Graphics Reference
In-Depth Information
(orhomogeneityforstratified data) intwo-waytables. heteststatistic isjustthesum
of the squared Pearson residuals
X
r
ij
,
=
(
.
)
i, j
which is known to have a limiting χ
distribution with
degrees of free-
dom under the null hypothesis. An important reason for using the unconditional
limiting distribution forthe X
statistic from(
.
)wasthe closed form result forthe
distribution. With the recent improvements in computing power, conditional infer-
ence (or permutation tests) - carried out either by simulation or by computation of
the(asymptotic) permutation distribution -havebeenreceiving increasing attention
(Ernst,
; Pesarin,
; Strasser and Weber,
).
he use of a permutation test is a particularly intuitive way of testing the inde-
pendence hypothesis from (
.
), due to the permutation invariance (given row and
column sums) of this problem. Consequently, all results in this paper are based on
conditional inference performed by simulating the permutation distribution of test
statistics of type λ
(
I
−
)(
J
−
)
.
Since the HCL space is three-dimensional and we have only used two 'degrees
of freedom' so far to code information (hue for the sign and a linear combination
of chroma and luminance for residual size), we can add a third piece of informa-
tion to the plot. For example, we can visualize the significance of some specified test
statistic (e.g.,the χ
teststatistic) using less colorful (“uninteresting”) colors fornon-
significant results. hesecan again bederived using the same proceduredescribed in
Sect.
.
.
but using a smaller amount of color, i.e., a smaller maximal chroma (e.g.,
instead of
).
he heuristic for choosing the cut-off points in the Friendly shading may lead to
wrong conclusions: especially in large tables, the test of independence may not be
significant, even though some of the residuals are “large.” On the other hand, the test
might be significant even though the residuals are “small.” In fact, the cut-off points
arereally data-dependent. Consider thecaseofthearthritis data (KochandEdwards,
), resulting from a double-blind clinical trial investigating a new treatment for
rheumatoidarthritis, stratified bygender(see Table
.
forthe femalepatients). Fig-
ure
.
visualizes the results for the female patients, again using a mosaic display.
Clearly, the hypothesis of independence is rejected by the χ
test, even at a
% level
(p
([
r
ij
])
the tiles re-
main uncolored.One solution tothis issueistouseadifferent teststatistic, forexam-
=
.
), but since all residuals are in the interval
[−
.
,
.
]
Table
.
.
he arthritis data (female patients)
Improvement
Treatment
None
Some
Marked Σ
Placebo
19
7
6
32
Treatment
6
5
16
27
Σ25
12
22
59
