NAG Library Routine Document

G11BBF

INTEGER	N, NFAC, ISF(NFAC), LFAC(NFAC), IFAC(LDF,NFAC), LDF, MAXT, NCELLS, NDIM, IDIM(NFAC), ICOUNT(MAXT), IWK(2*NFAC+N), IFAIL
REAL (KIND=nag_wp)	PERCNT, Y(N), WT(), TABLE(MAXT), WK(2N)
CHARACTER(1)	TYP, WEIGHT

3 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}^{'})

4 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

5 Parameters

1: $TYP$ – CHARACTER(1)Input

On entry: 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: $WEIGHT$ – CHARACTER(1)Input

On entry: indicates if there are weights associated with the variable to be tabulated.

$WEIGHT ='U'$: Weights are not input and unit weights are assumed.
$WEIGHT ='W'$: Weights must be supplied in WT.

Constraint:

WEIGHT ='U'

'W'

3: $N$ – INTEGERInput

On entry: the number of observations.

Constraint:

N \geq 2

4: $NFAC$ – INTEGERInput

On entry: the number of classifying factors in IFAC.

Constraint:

NFAC \geq 1

5: $ISF (NFAC)$ – INTEGER arrayInput

On entry: 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.

6: $LFAC (NFAC)$ – INTEGER arrayInput

On entry: 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

7: $IFAC (LDF, NFAC)$ – INTEGER arrayInput

On entry: 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

8: $LDF$ – INTEGERInput

On entry: the first dimension of the array IFAC as declared in the (sub)program from which G11BBF is called.

Constraint:

LDF \geq N

9: $PERCNT$ – REAL (KIND=nag_wp)Input

On entry:

p

, the percentile to be tabulated.

Constraint:

0.0 < p < 100.0

10: $Y (N)$ – REAL (KIND=nag_wp) arrayInput

On entry: the variable to be tabulated.

11: $WT (*)$ – REAL (KIND=nag_wp) arrayInput

Note: the dimension of the array WT must be at least

N

WEIGHT ='W'

, and at least

1

otherwise.

On entry: 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

12: $TABLE (MAXT)$ – 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: $MAXT$ – INTEGERInput

On entry: the maximum size of the table to be computed.

Constraint:

MAXT \geq

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

14: $NCELLS$ – INTEGEROutput

On exit: the number of cells in the table.

15: $NDIM$ – INTEGEROutput

On exit: the number of factors defining the table.

16: $IDIM (NFAC)$ – INTEGER arrayOutput

On exit: the first NDIM elements contain the number of levels for the factors defining the table.

17: $ICOUNT (MAXT)$ – INTEGER arrayOutput

On exit: a table containing the number of observations contributing to each cell of the table, stored identically to TABLE.

18: $IWK (2 \times NFAC + N)$ – INTEGER arrayWorkspace

19: $WK (2 \times N)$ – REAL (KIND=nag_wp) arrayWorkspace

20: $IFAIL$ – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ 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 parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

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.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

G11BBF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

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 implementation-specific information.

9 Further Comments

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:

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

10 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.

NAG Library Routine DocumentG11BBF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Routine Document

G11BBF