Multiple regression in action
- What's your life expectancy? Work it out
using this Life
- According to this multiple regression equation, what could you
do to improve your life expectancy?
- Estimate the unstandardised regression coefficients for each of
the variables you could change in order to increase your life
- A psychologist studying perceived "quality of life" in a large
number of cities (N = 150) came up with the following equation using
mean temperature (Temp), median income in $1000 (Income), per capita
expenditure on social services (SocSer), and population density (Popul)
Y (predicted) = 5.37 - 0.01Temp + 0.05Income + 0.003SocSer - 0.01Popul
- Interpret the regression equation in terms of the coefficients -
in other words, what is the effect of each of the IVs on the Y
- Assume a city has a mean temperature of 55 degrees, a median
income of $12,000, spends $500 per capita on social services, and
has a population density of 200 people per block. What is the
predicted Quality of Life score?
- What would we predict in a different city that is identical in
every way except that is spends $100 per capita on social services?
(see Howell, p.550-551 for answers)
The general recommended strategy for tackling Multiple Linear
Regression analyses is:
- Check assumptions (see below)
- Conduct a multiple linear regression (standard, hierarchical, stepwise,
forward, or backward)
- Interpret the technical and psychological meaning of the results,
- R, R2, Adjusted R2, the
statistical significance of R
- Changes in R and the significance of the changes if steps
(i.e., more than 1 model are used)
- Standardised and unstandardised regression coefficients for each
- Zero-order and partial correlations for each IV in each model
- If useful, interpret Y-intercept and write a regression equation
for predicting Y
- Check histograms of all variables in an analysis
(are the variables normally distributed?)
- Check scatterplots of the relation between each X variable and the Y
(are the relationships linear? is there homoscedasticity?)
- Check correlation table for linear relations between Xs and Y
(are the X-Y relationships linear? check for multicollinearity between Xs?)
- Check influential outlying cases using Mahalanobis
distance & Cook’s D.
- In the Linear Regression box, click on
Save and select Mahalanobis and Cooks. SPSS will
create new variables in your data file called mah_1 and coo_1 once you
run the analysis.
- In your output check the Residuals Statistics
table for the maximum Mahalanobis distance and Cook’s distance.
maximum Mahalanobis distance should not be greater than the critical
chi-squared value with degrees of freedom equal to number of
predictors & alpha =.001.
- Cook’s D should not be greater than 1. If
you detect any outliers on either measure, consider removing the case
from you analysis.
- In your output check the collinearity statistics in
the Coefficients table. The Variance Inflation Factor (VIF)
should be <3 and tolerance should be >.3.
5.1 (Worked example)
Multiple regression in excel