nag_contab_tabulate_percentile (g11bb) computes a table from a set of classification factors using a given percentile or quantile, for example the median.

Syntax

[table, ncells, ndim, idim, icount, ifail] = g11bb(typ, isf, lfac, ifac, percnt, y, maxt, 'n', n, 'nfac', nfac, 'wt', wt)

[table, ncells, ndim, idim, icount, ifail] = nag_contab_tabulate_percentile(typ, isf, lfac, ifac, percnt, y, maxt, 'n', n, 'nfac', nfac, 'wt', wt)

Note: the interface to this routine has changed since earlier releases of the toolbox:

At Mark 24:

weight was removed from the interface; wt was made optional

Description

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

Sex	Habitat
	Inner-city	Suburban	Rural
Male
Female

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:

Sex	Habitat
	Inner-city	Suburban	Rural
Male	24	31	37
Female	21.5	28.5	33

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_{(1)}

y_{(2)}, \dots, y_{(m)}

be the values for that cell sorted into ascending order. Also, associated with each value there is a weight,

w_{(1)}

w_{(2)}, \dots, w_{(m)}

, which could represent the observed frequency for that value, with

W_{j} = \sum_{i = 1}^{j} w_{(i)}

and

W_{j}^{'} = \sum_{i = 1}^{j} w_{(i)} - \frac{1}{2} w_{(j)}

. For the

p

percentile let

p_{w} = (p / 100) W_{m}

and

p_{w}^{'} = (p / 100) W_{m}^{'}

, 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}{l} y_{(j)} & if ​ W_{j - 1} < p_{W} < W_{j} \\ \frac{y_{(j + 1)} + y_{(j)}}{2} & if ​ p_{w} = W_{j} . \end{array}

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

\begin{array}{l} y_{(1)} & if ​ p_{w}^{'} \leq W_{1}^{'} \\ (1 - f) y_{(j - 1)} + f y_{(j)} & if ​ W_{j - 1}^{'} < p_{w}^{'} \leq W_{j}^{'} \\ y_{(m)} & if ​ p_{w}^{'} > W_{m}^{'}, \end{array}

where

f = (p_{w}^{'} - W_{j - 1}^{'}) / (W_{j}^{'} - W_{j - 1}^{'})

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

Parameters

Compulsory Input Parameters

1: $typ$ – string (length ≥ 1)

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

$typ ='D'$: The percentiles are computed for a discrete variable.
$typ ='C'$: The percentiles are computed for a continuous variable using linear interpolation.

Constraint:

typ ='D'

'C'

2: $isf (nfac)$ – int64int32nag_int array

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

isf (i) > 0

the

i

th factor in ifac is included in the tabulation.

Note that if

isf (i) \leq 0

, for

i = 1, 2, \dots, nfac

then the statistic for the whole sample is calculated and returned in a

1 \times 1

table.

3: $lfac (nfac)$ – int64int32nag_int array

The number of levels of the classifying factors in ifac.

Constraint: if

isf (i) > 0

lfac (i) \geq 2

, for

i = 1, 2, \dots, nfac

4: $ifac (ldf, nfac)$ – int64int32nag_int array

ldf, the first dimension of the array, must satisfy the constraint

ldf \geq n

The nfac coded classification factors for the n observations.

Constraint:

1 \leq ifac (i, j) \leq lfac (j)

, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, nfac

5: $percnt$ – double scalar

p

, the percentile to be tabulated.

Constraint:

0.0 < p < 100.0

6: $y (n)$ – double array

The variable to be tabulated.

7: $maxt$ – int64int32nag_int scalar

The maximum size of the table to be computed.

Constraint:

maxt \geq

product of the levels of the factors included in the tabulation.

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the dimension of the array y and the first dimension of the array ifac. (An error is raised if these dimensions are not equal.)
The number of observations.

Constraint: $n \geq 2$ .
2: $nfac$ – int64int32nag_int scalar: Default: the dimension of the arrays isf, lfac and the second dimension of the array ifac. (An error is raised if these dimensions are not equal.)
The number of classifying factors in ifac.

Constraint: $nfac \geq 1$ .
3: $wt (:)$ – double array: The dimension of the array wt must be at least $n$ if $weight ='W'$ , and at least $1$ otherwise

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

Constraint: if $weight ='W'$ , $wt (i) \geq 0.0$ , for $i = 1, 2, \dots, n$ .

Output Parameters

1: $table (maxt)$ – double array: 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.
2: $ncells$ – int64int32nag_int scalar: The number of cells in the table.
3: $ndim$ – int64int32nag_int scalar: The number of factors defining the table.
4: $idim (nfac)$ – int64int32nag_int array: The first ndim elements contain the number of levels for the factors defining the table.
5: $icount (maxt)$ – int64int32nag_int array: A table containing the number of observations contributing to each cell of the table, stored identically to table.
6: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

On entry,	$n < 2$ ,
or	$nfac < 1$ ,
or	$ldf < n$ ,
or	$typ \neq'D'$ or $'C'$ ,
or	$weight \neq'U'$ or $'W'$ ,
or	$percnt \leq 0.0$ ,
or	$percnt \geq 100.0$ .

$ifail = 2$

On entry,	$isf (i) > 0$ and $lfac (i) \leq 1$ , for some $i$ ,
or	$ifac (i, j) < 1$ , for some $i, j$ ,
or	$ifac (i, j) > lfac (j)$ , for some $i, j$ ,
or	maxt is too small,
or	$weight ='W'$ and $wt (i) < 0.0$ , for some $i$ .

$ifail = 3$: At least one cell is empty.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

Not applicable.

Further Comments

The tables created by nag_contab_tabulate_percentile (g11bb) and stored in table and icount are stored in the following way. Let there be

n

factors defining the table with factor

k

having

l_{k}

levels, then the cell defined by the levels

i_{1}

i_{2}, \dots, i_{n}

of the factors is stored in the

m

th cell given by:

m = 1 + \sum_{k = 1}^{n} [(i_{k} - 1) c_{k}],

where

c_{j} = \prod_{k = j + 1}^{n} l_{k}

, for

j = 1, 2, \dots, n - 1

and

c_{n} = 1

Example

The data, given by John and Quenouille (1977), is for a

3 \times 6

factorial experiment in

3

blocks of

18

units. The data is input in the order, blocks, factor with

3

levels, factor with

6

levels, yield, and the

3 \times 6

table of treatment medians for yield over blocks is computed and printed.

Open in the MATLAB editor: g11bb_example

function g11bb_example


fprintf('g11bb example results\n\n');

ifac = [int64(1),1,1; 1,2,1; 1,3,1; 1,1,2; 1,2,2; 1,3,2;
                 1,1,3; 1,2,3; 1,3,3; 1,1,4; 1,2,4; 1,3,4;
                 1,1,5; 1,2,5; 1,3,5; 1,1,6; 1,2,6; 1,3,6;
                 2,1,1; 2,2,1; 2,3,1; 2,1,2; 2,2,2; 2,3,2;
                 2,1,3; 2,2,3; 2,3,3; 2,1,4; 2,2,4; 2,3,4;
                 2,1,5; 2,2,5; 2,3,5; 2,1,6; 2,2,6; 2,3,6;
                 3,1,1; 3,2,1; 3,3,1; 3,1,2; 3,2,2; 3,3,2;
                 3,1,3; 3,2,3; 3,3,3; 3,1,4; 3,2,4; 3,3,4;
                 3,1,5; 3,2,5; 3,3,5; 3,1,6; 3,2,6; 3,3,6];
y   = [            274;   361;   253;   325;   317;   339;
                   326;   402;   336;   379;   345;   361;
                   352;   334;   318;   339;   393;   358;
                   350;   340;   203;   397;   356;   298;
                   382;   376;   355;   418;   387;   379;
                   432;   339;   293;   322;   417;   342;
                    82;   297;   133;   306;   352;   361;
                   220;   333;   270;   388;   379;   274;
                   336;   307;   266;   389;   333;   353];

lfac   = [int64(3); 3; 6];
isf    = [int64(0); 1; 1];
maxt   = prod(lfac(isf~=0));
maxt   = int64(maxt);

typ    = 'C';
percnt = 50;

% Compute classification table
[table, ncells, ndim, idim, icount, ifail] = ...
  g11bb( ...
         typ, isf, lfac, ifac, percnt, y, maxt);

% Display results
fprintf(' Table for %4dth percentile\n\n', percnt);
ncol = idim(ndim);
nrow = ncells/ncol;
table  = transpose(reshape(table,[ncol,nrow]));
icount = transpose(reshape(icount,[ncol,nrow]));
for i = 1:nrow
  row = [table(i,:); double(icount(i,:))];
  fprintf('%8.2f(%2d)', row);
  fprintf('\n');
end

g11bb example results

 Table for   50th percentile

  226.00( 3)  320.25( 3)  299.50( 3)  385.75( 3)  348.00( 3)  334.75( 3)
  329.25( 3)  343.25( 3)  365.25( 3)  370.50( 3)  327.25( 3)  378.00( 3)
  185.50( 3)  328.75( 3)  319.50( 3)  339.25( 3)  286.25( 3)  350.25( 3)

PDF version (NAG web site, 64-bit version, 64-bit version)

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