Next: Random Number Generation, Previous: Distributions, Up: Statistics [Contents][Index]

Octave can perform many 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.

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*[*`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.

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*[*`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.

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*[*`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 (`

degrees of freedom.`c`)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.

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*[*`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.

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**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 which correlation coefficient to use for testing. 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.

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*[*`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.

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*[*`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.

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*[*`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.

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*[*`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], usekolmogorov_smirnov_test (x, "unif", 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.

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*[*`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.

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*[*`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 by1 -

`sum_ties`/ (`n`^3 -`n`)where

`sum_ties`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 William H. Kruskal and W. Allen Wallis, Use of Ranks in One-Criterion Variance Analysis, Journal of the American Statistical Association, Vol. 47, No. 260 (Dec 1952).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.

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**manova***(*`x`,`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

`x`. 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.

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*[*`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.

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*[*`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.

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*[*`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.

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*[*`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

and`n`= sum (`x`!=`y`)`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.

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*[*`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 (`

. Under the null, the test statistic`x`) ==`m``t`follows a Student distribution with

degrees of freedom.`df`= length (`x`) - 1With 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 (`

. If`x`) !=`m``alt`is`">"`

, the one-sided alternative`mean (`

is considered. Similarly for`x`) >`m``"<"`, the one-sided alternative`mean (`

is considered. The default is the two-sided case.`x`) <`m`The p-value of the test is returned in

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

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*[*`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 (`

. If`x`) != mean (`y`)`alt`is`">"`

, the one-sided alternative`mean (`

is used. Similarly for`x`) > mean (`y`)`"<"`

, the one-sided alternative`mean (`

is used. The default is the two-sided case.`x`) < mean (`y`)The p-value of the test is returned in

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

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*[*`pval`,`t`,`df`] =**t_test_regression***(*`y`,`x`,`rr`,`r`,`alt`) Perform a t test for the null hypothesis

in a classical normal regression model`rr`*`b`=`r`

. Under the null, the test statistic`y`=`x`*`b`+`e``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

. If`rr`*`b`!=`r``alt`is`">"`

, the one-sided alternative

is used. Similarly for`rr`*`b`>`r``"<"`, the one-sided alternative

is used. The default is the two-sided case.`rr`*`b`<`r`The p-value of the test is returned in

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

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*[*`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.

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*[*`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 (`

. If`x`) != var (`y`)`alt`is`">"`

, the one-sided alternative`var (`

is used. Similarly for "<", the one-sided alternative`x`) > var (`y`)`var (`

is used. The default is the two-sided case.`x`) > var (`y`)The p-value of the test is returned in

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

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*[*`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 (`

. If`x`) !=`m``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.

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*[*`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.**Caution:**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.

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*[*`pval`,`z`] =**z_test***(*`x`,`m`,`v`,`alt`) Perform a Z-test of the null hypothesis

`mean (`

for a sample`x`) ==`m``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 (`

. If`x`) !=`m``alt`is`">"`

, the one-sided alternative`mean (`

is considered. Similarly for`x`) >`m``"<"`

, the one-sided alternative`mean (`

is considered. The default is the two-sided case.`x`) <`m`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.

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*[*`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 (`

. If alt is`x`) != mean (`y`)`">"`

, the one-sided alternative`mean (`

is used. Similarly for`x`) > mean (`y`)`"<"`

, the one-sided alternative`mean (`

is used. The default is the two-sided case.`x`) < mean (`y`)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.

Next: Random Number Generation, Previous: Distributions, Up: Statistics [Contents][Index]