Comparing Candidate Hospital Report Cards Page: 4 of 9
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hospital to its peer group. There are many possible
ways to define peer groups. We present results for
the following way. We searched for which of the nine
peer group categories (number of beds, geographic
region, etc.) exhibited the largest group-to-group
variation (compared to within-group variation) in p.
The two peer group categories that have significantly
largest between-group to within-group variation are
the percent Medicare and geographical area, which
each had five categories. To reduce the total number
of categories, we reduced the percent Medicare to
low (A or B) and high (C, D, or E), and we retained
the 5 geographical regions so our single peer group
variable had 10 groups with the following number of
hospitals in groups 1 to 10: 81, 238, 45, 211, 54, 104,
29, 28, 143, 149 (low Medicare: northeast, south-
east, central, northwest, southwest, and then high
Medicare with same regions). We report peer group
results here only for Score 2. Score 1 was difficult to
apply because the 3-by-3 covariance matrix was sin-
gular within peer groups for most of the 30 patient
subgroups.
For each scoring method we must somehow com-
bine scores over subgroups. Not all hospitals had
n > 0 for all patient subgroups. The number of
n > 0 subgroups ranged from 18 to 30 with an aver-
age of about 28. It is reasonable then to simply av-
erage the scores over the subgroups with n > 0, but
we have choices for how to average. We use three
weighted averaging methods: (1) wl: weights are
the average p scores for that subgroup (subgroups
having high p are weighted more heavily), (2) w2:
weights are the average charge scores for that sub-
group, (3) w3: weights are n for that subgroup, and
(4) w4: weights = 1 (unweighted). Weights 1 and 2
vary with subgroup but are the same for each hos-
pital. Weight 3 depends on both the subgroup and
the hospital.
3. Multivariate Outlier Detection
Our main goal is to identify unusually good or bad
hospitals and our second goal is to compare candi-
date report cards for hospitals. Our approaches can
all be viewed as multivariate outlier detection meth-
ods [2]. There are too many notions of what it means
to be an outlier to review here. Also, "the complex-
ity of the multivariate case suggests it would be fu-
tile to search for a truly omnibus outlier detection
procedure" [3]. Practical suggestions for detecting
multivariate outliers therefore include
(a) try several methods and compare them [3],
(b) reduce dimensionality somehow, such as usingprinciple components [2], and
(c) define a region where the outliers of interest
should lie [2].
In our analyses we apply several methods, all of
which reduce dimensionality because a scalar-valued
score results. Also, all of our methods look for either
"large" or "small" values of some of the features to
take advantage of an inherent preferred direction for
all three variables.
We cannot review the subject of multivariate out-
lier detection here, but we adopt the above practical
suggestions and consider three issues:
(1) outlier masking (example: the presence of one
outlier can make it difficult to detect a second out-
lier),
(2) the impact of having an a priori direction to
search for outliers, and
(3) how to compare our outlier detection methods.
Our comparison methods to compare our ranking
methods focus on the top nto, and bottom bottom
hospitals. We define two distance measures to com-
pare two ranking methods. Both distances lie be-
tween 0 and 1 with 0 being perfect agreement be-
tween two methods.
We present results here for ntop = 10 and 20
and the same for moottom. Distance 1 is 1 minus
the percent of cases that are in the ntoP by both
methods. The second distance finds the providers in
the ntoP ranks by method 1, and then records their
ranks by method 2. The Spearman correlation, pl,
is computed between those two sets of ranks, and
then the viewpoint is reversed; we compute p2 and
then symmetrize the comparison method by defining
p = 12 . Distance 2 is .5 x (1 - p) which ranges
from 0 to 1, where 0 (1) corresponds to an average
Spearman correlation (of the two viewpoints) of 1
(-1).
4. Data Analysis Results
In Fig. ld we plot the overall hospital scores versus
hospital number for Score 2 using n as the weighting
over subgroups method. We also plot in Figs. la-ic
the overall hospital scores using only (respectively)
p, charge, and los. This allows us to see that, for ex-
ample, Hospital 805 is an outlier according to Score
2 because it has large p and large los scores (though
its charge score is not large).
To show the effect of our four weighting methods,
in Fig. 2 we show Score 1 (MD) with each of the
four weighting methods. In Fig. 3 we show results
from several scoring methods, each combined across
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Burr, Tom L.; Rivenburgh, Reid D.; Scovel, James C. & White, James M. Comparing Candidate Hospital Report Cards, article, December 31, 1997; New Mexico. (https://digital.library.unt.edu/ark:/67531/metadc686576/m1/4/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.