Note: please be advised that this function is classed as ‘experimental’ and its interface may be developed further in the future. Please see
Section 4 in How to Use the NAG Library
for further information.
The function may be called by the names: g22ycc or nag_blgm_lm_design_matrix.
3Description
g22ycc generates a design matrix from a data matrix and a model description. Design matrices encapsulate the observed values of the independent variables and the required model in a form that can be used by many of the model fitting functions available in the NAG Library, for example those in Chapter G02.
3.1Notation
Let $D$ denote a data matrix with $n$ observations on ${m}_{d}$ independent variables, denoted by ${V}_{j}$, for $\mathit{j}=1,2,\dots ,{m}_{d}$. If ${V}_{j}$ is a categorical variable, let ${L}_{j}$ denote the number of levels associated with it. If ${V}_{j}$ is a binary, ordinal or continuous variable, let ${L}_{j}=1$.
Let ${V}_{ji}$ denote the $i$th value of ${V}_{j}$.
Let $\mathcal{M}$ denote a model made up of one or more terms, denoted by ${T}_{i}$. Each term consists of either a main effect or an interaction and hence can be described using one or more variable names ${V}_{j}$ and the interaction operator ‘$.$’. The operator ‘$+$’ is used to denote the addition of a term to the model. Therefore, $\mathcal{M}={T}_{1}+{T}_{2}+{T}_{3}={V}_{1}+{V}_{2}+{V}_{1}.{V}_{2}$ denotes a model with three terms, the first two terms being the main effects for variables ${V}_{1}$ and ${V}_{2}$ and the last term the interaction between them. For simplicity we reorder the terms of the model by the number of variables in them, so main effects come first, then two-way interactions, then three-way interactions etc. By default it is assumed that the model $\mathcal{M}$ contains a mean effect (or intercept term), if the mean effect is excluded, this will be denoted by ‘$-1$’, so $\mathcal{M}={T}_{1}$ is a model with one term and a mean effect and $\mathcal{M}={T}_{1}-1$ is the same model with the mean effect dropped.
g22ycc generates an $n\times {m}_{x}$ design matrix, $X$, from $D$ and $\mathcal{M}$.
3.2Dummy Variables
When constructing a design matrix, we cannot work directly with categorical variables. Categorical variables must first be recoded into dummy variables. A categorical variable ${V}_{j}$ requires ${L}_{j}$ dummy variables. Let ${\mathcal{D}}^{j}$ denote an $n\times {L}_{j}$ matrix of dummy variables for ${V}_{j}$ defined as
where ${\mathcal{D}}_{l}^{j}$ is the $l$th column of ${\mathcal{D}}^{j}$ and ${\mathcal{D}}_{li}^{j}$ is the $i$th element of ${\mathcal{D}}_{l}^{j}$.
For a binary, ordinal or continuous variable, ${\mathcal{D}}_{1i}^{j}={V}_{ji}$.
3.3Full Design Matrix
Given a model, $\mathcal{M}$, and the matrices of dummy variables constructing the full design matrix ${X}_{F}$ is trivial. Each term is processed in order and
1.If term $i$ is a main effect, that is ${T}_{i}={V}_{j}$ for some $j$, ${\mathcal{D}}^{j}$ is copied into ${X}_{F}$.
2.If term $i$ is a two-way interaction, that is ${T}_{i}={V}_{j}.{V}_{k}$, for some $j\ne k$, then
(i)Loop over ${l}_{j}=1,2,\dots {L}_{j}$.
(ii)Loop over ${l}_{k}=1,2,\dots {L}_{k}$.
(iii)Add a column to ${X}_{F}$ corresponding to the element-wise product of ${\mathcal{D}}_{{l}_{j}}^{j}$ and ${\mathcal{D}}_{{l}_{k}}^{k}$.
3.Higher interaction terms are handled in a similar manner as the two-way interactions by adding columns constructed from multiplying all combinations of the columns of the corresponding $\mathcal{D}$s that correspond to the variables involved. In all cases, the variables towards the right hand side of a term are iterated over the quickest.
3.4Contrasts
Using the full design matrix ${X}_{F}$ in an analysis can result in an overparameterized model. This is due to ${X}_{F}$ often not being of full rank as the sum of all the dummy variables for a particular variable is a vector of ones. This source of overparameterization can be alleviated by using a design matrix $X$ where (some) dummy variables are replaced by contrasts. For a categorical variable ${V}_{j}$ the contrasts are a set of ${L}_{j}-1$ functionally independent linear combinations of the dummy variables.
Whilst the choice of contrasts used in term ${T}_{i}$ will affect the individual model coefficients (parameters), it has no effect on the overall contribution of ${T}_{i}$.
For a given variable ${V}_{j}$, the contrasts can be represented by an ${L}_{j}\times {L}_{j}-1$ matrix, ${C}_{j}$. The rows of ${C}_{j}$ correspond to a particular value of ${V}_{j}$ and the columns correspond to the values to use in the design matrix.
Six types of contrast are available in g22ycc; two types of treatment contrasts, two types of sum contrasts, Helmert contrasts and polynomial contrasts. Unless specified otherwise, the contrasts used by g22ycc are treatment contrasts relative to the first level. See the description of the optional parameter ${\mathbf{Contrast}}$ in g22yac for ways of changing the contrasts used.
3.4.1Treatment Contrasts
Treatment contrasts are taken relative to either the first or last level of the variable. For example, if ${L}_{j}=4$,
would be the contrast matrix for ${V}_{j}$ using treatment contrasts relative to the first level. The contrast matrix obtained when using treatment contrasts relative to the last level is similar, but the row of zeros appears at the bottom and all other rows are shifted up one.
Strictly speaking, the term contrast implies that each row in the contrast matrix sums to zero. That is not the case for treatment contrasts, however they are included as this coding is commonly used in practice.
3.4.2Sum Contrasts
Sum contrasts are similar to treatment contrasts and again can be taken relative to the first or last level of the variable. Unlike treatment contrasts, sum contrasts effectively constrain the coefficients related to the variable to sum to zero. For example, if ${L}_{j}=4$,
would be the contrast matrix for ${V}_{j}$ using treatment contrasts relative to the last level. The contrast matrix obtained when using treatment contrasts relative to the first level is similar, but the row of $-1$s appears at the top and all other rows are shifted down one.
3.4.3Helmert Contrasts
With Helmert contrasts level $l$ of the variable is compared with the average effect of all previous levels. For example, if ${L}_{j}=4$,
would be the contrast matrix for ${V}_{j}$ using Helmert contrasts.
3.4.4Polynomial Contrasts
With polynomial contrasts the entries in the columns of ${C}_{j}$ correspond in linear, quadratic, cubic, quartic, etc. terms to a hypothetical underlying numeric variable that takes equally spaced values at each level. For example, if ${L}_{j}=4$,
would be the contrast matrix for ${V}_{j}$ using polynomial contrasts.
3.4.5When Contrasts Can Be Used
Depending on the specifics of the model, $\mathcal{M}$, it may not be possible to always replace the ${L}_{j}$ dummy variables with ${L}_{j}-1$ contrasts for all variables in all terms and retain the same model. A simple example of this is a data matrix, $D$, with four observations and two variables which have two and three levels respectively. This data matrix might look something like:
For the sake of argument, assume that our model contains the main effect for each variable, but does not contain a mean effect (or intercept term). So using the notation established earlier, $\mathcal{M}={V}_{1}+{V}_{2}-1$. The full design matrix, ${X}_{F}$, for this data matrix and model would be
However, ${X}_{F}$ is not of full rank (and hence $\mathcal{M}$ is overparameterized) because the sum of the first two columns is a vector of ones as is the sum of the last three columns.
In order to alleviate this we might try constructing ${X}_{C}$ where the dummy variables have been replaced by contrasts. Assuming treatment contrasts, relative to the first level, we would have
However, using ${X}_{C}$ makes an implicit assumption that the expected value of the dependent variable (the quantity being modelled) is zero when ${V}_{1}=1$ and ${V}_{2}=1$. This assumption was not made when we used ${X}_{F}$ and hence the two design matrices are not equivalent. One solution would be to use dummy variables for ${V}_{1}$ and contrasts for ${V}_{2}$, which would result in a design matrix, $X$ of
Using $X$ would give an equivalent model to using ${X}_{F}$.
The algorithm used by g22ycc to decide which variables, in which terms, can be coded as contrasts and which need to be coded as dummy variables is described below.
Suppose ${V}_{j}$ is any variable that appears in term ${T}_{i}$, let ${T}_{i\left(j\right)}$ denote the term obtained by dropping ${V}_{j}$ from ${T}_{i}$. For example, if ${T}_{3}={V}_{1}.{V}_{2}.{V}_{3}$, ${T}_{3\left(2\right)}={V}_{1}.{V}_{3}$. In this context, the empty term is taken to be the mean effect (or intercept term). We say that ${T}_{i\left(j\right)}$ appears in $\mathcal{M}$ if there exists a term ${T}_{k}$, $k<i$, that contains all of the variables appearing in ${T}_{i\left(j\right)}$. In most cases ${T}_{k}={T}_{i\left(j\right)}$, but this is not required. Note, as stated earlier, the terms in $\mathcal{M}$ are ordered by the number of variables in them.
A variable, ${V}_{j}$ in term ${T}_{i}$ is coded by contrasts if ${T}_{i\left(j\right)}$ appears in $\mathcal{M}$ and by dummy variables otherwise. It is, therefore, possible for variable ${V}_{j}$ to be coded by contrasts in some terms and dummy variables in others within the same $X$.
The above rule assumes the presence of a mean effect. If no such effect is present in the model, the main effect of the first categorical variable is coded by dummy variables to compensate. If no main effects appear in the model, the warning ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NW_POTENTIAL_PROBLEM is returned.
A longer description and informal proof that the resulting $X$ is a suitable design matrix for the model of interest can be found in chapter two of Chambers and Hastie (1992).
3.5Mean Effect
The mean effect (or intercept term) is included in a design matrix by adding a column of ones as the first column of $X$. However, many model fitting functions in the NAG Library handle the mean effect as a special case and do not require it to be explicitly added to the design matrix. Therefore, by default, g22ycc does not explicitly add the mean effect to the design matrix. This behaviour can be changed via the optional parameter ${\mathbf{Explicit\; Mean}}$ in g22yac.
4References
Chambers J M and Hastie T J (1992) Statistical Models in S Wadsworth and Brooks/Cole Computer Science Series
5Arguments
1: $\mathbf{hform}$ – void *Input
On entry: a G22 handle to the internal data structure containing a description of the model $\mathcal{M}$ as returned in hform by g22yac.
2: $\mathbf{hddesc}$ – void *Input
On entry: a G22 handle to the internal data structure containing a description of the data matrix, $D$ as returned in hddesc by g22ybc.
Note: the $(i,j)$th element of the matrix is stored in ${\mathbf{dat}}\left[(j-1)\times {\mathbf{pddat}}+i-1\right]$.
On entry: the data matrix, $D$. By default
${D}_{ij}$, the $\mathit{i}$th value for the $\mathit{j}$th variable, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,{m}_{d}$, should be supplied in ${\mathbf{dat}}\left[\left(j-1\right)\times {\mathbf{pddat}}+i-1\right]$.
If the optional parameter ${\mathbf{Storage\; Order}}$, described in g22ybc, is set to $\mathrm{VAROBS}$, ${D}_{ij}$ should be supplied in ${\mathbf{dat}}\left[\left(i-1\right)\times {\mathbf{pddat}}+j-1\right]$.
4: $\mathbf{pddat}$ – IntegerInput
On entry: the stride separating matrix row elements in the array dat.
Constraints:
if the optional parameter ${\mathbf{Storage\; Order}}$, described in g22ybc, is set to $\mathrm{VAROBS}$, ${\mathbf{pddat}}\ge {m}_{d}$;
if the optional parameter ${\mathbf{Storage\; Order}}$, described in g22ybc, is set to $\mathrm{VAROBS}$, ${\mathbf{sddat}}\ge n$;
otherwise ${\mathbf{sddat}}\ge {m}_{d}$.
6: $\mathbf{hxdesc}$ – void **Input/Output
On entry: must be set to NULL, alternatively an existing G22 handle may be supplied in which case this function will destroy the supplied G22 handle as if g22zac had been called.
On exit: holds a G22 handle to the internal data structure containing a description of the design matrix, $X$. You must not change the G22 handle other than through the functions in Chapter G22.
Note: the $(i,j)$th element of the matrix $X$ is stored in ${\mathbf{x}}\left[(j-1)\times {\mathbf{pdx}}+i-1\right]$.
On exit: the design matrix, $X$. By default
${X}_{ij}$, the $\mathit{i}$th value for the $\mathit{j}$th column, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,{m}_{x}$, is returned in ${\mathbf{x}}\left[\left(j-1\right)\times {\mathbf{pdx}}+i-1\right]$.
If the optional parameter ${\mathbf{Storage\; Order}}$, described in g22yac, is set to $\mathrm{VAROBS}$, ${X}_{ij}$ is returned in ${\mathbf{x}}\left[\left(i-1\right)\times {\mathbf{pdx}}+j-1\right]$.
If pdx or sdx are too small to hold x, the number of columns required to hold the design matrix is returned in mx.
Under some conditions it is possible to use the data matrix in place of the design matrix. Specifically, if $D$ has no categorical variables, $\mathcal{M}$ has only main effects and either has no mean effect or the mean effect does not need to be explicitly added to the design matrix. If pdx or sdx are too small under such circumstances, ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NW_ALTERNATIVE is returned and hxdesc is set up in such a way as to allow dat to be used as the design matrix.
If pdx and sdx are both zero, x is not referenced and may be NULL.
8: $\mathbf{pdx}$ – IntegerInput
On entry: the stride separating matrix row elements in the array x.
Constraints:
if the optional parameter ${\mathbf{Storage\; Order}}$, described in g22yac, is set to $\mathrm{VAROBS}$, ${\mathbf{pdx}}\ge {m}_{x}$;
if the optional parameter ${\mathbf{Storage\; Order}}$, described in g22yac, is set to $\mathrm{VAROBS}$, ${\mathbf{sdx}}\ge n$;
otherwise ${\mathbf{sdx}}\ge {m}_{x}$.
10: $\mathbf{mx}$ – Integer *Output
On exit: the minimum number of columns required to hold the design matrix.
In most cases ${\mathbf{mx}}={m}_{x}$. The one exception is when ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NW_ALTERNATIVE, that is the size of x was too small but the data matrix given in dat can be used as the design matrix. In this case mx holds the number of columns that would be required if only the relevant parts of dat were copied into a new array.
11: $\mathbf{fail}$ – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_ARRAY_SIZE
On entry, ${m}_{d}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{pddat}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{pddat}}\ge {m}_{d}$.
On entry, ${m}_{d}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{sddat}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{sddat}}\ge {m}_{d}$.
On entry, $n=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{pddat}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{pddat}}\ge n$.
On entry, $n=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{sddat}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{sddat}}\ge n$.
NE_BAD_PARAM
On entry, argument $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
NE_FIELD_UNKNOWN
A variable name used when creating hform is not present in hddesc. Variable name: $\u27e8\mathit{\text{value}}\u27e9$.
On entry, hxdesc is not NULL or a recognised G22 handle.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL_ARRAY
On entry, column $j$ of the data matrix, $D$, is not consistent with information supplied in hddesc, $j=\u27e8\mathit{\text{value}}\u27e9$.
NW_ALTERNATIVE
On entry, the size of x is too small to hold the design matrix. dat can be used instead.
NW_ARRAY_SIZE
On entry, ${m}_{x}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{pdx}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{pdx}}\ge {m}_{x}$.
On entry, ${m}_{x}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{sdx}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{sdx}}\ge {m}_{x}$.
On entry, $n=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{pdx}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{pdx}}\ge n$.
On entry, $n=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{sdx}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{sdx}}\ge n$.
NW_POTENTIAL_PROBLEM
Column $j$ of the data matrix, $D$, required rounding more than expected when being treated as a categorical variable, $j=\u27e8\mathit{\text{value}}\u27e9$.
All output is returned using the rounded value(s).
The model contains categorical variables, but no intercept or main effects terms have been requested. Please check the design matrix returned matches the model you require.
7Accuracy
Not applicable.
8Parallelism and Performance
g22ycc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g22ycc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
g22ydc can be used to obtain labels for the columns of the design matrix $X$.
Many of the analysis functions that require a design matrix to be supplied allow submodels to be defined through the use of a vector of ones or zeros indicating whether a column of $X$ should be included or excluded from the analyses (see for example sx in g02dacorg02gac). This allows nested models to be fit without having to reconstruct the design matrix for each analysis. g22ydc offers a mechanism for constructing these vectors using submodels specified using g22yac.
10Example
This example creates and outputs two design matrices for a simple linear regression model. The first design matrix uses sum contrasts for all variables and the second uses a combination of polynomial and Helmert contrasts. Column labels are generated using g22ydc.
As well as the optional parameters common to all G22 handles described in g22zmcandg22znc, a number of additional optional parameters can be specified for a G22 handle holding the description of a design matrix as returned by g22ycc in hxdesc.
The value of an optional parameter can be queried using g22znc.
The remainder of this section can be skipped if you wish to use the default values for all optional parameters.
The following is a list of the optional parameters available. A full description of each optional parameter is provided in Section 11.1.
For each option, we give a summary line, a description of the optional parameter and details of constraints.
The summary line contains:
a parameter value,
where the letters $a$, $i$ and $r$ denote options that take character, integer and real values respectively;
Keywords and character values are case and white space insensitive.
Formula
$a$
This optional parameter returns a verbose formula string describing the model, $\mathcal{M}$, used to create the design matrix. This formula will only contain variable names, the operators ‘$+$’ and ‘$.$’ and any contrast identifiers present.
Min Number of Columns
$i$
This optional parameter returns the minimum number of columns required to hold the design matrix, $X$. In most cases ${\mathbf{Min\; Number\; of\; Columns}}={\mathbf{Number\; of\; Columns}}$. The one exception is when ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NW_ALTERNATIVE, that is the size of x was too small but the data matrix given in dat can be used as the design matrix. In this case, ${\mathbf{Number\; of\; Columns}}={m}_{x}={m}_{d}$ and ${\mathbf{Min\; Number\; of\; Columns}}$ holds the number of columns that would be required if only the relevant parts of dat were copied into a new array.
Number of Columns
$i$
This optional parameter returns ${m}_{x}$, the number of columns in the design matrix.
Number of Observations
$i$
This optional parameter returns $n$, the number of observations in the design matrix.
Storage Order
$a$
This optional parameter returns how the design matrix, $X$, is stored in x.
If ${\mathbf{Storage\; Order}}=\mathrm{OBSVAR}$, ${X}_{ij}$, the value for the $j$th variable of the $i$th observation of the design matrix is stored in ${\mathbf{x}}\left[\left(j-1\right)\times {\mathbf{pdx}}+i-1\right]$.
If ${\mathbf{Storage\; Order}}=\mathrm{VAROBS}$, ${X}_{ij}$, the value for the $j$th variable of the $i$th observation of the design matrix is stored in ${\mathbf{x}}\left[\left(i-1\right)\times {\mathbf{pdx}}+j-1\right]$.
It should be noted that ${\mathbf{Storage\; Order}}$ is not writeable. If you wish to change the storage order of the design matrix you need to change ${\mathbf{Storage\; Order}}$ in hform as described in Section 11 in g22yac prior to calling g22ycc.