Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures used to analyze the differences among means. ANOVA was developed by the statistician Ronald Fisher. ANOVA is based on the law of total variance, where the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether two or more population means are equal, and therefore generalizes the t-test beyond two means. In other words, the ANOVA is used to test the difference between two or more means.
Nonparametric statistics is the branch of statistics that is not based solely on parametrized families of probability distributions. Nonparametric statistics is based on either being distribution-free or having a specified distribution but with the distribution's parameters unspecified. Nonparametric statistics includes both descriptive statistics and statistical inference. Nonparametric tests are often used when the assumptions of parametric tests are violated.
Pearson's chi-squared test is a statistical test applied to sets of categorical data to evaluate how likely it is that any observed difference between the sets arose by chance. It is the most widely used of many chi-squared tests – statistical procedures whose results are evaluated by reference to the chi-squared distribution. Its properties were first investigated by Karl Pearson in 1900. In contexts where it is important to improve a distinction between the test statistic and its distribution, names similar to Pearson χ-squared test or statistic are used.
In statistics, Spearman's rank correlation coefficient or Spearman's ρ, named after Charles Spearman and often denoted by the Greek letter (rho) or as , is a nonparametric measure of rank correlation. It assesses how well the relationship between two variables can be described using a monotonic function.
In statistics, the Mann–Whitney U test is a nonparametric test of the null hypothesis that, for randomly selected values X and Y from two populations, the probability of X being greater than Y is equal to the probability of Y being greater than X.
A t-test is any statistical hypothesis test in which the test statistic follows a Student's t-distribution under the null hypothesis. It is most commonly applied when the test statistic would follow a normal distribution if the value of a scaling term in the test statistic were known. When the scaling term is estimated based on the data, the test statistic—under certain conditions—follows a Student's t distribution. The t-test's most common application is to test whether the means of two populations are different.
The Kruskal–Wallis test by ranks, Kruskal–Wallis H test, or one-way ANOVA on ranks is a non-parametric method for testing whether samples originate from the same distribution. It is used for comparing two or more independent samples of equal or different sample sizes. It extends the Mann–Whitney U test, which is used for comparing only two groups. The parametric equivalent of the Kruskal–Wallis test is the one-way analysis of variance (ANOVA).
The Wilcoxon signed-rank test is a non-parametric statistical hypothesis test 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. The one-sample version serves a purpose similar to that of the one-sample Student's t-test. For two matched samples, it is a paired difference test like the paired Student's t-test. The Wilcoxon test can be a good alternative to the t-test when population means are not of interest; for example, when one wishes to test whether a population's median is nonzero, or whether there is a better than 50% chance that a sample from one population is greater than a sample from another population.
The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measures can be used in statistical hypothesis testing, e.g. to test for normality of residuals, to test whether two samples are drawn from identical distributions, or whether outcome frequencies follow a specified distribution. In the analysis of variance, one of the components into which the variance is partitioned may be a lack-of-fit sum of squares.
The Friedman test is a non-parametric statistical test developed by Milton Friedman. Similar to the parametric repeated measures ANOVA, it is used to detect differences in treatments across multiple test attempts. The procedure involves ranking each row together, then considering the values of ranks by columns. Applicable to complete block designs, it is thus a special case of the Durbin test.
In statistics, a rank correlation is any of several statistics that measure an ordinal association—the relationship between rankings of different ordinal variables or different rankings of the same variable, where a "ranking" is the assignment of the ordering labels "first", "second", "third", etc. to different observations of a particular variable. A rank correlation coefficient measures the degree of similarity between two rankings, and can be used to assess the significance of the relation between them. For example, two common nonparametric methods of significance that use rank correlation are the Mann–Whitney U test and the Wilcoxon signed-rank test.
The sign test is a statistical method to test for consistent differences between pairs of observations, such as the weight of subjects before and after treatment. Given pairs of observations for each subject, the sign test determines if one member of the pair tends to be greater than the other member of the pair.
In statistics, Levene's test is an inferential statistic used to assess the equality of variances for a variable calculated for two or more groups. Some common statistical procedures assume that variances of the populations from which different samples are drawn are equal. Levene's test assesses this assumption. It tests the null hypothesis that the population variances are equal. If the resulting p-value of Levene's test is less than some significance level (typically 0.05), the obtained differences in sample variances are unlikely to have occurred based on random sampling from a population with equal variances. Thus, the null hypothesis of equal variances is rejected and it is concluded that there is a difference between the variances in the population.
Kendall's W is a non-parametric statistic for rank correlation. It is a normalization of the statistic of the Friedman test, and can be used for assessing agreement among raters and in particular inter-rater reliability. Kendall's W ranges from 0 to 1.
In statistics, the Kendall rank correlation coefficient, commonly referred to as Kendall's τ coefficient, is a statistic used to measure the ordinal association between two measured quantities. A τ test is a non-parametric hypothesis test for statistical dependence based on the τ coefficient.
In probability theory, Dirichlet processes are a family of stochastic processes whose realizations are probability distributions. In other words, a Dirichlet process is a probability distribution whose range is itself a set of probability distributions. It is often used in Bayesian inference to describe the prior knowledge about the distribution of random variables—how likely it is that the random variables are distributed according to one or another particular distribution.
The logrank test, or log-rank test, is a hypothesis test to compare the survival distributions of two samples. It is a nonparametric test and appropriate to use when the data are right skewed and censored. It is widely used in clinical trials to establish the efficacy of a new treatment in comparison with a control treatment when the measurement is the time to event. The test is sometimes called the Mantel–Cox test. The logrank test can also be viewed as a time-stratified Cochran–Mantel–Haenszel test.
In statistics, the Siegel–Tukey test, named after Sidney Siegel and John Tukey, is a non-parametric test which may be applied to data measured at least on an ordinal scale. It tests for differences in scale between two groups.
Krippendorff's alpha coefficient, named after academic Klaus Krippendorff, is a statistical measure of the agreement achieved when coding a set of units of analysis. Since the 1970s, alpha has been used in content analysis where textual units are categorized by trained readers, in counseling and survey research where experts code open-ended interview data into analyzable terms, in psychological testing where alternative tests of the same phenomena need to be compared, or in observational studies where unstructured happenings are recorded for subsequent analysis.
In statistics, the Jonckheere trend test is a test for an ordered alternative hypothesis within an independent samples (between-participants) design. It is similar to the Kruskal–Wallis test in that the null hypothesis is that several independent samples are from the same population. However, with the Kruskal–Wallis test there is no a priori ordering of the populations from which the samples are drawn. When there is an a priori ordering, the Jonckheere test has more statistical power than the Kruskal–Wallis test. The test was developed by Aimable Robert Jonckheere, who was a psychologist and statistician at University College London.
