Statistical tests

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Nested hypothesis

Let be the set of all possibilities that satisfy hypothesis , and let be the set of all possibilities that satisfy hypothesis . Then is a nested hypothesis within iff , where denotes the proper subset.

Bonferroni correction

The Bonferroni correction is a multiple-comparison correction used when several dependent or independent statistical tests are being performed simultaneously (since while a given alpha value may be appropriate for each individual comparison, it is not for the set of all comparisons). In order to avoid a lot of spurious positives, the alpha value needs to be lowered to account for the number of comparisons being performed.The simplest and most conservative approach is the Bonferroni correction, which sets the alpha value for the entire set of comparisons equal to by taking the alpha value for each comparison equal to . Explicitly, given tests for hypotheses () under the assumption that all hypotheses are false, and if the individual test critical values are , then the experiment-wide critical value is . In equation form, iffor , thenwhich follows from the Bonferroni inequalities...

Bessel's statistical formula

Let and be the observed mean and variance of a sample of drawn from a normal universe with unknown mean and let and be the observed mean and variance of a sample of drawn from a normal universe with unknown mean . Assume the two universes have a common variance , and define(1)(2)(3)Then(4)is distributed as Student's t-distribution with .


Let . A value such that is considered "significant" (i.e., is not simply due to chance) is known as an alpha value. The probability that a variate would assume a value greater than or equal to the observed value strictly by chance, , is known as a P-value.Depending on the type of data and conventional practices of a given field of study, a variety of different alpha values may be used. One commonly used terminology takes as "not significant," , as "significant" (sometimes denoted *), and as "highly significant" (sometimes denoted **). Some authors use the term "almost significant" to refer to , although this practice is not recommended.


"Analysis of Variance." A statistical test for heterogeneity of means by analysis of group variances. ANOVA is implemented as ANOVA[data] in the Wolfram Language package ANOVA` .To apply the test, assume random sampling of a variate with equal variances, independent errors, and a normal distribution. Let be the number of replicates (sets of identical observations) within each of factor levels (treatment groups), and be the th observation within factor level . Also assume that the ANOVA is "balanced" by restricting to be the same for each factor level.Now define the sum of square terms(1)(2)(3)(4)(5)which are the total, treatment, and error sums of squares. Here, is the mean of observations within factor level , and is the "group" mean (i.e., mean of means). Compute the entries in the following table, obtaining the P-value corresponding to the calculated F-ratio of the mean squared values(6)category freedomSSmean..

Residual vs. predictor plot

A plot of versus the estimator . Random scatter indicates the model is probably good. A pattern indicates a problem with the model. If the spread in increases as increases, the errors are called heteroscedastic.

Likelihood ratio

A quantity used to test nested hypotheses. Let be a nested hypothesis with degrees of freedom within (which has degrees of freedom), then calculate the maximum likelihood of a given outcome, first given , then given . ThenComparison of to the critical value of the chi-squared distribution with degrees of freedom then gives the significance of the increase in likelihood.The term likelihood ratio is also used (especially in medicine) to test nonnested complementary hypotheses as follows,

Population comparison

Let and be the number of successes in variates taken from two populations. Define(1)(2)The estimator of the difference is then . Doing a so-called -transform,(3)where(4)The standard error is(5)(6)(7)

Fisher's exact test

Fisher's exact test is a statistical test used to determine if there are nonrandom associations between two categorical variables.Let there exist two such variables and , with and observed states, respectively. Now form an matrix in which the entries represent the number of observations in which and . Calculate the row and column sums and , respectively, and the total sum(1)of the matrix. Then calculate the conditional probability of getting the actual matrix given the particular row and column sums, given by(2)which is a multivariate generalization of the hypergeometric probability function. Now find all possible matrices of nonnegative integers consistent with the row and column sums and . For each one, calculate the associated conditional probability using (2), where the sum of these probabilities must be 1.To compute the P-value of the test, the tables must then be ordered by some criterion that measures dependence, and those tables..

Statistical test

A test used to determine the statistical significanceof an observation. Two main types of error can occur: 1. A type I error occurs when a false negative result is obtained in terms of the null hypothesis by obtaining a false positive measurement. 2. A type II error occurs when a false positive result is obtained in terms of the null hypothesis by obtaining a false negative measurement. The probability that a statistical test will be positive for a true statistic is sometimes called the test's sensitivity, and the probability that a test will be negative for a negative statistic is sometimes called the specificity. The following table summarizes the names given to the various combinations of the actual state of affairs and observed test results.resultnametrue positive resultsensitivityfalse negative result1-sensitivitytrue negative resultspecificityfalse positive result1-specificityMultiple-comparison corrections to statistical..

Hypothesis testing

Hypothesis testing is the use of statistics to determine the probability that a given hypothesis is true. The usual process of hypothesis testing consists of four steps.1. Formulate the null hypothesis (commonly, that the observations are the result of pure chance) and the alternative hypothesis (commonly, that the observations show a real effect combined with a component of chance variation). 2. Identify a test statistic that can be used toassess the truth of the null hypothesis. 3. Compute the P-value, which is the probability that a test statistic at least as significant as the one observed would be obtained assuming that the null hypothesis were true. The smaller the -value, the stronger the evidence against the null hypothesis. 4. Compare the -value to an acceptable significance value (sometimes called an alpha value). If , that the observed effect is statistically significant, the null hypothesis is ruled out, and the alternative hypothesis..

Wilcoxon signed rank test

A nonparametric alternative to the paired t-test which is similar to the Fisher sign test. This test assumes that there is information in the magnitudes of the differences between paired observations, as well as the signs. Take the paired observations, calculate the differences, and rank them from smallest to largest by absolute value. Add all the ranks associated with positive differences, giving the statistic. Finally, the P-value associated with this statistic is found from an appropriate table. The Wilcoxon test is an R-estimate.

Fisher sign test

A robust nonparametric test which is an alternative to the paired t-test. This test makes the basic assumption that there is information only in the signs of the differences between paired observations, not in their sizes. Take the paired observations, calculate the differences, and count the number of s and s , whereis the sample size. Calculate the binomial coefficientThen gives the probability of getting exactly this many s and s if positive and negative values are equally likely. Finally, to obtain the P-value for the test, sum all the coefficients that are and divide by .

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