naginterfaces.library.contab.tabulate_​percentile

naginterfaces.library.contab.tabulate_percentile(typ, isf, lfac, ifac, percnt, y, maxt, wt=None)

tabulate_percentile computes a table from a set of classification factors using a given percentile or quantile, for example the median.

For full information please refer to the NAG Library document for g11bb

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g11/g11bbf.html

Parameters

typstr, length 1

Indicates if the variable to be tabulated is discrete or continuous.

$typ=\text{'D'}$

The percentiles are computed for a discrete variable.

$typ=\text{'C'}$

The percentiles are computed for a continuous variable using linear interpolation.

isfint, array-like, shape $\left(\text{nfac}\right)$

Indicates which factors in $ifac$ are to be used in the tabulation.

If $isf[i-1]>0$ the $i$th factor in $ifac$ is included in the tabulation.

Note that if $isf[\text{i}-1]\le 0$, for $\text{i}=1,2,\dots ,\text{nfac}$ then the statistic for the whole sample is calculated and returned in a $1\times 1$ table.

lfacint, array-like, shape $\left(\text{nfac}\right)$

The number of levels of the classifying factors in $ifac$.

ifacint, array-like, shape $(n,\text{nfac})$

The $\text{nfac}$ coded classification factors for the $\text{n}$ observations.

percntfloat

$p$, the percentile to be tabulated.

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

The variable to be tabulated.

maxtint

The maximum size of the table to be computed.

wtNone or float, array-like, shape $\left(n\right)$, optional

If $\text{weight}=\text{'W'}$, $wt$ must contain the $\text{n}$ weights. Otherwise $wt$ is not referenced.

Returns

tablefloat, ndarray, shape $\left(maxt\right)$

The computed table. The $ncells$ cells of the table are stored so that for any two factors the index relating to the factor occurring later in $lfac$ and $ifac$ changes faster. For further details see Further Comments.

ncellsint

The number of cells in the table.

ndimint

The number of factors defining the table.

idimint, ndarray, shape $\left(\text{nfac}\right)$

The first $ndim$ elements contain the number of levels for the factors defining the table.

icountint, ndarray, shape $\left(maxt\right)$

A table containing the number of observations contributing to each cell of the table, stored identically to $table$.

Raises

NagValueError

(errno $1$)

On entry, $percnt=⟨value⟩$.

Constraint: $0.0<p<100.0$.

(errno $1$)

On entry, $typ=⟨value⟩$.

Constraint: $typ=\text{'D'}$ or $\text{'C'}$.

(errno $1$)

On entry, $\text{nfac}=⟨value⟩$.

Constraint: $\text{nfac}\ge 1$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 2$.

(errno $2$)

On entry, $maxt=⟨value⟩$ and minimum value for $maxt=⟨value⟩$.

Constraint: $maxt\ge \text{product}$ of the levels of the factors included in the tabulation.

(errno $2$)

On entry, $i=⟨value⟩$ and $wt[i-1]=⟨value⟩$.

Constraint: $wt[i-1]\ge 0.0$.

(errno $2$)

On entry, $i=⟨value⟩$, $j=⟨value⟩$, $lfac[j-1]=⟨value⟩$ and $ifac[i-1,j-1]=⟨value⟩$.

Constraint: $ifac[i-1,j-1]\le lfac[j-1]$.

(errno $2$)

On entry, $i=⟨value⟩$, $j=⟨value⟩$ and $ifac[i-1,j-1]=⟨value⟩$.

Constraint: $ifac[i-1,j-1]>0$.

(errno $2$)

On entry, $i=⟨value⟩$ and $lfac[i-1]=⟨value⟩$.

On entry, $lfac[i-1]>2$.

(errno $3$)

Some cells of the table are empty.

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.

A dataset may include both classification variables and general variables. The classification variables, known as factors, take a small number of values known as levels. For example, the factor sex would have the levels male and female. These can be coded as $1$ and $2$ respectively. Given several factors, a multi-way table can be constructed such that each cell of the table represents one level from each factor. For example, the two factors sex and habitat, habitat having three levels (inner-city, suburban and rural) define the $2\times 3$ contingency table

[table omitted]

For each cell statistics can be computed. If a third variable in the dataset was age then for each cell the median age could be computed:

[table omitted]

That is, the median age for all observations for males living in rural areas is $37$, the median being the 50% quantile. Other quantiles can also be computed: the $p$ percent quantile or percentile, $q_p$, is the estimate of the value such that $p$ percent of observations are less than $q_p$. This is calculated in two different ways depending on whether the tabulated variable is continuous or discrete. Let there be $m$ values in a cell and let $y_{\left(1\right)}$, $y_{\left(2\right)},\dots ,y_{\left(m\right)}$ be the values for that cell sorted into ascending order. Also, associated with each value there is a weight, $w_{\left(1\right)}$, $w_{\left(2\right)},\dots ,w_{\left(m\right)}$, which could represent the observed frequency for that value, with $W_j={\sum }_{i=1}^j{w}_{\left(i\right)}$ and $W_j^{\prime }={\sum }_{i=1}^j{w}_{\left(i\right)}-\frac{1}{2}{w}_{\left(j\right)}$. For the $p$ percentile let $p_w=\left(p/100\right)W_m$ and $p_w^{\prime }=\left(p/100\right)W_m^{\prime }$, then the percentiles for the two cases are as given below.

If the variable is discrete, that is, it takes only a limited number of (usually integer) values, then the percentile is defined as

$$\begin{array}{c}\begin{array}{cc}{y}_{\left(j\right)}& \text{if}{W}_{j-1}<p_W<W_j\\ & \\ \frac{y_{(j+1)}+y_{\left(j\right)}}{2}& \text{if}{p}_w=W_j\text{.}\end{array}\end{array}$$

If the data is continuous then the quantiles are estimated by linear interpolation.

$$\begin{array}{c}\begin{array}{cc}{y}_{\left(1\right)}& \text{if}{p}_w^{\prime }\le W_1^{\prime }\\ & \\ (1-f)y_{(j-1)}+f{y}_{\left(j\right)}& \text{if}{W}_{j-1}^{\prime }<p_w^{\prime }\le W_j^{\prime }\\ & \\ y_{\left(m\right)}& \text{if}{p}_w^{\prime }>W_m^{\prime }\text{,}\end{array}\end{array}$$

where $f=(p_w^{\prime }-W_{j-1}^{\prime })/(W_j^{\prime }-W_{j-1}^{\prime })$.

References

John, J A and Quenouille, M H, 1977, Experiments: Design and Analysis, Griffin

Kendall, M G and Stuart, A, 1969, The Advanced Theory of Statistics (Volume 1), (3rd Edition), Griffin