Multiple Regression

3/12/2002

Announcements

Exams not done

Basic Review

I’m going to start with the facts they know from last semester.

= bX + a

where b is the slope (the difference in for a one unit difference in X.

a is the intercept = the value of when X = 0.

r_YX  is = the correlation between Y and

This is true, but unimportant in simple regression, but becomes useful in multiple regression.

SS_Y = Sum of squares of Y = S(Y-)² = (N-1)s²_Y

SS_reg = sum of squares of = S( - )² = r²SS_Y

SS_error = residual sum of squares = S(Y - )² = (1-r²)SS_Y

df_error = N - p - 1 = N - 2

r² = percent of variation in Y accounted for by variation in X.

Each of these relationships has a counterpart in multiple regression, with only minor (and pretty much obvious) changes.

Multiple Regression

Here we try to use information on several predictors (X_j) to simultaneously predict Y.

We will form a linear combination of the X_j to yield , subject to the same least squares  
restriction that we had in Chapter 9.

Get them to remember what a linear combination is.

We want an equation of the form:

= b₁X₁ + b₂X₂ + b₃X₃ + ... + b_kX_k + b₀

where the b_j are chosen so as to minimize S(Y - )² = SS_residual

point out that the subscript j stands for the variable, and not the subject.

Point out b₀

Example

I’ll start out with the example from Esther Leerkes work. It only has two independent variables, which makes things a bit simpler. And it is a very nice study.

Esther was looking at Maternal Self Efficacy, measured at 5 months after mom became a mom (meq5). Esther wanted to know how that score was related to the care that mom received from her own mom when she was little (MCarem) and mom's own sense of self esteem (SE)

Define criterion variable = the dependent variable.

Define predictors = the independent variables.

Describe data file.

Variables are FamilyID, se, mcarem, and meq5

The first 12 cases follow. (These are the actual data she collected.)

familyid	se	mcarem	meq5
1	3.83	2.58	3.70
2	3.50	2.83	3.40
4	4.00	3.17	3.80
8	4.00	3.75	3.90
9	4.00	3.58	3.90
11	3.33	3.67	3.70
12	3.50	3.25	3.80
13	3.67	3.75	3.50
15	1.83	3.17	3.10
16	2.83	2.67	3.50
17	3.67	4.00	3.30
19	4.00	4.00	4.00

SPSS Printout

Descriptives:

Correlations:

 

 

3 - D Scatterplot

I rarely think that these are helpful, but some people really like them.

Regression:

REGRESSION

 

 

The following shows the Maternal Efficacy plotted against the predicted values

Discuss the above printout

Intercorrelation Matrix

"validities"

correlation among the variables

• discuss the meaning of these
• discuss multicollinearity

Missing values

Here we don’t have any, but if we did:

• pairwise deletion
• Casewise or listwise deletion

Scatterplot:

• Discuss how to get this.
• What does it tell us?

Regression solution:

Discuss overall F and R-squared

• Tie this to Anova and discuss similarities and differences
• Point out SS_regression and SS_residual
• Ask them why df_residual = 44.

Discuss parameter estimates:

• Intercept

• b_j

• Point out that these are just slopes

• Called regression coefficients

Predicted values

Optimal regression equation =

= .147*se + .0582*mcarem + 2.929

familyid	se	mcarem	meq5
1	3.83	2.58	3.70
2	3.50	2.83	3.40
4	4.00	3.17	3.80
8	4.00	3.75	3.90

For the first subject:

= .147*3.83 + .0582*2.58 + 2.929 = 3.642

For the second case = 3.608

For the third subject:

= .147*4.00 + .0582*3.17 + 2.929  = 3.70

 

Case	Y –	Residual
1	3.70 – 3.642	0.058
2	3.40 – 3.608	-.208
3	3.90 – 3.700	0.200

Standardized Regression Coefficients

Explain why we might want to standardize the variables

Get them to remember that this creates comparable standard deviations, and hence scales the coefficients by the standard deviations.

Last revised: 03/11/02