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### 24.4 Tests

Octave can perform several different statistical tests. The following table summarizes the available tests.

Hypothesis | Test Functions
Equal mean values | `anova`, `hotelling_test2`, `t_test_2`, `welch_test`, `wilcoxon_test`, `z_test_2`
Equal medians | `kruskal_wallis_test`, `sign_test`
Equal variances | `bartlett_test`, `manova`, `var_test`
Equal distributions | `chisquare_test_homogeneity`, `kolmogorov_smirnov_test_2`, `u_test`
Equal marginal frequencies | `mcnemar_test`
Equal success probabilities | `prop_test_2`
Independent observations | `chisquare_test_independence`, `run_test`
Uncorrelated observations | `cor_test`
Given mean value | `hotelling_test`, `t_test`, `z_test`
Observations from given distribution | `kolmogorov_smirnov_test`
Regression | `f_test_regression`, `t_test_regression`

The tests return a p-value that describes the outcome of the test. Assuming that the test hypothesis is true, the p-value is the probability of obtaining a worse result than the observed one. So large p-values corresponds to a successful test. Usually a test hypothesis is accepted if the p-value exceeds 0.05.

— Function File: [pval, f, df_b, df_w] = anova (y, g)

> Perform a one-way analysis of variance (ANOVA). The goal is to test whether the population means of data taken from k different groups are all equal.

> Data may be given in a single vector y with groups specified by a corresponding vector of group labels g (e.g., numbers from 1 to k). This is the general form which does not impose any restriction on the number of data in each group or the group labels.

> If y is a matrix and g is omitted, each column of y is treated as a group. This form is only appropriate for balanced ANOVA in which the numbers of samples from each group are all equal.

> Under the null of constant means, the statistic f follows an F distribution with df_b and df_w degrees of freedom.

> The p-value (1 minus the CDF of this distribution at f) is returned in pval.

> If no output argument is given, the standard one-way ANOVA table is printed.

— Function File: [pval, chisq, df] = bartlett_test (x1, ...)

> Perform a Bartlett test for the homogeneity of variances in the data vectors x1, x2, ..., xk, where k > 1.

> Under the null of equal variances, the test statistic chisq approximately follows a chi-square distribution with df degrees of freedom.

> The p-value (1 minus the CDF of this distribution at chisq) is returned in pval.

> If no output argument is given, the p-value is displayed.

— Function File: [pval, chisq, df] = chisquare_test_homogeneity (x, y, c)

> Given two samples x and y, perform a chisquare test for homogeneity of the null hypothesis that x and y come from the same distribution, based on the partition induced by the (strictly increasing) entries of c.

> For large samples, the test statistic chisq approximately follows a chisquare distribution with df = `length (`c`)` degrees of freedom.

> The p-value (1 minus the CDF of this distribution at chisq) is returned in pval.

> If no output argument is given, the p-value is displayed.

— Function File: [pval, chisq, df] = chisquare_test_independence (x)

> Perform a chi-square test for independence based on the contingency table x. Under the null hypothesis of independence, chisq approximately has a chi-square distribution with df degrees of freedom.

> The p-value (1 minus the CDF of this distribution at chisq) of the test is returned in pval.

> If no output argument is given, the p-value is displayed.

— Function File: cor_test (x, y, alt, method)

> Test whether two samples x and y come from uncorrelated populations.

> The optional argument string alt describes the alternative hypothesis, and can be `"!="` or `"<>"` (non-zero), `">"` (greater than 0), or `"<"` (less than 0). The default is the two-sided case.

> The optional argument string method specifies on which correlation coefficient the test should be based. If method is `"pearson"` (default), the (usual) Pearson's product moment correlation coefficient is used. In this case, the data should come from a bivariate normal distribution. Otherwise, the other two methods offer nonparametric alternatives. If method is `"kendall"`, then Kendall's rank correlation tau is used. If method is `"spearman"`, then Spearman's rank correlation rho is used. Only the first character is necessary.

> The output is a structure with the following elements:

> pval
>   The p-value of the test.
> stat
>   The value of the test statistic.
> dist
>   The distribution of the test statistic.
> params
>   The parameters of the null distribution of the test statistic.
> alternative
>   The alternative hypothesis.
> method
>   The method used for testing.

> If no output argument is given, the p-value is displayed.

— Function File: [pval, f, df_num, df_den] = f_test_regression (y, x, rr, r)

> Perform an F test for the null hypothesis rr * b = r in a classical normal regression model y = X * b + e.

> Under the null, the test statistic f follows an F distribution with df_num and df_den degrees of freedom.

> The p-value (1 minus the CDF of this distribution at f) is returned in pval.

> If not given explicitly, r = 0.

> If no output argument is given, the p-value is displayed.

— Function File: [pval, tsq] = hotelling_test (x, m)

> For a sample x from a multivariate normal distribution with unknown mean and covariance matrix, test the null hypothesis that `mean (`x`) ==` m.

> Hotelling's T^2 is returned in tsq. Under the null, (n-p) T^2 / (p(n-1)) has an F distribution with p and n-p degrees of freedom, where n and p are the numbers of samples and variables, respectively.

> The p-value of the test is returned in pval.

> If no output argument is given, the p-value of the test is displayed.

— Function File: [pval, tsq] = hotelling_test_2 (x, y)

> For two samples x from multivariate normal distributions with the same number of variables (columns), unknown means and unknown equal covariance matrices, test the null hypothesis `mean (`x`) == mean (`y`)`.

> Hotelling's two-sample T^2 is returned in tsq. Under the null,

>               (n_x+n_y-p-1) T^2 / (p(n_x+n_y-2))

> has an F distribution with p and n_x+n_y-p-1 degrees of freedom, where n_x and n_y are the sample sizes and p is the number of variables.

> The p-value of the test is returned in pval.

> If no output argument is given, the p-value of the test is displayed.

— Function File: [pval, ks] = kolmogorov_smirnov_test (x, dist, params, alt)

> Perform a Kolmogorov-Smirnov test of the null hypothesis that the sample x comes from the (continuous) distribution dist. I.e., if F and G are the CDFs corresponding to the sample and dist, respectively, then the null is that F == G.

> The optional argument params contains a list of parameters of dist. For example, to test whether a sample x comes from a uniform distribution on [2,4], use

>               kolmogorov_smirnov_test(x, "uniform", 2, 4)

> dist can be any string for which a function dist_cdf that calculates the CDF of distribution dist exists.

> With the optional argument string alt, the alternative of interest can be selected. If alt is `"!="` or `"<>"`, the null is tested against the two-sided alternative F != G. In this case, the test statistic ks follows a two-sided Kolmogorov-Smirnov distribution. If alt is `">"`, the one-sided alternative F > G is considered. Similarly for `"<"`, the one-sided alternative F > G is considered. In this case, the test statistic ks has a one-sided Kolmogorov-Smirnov distribution. The default is the two-sided case.

> The p-value of the test is returned in pval.

> If no output argument is given, the p-value is displayed.

— Function File: [pval, ks, d] = kolmogorov_smirnov_test_2 (x, y, alt)

> Perform a 2-sample Kolmogorov-Smirnov test of the null hypothesis that the samples x and y come from the same (continuous) distribution. I.e., if F and G are the CDFs corresponding to the x and y samples, respectively, then the null is that F == G.

> With the optional argument string alt, the alternative of interest can be selected. If alt is `"!="` or `"<>"`, the null is tested against the two-sided alternative F != G. In this case, the test statistic ks follows a two-sided Kolmogorov-Smirnov distribution. If alt is `">"`, the one-sided alternative F > G is considered. Similarly for `"<"`, the one-sided alternative F < G is considered. In this case, the test statistic ks has a one-sided Kolmogorov-Smirnov distribution. The default is the two-sided case.

> The p-value of the test is returned in pval.

> The third returned value, d, is the test statistic, the maximum vertical distance between the two cumulative distribution functions.

> If no output argument is given, the p-value is displayed.

— Function File: [pval, k, df] = kruskal_wallis_test (x1, ...)

> Perform a Kruskal-Wallis one-factor "analysis of variance".

> Suppose a variable is observed for k > 1 different groups, and let x1, ..., xk be the corresponding data vectors.

> Under the null hypothesis that the ranks in the pooled sample are not affected by the group memberships, the test statistic k is approximately chi-square with df = k - 1 degrees of freedom.

> If the data contains ties (some value appears more than once) k is divided by

> 1 - sumTies / ( n^3 - n )

> where sumTies is the sum of t^2 - t over each group of ties where t is the number of ties in the group and n is the total number of values in the input data. For more info on this adjustment see "Use of Ranks in One-Criterion Variance Analysis" in Journal of the American Statistical Association, Vol. 47, No. 260 (Dec 1952) by William H. Kruskal and W. Allen Wallis.

> The p-value (1 minus the CDF of this distribution at k) is returned in pval.

> If no output argument is given, the p-value is displayed.

— Function File: manova (y, g)

> Perform a one-way multivariate analysis of variance (MANOVA). The goal is to test whether the p-dimensional population means of data taken from k different groups are all equal. All data are assumed drawn independently from p-dimensional normal distributions with the same covariance matrix.

> The data matrix is given by y. As usual, rows are observations and columns are variables. The vector g specifies the corresponding group labels (e.g., numbers from 1 to k).

> The LR test statistic (Wilks' Lambda) and approximate p-values are computed and displayed.

— Function File: [pval, chisq, df] = mcnemar_test (x)

> For a square contingency table x of data cross-classified on the row and column variables, McNemar's test can be used for testing the null hypothesis of symmetry of the classification probabilities.

> Under the null, chisq is approximately distributed as chisquare with df degrees of freedom.

> The p-value (1 minus the CDF of this distribution at chisq) is returned in pval.

> If no output argument is given, the p-value of the test is displayed.

— Function File: [pval, z] = prop_test_2 (x1, n1, x2, n2, alt)

> If x1 and n1 are the counts of successes and trials in one sample, and x2 and n2 those in a second one, test the null hypothesis that the success probabilities p1 and p2 are the same. Under the null, the test statistic z approximately follows a standard normal distribution.

> With the optional argument string alt, the alternative of interest can be selected. If alt is `"!="` or `"<>"`, the null is tested against the two-sided alternative p1 != p2. If alt is `">"`, the one-sided alternative p1 > p2 is used. Similarly for `"<"`, the one-sided alternative p1 < p2 is used. The default is the two-sided case.

> The p-value of the test is returned in pval.

> If no output argument is given, the p-value of the test is displayed.

— Function File: [pval, chisq] = run_test (x)

> Perform a chi-square test with 6 degrees of freedom based on the upward runs in the columns of x. Can be used to test whether x contains independent data.

> The p-value of the test is returned in pval.

> If no output argument is given, the p-value is displayed.

— Function File: [pval, b, n] = sign_test (x, y, alt)

> For two matched-pair samples x and y, perform a sign test of the null hypothesis PROB (x > y) == PROB (x < y) == 1/2. Under the null, the test statistic b roughly follows a binomial distribution with parameters n`= sum (`x`!=` y`)` and p = 1/2.

> With the optional argument `alt`, the alternative of interest can be selected. If alt is `"!="` or `"<>"`, the null hypothesis is tested against the two-sided alternative PROB (x < y) != 1/2. If alt is `">"`, the one-sided alternative PROB (x > y) > 1/2 ("x is stochastically greater than y") is considered. Similarly for `"<"`, the one-sided alternative PROB (x > y) < 1/2 ("x is stochastically less than y") is considered. The default is the two-sided case.

> The p-value of the test is returned in pval.

> If no output argument is given, the p-value of the test is displayed.

— Function File: [pval, t, df] = t_test (x, m, alt)

> For a sample x from a normal distribution with unknown mean and variance, perform a t-test of the null hypothesis `mean (`x`) ==` m. Under the null, the test statistic t follows a Student distribution with df`= length (`x`) - 1` degrees of freedom.

> With the optional argument string alt, the alternative of interest can be selected. If alt is `"!="` or `"<>"`, the null is tested against the two-sided alternative `mean (`x`) !=` m. If alt is `">"`, the one-sided alternative `mean (`x`) >` m is considered. Similarly for "<", the one-sided alternative `mean (`x`) <` m is considered. The default is the two-sided case.

> The p-value of the test is returned in pval.

> If no output argument is given, the p-value of the test is displayed.

— Function File: [pval, t, df] = t_test_2 (x, y, alt)

> For two samples x and y from normal distributions with unknown means and unknown equal variances, perform a two-sample t-test of the null hypothesis of equal means. Under the null, the test statistic t follows a Student distribution with df degrees of freedom.

> With the optional argument string alt, the alternative of interest can be selected. If alt is `"!="` or `"<>"`, the null is tested against the two-sided alternative `mean (`x`) != mean (`y`)`. If alt is `">"`, the one-sided alternative `mean (`x`) > mean (`y`)` is used. Similarly for `"<"`, the one-sided alternative `mean (`x`) < mean (`y`)` is used. The default is the two-sided case.

> The p-value of the test is returned in pval.

> If no output argument is given, the p-value of the test is displayed.

— Function File: [pval, t, df] = t_test_regression (y, x, rr, r, alt)

> Perform an t test for the null hypothesis rr`*` b`=` r in a classical normal regression model y`=` x`*` b`+` e. Under the null, the test statistic t follows a t distribution with df degrees of freedom.

> If r is omitted, a value of 0 is assumed.

> With the optional argument string alt, the alternative of interest can be selected. If alt is `"!="` or `"<>"`, the null is tested against the two-sided alternative rr`*` b`!=` r. If alt is `">"`, the one-sided alternative rr`*` b`>` r is used. Similarly for "<", the one-sided alternative rr`*` b`<` r is used. The default is the two-sided case.

> The p-value of the test is returned in pval.

> If no output argument is given, the p-value of the test is displayed.

— Function File: [pval, z] = u_test (x, y, alt)

> For two samples x and y, perform a Mann-Whitney U-test of the null hypothesis PROB (x > y) == 1/2 == PROB (x < y). Under the null, the test statistic z approximately follows a standard normal distribution. Note that this test is equivalent to the Wilcoxon rank-sum test.

> With the optional argument string alt, the alternative of interest can be selected. If alt is `"!="` or `"<>"`, the null is tested against the two-sided alternative PROB (x > y) != 1/2. If alt is `">"`, the one-sided alternative PROB (x > y) > 1/2 is considered. Similarly for `"<"`, the one-sided alternative PROB (x > y) < 1/2 is considered. The default is the two-sided case.

> The p-value of the test is returned in pval.

> If no output argument is given, the p-value of the test is displayed.

— Function File: [pval, f, df_num, df_den] = var_test (x, y, alt)

> For two samples x and y from normal distributions with unknown means and unknown variances, perform an F-test of the null hypothesis of equal variances. Under the null, the test statistic f follows an F-distribution with df_num and df_den degrees of freedom.

> With the optional argument string alt, the alternative of interest can be selected. If alt is `"!="` or `"<>"`, the null is tested against the two-sided alternative `var (`x`) != var (`y`)`. If alt is `">"`, the one-sided alternative `var (`x`) > var (`y`)` is used. Similarly for "<", the one-sided alternative `var (`x`) > var (`y`)` is used. The default is the two-sided case.

> The p-value of the test is returned in pval.

> If no output argument is given, the p-value of the test is displayed.

— Function File: [pval, t, df] = welch_test (x, y, alt)

> For two samples x and y from normal distributions with unknown means and unknown and not necessarily equal variances, perform a Welch test of the null hypothesis of equal means. Under the null, the test statistic t approximately follows a Student distribution with df degrees of freedom.

> With the optional argument string alt, the alternative of interest can be selected. If alt is `"!="` or `"<>"`, the null is tested against the two-sided alternative `mean (`x`) !=` m. If alt is `">"`, the one-sided alternative mean(x) > m is considered. Similarly for `"<"`, the one-sided alternative mean(x) < m is considered. The default is the two-sided case.

> The p-value of the test is returned in pval.

> If no output argument is given, the p-value of the test is displayed.

— Function File: [pval, z] = wilcoxon_test (x, y, alt)

> For two matched-pair sample vectors x and y, perform a Wilcoxon signed-rank test of the null hypothesis PROB (x > y) == 1/2. Under the null, the test statistic z approximately follows a standard normal distribution when n > 25.

> Warning: This function assumes a normal distribution for z and thus is invalid for n <= 25.

> With the optional argument string alt, the alternative of interest can be selected. If alt is `"!="` or `"<>"`, the null is tested against the two-sided alternative PROB (x > y) != 1/2. If alt is `">"`, the one-sided alternative PROB (x > y) > 1/2 is considered. Similarly for `"<"`, the one-sided alternative PROB (x > y) < 1/2 is considered. The default is the two-sided case.

> The p-value of the test is returned in pval.

> If no output argument is given, the p-value of the test is displayed.

— Function File: [pval, z] = z_test (x, m, v, alt)

> Perform a Z-test of the null hypothesis `mean (`x`) ==` m for a sample x from a normal distribution with unknown mean and known variance v. Under the null, the test statistic z follows a standard normal distribution.

> With the optional argument string alt, the alternative of interest can be selected. If alt is `"!="` or `"<>"`, the null is tested against the two-sided alternative `mean (`x`) !=` m. If alt is `">"`, the one-sided alternative `mean (`x`) >` m is considered. Similarly for `"<"`, the one-sided alternative `mean (`x`) <` m is considered. The default is the two-sided case.

> The p-value of the test is returned in pval.

> If no output argument is given, the p-value of the test is displayed along with some information.

— Function File: [pval, z] = z_test_2 (x, y, v_x, v_y, alt)

> For two samples x and y from normal distributions with unknown means and known variances v_x and v_y, perform a Z-test of the hypothesis of equal means. Under the null, the test statistic z follows a standard normal distribution.

> With the optional argument string alt, the alternative of interest can be selected. If alt is `"!="` or `"<>"`, the null is tested against the two-sided alternative `mean (`x`) != mean (`y`)`. If alt is `">"`, the one-sided alternative `mean (`x`) > mean (`y`)` is used. Similarly for `"<"`, the one-sided alternative `mean (`x`) < mean (`y`)` is used. The default is the two-sided case.

> The p-value of the test is returned in pval.

> If no output argument is given, the p-value of the test is displayed along with some information.