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 multiway 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 (innercity, suburban and rural) define the
$2\times 3$ contingency table
Sex 
Habitat 

Innercity 
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 

Innercity 
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
${y}_{\left(1\right)}$,
${y}_{\left(2\right)},\dots ,{y}_{\left(m\right)}$ 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
${W}_{j}={\displaystyle \sum _{i=1}^{j}}{w}_{\left(i\right)}$ and
${W}_{j}^{\prime}={\displaystyle \sum _{i=1}^{j}}{w}_{\left(i\right)}\frac{1}{2}{w}_{\left(j\right)}$. 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
If the data is continuous then the quantiles are estimated by linear interpolation.
where
$f=\left({p}_{w}^{\prime}{W}_{j1}^{\prime}\right)/\left({W}_{j}^{\prime}{W}_{j1}^{\prime}\right)$.
 1: $\mathbf{typ}$ – Character(1)Input

On entry: indicates if the variable to be tabulated is discrete or continuous.
 ${\mathbf{typ}}=\text{'D'}$
 The percentiles are computed for a discrete variable.
 ${\mathbf{typ}}=\text{'C'}$
 The percentiles are computed for a continuous variable using linear interpolation.
Constraint:
${\mathbf{typ}}=\text{'D'}$ or $\text{'C'}$.
 2: $\mathbf{weight}$ – Character(1)Input

On entry: indicates if there are weights associated with the variable to be tabulated.
 ${\mathbf{weight}}=\text{'U'}$
 Weights are not input and unit weights are assumed.
 ${\mathbf{weight}}=\text{'W'}$
 Weights must be supplied in wt.
Constraint:
${\mathbf{weight}}=\text{'U'}$ or $\text{'W'}$.
 3: $\mathbf{n}$ – IntegerInput

On entry: the number of observations.
Constraint:
${\mathbf{n}}\ge 2$.
 4: $\mathbf{nfac}$ – IntegerInput

On entry: the number of classifying factors in
ifac.
Constraint:
${\mathbf{nfac}}\ge 1$.
 5: $\mathbf{isf}\left({\mathbf{nfac}}\right)$ – Integer arrayInput

On entry: indicates which factors in
ifac are to be used in the tabulation.
If
${\mathbf{isf}}\left(i\right)>0$ the
$i$th factor in
ifac is included in the tabulation.
Note that if
${\mathbf{isf}}\left(\mathit{i}\right)\le 0$, for $\mathit{i}=1,2,\dots ,{\mathbf{nfac}}$ then the statistic for the whole sample is calculated and returned in a $1\times 1$ table.
 6: $\mathbf{lfac}\left({\mathbf{nfac}}\right)$ – Integer arrayInput

On entry: the number of levels of the classifying factors in
ifac.
Constraint:
if ${\mathbf{isf}}\left(\mathit{i}\right)>0$, ${\mathbf{lfac}}\left(\mathit{i}\right)\ge 2$, for $\mathit{i}=1,2,\dots ,{\mathbf{nfac}}$.
 7: $\mathbf{ifac}\left({\mathbf{ldf}},{\mathbf{nfac}}\right)$ – Integer arrayInput

On entry: the
nfac coded classification factors for the
n observations.
Constraint:
$1\le {\mathbf{ifac}}\left(\mathit{i},\mathit{j}\right)\le {\mathbf{lfac}}\left(\mathit{j}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nfac}}$.
 8: $\mathbf{ldf}$ – IntegerInput

On entry: the first dimension of the array
ifac as declared in the (sub)program from which
g11bbf is called.
Constraint:
${\mathbf{ldf}}\ge {\mathbf{n}}$.
 9: $\mathbf{percnt}$ – Real (Kind=nag_wp)Input

On entry: $p$, the percentile to be tabulated.
Constraint:
$0.0<p<100.0$.
 10: $\mathbf{y}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the variable to be tabulated.
 11: $\mathbf{wt}\left(*\right)$ – Real (Kind=nag_wp) arrayInput

Note: the dimension of the array
wt
must be at least
${\mathbf{n}}$ if
${\mathbf{weight}}=\text{'W'}$, and at least
$1$ otherwise.
On entry: if
${\mathbf{weight}}=\text{'W'}$,
wt must contain the
n weights. Otherwise
wt is not referenced.
Constraint:
if ${\mathbf{weight}}=\text{'W'}$, ${\mathbf{wt}}\left(\mathit{i}\right)\ge 0.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
 12: $\mathbf{table}\left({\mathbf{maxt}}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: 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
Section 9.
 13: $\mathbf{maxt}$ – IntegerInput

On entry: the maximum size of the table to be computed.
Constraint:
${\mathbf{maxt}}\ge \text{}$ product of the levels of the factors included in the tabulation.
 14: $\mathbf{ncells}$ – IntegerOutput

On exit: the number of cells in the table.
 15: $\mathbf{ndim}$ – IntegerOutput

On exit: the number of factors defining the table.
 16: $\mathbf{idim}\left({\mathbf{nfac}}\right)$ – Integer arrayOutput

On exit: the first
ndim elements contain the number of levels for the factors defining the table.
 17: $\mathbf{icount}\left({\mathbf{maxt}}\right)$ – Integer arrayOutput

On exit: a table containing the number of observations contributing to each cell of the table, stored identically to
table.
 18: $\mathbf{iwk}\left(2\times {\mathbf{nfac}}+{\mathbf{n}}\right)$ – Integer arrayWorkspace
 19: $\mathbf{wk}\left(2\times {\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace

 20: $\mathbf{ifail}$ – IntegerInput/Output

On entry:
ifail must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this argument you should refer to
Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ 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).
If on entry
${\mathbf{ifail}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
Not applicable.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
The tables created by
g11bbf 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:
where
${c}_{\mathit{j}}={\displaystyle \prod _{k=\mathit{j}+1}^{n}}{l}_{k}$, for
$\mathit{j}=1,2,\dots ,n1$ and
${c}_{n}=1$.
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.