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SAS/INSIGHT Software |

In addition to the table summarizing the statistics for simultaneous
multiple comparison of means, SAS/INSIGHT software provides a
graphical technique to help visualize which groups are
significantly different from a selected group. Each test is
accompanied by a *comparison circles* plot that graphically
illustrates the comparisons (Sall, 1992).

There is a circle in the plot centered at each category's sample mean.
The radius of the *i*th circle is
, where *q* is a
quantile used to scale the circles according to the test being used.
For details on
how each quantile is computed, see (Hsu, 1996).

If the *j*th group is selected
(by clicking on its circle), then its circle will be
highlighted. This circle will be red on color monitors. You may
determine whether another group is significantly different than
the selected group based on how much their corresponding circles
overlap. If their circles are nested or nearly overlap so that the
external angle of intersection is greater than 90 degrees, then you
cannot claim that the means of the two groups are different. If,
however, the two circles are disjoint or just barely overlap so that
their external angle of intersection is less than 90 degrees, then you
may conclude that the means of the two groups are significantly
different at the given confidence level.

Circles corresponding to categories that are significantly different from the selected group are drawn in cyan on color monitors. Circles corresponding to categories that are not different are drawn with a dashed line, and are red on color monitors.

The geometry behind comparison circles is based on the Pythagorean
Theorem: since the radius of the *i*th circle is
, and since the circle is centered at
, then if the two circles meet at right
angles, the distance between centers
is the hypotenuse of the right triangle formed by the circles' radii.
Therefore, when the circles meet at right angles,
.
Statistically, this geometry corresponds to the critical case in which
zero happens to fall on the boundary of the confidence interval about
. If
, then the external intersection of the circles is less
than 90 degrees, and so zero is not contained in the confidence interval
about . Thus the circles are significantly
different.

The statistics for Hsu's Test for Best/Worst
are computed differently from the other tests. First,
the comparison circles are not selectable. The
Test for Best automatically selects the category with the largest sample mean;
the Test for Worst selects the category with the smallest sample mean.
Second, the quantile used to scale the comparison circles
is the maximum of the quantiles computed by running Dunnett's one-sided
test *k*-1 times, with each "non-best" (or "non-worst")
group serving in turn as the "control" for Dunnett's test.

Because Hsu's Test for Best does not provide symmetric intervals about , the comparison circle technique must be modified. While the statistical table reports exactly which groups can be inferred to not be the best, the comparison circles are more conservative because the quantile used to scale the circle radii is the maximum of all quantiles encountered during Hsu's test. The same is true for Hsu's Test for Worst.

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