The Wilcoxon signed rank test

Linear regression equations are used to fit a straight line to a set of 12 calibration points. The signs of the resulting residuals in order of increasing x value 

Here M = N = 6, and the number of runs is three. Table A. 10 shows that, at the T = 0.05 level, the number of runs must be 4 if the null hypothesis is to be rejected. So in this instance we can reject the null hypothesis, and conclude that the sequence of + and - signs is not a random one. The attempt to fit a straight line to the experimental points is therefore unsatisfactory, and a curvilinear regression plot is indicated instead. 

We may be concerned with imusually large numbers of short runs, as well as unusually small numbers of long runs. If six plus and six minus signs occurred in the order + - + - + - + - + - + -we would strongly suspect a non-random sequence. Table A.10 shows that, with N = M= 6, a total of 11 or 12 runs indicates that the null hypothesis of random order should be rejected, and some periodicity in the data suspected. 

Section 6.3 described the use of the sign test. Its value lies in the minimal assumptions it makes about the experimental data. The population from which the sample is taken is not assumed to be normal, or even to be s)mimetrical. On the other hand a disadvantage of the sign test is that it uses so little of the information provided. The only material point is whether an individual measurement is greater than or less than the median - the size of this deviation is not used at all. 

In many instances we will have every reason to believe that our measurements will be symmetrically distributed but will not wish to make the assumption that they are normally distributed. This assumption of symmetrical data, and the consequence 

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Because age is not normally distributed here, the Wilcoxon signed rank test is used to calculate the p-value and is placed into a data set called pvalue. (Inferential statistics are discussed further in Chapter 7.)... 

The p-value for the sign test or Wilcoxon signed rank test can be found in the pValue variable in the pvalue data set. If the variable is from a symmetric distribution, you can get the p-value from the Wilcoxon signed rank test, where the Test variable in the pvalue data set is Signed Rank. If the variable is from a skewed distribution, you can get the p-value from the sign test, where the Test variable in the pvalue data set is Sign. ... 

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. 

The Wilcoxon signed rank test (two-tailed) is used to compare baseline recordings and recordings after substance administration. 

Test for significance among the different treatments, using Friedman s test for related samples. The days are blocks for this test. If significant overall, examine differences between pairs of treatments by the Wilcoxon signed ranks test. Remember, the main question is whether predator odor reduces feeding. 

The Wilcoxon rank-sum test is a nonparametric test for assessing whether two samples of measurements come from the same distribution. That is, as an alternative to the two-sample f-test, this test can be used to discover differentially expressed candidates under two conditions. For example, again consider the measurements of the probe set used for the two-sample t-test. The gene expression values are 12.79, 12.53, and 12.46 for the naive condition and 11.12, 10.77, and 11.38 for the 48-h activated condition. Measurement 12.79 has rank 6, measurement 12.53 has 5, and measurement 12.46 has rank 4. The rank sum of the naive condition is 6 -I- 5 -I-4=15. Then after the sum is subtracted by ni(ni-I-l)/2 = 3 x 4/2 = 6, the Wilcoxon rank-sum test statistic becomes 9. Considering all of the combinations of the three measurements, we can compute the probability that the rank sum happens more extremely than 9. The probability becomes its p-value. This is the most extreme among the 20 combinations thus the p-value is 2 x Pr( W > 9) = 2 x = 0.1 for the two-sided test. It is hard to say that the probe set is differentially expressed since the p-value 0.1 > 0.05. This test is also called the Mann - Whitney- Wilcoxon test because this test was proposed initially by Wilcoxon for equal sample sizes and extended to arbitrary sample sizes by Mann and Whitney. As a nonparametric alternative to the paired t-test for the two related samples, the Wilcoxon signed-rank test can be used. The statistic is computed by ordering absolute values of differences of paired samples. For example, consider a peptide in the platelet study data. Their differences for each... 

While the results shown in Table 6.1 assume that the data resemble a normal probability distribution, some may argue the credibility of this assumption. Hence, a nonparametric hypothesis testing method (the Wilcoxon signed-rank test) was employed to confirm the significance of the results, assuming the distribution of data is not necessarily normal. The results for the nonparametric test as shown in Table 6.2 confirm that the results are statistically significant to the 0.01 level. 

Table 3 Posttest-pretest mean rank comparisons of the treatment group on the scientific attitude questionnaire using the Wilcoxon signed-ranks test (AT = 20)...

The Wilcoxon signed-rank tests were next conducted to compare the differences between the pretest and posttest rank mean scores for each of the three dimensions for the treatment and comparison groups as well as for the overall attitudes instrument (see Tables 6 and 7). 

Wilcoxon Signed Test n Also known as the Wilcoxon Signed Rank Test, the Wilcoxon Matched-Pairs Signed-Rank Test, or the Wilcoxon Rank Sum Test, and is a nonparametric test used to test the null hypothesis that the median of the difference between the values of pairs of elements from related samples or repeated measurements on the same sample, is zero. It is assumed that the observations are independent. The differences, Zi, between the pairs of random variables, (x,-, y,) are calculated as z,- = y,—x,- and the zero differences are disguarded. The remaining, M, differences are ranked... 

Table 2. Summary of the results of the basic versions in 10, 30, and 50 dimensions. The last column of each group indicates which version is better according to the Wilcoxon signed-rank test (p <0.01) for each problem dimension. 

To test the hypotheses, we used the McNemar Test for HI and the Wilcoxon signed-rank test (a non-parametric test used for dependent samples) for H2-H4 on a significance level of 0.95 (a = 0.05). 

In order to analyze differences in personal values across groups (first comparison), the Kruskal-Wallis test is used. To measure the changes in moral values from the beginning to the end of the course (second comparison), the Wilcoxon signed-ranked test is used. 

The tests performed were the same used for the analysis of individual motivational values. The Kruskal-Wallis test and the Dunn s post-test are used for the first comparison, and the Wilcoxon signed-ranked test is used for the second comparison. 

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