predicted values - scatterplots

Consequences of Violation of Assumptions

If samples are not random, validity of inferences about populations sampled is brought into question. If dependent variables are not measured on either interval or ratio scales results of the test are suspect.

Problematical Issues

(And How to Handle Them)

• Normality (Transform variables, e.g., square root, log, inverse, etc.)
• Linearity (Check scatterplots - transform variables, use nonlinear or logistic regression, sometimes test for nonlinear trends.)
• Homoscedasticity (Problem exists if ratio of largest to smallest SD > 3:1 - reduces statistical power - transform troublesome variables)
• Multicollinearity (Program Options - Check correlation matrix - delete redundant variables; use multicollinearity diagnostics (i.e., if a row condition index > 30 and 2 or more variance proportions in the row > .50 a multicollinearity problem exists (Belsely et al., (1980))
• Singularity - must have more cases than dependent variables, covariates and dependent variables must be linearly independent (Get larger samples for first, delete redundancies for second.).
• Reliability - reliable dependent variables and covariates - r > .80 (Obtain more reliable measures.)
• Outliers (Identify Univariate (generally p < .001; z > 3.29) & Multivariate (Mahalanobis Chi-square < .001 with df's = # of IV's) and then check for reasons and correct/change,scores transform variables, or eliminate from analysis)
• Categorical dependent variable (Use logistic regression or discriminant analysis)
• Sample Size - Have at least 10 times (Roscoe, 1974) as many cases as predictors to preferably 50 + 8m (where "m" is the number of predictors) for testing R and 104 + m for testing individual predictors (Tabachnick & Fidell, 2000,1996) or Cohen 1982, 1988 power tables.
• Missing Data (Replace blanks with a constant, preferably the mean and treat missingness as a variable to be controlled (Cohen & Cohen, 1983)

Procedural Checklist

(Adapted from Tabachnick & Fidell, 2000, 1996)

I. Handle Assumptions
• Assure adequate power & ratio of cases to IV's during design of study (use Cohen & Cohen, 1982; Cohen, 1988)
• Attend to normality, linearity & homoscedasticity of residuals using significance tests or appropriate graphs (use transformations where needed (e.g., square root, log, inverse, square, 1 - log, etc.)
• Examine outliers for anomalies, interpret & delete
• Assess multicollinarity & singularity
• Use a hierarchical strategy for controlling alpha inflation

II. Conduct Major Analyses of Importance
• Multiple R, R², overall (standard reg.) and/or incremental (sequential or hierarchical reg.) F ratio
• Adjusted multiple R² proportion of variance accounted for
• Significance of regression coefficients
• Squared semipartial correlations (equal R² changes)

III. Consider Additional Analyses
• Post hoc significance of correlations
• Unstandardized (B) & standardized (beta) weights regression coefficients, confidence limits
• Standardized (beta) weights, confidence limits
• Unique versus shared variability
• Suppressor variables
• Prediction equation
• Cross validation (for stepwise or setwise reg.)

Reporting Results

(Report what happened. Keep it simple and clear. Typically include the following.)

• Univariate F's, significance levels or source tables
• Measures of association (e.g., R, R², difference in R², correlations)
• Regression equations if relevant
• Regression coefficients, standard errors, and sample sizes
• Graphs (Use where it substantially helps communicate results.)

HI5353