The BGLIMM Procedure

MODEL Statement

MODEL response <(response-options)> = <fixed-effects> </ model-options>;

MODEL events / trials = <fixed-effects> </ model-options>;

The MODEL statement, which is required, names the dependent variable and the fixed effects. The fixed-effects determine the matrix of the model (see the section Notation for the Generalized Linear Mixed Model for details). You specify effects in the same way as in other SAS procedures.

An intercept is included in the fixed-effects model by default. You can remove it by using the NOINT option.

You can specify the dependent variable by using either the response syntax or the events/trials syntax. The events/trials syntax is specific to models for binomial data. A binomial(n,) variable is the sum of n independent Bernoulli trials with event probability . Each Bernoulli trial results in either an event or a nonevent (with probability ). You use the events/trials syntax to indicate to the BGLIMM procedure that the Bernoulli outcomes are grouped. The value of the second variable, trials, gives the number n of Bernoulli trials. The value of the first variable, events, is the number of events out of n. The values of both events and (trials–events) must be nonnegative, and the value of trials must be positive. Observations for which these conditions are not met are excluded from the analysis. If the events/trials syntax is used, PROC BGLIMM defaults to the binomial distribution. The response is then the events variable. The trials variable is accounted for in model fitting as an additional weight. If you use the response syntax, the procedure defaults to the normal distribution.

The MODEL statement uses two sets of options. The response-options determine how PROC BGLIMM models probabilities for binary, binomial, and multinomial data. The model-options control other aspects of model formation and inference. Table 3 summarizes these options, and subsequent sections describe them in detail.

Table 3: MODEL Statement Options

Option	Description
Response Variable Options
DESCENDING	Reverses the order of response categories
EVENT=	Specifies the event category in binary and binomial models
ORDER=	Specifies the sort order for the response variable
REF=	Specifies the reference category in generalized logit models
Model Options
COEFFPRIOR=	Specifies the prior of the fixed-effects coefficients
DIST=	Specifies the response distribution
SCALE=	Specifies a fixed value for the scale parameter
INIT=	Controls the generation of initial values of the fixed-effects coefficients
LINK=	Specifies the link function
NOINT	Excludes the fixed-effects intercept from the model
NOOUTPOST	Suppresses storing the posterior samples of missing responses
	in the OUTPOST= data set
OFFSET=	Specifies the offset variable
SCALEPRIOR=	Specifies the prior of the scale parameter

Response Variable Options

Response variable options determine how the BGLIMM procedure models probabilities for binary, binomial, and multinomial data.

You can specify the following response-options by enclosing them in parentheses after the response variable.

DESCENDING DESC

reverses the order of the response categories. If you specify both the DESCENDING and ORDER= options, PROC BGLIMM orders the response categories according to the ORDER= option and then reverses that order.

EVENT='category' | FIRST | LAST

specifies the event category for the binary or binomial response model. PROC BGLIMM models the probability of the event category. This option has no effect when there are more than two response categories.

You can specify any of the following values:

'category'

specifies that observations whose value matches category (formatted, if a format is applied) in quotation marks represent events in the data. For example, the following statements specify that observations that have a formatted value of '1' represent events in the data. The probability that is modeled by PROC BGLIMM is thus the probability that the variable def takes the (formatted) value '1'.

proc bglimm data=MyData;
    class A B C;
    model Def(event ='1') = A B C;
run;

FIRST

designates the first ordered category as the event.

LAST

designates the last ordered category as the event.

By default, EVENT=FIRST.

ORDER=FORMATTED | FREQ | INTERNAL

specifies the sort order of the levels of the response variable. When ORDER=FORMATTED (the default) for numeric variables for which you have supplied no explicit format (that is, for which there is no corresponding FORMAT statement in the current PROC BGLIMM run or in the DATA step that created the data set), the levels are ordered by their internal (numeric) value. Table 4 shows the interpretation of the ORDER= option.

Table 4: Sort Order

ORDER=	Levels Sorted By
FORMATTED	External formatted value, except for numeric variables that have no explicit format, which are sorted by their unformatted (internal) value
FREQ	Descending frequency count (levels that have the most observations come first in the order)
INTERNAL	Unformatted value

By default, ORDER=FORMATTED. For the FORMATTED and INTERNAL orders, the sort order is machine-dependent.

For more information about sort order, see the chapter about the SORT procedure in Base SAS Guide and the discussion of BY-group processing in SAS Language Reference: Concepts.

REFERENCE='category' | FIRST | LAST REF='category' | FIRST | LAST

specifies the reference category for the binary or binomial response model. Specifying one response category as the reference is the same as specifying the other response category as the event category. You can specify any of the following values:

'category': specifies that observations whose value matches category (formatted, if a format is applied) are designated as the reference.
FIRST: designates the first ordered category as the reference.
LAST: designates the last ordered category as the reference.

By default, REF=LAST.

Model Options

You can specify the following model-options in the MODEL statement after a slash (/):

COEFFPRIOR=NORMAL <(options)> | CONSTANT CPRIOR=NORMAL <(options)> | CONSTANT

specifies the prior distribution for the fixed-effects coefficients. The default is COEFFPRIOR=CONSTANT, which specifies the noninformative and improper prior of a constant. If you specify COEFFPRIOR=NORMAL, it is , where is the identity matrix. You can specify the following options for the normal prior, enclosed in parentheses:

INPUT=SAS-data-set: specifies a SAS data set that contains the mean and covariance information of the normal prior. The data set must have a _TYPE_ variable to represent the type of each observation and a variable for each regression coefficient. If the data set also contains a _NAME_ variable, the values of this variable are used to identify the covariance for the _TYPE_='COV' observations; otherwise, the _TYPE_='COV' observations are assumed to be in the same order as the explanatory variables in the MODEL statement. PROC BGLIMM reads the mean vector from the observation for which _TYPE_='MEAN' and reads the covariance matrix from observations for which _TYPE_='COV'. For an independent normal prior, the variances can be specified using _TYPE_='VAR'; alternatively, the precision (inverse of the variances) can be specified using _TYPE_='PRECISION'.
VAR <=c>: specifies the normal prior , where is the identity matrix and c is a scalar. By default, c=1e4.

This VAR=c option specifies a normal prior with a single variance value that is equal to "c", hence the covariance matrix is uncorrelated in the multivariate normal cases. If you want to have a specific mean and variance that are different from the default values, you can provide a SAS data set that contains the mean and variance of the normal prior through the INPUT=SAS-data-set in the CPRIOR= option. See an example below:

data Prior1;
   length _TYPE_ $4;
   input _TYPE_ $ Age Duration;
   datalines;
   Mean  1.0 -2.0
   Var   10  9
   ;

proc bglimm data=Neuralgia seed=123;
   model numPain = Age Duration / dist=binary cprior=normal(input=Prior1);
run;

Instead of just providing values for the variances, sometimes you may want to specify for the whole covariance matrix. Here is an example:

data Prior2;
   length _TYPE_ $4 _NAME_ $13;
   input _TYPE_ $ _NAME_ $ Intercept Age Treatment_A;
   datalines;
   Mean .           1.0 -2.0  1.0
   Cov Intercept    1    0.5  0.2
   Cov Age          0.5  2    0.5
   Cov Treatment_A  0.2  0.5  3
   ;

proc bglimm data=Neuralgia seed=123;
   class Treatment;
   model numPain = Age Duration Treatment / dist=binary
         cprior=normal(input=Prior2);
run;

If there exists a classification variable for which you want to specify a prior value, you need to name it by both the variable name and one of its levels. For example, The last parameter is called "Treatment_A" instead of "Treatment". If any model parameter is not specified in the prior data, its prior mean and variance use the default values.

DISTRIBUTION=keyword DIST=keyword ERROR=keyword ERR=keyword

specifies the response distribution for the model. The keywords and their associated distributions are shown in Table 5.

Table 5: Built-In Distribution Functions

	Distribution
DISTRIBUTION=	Function
BINARY	Binary
BINOMIAL	Binary or binomial
EXPONENTIAL \| EXPO	Exponential
GAMMA \| GAM	Gamma
GEOMETRIC \| GEOM	Geometric
INVGAUSS \| IG	Inverse Gaussian
MULTINOMIAL	Multinomial
NEGBINOMIAL \| NEGBIN \| NB	Negative binomial
NORMAL \| GAUSSIAN \| GAUSS	Normal
POISSON \| POI	Poisson

If you do not specify a distribution, PROC BGLIMM defaults to the normal distribution for continuous response variables and to the binomial or multinomial distribution for classification or character variables, unless you use the events/trial syntax in the MODEL statement. If you choose the events/trial syntax, the procedure defaults to the binomial distribution.

If you do not specify a link function in the LINK= option, a default link function is used. The default link function for each distribution and other commonly used link functions are shown in Table 6. You can also use the LINK= option to specify any link function shown in Table 7.

Table 6: Default and Commonly Used Link Functions

	Default	Other Commonly Used
DISTRIBUTION=	Link Function	Link Functions
BINARY	Logit	Probit, complementary log-log, log-log
BINOMIAL	Logit	Probit, complementary log-log, log-log
EXPONENTIAL \| EXPO	Log	Reciprocal
GAMMA \| GAM	Log	Reciprocal
GEOMETRIC \| GEOM	Log
INVGAUSS \| IG	Reciprocal square
MULTINOMIAL	Cumulative logit	Cumulative probit, cumulative log-log, generalized logit
NEGBINOMIAL \| NEGBIN \| NB	Log
NORMAL \| GAUSSIAN \| GAUSS	Identity	Log
POISSON \| POI	Log

SCALE=options

specifies a fixed value for the scale parameter that is denoted by in the log likelihood functions. Not all exponential family distributions have a scale parameter. For the normal distribution, corresponds to the variance of the response. This option is only available to the normal distribution that has the identity link and has no repeated measurements.

You can specify the following options:

SCALE=variable: enables you to use a variable to specify what value the scale parameter takes for each observation in the INPUT= data. The variable value can be different from observation to observation.
SCALE=c: specifies a numerical value for the scale parameter of a distribution. The value is the same for all the observation lines in the INPUT= data. The specified value must be in the right range of the scale parameter. By default, c=1.

The default is SCALE=1, which specifies the scale parameter to be fixed at 1 for all observations.

INIT=keyword-list | (numeric-list) INITIAL=keyword-list | (numeric-list)

specifies options for generating the initial values for the coefficients parameters that you specify as fixed-effects in the MODEL statement. You can specify the following keywords:

LIST=numeric-list

assigns the numbers to use as the initial values of the fixed effects in the corresponding list order, including the intercept. The length of the numeric-list must be the same as the number of fixed effects. For example, the following statement assigns the values 1, 2, and 3 to the first, second, and third coefficients in the model and prints the table of initial values:

model y = x / init=(list=(1 2 3) pinit);

If the number of items in the numeric-list is less than the number of fixed effects, the initial value of each remaining parameter is replaced by the corresponding default initial value. For example, the corresponding mode of the posterior density is used. If the number of items in the numeric-list is greater than the number of fixed effects, the extra numbers are ignored.

PINIT

tabulates initial values for the fixed effects. (By default, PROC BGLIMM does not display the initial values.)

POSTMODE

uses the mode of the posterior density as the initial value of the parameter.

PRIORMODE

uses the mode of the prior density as the initial value of the parameter.

By default, INIT=POSTMODE.

LINK=keyword

specifies the link function for the model. The keywords and the associated link functions are shown in Table 7. Default and commonly used link functions for the available distributions are shown in Table 6.

Table 7: Built-In Link Functions

LINK=	Link Function
CUMCLL \| CCLL	Cumulative
	Complementary log-log
CUMLOGIT \| CLOGIT	Cumulative logit
CUMLOGLOG	Cumulative log-log
CUMPROBIT \| CPROBIT	Cumulative probit
CLOGLOG \| CLL	Complementary log-log
GLOGIT \| GENLOGIT	Generalized logit
IDENTITY \| ID	Identity
INVERSE \| RECIPROCAL	Reciprocal
LOG	Logarithm
LOGIT	Logit
LOGLOG	Log-log
POWERMINUS2	Power with exponent –2
PROBIT	Probit

in the probit and cumulative probit links denotes the quantile function of the standard normal distribution. For the other cumulative links, denotes a cumulative category probability. The cumulative and generalized logit link functions are appropriate only for the multinomial distribution. When you choose a cumulative link function, PROC BGLIMM assumes that the data are ordinal. When you specify LINK=GLOGIT, the procedure assumes that the data are nominal (not ordered).

NOINT

excludes the intercept from the fixed-effects model. An intercept is included by default.

NOOUTPOST

suppresses storing the posterior samples of missing responses in the OUTPOST= data set. By default, PROC BGLIMM outputs the posterior samples of all missing responses to the OUTPOST= data set.

OFFSET=variable

specifies a variable to use as an offset to the linear predictor. An offset plays the role of an effect whose coefficient is known to be 1. The offset variable cannot be a classification variable or appear elsewhere in the MODEL statement. Observations that have missing values for the offset variable are excluded from the analysis.

SCALEPRIOR=prior-distribution SPRIOR=prior-distribution

specifies the prior distribution for the scale parameter, if the model has a scale parameter. For models that do not have a dispersion parameter (the Poisson and binomial), this option is ignored.

You can specify the one of following prior-distributions:

GAMMA<(options)>

specifies a gamma prior with the density . The hyperparameters a and b are the shape and inverse-scale parameters of the gamma distribution, respectively. The default is . You can set the parameters of the gamma distribution by specifying one or both of the following options, separated by a comma or a space:

SHAPE=a: specifies the shape parameter a of the gamma distribution. By default, SHAPE=0.0001.
ISCALE=b: specifies the inverse-scale parameter b of the gamma distribution. By default, ISCALE=0.0001.

IGAMMA<(options)>

specifies an inverse gamma prior with the density . The hyperparameters a and b are the shape and scale parameters of the inverse gamma distribution, respectively. The default is for the normal likelihood with an identity link, and for all other models. You can set the parameters of the inverse gamma distribution by specifying one or both of the following options, separated by a comma or a space:

SHAPE=a: specifies the shape parameter a of the inverse gamma distribution. By default, SHAPE=2 for the normal likelihood with an identity link, and SHAPE=2.001 for all other models.
SCALE=b: specifies the scale parameter b of the inverse gamma distribution. By default, SCALE=2 for the normal likelihood with an identity link, and SCALE=0.001 for all other models.

IMPROPER

specifies an improper prior with the density proportional to .

UNIFORM<(options)> UNIF<(options)>

specifies a uniform prior, , for the scale parameter. You can set the lower and upper bounds of the uniform distribution by specifying one or both of the following options, separated by a comma or a space:

LOWER=a: specifies the lower bound a of the uniform distribution. The lower bound must be nonnegative. By default, LOWER=0.
UPPER=b: specifies the upper bound b of the uniform distribution. The upper bound must be positive. By default, UPPER=1E10.

For the normal likelihood with an identity link, the prior for the scale parameter is always an inverse gamma, and the default is . Any other prior specification is ignored if the prior is not an inverse gamma.

For any other likelihood or link function, you can choose a gamma prior, an inverse gamma prior, an improper prior, or a uniform prior. The default is the improper prior.

Last updated: December 09, 2022