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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 Classa Levelsb Valuesc female 2 0 1 prog 3 1 2 3 Number of Observations Readd 200 Number of Observations Usedd 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 Variablee: write
Sum of
Sourcef DFg Squaresh Mean Squarei F Valuej Pr > Fj
Model 5 4630.36091 926.07218 13.56 <.0001
Error 194 13248.51409 68.29131
Corrected Total 199 17878.87500
R-Squarek Coeff Varl Root MSEm write Meann
0.258985 15.65866 8.263856 52.77500
Sourceo DFp Type III SSq Mean Squarer F Values Pr > Fs
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, SSCorrected Total = SSModel + SSError.
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, DFCorrected Source = DFModel + DFError. The DFCorrected 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 DFModel. Hence, DFError =DFCorrected Total - DFModel. The DFModel and DFError 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.
SSModel - 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 SSModel, and
SSError would nearly be equal to SSCorrected Total.
SSError - The Error sum of squares is the
squared difference of the observed value from the predicted value summed over all
observations.
SSCorrected 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 MSModel / MSError, and under the null hypothesis, F Value follows a central F-distribution with numerator DF = DFModel and denominator DF =DFError. 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 = SSModel/SSCorrected 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 MSError 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 DFfemale = 2-1=1. Also, prog has three
levels and DFprog = 3-1=2. For the interaction term, DFfemale*prog
= DFprog* DFfemale = 1*2 =2. The DF of the predictor
variables, along with the DFError, 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, SSfemale
is calculated with respect to the other variables in the model, prog and female*prog. Also, we showed
earlier that SSCorrected Total = SSModel + SSError,
and we might expect that SSModel = SSfemale + SSprog+
SSprog*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 MSError, 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 MSSource Var
/ MSError. Under the null
hypothesis, F Value follows a central F-distribution with numerator DF = DFSource
Var, where Source Var is the predictor variable of interest, and denominator DF =DFError.
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 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.
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