WHAT KIND OF CONTRASTS ARE THESE?

 

                          David P. Nichols

                     Senior Support Statistician

                             SPSS, Inc.

                 From SPSS Keywords, Number 63, 1997

 

 

Interpretation of parameter estimates is an essential part of the predictive

modeling process. Estimates of interest often represent contrasts among the

levels of a categorical predictor variable. A contrast is defined by a set of

coefficients that sum to 0 over the levels of the categorical variable of

interest.

 

In SPSS, issues of interpretation of contrast results arise in several

procedures, including LOGISTIC REGRESSION and COX REGRESSION. Both procedures

have facilities for automatically treating predictors (or covariates) as

as categorical variables. When a covariate with K levels is declared to be

categorical in either one of these procedures, a set of K-1 variables is

produced internally, and these variables are used as a set in the analysis.

 

The values of the K-1 variables are determined by the choice of contrasts

made by the user. The default contrasts in the current 7.5 release of SPSS

for Windows have been changed in both procedures to INDICATOR, with the

last category as the reference group. These contrasts produce estimates

comparing each other group to the reference group.

 

A point of considerable confusion among SPSS users is the relationship between

the values of the internally created variables and the interpretation of the

resulting parameter estimates. The output for the LOGISTIC REGRESSION and COX

REGRESSION procedures provides the values of the internal variables used to

estimate the desired contrasts. For example, suppose we have a three level

categorical covariate. The new default INDICATOR contrasts would produce a

set of "parameter codings" like those in Figure 1.

 

Figure 1: Parameter codings for INDICATOR contrasts

-------------------------------------------------------------------------------

                            Parameter

              Value   Freq  Coding

                              (1)    (2)

GROUP

                  1    106  1.000   .000

                  2    116   .000  1.000

                  3    107   .000   .000

-------------------------------------------------------------------------------

End Figure 1

 

The predictor here is called simply GROUP. It takes on the values 1-3, with

frequencies listed in the "Freq" column. The columns on the right (what are

being called parameter codings) give the values of the internal variables

created to represent the original categorical covariate. In this case there

are two internal variables created. For the first variable, cases with a

value of 1 for GROUP get a 1, while all other cases get a 0. For the second,

cases with a 2 for GROUP get a 1, with all other cases getting a 0.

 

The question that this output often elicits from SPSS users is how does this

coding produce the contrasts claimed in our documentation? The reason is that

one must distinguish between the values of the contrast coefficients defining

contrasts of interest and the values of the variables in the data that will

produce such a set of contrasts. The columns in the data that produce certain

contrasts will resemble the contrast coefficients only when the matrix of

contrast coefficients is orthogonal (the inner product of any two row vectors

in the contrast matrix is 0). INDICATOR contrasts are not orthogonal, nor are

the other most commonly used types in logistic or Cox regression models. Thus

it is important to understand the following relationship between the columns

of the data and the contrast results.

 

If we append a constant unit (1) column onto the beginning of the two columns

given above, we get what we call a basis or design matrix for generating the

desired contrasts. If we call this matrix X, then for any model that uses a

linear combination of the predictors in generating it's prediction function,

we can compute C, the matrix of contrast coefficients, as:

 

         -1

C = (X'X)  X'

 

For the example given here, the basis matrix for INDICATOR contrasts given

in Figure 2 produces the contrast matrix given in Figure 2.

 

Figure 2: Basis and contrast matrices for INDICATOR contrasts

-------------------------------------------------------------------------------

Basis:   1  1  0           Contrast:   0  0  1

         1  0  1                       1  0 -1

         1  0  0                       0  1 -1

-------------------------------------------------------------------------------

 

The first row of the contrast matrix gives the coefficients for the constant

or intercept term, which with INDICATOR contrasts estimates the predicted

value for the reference group (here, the last one). The other two rows give

the contrasts estimated by the GROUP(1) and GROUP(2) parameter estimates,

which are, respectively, the first group minus the last and the second minus

the last.

 

Earlier releases of SPSS used DEVIATION as the default contrast type, with the

last category as the reference or excluded out category. DEVIATION contrasts

compare each group other than the excluded group to the unweighted average of

all groups. The value for the left out group is then by definition the

negative of the sum of the given parameter estimates. Considerable confusion

has resulted from the fact that the basis or design matrix for DEVIATION

contrasts resembles the contrast matrix for SIMPLE contrasts, which compare

each group to a reference category (like INDICATOR contrasts). It turns out

that DEVIATION and SIMPLE contrasts are in a sense mirror images of one

another, in that the variable codings required to produce one type of

contrasts look like the transpose of the contrast matrix for the other type

of contrasts.

 

These relationships are illustrated for the three level case in Figures 3 and

4 (using fractions for precision; SPSS output shows decimal values). Note that

the contrasts estimated for GROUP(1) and GROUP(2) are the same for SIMPLE

contrasts as for INDICATOR, but that the intercept is now an unweighted

average of all levels rather than the value for the last (or more generally,

the reference) group.

 

Figure 3: Basis and contrast matrices for DEVIATION contrasts

-------------------------------------------------------------------------------

Basis:   1  1  0           Contrast:   1/3  1/3  1/3

         1  0  1                       2/3 -1/3 -1/3

         1 -1 -1                      -1/3  2/3 -1/3

-------------------------------------------------------------------------------

 

Figure 4: Basis and contrast matrices for SIMPLE contrasts

-------------------------------------------------------------------------------

Basis:   1  2/3 -1/3       Contrast:   1/3  1/3  1/3

         1 -1/3  2/3                    1    0   -1

         1 -1/3 -1/3                    0    1   -1

-------------------------------------------------------------------------------