void	nag_dummy_vars (Nag_DummyType type, Integer n, Integer levels, const Integer factor[], double x[], Integer tdx, const double v[], double num_reps[], NagError *fail)

3 Description

In the analysis of an experimental design using a general linear model the factors or classification variables that specify the design have to be coded as dummy variables. nag_dummy_vars (g04eac) computes dummy variables that can then be used in the fitting of the general linear model using nag_regsn_mult_linear (g02dac).

If the factor of length

n

has

k

levels then the simplest representation is to define

k

dummy variables,

X_{j}

such that

X_{j} = 1

if the factor is at level

j

and 0 otherwise, for

j = 1, 2, \dots, k

. However, there is usually a mean included in the model and the sum of the dummy variables will be aliased with the mean. To avoid the extra redundant argument,

k - 1

dummy variables can be defined as the contrasts between one level of the factor, the reference level and the remaining levels. If the reference level is the first level then the dummy variables can be defined as

X_{j} = 1

if the factor is at level

j

and 0 otherwise, for

j = 2, 3, \dots, k

. Alternatively, the last level can be used as the reference level.

A second way of defining the

k - 1

dummy variables is to use a Helmert matrix in which levels

2, 3, \dots, k

are compared with the average effect of the previous levels. For example if

k = 4

then the contrasts would be:

\begin{array}{r} 1 & - 1 & - 1 & - 1 \\ 2 & 1 & - 1 & - 1 \\ 3 & 0 & 2 & - 1 \\ 4 & 0 & 0 & 3 \end{array}

Thus variable

j

, for

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

, is given by

$X_{j} = - 1$ if factor is at level less than $j + 1$
$X_{j} = \sum_{i = 1}^{j} r_{i} / r_{j + 1}$ if factor is at level $j + 1$
$X_{j} = 0$ if factor is at level greater than $j + 1$

where

r_{j}

is the number of replicates of level

j

If the factor can be considered as a set of values from an underlying continuous variable then the factor can be represented by a set of

k - 1

orthogonal polynomials representing the linear, quadratic, etc. effects of the underlying variable. The orthogonal polynomial is computed using Forsythe's algorithm (see Forsythe (1957) and Cooper (1968)). The values of the underlying continuous variable represented by the factor levels have to be supplied to the function.

The orthogonal polynomials are standardized so that the sum of squares for each dummy variable is one. For the other methods integer

(\pm 1)

representations are retained except that in the Helmert representation the code of level

j + 1

in dummy variable

j

will be a fraction.

4 References

Cooper B E (1968) Algorithm AS 10. The use of orthogonal polynomials Appl. Statist. 17 283–287

Forsythe G E (1957) Generation and use of orthogonal polynomials for data fitting with a digital computer J. Soc. Indust. Appl. Math. 5 74–88

5 Arguments

1: $type$ – Nag_DummyTypeInput

On entry: the type of dummy variable to be computed.

$type = Nag_Poly$: An orthogonal Polynomial representation is computed.
$type = Nag_Helmert$: A Helmert matrix representation is computed.
$type = Nag_FirstLevel$: The contrasts relative to the First level are computed.
$type = Nag_LastLevel$: The contrasts relative to the Last level are computed.
$type = Nag_AllLevels$: A complete set of dummy variables is computed.

Constraint:

type = Nag_Poly

Nag_Helmert

Nag_FirstLevel

Nag_LastLevel

Nag_AllLevels

2: $n$ – IntegerInput

On entry: the number of observations for which the dummy variables are to be computed,

n

Constraint:

n \geq levels

3: $levels$ – IntegerInput

On entry: the number of levels of the factor,

k

Constraint:

levels \geq 2

4: $factor [n]$ – const IntegerInput

On entry: the

n

values of the factor.

Constraint:

1 \leq factor [i - 1] \leq levels

, for

i = 1, 2, \dots, n

5: $x [n \times tdx]$ – doubleOutput

Note: the

(i, j)

th element of the matrix

X

is stored in

x [(i - 1) \times tdx + j - 1]

On exit: the

n

k^{*}

matrix of dummy variables, where

k^{*} = k - 1

type = Nag_Poly

Nag_Helmert

Nag_FirstLevel

Nag_LastLevel

and

k^{*} = k

type = Nag_AllLevels

6: $tdx$ – IntegerInput

On entry: the stride separating matrix column elements in the array x.

Constraints:

if $type = Nag_Poly$ , $Nag_Helmert$ , $Nag_FirstLevel$ or $Nag_LastLevel$ , $tdx \geq levels - 1$ ;
if $type = Nag_AllLevels$ , $tdx \geq levels$ .

7: $v [\dim]$ – const doubleInput

Note: the dimension, dim, of the array v must be at least

$levels$ when $type = Nag_Poly$ ;
$1$ otherwise.

On entry: if

type = Nag_Poly

, the

k

distinct values of the underlying variable for which the orthogonal polynomial is to be computed. If

type \neq Nag_Poly

, v is not referenced.

Constraint: if

type = Nag_Poly

, then the

k

values of v must be distinct.

8: $num_reps [levels]$ – doubleOutput

On exit:

num_reps [i - 1]

contains the number of replications for each level of the factor,

r_{i}

, for

i = 1, 2, \dots, k

9: $fail$ – NagError *Input/Output

The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6 Error Indicators and Warnings

NE_2_INT_ARG_LT: On entry, $n = 〈value〉$ while $levels = 〈value〉$ . These arguments must satisfy $n \geq levels$ .

On entry, $tdx = 〈value〉$ while $levels = 〈value〉$ . These arguments must satisfy $tdx \geq levels$ .

On entry, $tdx = 〈value〉$ while $levels - 1 = 〈value〉$ . These arguments must satisfy $tdx \geq levels - 1$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_ARRAY_CONS: The contents of array v are not valid.
Constraint: all values of v must be distinct.
NE_BAD_PARAM: On entry, argument type had an illegal value.
NE_G04EA_LEVELS: All levels are not represented in array factor.
NE_G04EA_ORTHO_POLY: An orthogonal polynomial has all values zero. This will be due to some values of v being close together. This can only occur if $type = Nag_Poly$ .
NE_INT_ARG_LT: On entry, levels must not be less than 2: $levels = 〈value〉$ .
NE_INT_ARRAY_CONS: On entry, $factor [0] = 〈value〉$ .
Constraint: $1 \leq factor [0] \leq levels$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

7 Accuracy

The computations are stable.

8 Parallelism and Performance

nag_dummy_vars (g04eac) is not threaded in any implementation.

9 Further Comments

Other functions for fitting polynomials can be found in Chapter e02.

10 Example

Data are read in from an experiment with four treatments and three observations per treatment with the treatment coded as a factor. nag_dummy_vars (g04eac) is used to compute the required dummy variables and the model is then fitted by nag_regsn_mult_linear (g02dac).

NAG Library Function Documentnag_dummy_vars (g04eac)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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 Function Document

nag_dummy_vars (g04eac)