Consider this simple data file that has nine subjects
(sub)
in three groups (iv) with a score on the outcome or dependent variable (dv).
data list list / sub iv dv. begin data 1 1 48 2 1 49 3 1 50 4 2 17 5 2 20 6 2 23 7 3 28 8 3 30 9 3 32 end data.
Below we use the means command to find the
overall mean and the means for the three groups.
means tables = dv by iv.
As we see below, the overall mean is 33, and the means for groups 1, 2 and 3 are 49, 20 and 30 respectively.
Case Processing Summary
Cases Included Excluded Total N Percent N Percent N Percent DV * IV 9 100.0% 0 .0% 9 100.0%
Report
DVIV Mean N Std. Deviation 1.00 49.0000 3 1.00000 2.00 20.0000 3 3.00000 3.00 30.0000 3 2.00000 Total 33.0000 9 12.89380
Let's run a standard ANOVA on these data using glm.
glm dv by iv.
The results of the ANOVA are shown below.
Between-Subjects Factors
N
IV
1.00
3
2.00
3
3.00
3
Tests of Between-Subjects Effects
Dependent Variable: DV
Source
Type III Sum of Squares
df
Mean Square
F
Sig.
Corrected Model
1302.000(a)
2
651.000
139.500
.000
Intercept
9801.000
1
9801.000
2100.214
.000
IV
1302.000
2
651.000
139.500
.000
Error
28.000
6
4.667
Total
11131.000
9
Corrected Total
1330.000
8
a R Squared = .979 (Adjusted R Squared = .972)
Now, let's take this information we have found and
relate it to the results that we get when we run a similar analysis using dummy coding. Let's make a data file called dummy2 that has dummy variables called
iv1
(1 if iv=1), iv2 (1 if iv=2) and iv3 (1 if iv=3).
Note that iv3 is not really necessary, but it could be useful for further
exploring the meaning of dummy variables. We will then use the
regression command to
predict dv from iv1 and iv2.
compute iv1 = 0. if iv = 1 iv1 = 1. compute iv2 = 0. if iv = 2 iv2 = 1. compute iv3 = 0. if iv = 3 iv3 = 1. execute. regression /dependent = dv /method = enter iv1 iv2.
The output is shown below.
Variables Entered/Removed(b)
Model
Variables Entered
Variables Removed
Method
1
IV2, IV1(a)
.
Enter
a All requested variables entered.
b Dependent Variable: DV
Model Summary
Model
R
R Square
Adjusted R Square
Std. Error of the Estimate
1
.989(a)
.979
.972
2.16025
a Predictors: (Constant), IV2, IV1
ANOVA(b)
Model
Sum of Squares
df
Mean Square
F
Sig.
1
Regression
1302.000
2
651.000
139.500
.000(a)
Residual
28.000
6
4.667
Total
1330.000
8
a Predictors: (Constant), IV2, IV1
b Dependent Variable: DV
Coefficients(a)
Unstandardized Coefficients
Standardized Coefficients
t
Sig.
Model
B
Std. Error
Beta
1
(Constant)
30.000
1.247
24.054
.000
IV1
19.000
1.764
.737
10.772
.000
IV2
-10.000
1.764
-.388
-5.669
.001
a Dependent Variable: DV
First, note that from the ANOVA using the glm
command
that the F-value was 139.5 and for the regression using the regression
command the F-value (for the model) is also 139.5. This illustrates that the overall test of the model
using regression is really the same as doing an ANOVA.
After the
ANOVA table, there is a table entitled Coefficients. What is the interpretation of the values listed there, the 30, 19 and
-10? Notice how we have iv1 and iv2 that refer to group
1 and group 2, but we did not include any dummy variable referring to group 3.
Group 3 is
often called the omitted group or reference group.
Recall that the means of the 3 groups were 49, 20 and 30 respectively. The
intercept
term is the mean of the dependent variable, which we called dv, for the
omitted group, and indeed the parameter estimate
(in the column B) from the output is the mean of group 3, 30. The parameter estimate for
iv1
is the mean of the dependent variable, dv, for group 1 minus the mean of
the dependent variable for group 3, 49 - 30 = 19, and indeed that is the
parameter estimate for iv1. Likewise, the parameter estimate for
iv2
is the mean of the dependent variable for group 2 minus the mean of the
dependent variable for group 3, 20 - 30 = -10, the parameter estimate for
iv2.
So, in summary:
Intercept mean of group 3 (mean of omitted group) iv1 mean of group 1 - group 3 (omitted group) iv2 mean of group 2 - group 3 (omitted group)
Try running this example, but use iv2 and iv3
using regression (making group 1 the omitted group) and see what happens.
Finally, consider how the parameter estimates can be
used in the regression model to obtain the means for the groups (the predicted values).
The regression model is
Ypredicted = 30 + iv1*19 + iv2*-10For group 1: Ypredicted = 30 + 1 * 19 + 0 * -10 = 49 For group 2: Ypredicted = 30 + 0 * 19 + 1 * -10 = 20 For group 3: Ypredicted = 30 + 0 * 19 + 0 * -10 = 30
As you see, the regression formula predicts that each
group will have the mean value of its group.
You can also perform the same analysis using
glm. The print = parameter
subcommand tells SPSS
to print the regression coefficients.
glm dv with iv1 iv2
/print = parameter.
Tests of Between-Subjects Effects
Dependent Variable: DV
Source
Type III Sum of Squares
df
Mean Square
F
Sig.
Corrected Model
1302.000(a)
2
651.000
139.500
.000
Intercept
2700.000
1
2700.000
578.571
.000
IV1
541.500
1
541.500
116.036
.000
IV2
150.000
1
150.000
32.143
.001
Error
28.000
6
4.667
Total
11131.000
9
Corrected Total
1330.000
8
a R Squared = .979 (Adjusted R Squared = .972)
Parameter Estimates
Dependent Variable: DV
B
Std. Error
t
Sig.
95% Confidence Interval
Parameter
Lower Bound
Upper Bound
Intercept
30.000
1.247
24.054
.000
26.948
33.052
IV1
19.000
1.764
10.772
.000
14.684
23.316
IV2
-10.000
1.764
-5.669
.001
-14.316
-5.684