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<a name="Tests"></a>
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<p>
Next: <a href="Random-Number-Generation.html#Random-Number-Generation" accesskey="n" rel="next">Random Number Generation</a>, Previous: <a href="Distributions.html#Distributions" accesskey="p" rel="prev">Distributions</a>, Up: <a href="Statistics.html#Statistics" accesskey="u" rel="up">Statistics</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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<hr>
<a name="Tests-1"></a>
<h3 class="section">26.6 Tests</h3>
<p>Octave can perform many different statistical tests. The following
table summarizes the available tests.
</p>
<table>
<thead><tr><th width="40%">Hypothesis</th><th width="50%">Test Functions</th></tr></thead>
<tr><td width="40%">Equal mean values</td><td width="50%"><code>anova</code>, <code>hotelling_test2</code>, <code>t_test_2</code>,
<code>welch_test</code>, <code>wilcoxon_test</code>, <code>z_test_2</code></td></tr>
<tr><td width="40%">Equal medians</td><td width="50%"><code>kruskal_wallis_test</code>, <code>sign_test</code></td></tr>
<tr><td width="40%">Equal variances</td><td width="50%"><code>bartlett_test</code>, <code>manova</code>, <code>var_test</code></td></tr>
<tr><td width="40%">Equal distributions</td><td width="50%"><code>chisquare_test_homogeneity</code>, <code>kolmogorov_smirnov_test_2</code>,
<code>u_test</code></td></tr>
<tr><td width="40%">Equal marginal frequencies</td><td width="50%"><code>mcnemar_test</code></td></tr>
<tr><td width="40%">Equal success probabilities</td><td width="50%"><code>prop_test_2</code></td></tr>
<tr><td width="40%">Independent observations</td><td width="50%"><code>chisquare_test_independence</code>, <code>run_test</code></td></tr>
<tr><td width="40%">Uncorrelated observations</td><td width="50%"><code>cor_test</code></td></tr>
<tr><td width="40%">Given mean value</td><td width="50%"><code>hotelling_test</code>, <code>t_test</code>, <code>z_test</code></td></tr>
<tr><td width="40%">Observations from given distribution</td><td width="50%"><code>kolmogorov_smirnov_test</code></td></tr>
<tr><td width="40%">Regression</td><td width="50%"><code>f_test_regression</code>, <code>t_test_regression</code></td></tr>
</table>
<p>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.
</p>
<a name="XREFanova"></a><dl>
<dt><a name="index-anova"></a>Function File: <em>[<var>pval</var>, <var>f</var>, <var>df_b</var>, <var>df_w</var>] =</em> <strong>anova</strong> <em>(<var>y</var>, <var>g</var>)</em></dt>
<dd><p>Perform a one-way analysis of variance (ANOVA). The goal is to test
whether the population means of data taken from <var>k</var> different
groups are all equal.
</p>
<p>Data may be given in a single vector <var>y</var> with groups specified by
a corresponding vector of group labels <var>g</var> (e.g., numbers from 1
to <var>k</var>). This is the general form which does not impose any
restriction on the number of data in each group or the group labels.
</p>
<p>If <var>y</var> is a matrix and <var>g</var> is omitted, each column of <var>y</var>
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.
</p>
<p>Under the null of constant means, the statistic <var>f</var> follows an F
distribution with <var>df_b</var> and <var>df_w</var> degrees of freedom.
</p>
<p>The p-value (1 minus the CDF of this distribution at <var>f</var>) is
returned in <var>pval</var>.
</p>
<p>If no output argument is given, the standard one-way ANOVA table is
printed.
</p></dd></dl>
<a name="XREFbartlett_005ftest"></a><dl>
<dt><a name="index-bartlett_005ftest"></a>Function File: <em>[<var>pval</var>, <var>chisq</var>, <var>df</var>] =</em> <strong>bartlett_test</strong> <em>(<var>x1</var>, …)</em></dt>
<dd><p>Perform a Bartlett test for the homogeneity of variances in the data
vectors <var>x1</var>, <var>x2</var>, …, <var>xk</var>, where <var>k</var> > 1.
</p>
<p>Under the null of equal variances, the test statistic <var>chisq</var>
approximately follows a chi-square distribution with <var>df</var> degrees of
freedom.
</p>
<p>The p-value (1 minus the CDF of this distribution at <var>chisq</var>) is
returned in <var>pval</var>.
</p>
<p>If no output argument is given, the p-value is displayed.
</p></dd></dl>
<a name="XREFchisquare_005ftest_005fhomogeneity"></a><dl>
<dt><a name="index-chisquare_005ftest_005fhomogeneity"></a>Function File: <em>[<var>pval</var>, <var>chisq</var>, <var>df</var>] =</em> <strong>chisquare_test_homogeneity</strong> <em>(<var>x</var>, <var>y</var>, <var>c</var>)</em></dt>
<dd><p>Given two samples <var>x</var> and <var>y</var>, perform a chisquare test for
homogeneity of the null hypothesis that <var>x</var> and <var>y</var> come from
the same distribution, based on the partition induced by the
(strictly increasing) entries of <var>c</var>.
</p>
<p>For large samples, the test statistic <var>chisq</var> approximately follows a
chisquare distribution with <var>df</var> = <code>length (<var>c</var>)</code>
degrees of freedom.
</p>
<p>The p-value (1 minus the CDF of this distribution at <var>chisq</var>) is
returned in <var>pval</var>.
</p>
<p>If no output argument is given, the p-value is displayed.
</p></dd></dl>
<a name="XREFchisquare_005ftest_005findependence"></a><dl>
<dt><a name="index-chisquare_005ftest_005findependence"></a>Function File: <em>[<var>pval</var>, <var>chisq</var>, <var>df</var>] =</em> <strong>chisquare_test_independence</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Perform a chi-square test for independence based on the contingency
table <var>x</var>. Under the null hypothesis of independence,
<var>chisq</var> approximately has a chi-square distribution with
<var>df</var> degrees of freedom.
</p>
<p>The p-value (1 minus the CDF of this distribution at chisq) of the
test is returned in <var>pval</var>.
</p>
<p>If no output argument is given, the p-value is displayed.
</p></dd></dl>
<a name="XREFcor_005ftest"></a><dl>
<dt><a name="index-cor_005ftest"></a>Function File: <em></em> <strong>cor_test</strong> <em>(<var>x</var>, <var>y</var>, <var>alt</var>, <var>method</var>)</em></dt>
<dd><p>Test whether two samples <var>x</var> and <var>y</var> come from uncorrelated
populations.
</p>
<p>The optional argument string <var>alt</var> describes the alternative
hypothesis, and can be <code>"!="</code> or <code>"<>"</code> (non-zero),
<code>">"</code> (greater than 0), or <code>"<"</code> (less than 0). The
default is the two-sided case.
</p>
<p>The optional argument string <var>method</var> specifies which
correlation coefficient to use for testing. If <var>method</var> is
<code>"pearson"</code> (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 <var>method</var> is
<code>"kendall"</code>, then Kendall’s rank correlation tau is used. If
<var>method</var> is <code>"spearman"</code>, then Spearman’s rank correlation
rho is used. Only the first character is necessary.
</p>
<p>The output is a structure with the following elements:
</p>
<dl compact="compact">
<dt><var>pval</var></dt>
<dd><p>The p-value of the test.
</p>
</dd>
<dt><var>stat</var></dt>
<dd><p>The value of the test statistic.
</p>
</dd>
<dt><var>dist</var></dt>
<dd><p>The distribution of the test statistic.
</p>
</dd>
<dt><var>params</var></dt>
<dd><p>The parameters of the null distribution of the test statistic.
</p>
</dd>
<dt><var>alternative</var></dt>
<dd><p>The alternative hypothesis.
</p>
</dd>
<dt><var>method</var></dt>
<dd><p>The method used for testing.
</p></dd>
</dl>
<p>If no output argument is given, the p-value is displayed.
</p></dd></dl>
<a name="XREFf_005ftest_005fregression"></a><dl>
<dt><a name="index-f_005ftest_005fregression"></a>Function File: <em>[<var>pval</var>, <var>f</var>, <var>df_num</var>, <var>df_den</var>] =</em> <strong>f_test_regression</strong> <em>(<var>y</var>, <var>x</var>, <var>rr</var>, <var>r</var>)</em></dt>
<dd><p>Perform an F test for the null hypothesis rr * b = r in a
classical normal regression model y = X * b + e.
</p>
<p>Under the null, the test statistic <var>f</var> follows an F distribution
with <var>df_num</var> and <var>df_den</var> degrees of freedom.
</p>
<p>The p-value (1 minus the CDF of this distribution at <var>f</var>) is
returned in <var>pval</var>.
</p>
<p>If not given explicitly, <var>r</var> = 0.
</p>
<p>If no output argument is given, the p-value is displayed.
</p></dd></dl>
<a name="XREFhotelling_005ftest"></a><dl>
<dt><a name="index-hotelling_005ftest"></a>Function File: <em>[<var>pval</var>, <var>tsq</var>] =</em> <strong>hotelling_test</strong> <em>(<var>x</var>, <var>m</var>)</em></dt>
<dd><p>For a sample <var>x</var> from a multivariate normal distribution with unknown
mean and covariance matrix, test the null hypothesis that <code>mean
(<var>x</var>) == <var>m</var></code>.
</p>
<p>Hotelling’s <em>T^2</em> is returned in <var>tsq</var>. Under the null,
<em>(n-p) T^2 / (p(n-1))</em> has an F distribution with <em>p</em> and
<em>n-p</em> degrees of freedom, where <em>n</em> and <em>p</em> are the
numbers of samples and variables, respectively.
</p>
<p>The p-value of the test is returned in <var>pval</var>.
</p>
<p>If no output argument is given, the p-value of the test is displayed.
</p></dd></dl>
<a name="XREFhotelling_005ftest_005f2"></a><dl>
<dt><a name="index-hotelling_005ftest_005f2"></a>Function File: <em>[<var>pval</var>, <var>tsq</var>] =</em> <strong>hotelling_test_2</strong> <em>(<var>x</var>, <var>y</var>)</em></dt>
<dd><p>For two samples <var>x</var> from multivariate normal distributions with
the same number of variables (columns), unknown means and unknown
equal covariance matrices, test the null hypothesis <code>mean
(<var>x</var>) == mean (<var>y</var>)</code>.
</p>
<p>Hotelling’s two-sample <em>T^2</em> is returned in <var>tsq</var>. Under the null,
</p>
<div class="example">
<pre class="example">(n_x+n_y-p-1) T^2 / (p(n_x+n_y-2))
</pre></div>
<p>has an F distribution with <em>p</em> and <em>n_x+n_y-p-1</em> degrees of
freedom, where <em>n_x</em> and <em>n_y</em> are the sample sizes and
<em>p</em> is the number of variables.
</p>
<p>The p-value of the test is returned in <var>pval</var>.
</p>
<p>If no output argument is given, the p-value of the test is displayed.
</p></dd></dl>
<a name="XREFkolmogorov_005fsmirnov_005ftest"></a><dl>
<dt><a name="index-kolmogorov_005fsmirnov_005ftest"></a>Function File: <em>[<var>pval</var>, <var>ks</var>] =</em> <strong>kolmogorov_smirnov_test</strong> <em>(<var>x</var>, <var>dist</var>, <var>params</var>, <var>alt</var>)</em></dt>
<dd><p>Perform a Kolmogorov-Smirnov test of the null hypothesis that the
sample <var>x</var> 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.
</p>
<p>The optional argument <var>params</var> contains a list of parameters of
<var>dist</var>. For example, to test whether a sample <var>x</var> comes from
a uniform distribution on [2,4], use
</p>
<div class="example">
<pre class="example">kolmogorov_smirnov_test (x, "unif", 2, 4)
</pre></div>
<p><var>dist</var> can be any string for which a function <var>dist_cdf</var>
that calculates the CDF of distribution <var>dist</var> exists.
</p>
<p>With the optional argument string <var>alt</var>, the alternative of
interest can be selected. If <var>alt</var> is <code>"!="</code> or
<code>"<>"</code>, the null is tested against the two-sided alternative F
!= G. In this case, the test statistic <var>ks</var> follows a two-sided
Kolmogorov-Smirnov distribution. If <var>alt</var> is <code>">"</code>, the
one-sided alternative F > G is considered. Similarly for <code>"<"</code>,
the one-sided alternative F > G is considered. In this case, the
test statistic <var>ks</var> has a one-sided Kolmogorov-Smirnov
distribution. The default is the two-sided case.
</p>
<p>The p-value of the test is returned in <var>pval</var>.
</p>
<p>If no output argument is given, the p-value is displayed.
</p></dd></dl>
<a name="XREFkolmogorov_005fsmirnov_005ftest_005f2"></a><dl>
<dt><a name="index-kolmogorov_005fsmirnov_005ftest_005f2"></a>Function File: <em>[<var>pval</var>, <var>ks</var>, <var>d</var>] =</em> <strong>kolmogorov_smirnov_test_2</strong> <em>(<var>x</var>, <var>y</var>, <var>alt</var>)</em></dt>
<dd><p>Perform a 2-sample Kolmogorov-Smirnov test of the null hypothesis
that the samples <var>x</var> and <var>y</var> come from the same (continuous)
distribution. I.e., if F and G are the CDFs corresponding to the
<var>x</var> and <var>y</var> samples, respectively, then the null is that F ==
G.
</p>
<p>With the optional argument string <var>alt</var>, the alternative of
interest can be selected. If <var>alt</var> is <code>"!="</code> or
<code>"<>"</code>, the null is tested against the two-sided alternative F
!= G. In this case, the test statistic <var>ks</var> follows a two-sided
Kolmogorov-Smirnov distribution. If <var>alt</var> is <code>">"</code>, the
one-sided alternative F > G is considered. Similarly for <code>"<"</code>,
the one-sided alternative F < G is considered. In this case, the
test statistic <var>ks</var> has a one-sided Kolmogorov-Smirnov
distribution. The default is the two-sided case.
</p>
<p>The p-value of the test is returned in <var>pval</var>.
</p>
<p>The third returned value, <var>d</var>, is the test statistic, the maximum
vertical distance between the two cumulative distribution functions.
</p>
<p>If no output argument is given, the p-value is displayed.
</p></dd></dl>
<a name="XREFkruskal_005fwallis_005ftest"></a><dl>
<dt><a name="index-kruskal_005fwallis_005ftest"></a>Function File: <em>[<var>pval</var>, <var>k</var>, <var>df</var>] =</em> <strong>kruskal_wallis_test</strong> <em>(<var>x1</var>, …)</em></dt>
<dd><p>Perform a Kruskal-Wallis one-factor analysis of variance.
</p>
<p>Suppose a variable is observed for <var>k</var> > 1 different groups, and
let <var>x1</var>, …, <var>xk</var> be the corresponding data vectors.
</p>
<p>Under the null hypothesis that the ranks in the pooled sample are not
affected by the group memberships, the test statistic <var>k</var> is
approximately chi-square with <var>df</var> = <var>k</var> - 1 degrees of
freedom.
</p>
<p>If the data contains ties (some value appears more than once)
<var>k</var> is divided by
</p>
<p>1 - <var>sum_ties</var> / (<var>n</var>^3 - <var>n</var>)
</p>
<p>where <var>sum_ties</var> is the sum of <var>t</var>^2 - <var>t</var> over each group
of ties where <var>t</var> is the number of ties in the group and <var>n</var>
is the total number of values in the input data. For more info on
this adjustment see William H. Kruskal and W. Allen Wallis,
<cite>Use of Ranks in One-Criterion Variance Analysis</cite>,
Journal of the American Statistical Association, Vol. 47,
No. 260 (Dec 1952).
</p>
<p>The p-value (1 minus the CDF of this distribution at <var>k</var>) is
returned in <var>pval</var>.
</p>
<p>If no output argument is given, the p-value is displayed.
</p></dd></dl>
<a name="XREFmanova"></a><dl>
<dt><a name="index-manova"></a>Function File: <em></em> <strong>manova</strong> <em>(<var>x</var>, <var>g</var>)</em></dt>
<dd><p>Perform a one-way multivariate analysis of variance (MANOVA). The
goal is to test whether the p-dimensional population means of data
taken from <var>k</var> different groups are all equal. All data are
assumed drawn independently from p-dimensional normal distributions
with the same covariance matrix.
</p>
<p>The data matrix is given by <var>x</var>. As usual, rows are observations
and columns are variables. The vector <var>g</var> specifies the
corresponding group labels (e.g., numbers from 1 to <var>k</var>).
</p>
<p>The LR test statistic (Wilks’ Lambda) and approximate p-values are
computed and displayed.
</p></dd></dl>
<a name="XREFmcnemar_005ftest"></a><dl>
<dt><a name="index-mcnemar_005ftest"></a>Function File: <em>[<var>pval</var>, <var>chisq</var>, <var>df</var>] =</em> <strong>mcnemar_test</strong> <em>(<var>x</var>)</em></dt>
<dd><p>For a square contingency table <var>x</var> 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.
</p>
<p>Under the null, <var>chisq</var> is approximately distributed as chisquare
with <var>df</var> degrees of freedom.
</p>
<p>The p-value (1 minus the CDF of this distribution at <var>chisq</var>) is
returned in <var>pval</var>.
</p>
<p>If no output argument is given, the p-value of the test is displayed.
</p></dd></dl>
<a name="XREFprop_005ftest_005f2"></a><dl>
<dt><a name="index-prop_005ftest_005f2"></a>Function File: <em>[<var>pval</var>, <var>z</var>] =</em> <strong>prop_test_2</strong> <em>(<var>x1</var>, <var>n1</var>, <var>x2</var>, <var>n2</var>, <var>alt</var>)</em></dt>
<dd><p>If <var>x1</var> and <var>n1</var> are the counts of successes and trials in
one sample, and <var>x2</var> and <var>n2</var> those in a second one, test the
null hypothesis that the success probabilities <var>p1</var> and <var>p2</var>
are the same. Under the null, the test statistic <var>z</var>
approximately follows a standard normal distribution.
</p>
<p>With the optional argument string <var>alt</var>, the alternative of
interest can be selected. If <var>alt</var> is <code>"!="</code> or
<code>"<>"</code>, the null is tested against the two-sided alternative
<var>p1</var> != <var>p2</var>. If <var>alt</var> is <code>">"</code>, the one-sided
alternative <var>p1</var> > <var>p2</var> is used. Similarly for <code>"<"</code>,
the one-sided alternative <var>p1</var> < <var>p2</var> is used.
The default is the two-sided case.
</p>
<p>The p-value of the test is returned in <var>pval</var>.
</p>
<p>If no output argument is given, the p-value of the test is displayed.
</p></dd></dl>
<a name="XREFrun_005ftest"></a><dl>
<dt><a name="index-run_005ftest"></a>Function File: <em>[<var>pval</var>, <var>chisq</var>] =</em> <strong>run_test</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Perform a chi-square test with 6 degrees of freedom based on the
upward runs in the columns of <var>x</var>. Can be used to test whether
<var>x</var> contains independent data.
</p>
<p>The p-value of the test is returned in <var>pval</var>.
</p>
<p>If no output argument is given, the p-value is displayed.
</p></dd></dl>
<a name="XREFsign_005ftest"></a><dl>
<dt><a name="index-sign_005ftest"></a>Function File: <em>[<var>pval</var>, <var>b</var>, <var>n</var>] =</em> <strong>sign_test</strong> <em>(<var>x</var>, <var>y</var>, <var>alt</var>)</em></dt>
<dd><p>For two matched-pair samples <var>x</var> and <var>y</var>, perform a sign test
of the null hypothesis PROB (<var>x</var> > <var>y</var>) == PROB (<var>x</var> <
<var>y</var>) == 1/2. Under the null, the test statistic <var>b</var> roughly
follows a binomial distribution with parameters <code><var>n</var> = sum
(<var>x</var> != <var>y</var>)</code> and <var>p</var> = 1/2.
</p>
<p>With the optional argument <code>alt</code>, the alternative of interest
can be selected. If <var>alt</var> is <code>"!="</code> or <code>"<>"</code>, the
null hypothesis is tested against the two-sided alternative PROB
(<var>x</var> < <var>y</var>) != 1/2. If <var>alt</var> is <code>">"</code>, the
one-sided alternative PROB (<var>x</var> > <var>y</var>) > 1/2 ("x is
stochastically greater than y") is considered. Similarly for
<code>"<"</code>, the one-sided alternative PROB (<var>x</var> > <var>y</var>) < 1/2
("x is stochastically less than y") is considered. The default is
the two-sided case.
</p>
<p>The p-value of the test is returned in <var>pval</var>.
</p>
<p>If no output argument is given, the p-value of the test is displayed.
</p></dd></dl>
<a name="XREFt_005ftest"></a><dl>
<dt><a name="index-t_005ftest"></a>Function File: <em>[<var>pval</var>, <var>t</var>, <var>df</var>] =</em> <strong>t_test</strong> <em>(<var>x</var>, <var>m</var>, <var>alt</var>)</em></dt>
<dd><p>For a sample <var>x</var> from a normal distribution with unknown mean and
variance, perform a t-test of the null hypothesis <code>mean
(<var>x</var>) == <var>m</var></code>. Under the null, the test statistic <var>t</var>
follows a Student distribution with <code><var>df</var> = length (<var>x</var>)
- 1</code> degrees of freedom.
</p>
<p>With the optional argument string <var>alt</var>, the alternative of
interest can be selected. If <var>alt</var> is <code>"!="</code> or
<code>"<>"</code>, the null is tested against the two-sided alternative
<code>mean (<var>x</var>) != <var>m</var></code>. If <var>alt</var> is <code>">"</code>, the
one-sided alternative <code>mean (<var>x</var>) > <var>m</var></code> is considered.
Similarly for <var>"<"</var>, the one-sided alternative <code>mean
(<var>x</var>) < <var>m</var></code> is considered. The default is the two-sided
case.
</p>
<p>The p-value of the test is returned in <var>pval</var>.
</p>
<p>If no output argument is given, the p-value of the test is displayed.
</p></dd></dl>
<a name="XREFt_005ftest_005f2"></a><dl>
<dt><a name="index-t_005ftest_005f2"></a>Function File: <em>[<var>pval</var>, <var>t</var>, <var>df</var>] =</em> <strong>t_test_2</strong> <em>(<var>x</var>, <var>y</var>, <var>alt</var>)</em></dt>
<dd><p>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
<var>t</var> follows a Student distribution with <var>df</var> degrees of
freedom.
</p>
<p>With the optional argument string <var>alt</var>, the alternative of
interest can be selected. If <var>alt</var> is <code>"!="</code> or
<code>"<>"</code>, the null is tested against the two-sided alternative
<code>mean (<var>x</var>) != mean (<var>y</var>)</code>. If <var>alt</var> is <code>">"</code>,
the one-sided alternative <code>mean (<var>x</var>) > mean (<var>y</var>)</code> is
used. Similarly for <code>"<"</code>, the one-sided alternative <code>mean
(<var>x</var>) < mean (<var>y</var>)</code> is used. The default is the two-sided
case.
</p>
<p>The p-value of the test is returned in <var>pval</var>.
</p>
<p>If no output argument is given, the p-value of the test is displayed.
</p></dd></dl>
<a name="XREFt_005ftest_005fregression"></a><dl>
<dt><a name="index-t_005ftest_005fregression"></a>Function File: <em>[<var>pval</var>, <var>t</var>, <var>df</var>] =</em> <strong>t_test_regression</strong> <em>(<var>y</var>, <var>x</var>, <var>rr</var>, <var>r</var>, <var>alt</var>)</em></dt>
<dd><p>Perform a t test for the null hypothesis
<code><var>rr</var> * <var>b</var> = <var>r</var></code> in a classical normal
regression model <code><var>y</var> = <var>x</var> * <var>b</var> + <var>e</var></code>. Under the
null, the test statistic <var>t</var> follows a <var>t</var> distribution with
<var>df</var> degrees of freedom.
</p>
<p>If <var>r</var> is omitted, a value of 0 is assumed.
</p>
<p>With the optional argument string <var>alt</var>, the alternative of
interest can be selected. If <var>alt</var> is <code>"!="</code> or
<code>"<>"</code>, the null is tested against the two-sided alternative
<code><var>rr</var> * <var>b</var> != <var>r</var></code>. If <var>alt</var> is <code>">"</code>,
the one-sided alternative
<code><var>rr</var> * <var>b</var> > <var>r</var></code> is used. Similarly for
<var>"<"</var>, the one-sided alternative
<code><var>rr</var> * <var>b</var> < <var>r</var></code> is used. The default is the
two-sided case.
</p>
<p>The p-value of the test is returned in <var>pval</var>.
</p>
<p>If no output argument is given, the p-value of the test is displayed.
</p></dd></dl>
<a name="XREFu_005ftest"></a><dl>
<dt><a name="index-u_005ftest"></a>Function File: <em>[<var>pval</var>, <var>z</var>] =</em> <strong>u_test</strong> <em>(<var>x</var>, <var>y</var>, <var>alt</var>)</em></dt>
<dd><p>For two samples <var>x</var> and <var>y</var>, perform a Mann-Whitney U-test of
the null hypothesis PROB (<var>x</var> > <var>y</var>) == 1/2 == PROB (<var>x</var>
< <var>y</var>). Under the null, the test statistic <var>z</var> approximately
follows a standard normal distribution. Note that this test is
equivalent to the Wilcoxon rank-sum test.
</p>
<p>With the optional argument string <var>alt</var>, the alternative of
interest can be selected. If <var>alt</var> is <code>"!="</code> or
<code>"<>"</code>, the null is tested against the two-sided alternative
PROB (<var>x</var> > <var>y</var>) != 1/2. If <var>alt</var> is <code>">"</code>, the
one-sided alternative PROB (<var>x</var> > <var>y</var>) > 1/2 is considered.
Similarly for <code>"<"</code>, the one-sided alternative PROB (<var>x</var> >
<var>y</var>) < 1/2 is considered. The default is the two-sided case.
</p>
<p>The p-value of the test is returned in <var>pval</var>.
</p>
<p>If no output argument is given, the p-value of the test is displayed.
</p></dd></dl>
<a name="XREFvar_005ftest"></a><dl>
<dt><a name="index-var_005ftest"></a>Function File: <em>[<var>pval</var>, <var>f</var>, <var>df_num</var>, <var>df_den</var>] =</em> <strong>var_test</strong> <em>(<var>x</var>, <var>y</var>, <var>alt</var>)</em></dt>
<dd><p>For two samples <var>x</var> and <var>y</var> 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
<var>f</var> follows an F-distribution with <var>df_num</var> and <var>df_den</var>
degrees of freedom.
</p>
<p>With the optional argument string <var>alt</var>, the alternative of
interest can be selected. If <var>alt</var> is <code>"!="</code> or
<code>"<>"</code>, the null is tested against the two-sided alternative
<code>var (<var>x</var>) != var (<var>y</var>)</code>. If <var>alt</var> is <code>">"</code>,
the one-sided alternative <code>var (<var>x</var>) > var (<var>y</var>)</code> is
used. Similarly for "<", the one-sided alternative <code>var
(<var>x</var>) > var (<var>y</var>)</code> is used. The default is the two-sided
case.
</p>
<p>The p-value of the test is returned in <var>pval</var>.
</p>
<p>If no output argument is given, the p-value of the test is displayed.
</p></dd></dl>
<a name="XREFwelch_005ftest"></a><dl>
<dt><a name="index-welch_005ftest"></a>Function File: <em>[<var>pval</var>, <var>t</var>, <var>df</var>] =</em> <strong>welch_test</strong> <em>(<var>x</var>, <var>y</var>, <var>alt</var>)</em></dt>
<dd><p>For two samples <var>x</var> and <var>y</var> 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 <var>t</var> approximately follows a
Student distribution with <var>df</var> degrees of freedom.
</p>
<p>With the optional argument string <var>alt</var>, the alternative of
interest can be selected. If <var>alt</var> is <code>"!="</code> or
<code>"<>"</code>, the null is tested against the two-sided alternative
<code>mean (<var>x</var>) != <var>m</var></code>. If <var>alt</var> is <code>">"</code>, the
one-sided alternative mean(x) > <var>m</var> is considered. Similarly for
<code>"<"</code>, the one-sided alternative mean(x) < <var>m</var> is
considered. The default is the two-sided case.
</p>
<p>The p-value of the test is returned in <var>pval</var>.
</p>
<p>If no output argument is given, the p-value of the test is displayed.
</p></dd></dl>
<a name="XREFwilcoxon_005ftest"></a><dl>
<dt><a name="index-wilcoxon_005ftest"></a>Function File: <em>[<var>pval</var>, <var>z</var>] =</em> <strong>wilcoxon_test</strong> <em>(<var>x</var>, <var>y</var>, <var>alt</var>)</em></dt>
<dd><p>For two matched-pair sample vectors <var>x</var> and <var>y</var>, perform a
Wilcoxon signed-rank test of the null hypothesis PROB (<var>x</var> >
<var>y</var>) == 1/2. Under the null, the test statistic <var>z</var>
approximately follows a standard normal distribution when <var>n</var> > 25.
</p>
<p><strong>Caution:</strong> This function assumes a normal distribution for <var>z</var>
and thus is invalid for <var>n</var> ≤ 25.
</p>
<p>With the optional argument string <var>alt</var>, the alternative of
interest can be selected. If <var>alt</var> is <code>"!="</code> or
<code>"<>"</code>, the null is tested against the two-sided alternative
PROB (<var>x</var> > <var>y</var>) != 1/2. If alt is <code>">"</code>, the one-sided
alternative PROB (<var>x</var> > <var>y</var>) > 1/2 is considered. Similarly
for <code>"<"</code>, the one-sided alternative PROB (<var>x</var> > <var>y</var>) <
1/2 is considered. The default is the two-sided case.
</p>
<p>The p-value of the test is returned in <var>pval</var>.
</p>
<p>If no output argument is given, the p-value of the test is displayed.
</p></dd></dl>
<a name="XREFz_005ftest"></a><dl>
<dt><a name="index-z_005ftest"></a>Function File: <em>[<var>pval</var>, <var>z</var>] =</em> <strong>z_test</strong> <em>(<var>x</var>, <var>m</var>, <var>v</var>, <var>alt</var>)</em></dt>
<dd><p>Perform a Z-test of the null hypothesis <code>mean (<var>x</var>) ==
<var>m</var></code> for a sample <var>x</var> from a normal distribution with unknown
mean and known variance <var>v</var>. Under the null, the test statistic
<var>z</var> follows a standard normal distribution.
</p>
<p>With the optional argument string <var>alt</var>, the alternative of
interest can be selected. If <var>alt</var> is <code>"!="</code> or
<code>"<>"</code>, the null is tested against the two-sided alternative
<code>mean (<var>x</var>) != <var>m</var></code>. If <var>alt</var> is <code>">"</code>, the
one-sided alternative <code>mean (<var>x</var>) > <var>m</var></code> is considered.
Similarly for <code>"<"</code>, the one-sided alternative <code>mean
(<var>x</var>) < <var>m</var></code> is considered. The default is the two-sided
case.
</p>
<p>The p-value of the test is returned in <var>pval</var>.
</p>
<p>If no output argument is given, the p-value of the test is displayed
along with some information.
</p></dd></dl>
<a name="XREFz_005ftest_005f2"></a><dl>
<dt><a name="index-z_005ftest_005f2"></a>Function File: <em>[<var>pval</var>, <var>z</var>] =</em> <strong>z_test_2</strong> <em>(<var>x</var>, <var>y</var>, <var>v_x</var>, <var>v_y</var>, <var>alt</var>)</em></dt>
<dd><p>For two samples <var>x</var> and <var>y</var> from normal distributions with
unknown means and known variances <var>v_x</var> and <var>v_y</var>, perform a
Z-test of the hypothesis of equal means. Under the null, the test
statistic <var>z</var> follows a standard normal distribution.
</p>
<p>With the optional argument string <var>alt</var>, the alternative of
interest can be selected. If <var>alt</var> is <code>"!="</code> or
<code>"<>"</code>, the null is tested against the two-sided alternative
<code>mean (<var>x</var>) != mean (<var>y</var>)</code>. If alt is <code>">"</code>, the
one-sided alternative <code>mean (<var>x</var>) > mean (<var>y</var>)</code> is used.
Similarly for <code>"<"</code>, the one-sided alternative <code>mean
(<var>x</var>) < mean (<var>y</var>)</code> is used. The default is the two-sided
case.
</p>
<p>The p-value of the test is returned in <var>pval</var>.
</p>
<p>If no output argument is given, the p-value of the test is displayed
along with some information.
</p></dd></dl>
<hr>
<div class="header">
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Next: <a href="Random-Number-Generation.html#Random-Number-Generation" accesskey="n" rel="next">Random Number Generation</a>, Previous: <a href="Distributions.html#Distributions" accesskey="p" rel="prev">Distributions</a>, Up: <a href="Statistics.html#Statistics" accesskey="u" rel="up">Statistics</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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