naginterfaces.library.nonpar.test_​sign

naginterfaces.library.nonpar.test_sign(x, y)

test_sign performs the Sign test on two related samples of size $n$.

For full information please refer to the NAG Library document for g08aa

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g08/g08aaf.html

Parameters

xfloat, array-like, shape $\left(n\right)$

$x[\text{i}-1]$ and $y[\text{i}-1]$ must be set to the $\text{i}$th pair of data values, $\{x_{\text{i}},y_{\text{i}}\}$, for $\text{i}=1,2,\dots ,n$.

yfloat, array-like, shape $\left(n\right)$

$x[\text{i}-1]$ and $y[\text{i}-1]$ must be set to the $\text{i}$th pair of data values, $\{x_{\text{i}},y_{\text{i}}\}$, for $\text{i}=1,2,\dots ,n$.

Returns

isgnint

The Sign test statistic, $S$.

n1int

The number of non-tied pairs, $n_1$.

pfloat

The lower tail probability, $p$, corresponding to $S$.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

(errno $2$)

On entry, the samples are identical, i.e., $n_1=0$.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

The Sign test investigates the median difference between pairs of scores from two matched samples of size $n$, denoted by $\{x_{\text{i}},y_{\text{i}}\}$, for $\text{i}=1,2,\dots ,n$. The hypothesis under test, $H_0$, often called the null hypothesis, is that the medians are the same, and this is to be tested against a one - or two-sided alternative $H_1$ (see below).

test_sign computes:

1. the test statistic $S$, which is the number of pairs for which $x_i<y_i$;

2. the number $n_1$ of non-tied pairs $(x_i\ne y_i)$;

3. the lower tail probability $p$ corresponding to $S$ (adjusted to allow the complement $(1-p)$ to be used in an upper one tailed or a two tailed test). $p$ is the probability of observing a value $\le S$ if $S<\frac{1}{2}{n}_1$, or of observing a value $<S$ if $S>\frac{1}{2}{n}_1$, given that $H_0$ is true. If $S=\frac{1}{2}{n}_1$, $p$ is set to $0.5$.

Suppose that a significance test of a chosen size $\alpha$ is to be performed (i.e., $\alpha$ is the probability of rejecting $H_0$ when $H_0$ is true; typically $\alpha$ is a small quantity such as $0.05$ or $0.01$). The returned value of $p$ can be used to perform a significance test on the median difference, against various alternative hypotheses $H_1$, as follows

1. $H_1$: median of $x\ne$ median of $y$. $H_0$ is rejected if $2\times min(p,1-p)<\alpha$.

2. $H_1$: median of $x>$ median of $y$. $H_0$ is rejected if $p<\alpha$.

3. $H_1$: median of $x<$ median of $y$. $H_0$ is rejected if $1-p<\alpha$.

References

Siegel, S, 1956, Non-parametric Statistics for the Behavioral Sciences, McGraw–Hill