Parametric tests

# HypothesisTests.PowerDivergenceTest — Type

Perform a Power Divergence test.

If y is not given and x is a matrix with one row or column, or x is a vector, then a goodness-of-fit test is performed (x is treated as a one-dimensional contingency table). In this case, the hypothesis tested is whether the population probabilities equal those in theta0, or are all equal if theta0 is not given.

If x is a matrix with at least two rows and columns, it is taken as a two-dimensional contingency table. Otherwise, x and y must be vectors of the same length. The contingency table is calculated using the counts function from the StatsBase package. Then the power divergence test is conducted under the null hypothesis that the joint distribution of the cell counts in a 2-dimensional contingency table is the product of the row and column marginals.

Note that the entries of x (and y if provided) must be non-negative integers.

Computed confidence intervals by default are Quesenberry-Hurst intervals if the minimum of the expected cell counts exceeds 100, and Sison-Glaz intervals otherwise. See the confint(::PowerDivergenceTest) documentation for a list of supported methods to compute confidence intervals.

The power divergence test is given by

where is the cell count in the th row and th column and is a real number determining the nature of the test to be performed:

Under regularity conditions, the asymptotic distributions are identical (see Drost et. al. 1989). The null approximation works best for near .

Implements: pvalue, confint(::PowerDivergenceTest)

References

# StatsAPI.confint — Method

Compute a confidence interval with coverage level for multinomial proportions using one of the following methods. Possible values for method are:

References

# HypothesisTests.ChisqTest — Function

Perform a Pearson chi-squared test (equivalent to a PowerDivergenceTest with ).

If y is not given and x is a matrix with one row or column, or x is a vector, then a goodness-of-fit test is performed (x is treated as a one-dimensional contingency table). In this case, the hypothesis tested is whether the population probabilities equal those in theta0, or are all equal if theta0 is not given.

If only y and x are given and both are vectors of integer type, then once again a goodness-of-fit test is performed. In this case, theta0 is calculated by the proportion of each individual values in y. Here, the hypothesis tested is whether the two samples x and y come from the same population or not.

If x is a matrix with at least two rows and columns, it is taken as a two-dimensional contingency table. Otherwise, x and y must be vectors of the same length. The contingency table is calculated using counts function from the StatsBase package. Then the power divergence test is conducted under the null hypothesis that the joint distribution of the cell counts in a 2-dimensional contingency table is the product of the row and column marginals.

Note that the entries of x (and y if provided) must be non-negative integers.

Implements: pvalue, confint

# HypothesisTests.MultinomialLRTest — Function

Perform a multinomial likelihood ratio test (equivalent to a PowerDivergenceTest with ).

If y is not given and x is a matrix with one row or column, or x is a vector, then a goodness-of-fit test is performed (x is treated as a one-dimensional contingency table). In this case, the hypothesis tested is whether the population probabilities equal those in theta0, or are all equal if theta0 is not given.

If x is a matrix with at least two rows and columns, it is taken as a two-dimensional contingency table. Otherwise, x and y must be vectors of the same length. The contingency table is calculated using counts function from the StatsBase package. Then the power divergence test is conducted under the null hypothesis that the joint distribution of the cell counts in a 2-dimensional contingency table is the product of the row and column marginals.

Note that the entries of x (and y if provided) must be non-negative integers.

Implements: pvalue, confint

# HypothesisTests.OneSampleTTest — Type

Perform a one sample t-test of the null hypothesis that n values with mean xbar and sample standard deviation stddev come from a distribution with mean μ0 against the alternative hypothesis that the distribution does not have mean μ0.

Implements: pvalue, confint

Perform a one sample t-test of the null hypothesis that the data in vector v comes from a distribution with mean μ0 against the alternative hypothesis that the distribution does not have mean μ0.

Implements: pvalue, confint

Perform a paired sample t-test of the null hypothesis that the differences between pairs of values in vectors x and y come from a distribution with mean μ0 against the alternative hypothesis that the distribution does not have mean μ0.

Implements: pvalue, confint

# HypothesisTests.EqualVarianceTTest — Type

Perform a two-sample t-test of the null hypothesis that samples x and y described by the number of elements nx and ny, the mean mx and my, and variance vx and vy come from distributions with equals means and variances. The alternative hypothesis is that the distributions have different means but equal variances.

Implements: pvalue, confint

Perform a two-sample t-test of the null hypothesis that x and y come from distributions with equal means and variances against the alternative hypothesis that the distributions have different means but equal variances.

Implements: pvalue, confint

# HypothesisTests.UnequalVarianceTTest — Type

Perform an unequal variance two-sample t-test of the null hypothesis that x and y come from distributions with equal means against the alternative hypothesis that the distributions have different means.

This test is sometimes known as Welch’s t-test. It differs from the equal variance t-test in that it computes the number of degrees of freedom of the test using the Welch-Satterthwaite equation:

Implements: pvalue, confint

# HypothesisTests.OneSampleZTest — Type

Perform a one sample z-test of the null hypothesis that n values with mean xbar and population standard deviation stddev come from a distribution with mean μ0 against the alternative hypothesis that the distribution does not have mean μ0.

Implements: pvalue, confint

Perform a one sample z-test of the null hypothesis that the data in vector v comes from a distribution with mean μ0 against the alternative hypothesis that the distribution does not have mean μ0.

Implements: pvalue, confint

Perform a paired sample z-test of the null hypothesis that the differences between pairs of values in vectors x and y come from a distribution with mean μ0 against the alternative hypothesis that the distribution does not have mean μ0.

Implements: pvalue, confint

# HypothesisTests.EqualVarianceZTest — Type

Perform a two-sample z-test of the null hypothesis that x and y come from distributions with equal means and variances against the alternative hypothesis that the distributions have different means but equal variances.

Implements: pvalue, confint

# HypothesisTests.UnequalVarianceZTest — Type

Perform an unequal variance two-sample z-test of the null hypothesis that x and y come from distributions with equal means against the alternative hypothesis that the distributions have different means.

Implements: pvalue, confint

# HypothesisTests.VarianceFTest — Type

Perform an F-test of the null hypothesis that two real-valued vectors x and y have equal variances.

Implements: pvalue

References

External links

# HypothesisTests.OneWayANOVATest — Function

Perform one-way analysis of variance test of the hypothesis that that the groups means are equal.

The one-way analysis of variance (one-way ANOVA) is a technique that can be used to compare means of two or more samples. The ANOVA tests the null hypothesis, which states that samples in all groups are drawn from populations with the same mean values. To do this, two estimates are made of the population variance. The ANOVA produces an F-statistic, the ratio of the variance calculated among the means to the variance within the samples.

Implements: pvalue

External links

# HypothesisTests.LeveneTest — Function

Perform Levene’s test of the hypothesis that that the groups variances are equal. By default the mean statistic is used for centering in each of the groups, but other statistics are accepted: median or truncated mean, see BrownForsytheTest. By default the absolute value of the score difference, scorediff, is used, but other functions are accepted: x² or √|x|.

The test statistic, , is equivalent to the statistic, and is defined as follows:

where

The test statistic is approximately -distributed with and degrees of freedom.

Implements: pvalue

References

External links

# HypothesisTests.BrownForsytheTest — Function

The Brown—​Forsythe test is a statistical test for the equality of groups variances.

The Brown—​Forsythe test is a modification of the Levene’s test with the median instead of the mean statistic for computing the spread within each group.

Implements: pvalue

References

External links