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The Wilcoxon signed-rank test is a non-parametric rank test for statistical hypothesis testing used either to test the location of a population based on a sample of data, or to compare the locations of two populations using two matched samples. [1] The one-sample version serves a purpose similar to that of the one-sample Student's t-test. [2]
Suppose that we take a sample of size n from each of k populations with the same normal distribution N(μ, σ 2) and suppose that ¯ is the smallest of these sample means and ¯ is the largest of these sample means, and suppose S 2 is the pooled sample variance from these samples. Then the following random variable has a Studentized range ...
The difference between the two sample means, each denoted by X i, which appears in the numerator for all the two-sample testing approaches discussed above, is ¯ ¯ = The sample standard deviations for the two samples are approximately 0.05 and 0.11, respectively. For such small samples, a test of equality between the two population variances ...
For example, if participants completed a specific measure at three time points, C = 3, and df WS = 2. The degrees of freedom for the interaction term of between-subjects by within-subjects term(s), df BSXWS = (R – 1)(C – 1), where again R refers to the number of levels of the between-subject groups, and C is the number of within-subject tests.
The demonstration of the t and chi-squared distributions for one-sample problems above is the simplest example where degrees-of-freedom arise. However, similar geometry and vector decompositions underlie much of the theory of linear models , including linear regression and analysis of variance .
The Dice-Sørensen coefficient (see below for other names) is a statistic used to gauge the similarity of two samples. It was independently developed by the botanists Lee Raymond Dice [ 1 ] and Thorvald Sørensen, [ 2 ] who published in 1945 and 1948 respectively.
The one-sample test statistic, , for Kuiper's test is defined as follows. Let F be the continuous cumulative distribution function which is to be the null hypothesis . Denote by F n the empirical distribution function for n independent and identically distributed (i.i.d.) observations X i , which is defined as
The logrank test statistic compares estimates of the hazard functions of the two groups at each observed event time. It is constructed by computing the observed and expected number of events in one of the groups at each observed event time and then adding these to obtain an overall summary across all-time points where there is an event.