Type I and Type III Sums of Squares

Type I and Type III Sums of Squares

Supplement to Section 8.3

Brian Habing ¨C University of South Carolina

Last Updated: February 4, 2003

PROC REG, PROC GLM, and PROC INSIGHT all calculate three types of F tests:

? The omnibus test: The omnibus test is the test that is found in the ANOVA table. Its F statistic is

found by dividing the Sum of Squares for the Model (the SSR in the case of regression) by the SSE. It

is called the Test for the Model by the text and is discussed on pages 355-356.

? The Type III Tests: The Type III tests are the ones that the text calls the Tests for Individual

Coefficients and describes on page 357. The p-values and F statistics for these tests are found in the box

labeled Type III Sum of Squares on the output.

? The Type I Tests: The Type I tests are also called the Sequential Tests. They are discussed briefly on

page 372-374.

The following code uses PROC GLM to analyze the data in Table 8.2 (pg. 346-347) and to produce the

output discussed on pages 353-358. Notice that even though there are 59 observations in the data set,

seven of them have missing values so there are only 52 observations used for the multiple regression.

DATA fw08x02;

INPUT Obs

age

bed

bath

size

lot

price;

CARDS;

1

21

3

3.0

0.951

64.904

30.000

2

21

3

2.0

1.036

217.800

39.900

3

7

1

1.0

0.676

54.450

46.500

58

1

3

2.0

2.510

.

189.500

59

33

3

4.0

3.627

17.760

199.000

;

PROC GLM DATA=fw08x02;

MODEL price = age bed bath size lot;

RUN;

With five independent variables, this procedure produces seventeen p-values for testing hypotheses.

A ¨C This is the test associated with the ANOVA table. It always (even for cases that aren¡¯t regression)

tests the null hypothesis that none of the independent variables linearly predict the dependent variable.

The alternate hypothesis is that at least one of the independent variables does linearly predict the

dependent variables. Because it tests all of these at once it is sometimes called the omnibus test. For this

example it is testing the null hypothesis H0: all of ¦Âage=0, ¦Âbed=0, ¦Âbath=0, ¦Âsize=0 and ¦Âlot=0.

The alternate hypothesis is that at least one of them is not zero. In this example, the p-value is less than

0.0001 and we would reject the null hypothesis. (You can check these SS, MS, and F values with what

the text gives on page 356.)

The GLM Procedure

Dependent Variable: price

Source

DF

Sum of

Squares

Model

Error

Corrected Total

5

45

50

65695.79292

13774.04972

79469.84265

R-Square

0.826676

Source

Coeff Var

15.98770

Mean Square

F Value

Pr > F

13139.15858

306.08999

42.93

F

age

1

526.13923

526.13923

1.72

0.1965

bed

1

10713.09272

10713.09272

35.00

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