Table 4.2 is a contingency table containing (fictitious) data of the patients at a local counselling service over the past year.
Type of Counselling Stress Post Relationship Loss of Other Abortion Breakdown Loved One Types Trauma Male 65 n/a 21 20 39 Female 31 26 49 52 29 Total 96 26 70 72 68 Counselling Service Data
Suppose a log-linear model is to be fitted to the data. This problem concerns count data and can therefore be modelled by the Poisson distribution. We first define the variables in the problem:
>> Count=[65 31 0 26 21 49 20 52 39 29]';The makefac command can be used to define Gender and Type (see Section 3.4.2). Count has been defined to contain the first column of the table, then the second column, then the third, etc. Gender, then, should be a variable of length 10 (the same size as Count). There are two levels of this variable ("male" and "female") and they occur one at a time. (That is, according to the ordering of the variable Count, Gender is listed as "male", "female", "male", "female"...so that each gender is listed as a block of size one.) Therefore, Gender can be specified as
>> Gender=makefac(10,2,1);
>> Gender'
ans =
1 2 1 2 1 2 1 2 1 2
Similarly, makefac can be used for the type of
counselling, Type.
>> Type=makefac(10,5,2);
>> Type'
ans =
1 1 2 2 3 3 4 4 5 5
There is one further problem with the data: That males cannot be exposed to post-abortion trauma. This is called a structural zero. To circumvent this problem, define a vector of prior weights that effectively ignores the cell corresponding to male post abortion trauma:
>> Priorw=[1 1 0 1 1 1 1 1 1 1]';The prior weights could also have been defined in this manner:
>> Priorw=ones(size(Count)); Priorw(3)=0;Having defined the variables, glmlab can be started. As discussed before, if glmlab is already running, declare a new model (in the Options menu).
For this problem, the default normal distribution is no longer adequate. To change the distribution, press the Distribution menu item on the main glmlab window and choose poisson, as shown in Figure 4.2. This will cause the link function and the scale parameter to change (to the Poisson defaults), but no changes will be obvious in the glmlab window. Click on the Link menu item and the logarithm link should be selected; this is the default link function for the Poisson distribution. The variables can now be typed into the main glmlab window. First, type the name of the response variable (Count) and the prior weights (Priorw) as shown in Figure 4.2. After pressing the FIT MODEL button, the following parameter estimates are produced:
Selecting the Poisson Distribution
Response and Prior Weights Only Entered
----------------------------------------- Estimate S.E. Variable ----------------------------------------- 3.607910 0.054882 Constant ----------------------------------------- Scaled deviance: 50.434008 Link: LOG Residual df: 8 Distribution: POISSON Scale parameter (dispersion parameter): 1.000000To add Gender to the covariate list, remember to use the fac command (as Gender is qualitative). Add fac(Gender) to the covariate list and press FIT MODEL again to produce new estimates:
----------------------------------------- Estimate S.E. Variable ----------------------------------------- 3.590439 0.083044 Constant 0.031231 0.110653 Gender(2) ----------------------------------------- Scaled Deviance: 50.354244 (change: -0.079764) Residual df: 7 (change: -1) Scale parameter (dispersion parameter): 1.000000Now add the last variable, Type, to the covariate list so the list of variables in the Covariate area includes
fac(Gender) and fac(Type).
Remember to use the fac
command again because Type is qualitative.
Press the FIT MODEL button to produces new estimates.
----------------------------------------- Estimate S.E. Variable ----------------------------------------- 3.817497 0.118512 Constant 0.104671 0.114489 Gender(2) -0.664071 0.227643 Type(2) -0.315853 0.157170 Type(3) -0.287682 0.155902 Type(4) -0.344840 0.158501 Type(5) ----------------------------------------- Scaled Deviance: 39.197423 (change: -11.156820) Residual df: 3 (change: -4) Scale parameter (dispersion parameter): 1.000000
The variable Type appears four times to indicate that there are five levels of this qualitative variable (only four variables are needed for a five level factor to maintain full rank). These four variables correspond to levels two, three, four and five of the Type variable.
It is common to want to fit interaction terms during modelling. To fit interaction terms, glmlab uses the character @ (usually SHIFT-2 on the keyboard). See Section 3.4.3. In this problem, the interaction between the two qualitative variables Gender and Type could be included. This can be done in glmlab by entering the following string in the covariate section of the main glmlab window:
fac(Gender), fac(Type), fac(Gender)@fac(Type)Notice that the fac command has been used again. Thus, interactions between variables can be specified by separating the variables with the character @.
To define the interaction between variables in glmlab, use
the @ character between the interacting variables. Remember
to still use fac for qualitative variables.
Including an Interaction Term
Pressing the FIT MODEL button then gives the following parameter estimates:
----------------------------------------- Estimate S.E. Variable ----------------------------------------- 4.174387 0.124035 Constant -0.740400 0.218272 Gender(2) -0.175891 0.265932 Type(2) -1.129865 0.251005 Type(3) -1.178655 0.255704 Type(4) -0.510826 0.202548 Type(5) 0.000000 aliased Gender(2)@Type(2) 1.587698 0.340103 Gender(2)@Type(3) 1.695912 0.341868 Gender(2)@Type(4) 0.444134 0.328278 Gender(2)@Type(5) ----------------------------------------- Scaled Deviance: 0.000000 (change: -39.197423) Residual df: 0 (change: -3) Scale parameter (dispersion parameter): 1.000000
Notice that the interaction between Gender and Type produces four terms in the model. There is no residual deviance or degrees of freedom as there are more parameters than there are observations to estimate. The presence of the term aliased indicates that the variable Gender(2)@Type(2) contains no new information.