Tutorial 4

Multiple Linear Regression

qfsall.sav

The general recommended strategy for tackling an MLR analysis is:

Conceptualise model (e.g., draw diagram - Path or Venn)
Recode predictors (if necessary) as dummy variables
Check assumptions
Conduct standard, hierarchical, stepwise, forward, or backward MLR
Interpret statistical findings and psychological meaning of results, taking into account:
1. R, R², Adjusted R², sig. of R
2. ∆ in R and its sig. (i.e., if more than one model)
3. Standardised (B) and unstandardised (β (beta)) regression coefficients
4. Zero-order and partial correlations for each IV in each model
If relevant and useful, interpret Y-intercept and write a regression equation for predicting Y

IVs: Two or more interval/ratio or dichotomous variables

DV: One interval/ratio variable

Do you have enough data?
(min. 5 cases per predictor, ideally 20 cases per predictor, with an overall N of at least 100; this allows sufficient power to detect a medium ES of R square of .13 (Francis, p. 128))
Are the variables normally distributed?
(Check histograms of all variables in an analysis; MLR is reasonably robust to violations of this assumption)
Are the bivariate relationships linear?
(Check scatterplots and correlations between Y and each of Xs)
Is there homoscedasticity?
(Check scatterplots between Y and each of Xs and/or check scatterplot of the residuals (ZRESID) and predicted values (ZPRED))
Is there multicollinearity between Xs?
(Check scatterplots and correlations between each of the Xs and/or check the collinearity statistics in the Coefficients table. The Variance Inflation Factor (VIF) should be < 3 and tolerance should be > .3.)
Are there multivariate outliers?
(Check Mahalanobis' Distance and Cook's D)

See also Francis 5.1.4 Practical Issues and Assumptions (pp. 126-128)

To check whether there are influential multivariate outlying cases using Mahalanobis distance & Cook’s D:

Linear Regression - Save - Mahalanobis and Cook's D - OK
SPSS will create new variables called mah_1 and coo_1.
Check the Residuals Statistics table in the output for the maximum Mahalanobis and Cook’s distances.
The max Mahalanobis distance should not be greater than the critical chi-square value with degrees of freedom equal to number of predictors, with critical alpha =.001.
Cook’s D should not be greater than 1.
If outliers are detected, check each case, and consider removing the case from the analysis.

Follow the general steps to conduct a meaningful MLR analysis of your own design (qfsall.sav - or any other data set).
Interpret the analysis - explain it to someone next to you or your tutor.

Conduct a meaningful MLR analysis which requires you to recode one or more of the variables as dummy variables.

Re-do any of your previous MLRs using each of Direct, Forward, Backward, Stepwise, and Hierarchical methods.
Why or why there aren't differences between the results for each method?

For any previous MLR analysis, create stand-alone correlations and depict the relationships in a Venn Diagram.

If you're curious about how multiple regression can be used for prediction, then try using a Life Expectancy Calculator (this one takes about 15 minutes):
1. What could you do to improve your life expectancy?
2. What do you estimate the unstandardised regression coefficients to be for each of the variables the calculator recommends you could change in order to increase your life expectancy?
3. What do you think the an 80% confidence interval would be for your life expectancy estimate? (can be estimated via this very short Life Expectancy Calculator (takes <1 minute).