The routine may be called by the names g08aaf or nagf_nonpar_test_sign.
3Description
The Sign test investigates the median difference between pairs of scores from two matched samples of size $n$, denoted by $\{{x}_{\mathit{i}},{y}_{\mathit{i}}\}$, for $\mathit{i}=1,2,\dots ,n$. 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).
g08aaf computes:
(a)the test statistic $S$, which is the number of pairs for which ${x}_{i}<{y}_{i}$;
(b)the number ${n}_{1}$ of non-tied pairs $({x}_{i}\ne {y}_{i})$;
(c)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 $\text{}\le S$ if $S<\frac{1}{2}{n}_{1}$, or of observing a value $\text{}<S$ 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
(i)${H}_{1}$: median of $x\ne \text{}$ median of $y$. ${H}_{0}$ is rejected if $2\times \mathrm{min}\phantom{\rule{0.125em}{0ex}}(p,1-p)<\alpha $.
(ii)${H}_{1}$: median of $x>\text{}$ median of $y$. ${H}_{0}$ is rejected if $p<\alpha $.
(iii)${H}_{1}$: median of $x<\text{}$ median of $y$. ${H}_{0}$ is rejected if $1-p<\alpha $.
4References
Siegel S (1956) Non-parametric Statistics for the Behavioral Sciences McGraw–Hill
5Arguments
1: $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
2: $\mathbf{y}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{x}}\left(\mathit{i}\right)$ and ${\mathbf{y}}\left(\mathit{i}\right)$ must be set to the $\mathit{i}$th pair of data values, $\{{x}_{\mathit{i}},{y}_{\mathit{i}}\}$, for $\mathit{i}=1,2,\dots ,n$.
3: $\mathbf{n}$ – IntegerInput
On entry: $n$, the size of each sample.
Constraint:
${\mathbf{n}}\ge 1$.
4: $\mathbf{isgn}$ – IntegerOutput
On exit: the Sign test statistic, $S$.
5: $\mathbf{n1}$ – IntegerOutput
On exit: the number of non-tied pairs, ${n}_{1}$.
6: $\mathbf{p}$ – Real (Kind=nag_wp)Output
On exit: the lower tail probability, $p$, corresponding to $S$.
7: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=2$
On entry, the samples are identical, i.e., ${n}_{1}=0$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
The tail probability, $p$, is computed using the relationship between the binomial and beta distributions. For ${n}_{1}<120$, $p$ should be accurate to at least $4$ significant figures, assuming that the machine has a precision of $7$ or more digits. For ${n}_{1}\ge 120$, $p$ should be computed with an absolute error of less than $0.005$. For further details see g01eef.
8Parallelism and Performance
g08aaf is not threaded in any implementation.
9Further Comments
The time taken by g08aaf is small, and increases with $n$.
10Example
This example is taken from page 69 of Siegel (1956). The data relates to ratings of ‘insight into paternal discipline’ for $17$ sets of parents, recorded on a scale from $1$ to $5$.