Page 225 - Intermediate Statistics for Dummies

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Part III: Comparing Many Means with ANOVA
In Figure 12-1, you can see the degrees of freedom for each sums of squares
listed under the DF column of the ANOVA part of the output. You see SSR
has 2 – 1 = 1 degree of freedom, SSE has 250 – 2 = 248 degrees of freedom
(because n = 250 observations were in the data set and k = 2 and you find
n – k to get degrees of freedom for SSE). The degrees of freedom for SSTO is
250 – 1 = 249.
Bringing regression to the ANOVA table
In ANOVA, you test your model Ho: All k population means are equal versus
Ha: At least two population means are different by using a F-test. You build
your F-test statistic by relating the sums of squares for treatment to the sum
of squares for error. To do this, you divide SSE and SST by their degrees of
freedom (n – k and k – 1, respectively, where n is the sample size and k is the
number of treatments) to get the mean sums of squares for error (MSE) and
mean sums of squares for treatment (MST). In general, you want MST to be
large compared to MSE, which would indicate that the model fits well. The
results of all these statistical gymnastics are summarized by Minitab in a
table called (cleverly) the ANOVA table.
The ANOVA table shown in the bottom part of Figure 12-1 for the Internet-use
data represents the ANOVA table you get from using the regression line as
your model. Under the Source column, you may be used to seeing treatment,
error, and total. For regression, the treatment is the regression line, so you
see regression instead of treatment. The error term in ANOVA is labeled resid-
ual error, because in regression, you measure error in terms of residuals.
Finally you see total, which is the same the world around.
The SS column represents the sums of squares for the regression model. The
three sums of squares listed in the SS column are SSR (for regression), SSE
(for residuals), and SST (total). These sums of squares are calculated using
the formulas from the previous section; the degrees of freedom, DF in the
table, are found by using the formulas from the previous section also.
The MS column takes the value of SS “whatever”(you fill in the blank) and
divides it by the respective degrees of freedom, just like ANOVA. For example
in Figure 12-1, SSE is 12,968.5, and the degrees of freedom is 248. Take the first
value divided by the second one to get 52.29 or 52.3, which is listed in the
ANOVA table for MSE.
MST , 9 085 .6
The value of the F-statistic, using the ANOVA method, is F = = =
MSE 52 .3
173.7 in the Internet example, which you can see in column five of the ANOVA
part of Figure 12-1 (subject to rounding). The F-statistics’s p-value is calcu-
lated based on an F-distribution with 2 – 1 = 1 and 250 – 2 = 248 degrees of