Kruskal—Wallis tests

If the populations to be compared across treatments are not normally distributed, you can use the nonparametric Kruskal-Wallis test of the distributions by running PROC NPAR1WAY as follows ... 

Analyses were conducted with SPSS version 13.0, and data satisfied the requirements of the statistical tests used non-normally distributed data were analysed with the Kruskal-Wallis test. We compared how highly the males were rated in terms of their attractiveness by the women in each of the three experimental conditions. 

These methods are essential when there is any significant degree of mortality in a bioassay. They seek to adjust for the differences in periods of risk individual animals undergo. Life table techniques can be used for those data where there are observable or palpable tumors. Specifically, one should use Kaplan-Meier product limit estimates from censored data graphically, Cox-Tarone binary regression (log-rank test), and Gehan-Breslow modification of Kruskal-Wallis tests (Thomas et al., 1977 Portier and Bailer, 1989) on censored data. 

Gehan-Breslow Modification of Kruskal-Wallis Test is a nonparametric test on censored observations. It assigns more weight to early incidences compared to Cox-Tarone test. 

A table of data is set up with each of the two variables being ranked separately. Tied ranks are assigned as demonstrated earlier under the Kruskall Wallis test. From... 

For statistical analysis, fetal abnormality values belong to two types those where at least 50% of litters have one or more fetuses affected, and those where most litters have no affected fetuses. For the first type, the incidences (percentage of affected fetuses within that litter) are analyzed by the Kruskal-Wallis test (13) for the second type, the number of litters with affected fetuses is compared with the number with no affected fetuses by Fisher s Exact test (14). 

Having calculated the level of significance can be obtained from appropriate tables. The Wilcoxon signed rank test is the non-parametric equivalent of the paired t-test. The Kruskal-Wallis test is another rank sums test that is used to test the null hypothesis that k independent samples come from identical populations against the alternative that the means of the populations are unequal. It provides a non-parametric alternative to the one-way analysis of variance. 

Fig. 6 Effect of methylphenidate on Acquisition of the PAR in juvenile rat pups. Juvenile rat pups (day 15-16) were tested for acquisition of a multi-trial PAR. Littermates were equally divided into vehide or drug treatment groups. Methylphenidate salt was given ip at a dose of 3 mg/kg (base), 30 mins prior to training. Animals were returned to their home cage with their littermates for the intertrial time period. indicates statistically significant differences between drug-treatment group and vehide-treatment group at the specific trial. Non-parametric statistical analysis (Kruskal-Wallis test) was conducted on median latencies (sec). Mean + SEM entry latendes (sec) are presented (n = 12-18/group).

The data are ordinal and, with only 20 observations for each drug, a histogram would provide little guidance as to whether the data are normally distributed. Under these circumstances, it would be risky to assume a normal distribution. The non-parametric Kruskal-Wallis test is preferable to a one-way analysis of variance. 

Table 17.8 Generic output from a Kruskal-Wallis test of pain scores with different analgesics...

With the one-way ANOVA, most statistical packages implement a series of followup tests to determine exactly where any differences lie. Similar procedures exist to allow follow-up after a significant Kruskal-Wallis test, but unfortunately they are not widely implemented in statistical packages. There would be no point in doing so in the present case, but if another data set proves significant and you want to perform follow-up tests, you will either have to resort to a very powerful (and probably not very friendly) statistical package, or do the calculation manually. The latter is tedious, but recipes are available. (A clear account is available in Zar J.H., 1999, Biostatistical Analysis, Prentice Hall, NJ pp. 223-226.)... 

Non-parametric Mann-Whitney test Wilcoxon paired samples test Kruskal-Wallis test Spearman correlation... 

Parametric data were presented as mean SD. To determine differences in glutamate concentrations, a repeated-measures analysis of variance was performed. The cutaneous sensation, hind-limb motor function, and morphological changes of the spinal cord were analyzed with a non-parametric method (Kruskal-Wallis test) followed by the Mann-Whitney U-test. 

Time until occlusion and patency are expressed as median and the interquartile range/2 (IQR/2). Significant differences (p < 0.05) are calculated by the non-parametric Kruskal-Wallis test. 

To check the assumptions of the model, Bartlett s or Levene s tests can be used to assess the assumption of equality of variance, and the normal probability plot of the residuals (etj = Xij - Xj) to assess the assumption of normality. If either equality or normality are inappropriate, we can transform the data, or we can use the nonparametric Kruskal-Wallis test to compare the k groups. In any case, the ANOVA procedure is insensitive to moderate departures from the assumptions (Massart et al. 1990). 

Non-parametric comparisons of location for three or more samples include the Kruskal-Wallis //-test. Here, the two data sets can be unequal in size, but again the underlying distributions are assumed to be similar. 

The grading of reflection reports from the three student populations is analyzed and correlated with a number of other statistics and performance indicators. All data were collected in Microsoft Excel and unported into Statgraphics Centurion XV for further analysis [7], We performed ANOVA tests, multiple range test or Kruskal-Wallis tests, and f-tests on the gathered information. 

Table 1 Results of ANOVA and multiple range test or Kruskal-Wallis test of significant differences (SD) between three groups at the 95.0% confidence level. See text for further details...

There are times when the required assumptions for ANOVA, a parametric test, are not met. One example would be if the underlying distributions are non-normal. In these cases, nonparametric tests are very useful and informative. For example, we saw in Section 11.3 that a nonparametric analog to the two-sample ttest, Wilcoxon s rank sum test, makes use of the ranks of observations rather than the scores themselves. When a one-factor ANOVA is not appropriate in a particular case a corresponding nonparametric approach called the Kruskal-Wallis test can be used. This test is a hypothesis test of the location of (more than) two distributions. 

In the Kruskal-Wallis test the original scores are first ranked and an ANOVA analysis is then carried out on the ranks. As with Wilcoxon s rank sum test, ranking of the observations must deal with ties. The sums of squares are based on... 

For the sake of this example, we use the data from the parametric ANOVA example to illustrate the Kruskal-Wallis test. If it seems at all strange to use the same data for both examples, a parametric analysis and a nonparametric analysis, it is worth noting that a nonparametric analysis is always appropriate for a given dataset meeting the requirements at the start of the chapter. Parametric analyses are not always appropriate for all datasets. 

Table 11.8 General one-way ANOVA table for ranks (Kruskal-Wallis test) ...

In what situations would the Kruskal-Wallis test be appropriate ... 

Figure 44.2 is based on the same data as (Andersen et ai, 2001), and illustrates that individual mean values vary (Andersen et ai, 2003). The difference between individual mean urinary iodine concentrations is highly significant (Kruskal—Wallis test p < 0.001 for all variables), compared to that of TSH in serum (Andersen et al., 2001). This is consistent with other findings of marked differences in urinary iodine excretion between individuals (Rasmussen et ai, 1999 Busnardo et ai, 2006). 

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