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<a name="Tests"></a>
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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>: <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).
</p>
<p>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>
<p><strong>See also:</strong> <a href="#XREFmanova">manova</a>.
</p></dd></dl>


<a name="XREFbartlett_005ftest"></a><dl>
<dt><a name="index-bartlett_005ftest"></a>: <em>[<var>pval</var>, <var>chisq</var>, <var>df</var>] =</em> <strong>bartlett_test</strong> <em>(<var>x1</var>, &hellip;)</em></dt>
<dd><p>Perform a Bartlett test for the homogeneity of variances in the data
vectors <var>x1</var>, <var>x2</var>, &hellip;, <var>xk</var>, where <var>k</var> &gt; 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>: <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>: <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>.
</p>
<p>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>: <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>&quot;!=&quot;</code> or <code>&quot;&lt;&gt;&quot;</code> (nonzero), <code>&quot;&gt;&quot;</code>
(greater than 0), or <code>&quot;&lt;&quot;</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>&quot;pearson&quot;</code>
(default), the (usual) Pearson&rsquo;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>&quot;kendall&quot;</code>, then Kendall&rsquo;s rank
correlation tau is used.  If <var>method</var> is <code>&quot;spearman&quot;</code>, then
Spearman&rsquo;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>: <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>: <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&rsquo;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>: <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&rsquo;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>: <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 <var>dist</var>.
</p>
<p>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, &quot;unif&quot;, 2, 4)
</pre></div>

<p><var>dist</var> can be any string for which a function <var>distcdf</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>&quot;!=&quot;</code> or <code>&quot;&lt;&gt;&quot;</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>&quot;&gt;&quot;</code>, the one-sided alternative F &gt;
G is considered.  Similarly for <code>&quot;&lt;&quot;</code>, the one-sided alternative F &gt;
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>: <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.
</p>
<p>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>&quot;!=&quot;</code> or <code>&quot;&lt;&gt;&quot;</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>&quot;&gt;&quot;</code>, the one-sided alternative F &gt;
G is considered.  Similarly for <code>&quot;&lt;&quot;</code>, the one-sided alternative F &lt;
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>: <em>[<var>pval</var>, <var>k</var>, <var>df</var>] =</em> <strong>kruskal_wallis_test</strong> <em>(<var>x1</var>, &hellip;)</em></dt>
<dd><p>Perform a Kruskal-Wallis one-factor analysis of variance.
</p>
<p>Suppose a variable is observed for <var>k</var> &gt; 1 different groups, and let
<var>x1</var>, &hellip;, <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>: <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).
</p>
<p>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&rsquo; Lambda) and approximate p-values are
computed and displayed.
</p>
<p><strong>See also:</strong> <a href="#XREFanova">anova</a>.
</p></dd></dl>


<a name="XREFmcnemar_005ftest"></a><dl>
<dt><a name="index-mcnemar_005ftest"></a>: <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&rsquo;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>: <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.
</p>
<p>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>&quot;!=&quot;</code> or <code>&quot;&lt;&gt;&quot;</code>, the null
is tested against the two-sided alternative <var>p1</var> != <var>p2</var>.  If
<var>alt</var> is <code>&quot;&gt;&quot;</code>, the one-sided alternative <var>p1</var> &gt; <var>p2</var> is
used.  Similarly for <code>&quot;&lt;&quot;</code>, the one-sided alternative
<var>p1</var> &lt; <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>: <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>.
</p>
<p><code>run_test</code> can be used to decide 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>: <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> &gt; <var>y</var>) == PROB (<var>x</var> &lt; <var>y</var>) == 1/2.
</p>
<p>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>&quot;!=&quot;</code> or <code>&quot;&lt;&gt;&quot;</code>, the null
hypothesis is tested against the two-sided alternative
PROB (<var>x</var> &lt; <var>y</var>) != 1/2.  If <var>alt</var> is <code>&quot;&gt;&quot;</code>, the one-sided
alternative PROB (<var>x</var> &gt; <var>y</var>) &gt; 1/2 (&quot;x is stochastically greater
than y&quot;) is considered.  Similarly for <code>&quot;&lt;&quot;</code>, the one-sided
alternative PROB (<var>x</var> &gt; <var>y</var>) &lt; 1/2 (&quot;x is stochastically less than
y&quot;) 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>: <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>.
</p>
<p>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>&quot;!=&quot;</code> or <code>&quot;&lt;&gt;&quot;</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>&quot;&gt;&quot;</code>, the one-sided alternative
<code>mean (<var>x</var>) &gt; <var>m</var></code> is considered.  Similarly for <var>&quot;&lt;&quot;</var>,
the one-sided alternative <code>mean (<var>x</var>) &lt; <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>: <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.
</p>
<p>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>&quot;!=&quot;</code> or <code>&quot;&lt;&gt;&quot;</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>&quot;&gt;&quot;</code>, the one-sided alternative
<code>mean (<var>x</var>) &gt; mean (<var>y</var>)</code> is used.  Similarly for
<code>&quot;&lt;&quot;</code>, the one-sided alternative <code>mean (<var>x</var>) &lt; 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>: <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>.
</p>
<p>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>&quot;!=&quot;</code> or <code>&quot;&lt;&gt;&quot;</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>&quot;&gt;&quot;</code>, the one-sided
alternative <code><var>rr</var> * <var>b</var> &gt; <var>r</var></code> is used.
Similarly for <var>&quot;&lt;&quot;</var>, the one-sided alternative <code><var>rr</var>
* <var>b</var> &lt; <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>: <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> &gt; <var>y</var>) == 1/2 == PROB (<var>x</var> &lt; <var>y</var>).
</p>
<p>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>&quot;!=&quot;</code> or <code>&quot;&lt;&gt;&quot;</code>, the null
is tested against the two-sided alternative
PROB (<var>x</var> &gt; <var>y</var>) != 1/2.  If <var>alt</var> is <code>&quot;&gt;&quot;</code>, the one-sided
alternative PROB (<var>x</var> &gt; <var>y</var>) &gt; 1/2 is considered.  Similarly for
<code>&quot;&lt;&quot;</code>, the one-sided alternative PROB (<var>x</var> &gt; <var>y</var>) &lt; 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>: <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.
</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>With the optional argument string <var>alt</var>, the alternative of interest
can be selected.  If <var>alt</var> is <code>&quot;!=&quot;</code> or <code>&quot;&lt;&gt;&quot;</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>&quot;&gt;&quot;</code>, the one-sided alternative
<code>var (<var>x</var>) &gt; var (<var>y</var>)</code> is used.  Similarly for &quot;&lt;&quot;, the
one-sided alternative <code>var (<var>x</var>) &gt; 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>: <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.
</p>
<p>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>&quot;!=&quot;</code> or <code>&quot;&lt;&gt;&quot;</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>&quot;&gt;&quot;</code>, the
one-sided alternative mean(x) &gt; <var>m</var> is considered.  Similarly for
<code>&quot;&lt;&quot;</code>, the one-sided alternative mean(x) &lt; <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>: <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> &gt; <var>y</var>) == 1/2.
</p>
<p>Under the null, the test statistic <var>z</var> approximately follows a
standard normal distribution when <var>n</var> &gt; 25.
</p>
<p><strong>Caution:</strong> This function assumes a normal distribution for <var>z</var>
and thus is invalid for <var>n</var> &le; 25.
</p>
<p>With the optional argument string <var>alt</var>, the alternative of interest
can be selected.  If <var>alt</var> is <code>&quot;!=&quot;</code> or <code>&quot;&lt;&gt;&quot;</code>, the null
is tested against the two-sided alternative
PROB (<var>x</var> &gt; <var>y</var>) != 1/2.  If alt is <code>&quot;&gt;&quot;</code>, the one-sided
alternative PROB (<var>x</var> &gt; <var>y</var>) &gt; 1/2 is considered.  Similarly for
<code>&quot;&lt;&quot;</code>, the one-sided alternative PROB (<var>x</var> &gt; <var>y</var>) &lt; 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>: <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>.
</p>
<p>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>&quot;!=&quot;</code> or <code>&quot;&lt;&gt;&quot;</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>&quot;&gt;&quot;</code>, the
one-sided alternative <code>mean (<var>x</var>) &gt; <var>m</var></code> is considered.
Similarly for <code>&quot;&lt;&quot;</code>, the one-sided alternative
<code>mean (<var>x</var>) &lt; <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>: <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.
</p>
<p>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>&quot;!=&quot;</code> or <code>&quot;&lt;&gt;&quot;</code>, the null
is tested against the two-sided alternative
<code>mean (<var>x</var>) != mean (<var>y</var>)</code>.  If alt is <code>&quot;&gt;&quot;</code>, the
one-sided alternative <code>mean (<var>x</var>) &gt; mean (<var>y</var>)</code> is used.
Similarly for <code>&quot;&lt;&quot;</code>, the one-sided alternative
<code>mean (<var>x</var>) &lt; 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>


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