Hierarchical Regression Analysis
In a hierarchical multiple regression, the researcher decides not only how
many predictors to enter but also the order in which they enter. Usually, the
order of entry is based on logical or theoretical considerations.
There are three predictor variables and one criterion variable in the
following data set. A researcher decided the order of entry is X1,
X2, and X3.
SPSS for Windows
1. Enter Data.
2. Choose Analyze / Regression / Linear.
Dependent: Select "y" and move it to the Dependent variable list. First,
click on the variable y. Next, click on the right arrow.
Block 1 of 1
Independent(s): Choose the first predictor variable x1 and move it to the
Independent(s) box. Next, click the Next button as shown below.
Block 2 of 2
Click the predictor variable x2 and move it to the Independent(s) box. Next,
click the Next button as shown below.
Block 3 of 3
Click the predictor variable x3 and move it to the Independent(s) box.
3. Click the Statistics button. Check R squared change.
Click Continue and OK.
SPSS Output
1. R square Change
R Square and R Square Change
Order
of Entry
Model 1 : Enter
X1
Model 1: R square = .25
The predictor X1 alone accounts for 25% of the variance in
Y.
R2 = .25
Model 2 : Enter
X2
next
.Model 2: R square = .582
The Increase in R square: .
582 - .25 = .332
The predictor X2
accounts for 33% of the variance in Y after controlling for X1.
R2 = .25 +
.332 = .582
Model Three: Enter X3
third
Model 3: R square = .835
The Increase in R square: .
835 - .582 = .253
The predictor X3
accounts for 25% of the variance in Y, after X1 and X2
were partialed out from X3.
R2 =
.25 + .332 +
.253 = .835
About 84% of the variance in the criterion variable was explained by the first
(25%), second (33%) and third (25%) predictor variables.
2. Adjusted R Square
For our example, there are only five
subjects. However, there are three predictors. Recall that
R square may be overestimated when the data
sets have few cases (n) relative to number of
predictors (k).
Data sets with a
small sample size and a large number of predictors will have a
greater difference between the obtained and adjusted R square (.25 vs. .000,
.582 vs. .165, and .835 vs. .338).
3. F Change and Sig. F Change
If the R square change
associated with a predictor variable in question is large, it means that the
predictor variable is a good predictor of the criterion variable.
In the first step, enter the predictor variable x1 first. This resulted in an
R square of .25, which was not statistically significant (F Change = 1.00, p >
.05). In the second step, we add x2. This increased the R square by 33%,
which was not statistically significant (F Change = 1.592, p > .05). In the
third step, we add x3. This increased the R square by an additional 25%,
which was not statistically significant (F Change = 1.592, p > .05).
4. ANOVA Table
Model1:
About 25% (2.5/10 = .25) of the variance in the criterion
variable (Y) can be accounted for by X1. The
first model, which includes one predictor variable ( X1), resulted in an F ratio of
1.000 with a p > .05.
Model 2
About 58% (5.82/10 = .58) of the variance in the criterion
variable (Y) can be accounted for by X1 and X2. The
second model, which includes two predictors (X1 and X2), resulted in an F ratio of
1.395 with a p > .05.
Model 3:
About 84% (8.346/10 = .84) of the variance in the criterion
variable (Y) can be accounted for by all three predictors (X1, X2 and X3). The
third model, which includes all three predictors, resulted in an F ratio of
1.681 with a p > .05.
