Response level ordering is important because, by default, PROC SURVEYLOGISTIC models the probabilities of response levels with lower Ordered Values. Ordered Values, displayed in the "Response Profile" table, are assigned to response levels in ascending sorted order. That is, the lowest response level is assigned Ordered Value 1, the next lowest is assigned Ordered Value 2, and so on. For example, if your response variable Y takes values in , then the functions of the response probabilities modeled with the cumulative model are
and for the generalized logit model they are
where the highest Ordered Value is the reference level. You can change these default functions by specifying the EVENT= , REF= , DESCENDING , or ORDER= response variable options in the MODEL statement.
For binary response data with event and nonevent categories, the procedure models the function
where p is the probability of the response level assigned to Ordered Value 1 in the "Response Profiles" table. Since
the effect of reversing the order of the two response levels is to change the signs of and in the model .
If your event category has a higher Ordered Value than the nonevent category, the procedure models the nonevent probability.
You can use response variable options to model the event probability. For example, suppose the binary response variable Y
takes the values 1 and 0 for event and nonevent, respectively, and Exposure
is the explanatory variable. By default, the procedure assigns Ordered Value 1 to response level Y
=0, and Ordered Value 2 to response level Y
=1. Therefore, the procedure models the probability of the nonevent (Ordered Value=1) category. To model the event probability,
you can do the following:
Explicitly state which response level is to be modeled by using the response variable option EVENT= in the MODEL statement:
model Y(event='1') = Exposure;
Specify the response variable option DESCENDING in the MODEL statement:
model Y(descending)=Exposure;
Specify the response variable option REF= in the MODEL statement as the nonevent category for the response variable. This option is most useful when you are fitting a generalized logit model.
model Y(ref='0') = Exposure;
Assign a format to Y
such that the first formatted value (when the formatted values are put in sorted order) corresponds to the event. For this
example, Y
=1 is assigned formatted value 'event' and Y
=0 is assigned formatted value 'nonevent.' Since ORDER=
FORMATTED by default, Ordered Value 1 is assigned to response level Y
=1 so the procedure models the event.
proc format; value Disease 1='event' 0='nonevent'; run; proc surveylogistic; format Y Disease.; model Y=Exposure; run;
Consider a model with one CLASS
variable A
with four levels: 1, 2, 5, and 7. Details of the possible choices for the PARAM= option follow.
Three columns are created to indicate group membership of the nonreference levels. For the reference level, all three dummy
variables have a value of –1. For instance, if the reference level is 7 (REF=7), the design matrix columns for A
are as follows.
Design Matrix 


A 
A1 
A2 
A5 
1 
1 
0 
0 
2 
0 
1 
0 
5 
0 
0 
1 
7 
–1 
–1 
–1 
For CLASS main effects that use the EFFECT coding scheme, individual parameters correspond to the difference between the effect of each nonreference level and the average over all four levels.
As in PROC GLM, four columns are created to indicate group membership. The design matrix columns for A
are as follows.
Design Matrix 


A 
A1 
A2 
A5 
A7 
1 
1 
0 
0 
0 
2 
0 
1 
0 
0 
5 
0 
0 
1 
0 
7 
0 
0 
0 
1 
For CLASS main effects that use the GLM coding scheme, individual parameters correspond to the difference between the effect of each level and the last level.
Three columns are created to indicate group membership of the higher levels of the effect. For the first level of the effect
(which for A
is 1), all three dummy variables have a value of 0. The design matrix columns for A
are as follows.
Design Matrix 


A 
A2 
A5 
A7 
1 
0 
0 
0 
2 
1 
0 
0 
5 
1 
1 
0 
7 
1 
1 
1 
The first level of the effect is a control or baseline level.
For CLASS main effects that use the ORDINAL coding scheme, the first level of the effect is a control or baseline level; individual parameters correspond to the difference between effects of the current level and the preceding level. When the parameters for an ordinal main effect have the same sign, the response effect is monotonic across the levels.
Three columns are created. The first represents the linear term (x), the second represents the quadratic term (), and the third represents the cubic term (), where x is the level value. If the CLASS levels are not numeric, they are translated into 1, 2, 3, according to their sort order. The design matrix columns for A
are as follows.
Design Matrix 


A 
APOLY1 
APOLY2 
APOLY3 
1 
1 
1 
1 
2 
2 
4 
8 
5 
5 
25 
125 
7 
7 
49 
343 
Three columns are created to indicate group membership of the nonreference levels. For the reference level, all three dummy
variables have a value of 0. For instance, if the reference level is 7 (REF=7), the design matrix columns for A
are as follows.
Design Matrix 


A 
A1 
A2 
A5 
1 
1 
0 
0 
2 
0 
1 
0 
5 
0 
0 
1 
7 
0 
0 
0 
For CLASS main effects that use the REFERENCE coding scheme, individual parameters correspond to the difference between the effect of each nonreference level and the reference level.
The columns are obtained by applying the GramSchmidt orthogonalization to the columns for PARAM=EFFECT. The design matrix
columns for A
are as follows.
Design Matrix 


A 
AOEFF1 
AOEFF2 
AOEFF3 
1 
1.41421 
–0.81650 
–0.57735 
2 
0.00000 
1.63299 
–0.57735 
5 
0.00000 
0.00000 
1.73205 
7 
–1.41421 
–0.81649 
–0.57735 
The columns are obtained by applying the GramSchmidt orthogonalization to the columns for PARAM=ORDINAL. The design matrix
columns for A
are as follows.
Design Matrix 


A 
AOORD1 
AOORD2 
AOORD3 
1 
–1.73205 
0.00000 
0.00000 
2 
0.57735 
–1.63299 
0.00000 
5 
0.57735 
0.81650 
–1.41421 
7 
0.57735 
0.81650 
1.41421 
The columns are obtained by applying the GramSchmidt orthogonalization to the columns for PARAM=POLY. The design matrix columns
for A
are as follows.
Design Matrix 


A 
AOPOLY1 
AOPOLY2 
AOPOLY5 
1 
–1.153 
0.907 
–0.921 
2 
–0.734 
–0.540 
1.473 
5 
0.524 
–1.370 
–0.921 
7 
1.363 
1.004 
0.368 
The columns are obtained by applying the GramSchmidt orthogonalization to the columns for PARAM=REFERENCE. The design matrix
columns for A
are as follows.
Design Matrix 


A 
AOREF1 
AOREF2 
AOREF3 
1 
1.73205 
0.00000 
0.00000 
2 
–0.57735 
1.63299 
0.00000 
5 
–0.57735 
–0.81650 
1.41421 
7 
–0.57735 
–0.81650 
–1.41421 
Four link functions are available in the SURVEYLOGISTIC procedure. The logit function is the default. To specify a different link function, use the LINK= option in the MODEL statement. The link functions and the corresponding distributions are as follows:
The logit function
is the inverse of the cumulative logistic distribution function, which is
The probit (or normit) function
is the inverse of the cumulative standard normal distribution function, which is
Traditionally, the probit function includes an additive constant 5, but throughout PROC SURVEYLOGISTIC, the terms probit and normit are used interchangeably, previously defined as .
The complementary loglog function
is the inverse of the cumulative extremevalue function (also called the Gompertz distribution), which is
The generalized logit function extends the binary logit link to a vector of levels by contrasting each level with a fixed level
The variances of the normal, logistic, and extremevalue distributions are not the same. Their respective means and variances are
Distribution 
Mean 
Variance 

Normal 
0 
1 
Logistic 
0 

Extremevalue 


where is the Euler constant. In comparing parameter estimates that use different link functions, you need to take into account the different scalings of the corresponding distributions and, for the complementary loglog function, a possible shift in location. For example, if the fitted probabilities are in the neighborhood of 0.1 to 0.9, then the parameter estimates from using the logit link function should be about larger than the estimates from the probit link function.