Help the Stat Consulting Group by giving a gift

GLM

This page shows an example of analysis of variance run through a general linear model (glm) with footnotes
explaining the output. The data were collected on 200 high school students, with
measurements on various tests, including science, math, reading and social
studies. The response variable is writing test score (**write**), from which we explore its relationship with
gender (**female**) and academic program (**prog**). The model examined
has the main effects of female and program type, as well as their interaction. The
dataset used in this page can be downloaded from
http://www.ats.ucla.edu/stat/sas/webbooks/reg/default.htm.

The syntax for the page is provided below. The **class** statement defines
which variables are to be treated as categorical variables in the **model**
statement. The **model** statement has the main effects of **female** and **
prog**, as well as their interaction; the interaction is specified by taking
the product of the two main effect terms. The option **ss3** tells SAS we
want type 3 sums of squares; an explanation of type 3 sums of squares is
provided below.

proc glm data = "c:\temp\hsb2"; class female prog; model write = female prog female*prog /ss3; run; quit;The GLM Procedure Class Level Information Class Levels Values female 2 0 1 prog 3 1 2 3 Number of Observations Read 200 Number of Observations Used 200 Dependent Variable: write Sum of Source DF Squares Mean Square F Value Pr > F Model 5 4630.36091 926.07218 13.56 <.0001 Error 194 13248.51409 68.29131 Corrected Total 199 17878.87500 R-Square Coeff Var Root MSE write Mean 0.258985 15.65866 8.263856 52.77500 Source DF Type III SS Mean Square F Value Pr > F female 1 1261.853291 1261.853291 18.48 <.0001 prog 2 3274.350821 1637.175410 23.97 <.0001 female*prog 2 325.958189 162.979094 2.39 0.0946

Class Level Information Class^{a}Levels^{b}Values^{c}female 2 0 1 prog 3 1 2 3 Number of Observations Read^{d}200 Number of Observations Used^{d}200

a. **Class** - Underneath are the categorical (factor) variables, which
were defined as such in the **class** statement. Had the categorical variables not
been defined in the **class** statement and just entered in the **model** statement, the
respective variables would be treated as continuous variables, which would be
inappropriate.

b.** Levels** - Underneath are the respective number of levels
(categories) of the factor variables defined in the **class** statement.

c.** Values** - Underneath are the respective values of the levels for the
factor variables defined in the **class** statement.

d.** Number of Observations Read** and **Number of Observations Used** -
This is the number of observations read and the number of observation used in
the analysis. The **Number of Observations Used** may be less than the **
Number of Observations Read** if there are missing values for any variables
in the equation. By default, SAS does a listwise deletion of incomplete cases.

Dependent Variable^{e}: write Sum of Source^{f}DF^{g}Squares^{h}Mean Square^{i}F Value^{j}Pr > F^{j}Model 5 4630.36091 926.07218 13.56 <.0001 Error 194 13248.51409 68.29131 Corrected Total 199 17878.87500 R-Square^{k}Coeff Var^{l}Root MSE^{m}write Mean^{n}0.258985 15.65866 8.263856 52.77500 Source^{o}DF^{p}Type III SS^{q}Mean Square^{r}F Value^{s}Pr > F^{s}female 1 1261.853291 1261.853291 18.48 <.0001 prog 2 3274.350821 1637.175410 23.97 <.0001 female*prog 2 325.958189 162.979094 2.39 0.0946

e. **Dependent Variable** - This is the dependent variable in our glm
model.

f. **Source** - Underneath are the sources of variation of the dependent
variable. There
are three parts, Model, Error, and Corrected Total. With glm, you must think in
terms of the variation of the response variable (sums of squares), and
partitioning this variation. The variation in the response variable, denoted by Corrected Total,
can be partitioned into two unique parts. The first partition, Model, is the variance in
the response accounted by our model (**female prog female*prog**). The
second source, Error, is the variation not explained by the Model. These two
sources, the explained (Model), and unexplained (Error), add up to the
Corrected Total, SS_{Corrected Total }= SS_{Model }+ SS_{Error}.

The term "Corrected Total" is called such, as compared to "Total", or more correctly, "Uncorrected Total," because the "Corrected Total" adjusts the sums of squares to incorporate information on the intercept. Specifically, the Corrected Total is the sum of the squared difference between the response variable and the mean of the response variable, whereas the Uncorrected Total is the sum of the squared values of just the response variable.

g. **DF** - These are the degrees of freedom associated with the respective sources of
variance. As with the additive nature of the sums of squares, the degrees of
freedom are also additve, DF_{Corrected Source }= DF_{Model
}+ DF_{Error}. The DF_{Corrected Total }has N-1 degrees of
freedom, where N is the total sample size. See DF, superscript p, for
the calculation of the DF for each individual predictor variable, which add up to DF_{Model}.
Hence, DF_{Error }=DF_{Corrected Total }- DF_{Model}.
The DF_{Model }and DF_{Error} define the parameters of the
F-distribution used to test F Value, superscript j.

h. **Sum of Squares** - These are the sums of squares that correspond to
the three sources of variation.

SS_{Model} - The Model sum of squares is the squared difference of the predicted value and the grand mean
summed over all
observations. Suppose our model did not explain a significant proportion of variance,
then the
predicted value would be near the grand mean, which would result with a small SS_{Model}, and
SS_{Error} would nearly be equal to SS_{Corrected Total}.

SS_{Error} - The Error sum of squares is the
squared difference of the observed value from the predicted value summed over all
observations.

SS_{Corrected Total} - The Corrected Total sum of
squares is the squared difference of the observed value from the
grand mean summed over all observations.

i. **Mean Square** - These are the Mean Squares (MS) that correspond to the
partitions of the total variance. The MS is defined as SS/DF.

j. **F Value** and **Pr > F** - These are the F Value and p-value,
respectively, testing the null hypothesis that the Model does not explain the variance
of our response variable. F Value is
computed as MS_{Model }/ MS_{Error}, and under the null
hypothesis, F Value follows a central F-distribution with numerator DF = DF_{Model
}and denominator DF =DF_{Error}. The probability of observing an F
Value as large as, or larger, than 13.56 under the null hypothesis is < 0.0001.
If we set our alpha level at 0.05, our willingness to accept a Type I error,
we'd reject the null hypothesis and conclude that our model explains a
statistically significant proportion of the variance.

k. **R-Square** - This is the R-Square value for the model. R-Square
defines the proportion of the total variance explained by the Model and
is calculated as R-Square = SS_{Model}/SS_{Corrected Total} = 4630.36/17878.88=0.259.

l. **Coeff Var** - This is the Coefficient of Variation (CV). The coefficient
of variation is defined as the 100 times root MSE divided by the mean of
response variable; CV = 100*8.26/52.775 = 15.659. The CV is a dimensionless
quantity and allows the comparison of the variation of populations.

m. **Root MSE** - This is the root mean square error. It is the square
root of the MS_{Error }and defines the standard deviation of an
observation about the predicted value.

n. **write Mean** - This is the grand mean of the response variable.

o. **Source** - Underneath are the variables in the model. Our model has **
female**, **prog**, and the interaction of **female** and **prog**.
The interaction disallows the effect of, say, **prog,** over the levels of **female** to
be additive. Also, our model follows the
hierarchical principal, i.e., if an interaction term is in the model (**female*prog**),
the lower order terms (**female** and **prog**) must be included. Further, when there
is a significant interaction in the model, the main effects (the
lower order terms) are difficult to
interpret. If the interaction term is not statistically significant, some would advise dropping
the term and rerunning the model with just the main effects, so that the main
effects would have an unambiguous meaning. The traditional anova approach would leave the nonsignificant
interaction in the model and interpret the main effects in the normal manner.
If the interaction term is found statistically
significant, one would leave the model as is and evaluate the simple main effects.

p. **DF** - These are the degrees of freedom for the individual predictor
variables in the model. From the class level information section, the
lower order term DF is given by the number of levels minus one. For example, **female** as two levels,
therefore DF_{female} = 2-1=1. Also, **prog** has three
levels and DF_{prog} = 3-1=2. For the interaction term, DF_{female*prog}
= DF_{prog}* DF_{female} = 1*2 =2. The DF of the predictor
variables, along with the DF_{Error}, define the parameters of the F-distribution used to test the significance of F
Value, superscript s.

q. **Type III SS** - These are the type III sum of squares, which are referred
to as partial sum of squares. For a particular variable, say **female**, SS_{female}
is calculated with respect to the other variables in the model, **prog** and **female*prog**. Also, we showed
earlier that SS_{Corrected Total }= SS_{Model }+ SS_{Error},
and we might expect that SS_{Model }= SS_{female }+ SS_{prog}+
SS_{prog*female}; however, this is generally not the case (this is only
true for a balanced design).

r. **Mean Square** - These are the mean squares for the individual
predictor variables in the model. They are calculated as SS/DF, and along MS_{Error}, they are used to calculate F Value, superscript s.

s. **F Value** and **Pr > F** - These are the F Value and p-value,
respectively, testing the null hypothesis that an individual predictor
in the model does not explain a significant proportion of the variance, given
the other variables are in the model.
F Value is computed as MS_{Source Var
}/ MS_{Error}. Under the null
hypothesis, F Value follows a central F-distribution with numerator DF = DF_{Source
Var}, where Source Var is the predictor variable of interest, and denominator DF =DF_{Error}.
Following the point made in Source, superscript o, we focus only on the
interaction term.

**female*prog** - This is the F Value and p-value testing
the interaction of **female** and **prog** on the response variable, given
the other variables are in the model. The probability of observing an F Value,
as large as, or larger, than 2.39 under the null hypothesis that there is not an
interaction of **female** and **prog**, given the other variables are in
the model, is 0.0946. If we set our alpha level at 0.05, the probability of a
Type I error, we would fail to reject the null hypothesis that **female** and
**prog ** do not interact. Based on this finding, some would advise rerunning the
model without the interaction term, including only the main effects in the model
(and the intercept). This
would in turn permit a valid interpretation of the main effects of **female**
and **prog**.

The content of this web site should not be construed as an endorsement of any particular web site, book, or software product by the University of California.