This article considers the problem of comparing two independent groups in terms of some measure of location. It is well known that with Student's two-independent-sample t test, the actual level of significance can be well above or below the nominal level, confidence intervals can have inaccurate probability coverage, and power can be low relative to other methods. A solution to deal with heterogeneity is Welch's (1938) test. Welch's test deals with heteroscedasticity but can have poor power under arbitrarily small departures from normality. Yuen (1974) generalized Welch's test to trimmed means; her method provides improved control over the probability of a Type I error, but problems remain. Transformations for skewness improve matters, but the probability of a Type I error remains unsatisfactory in some situations. We find that a transformation for skewness combined with a bootstrap method improves Type I error control and probability coverage even if sample sizes are small.
Recently some have
called for randomized controlled trials
comparing RFA to hepatic resection, particularly
for patients with only a few small metastases.
The objectives were to compare local
recurrence and survival following RFA and
hepatic resection for colorectal liver
metastases. This was a retrospective review of
open RFA and hepatic resection for colorectal
liver metastases between January 1998 and May
2007. All patients who had RFA were considered
to have unresectable disease. 58 patients had
hepatic resection and 43 had RFA. A 5-year
survival after resection was 43% compared to
23% after RFA. For patients with solitary
lesions, a 5-year survival was 48% after
resection and 15% after RFA. Sixty percent
of patients suffered local recurrences after RFA
compared to 7% after hepatic resection. RFA
is inferior to resection. The results observed
in this study support the consensus that RFA
cannot be considered an equivalent procedure to
hepatic resection.
Researchers can adopt different measures of central tendency and test statistics to examine the effect of a treatment variable across groups (e.g., means, trimmed means, M-estimators, & medians. Recently developed statistics are compared with respect to their ability to control Type I errors when data were nonnormal, heterogeneous, and the design was unbalanced: (1) a preliminary test for symmetry which determines whether data should be trimmed symmetrically or asymmetrically, (2) two different transformations to eliminate skewness, (3) the accuracy of assessing statistical significance with a bootstrap methodology was examined, and (4) statistics that use a robust measure of the typical score that empirically determined whether data should be trimmed, and, if so, in which direction, and by what amount were examined. The 56 procedures considered were remarkably robust to extreme forms of heterogeneity and nonnormality. However, we recommend a number of Welch-James heteroscedastic statistics which are preceded by the Babu, Padmanaban, and Puri (1999) test for symmetry that either symmetrically trimmed 10% of the data per group, or asymmetrically trimmed 20% of the data per group, after which either Johnson's (1978) or Hall's (1992 transformation was applied to the statistic and where significance was assessed through bootstrapping. Close competitors to the best methods were found that did not involve a transformation.
We examined nine adaptive methods of trimming, that is, methods that empirically determine when data should be trimmed and the amount to be trimmed from the tails of the empirical distribution. Over the 240 empirical values collected for each method investigated, in which we varied the total percentage of data trimmed, sample size, degree of variance heterogeneity, pairing of variances and group sizes, and population shape, one method resulted in exceptionally good control of Type I errors. However, under less extreme cases of non-normality and variance heterogeneity a number of methods exhibited reasonably good Type I error control. With regard to the power to detect non-null treatment effects, we found that the choice among the methods depended on the degree of non-normality and variance heterogeneity. Recommendations are offered.
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