The SEVERITY Procedure

Custom Objective Functions

You can use a series of programming statements that use variables in the DATA= data set to assign a value to an objective function symbol. You must specify the objective function symbol by using the OBJECTIVE= option in the PROC SEVERITY statement.

The objective function can be programmed such that it is applicable to any distribution that is used in the model. For that purpose, PROC SEVERITY recognizes the following keyword functions in the programming statements:

_PDF_(x)

returns the probability density function (PDF) of a distribution evaluated at the current value of a data set variable `x`.

_CDF_(x)

returns the cumulative distribution function (CDF) of a distribution evaluated at the current value of a data set variable `x`.

_SDF_(x)

returns the survival distribution function (SDF) of a distribution evaluated at the current value of a data set variable `x`.

_LOGPDF_(x)

returns the natural logarithm of the PDF of a distribution evaluated at the current value of a data set variable `x`.

_LOGCDF_(x)

returns the natural logarithm of the CDF of a distribution evaluated at the current value of a data set variable `x`.

_LOGSDF_(x)

returns the natural logarithm of the SDF of a distribution evaluated at the current value of a data set variable `x`.

_EDF_(x)

returns the empirical distribution function (EDF) estimate evaluated at the current value of a data set variable `x`. Internally, PROC SEVERITY computes the estimate using the SVRTUTIL_EDF function as described in the section Predefined Utility Functions. The EDF estimate that is required by the SVRTUTIL_EDF function is computed by using the response variable values in the current BY group or in the entire input data set if you do not specify the BY statement.

_EMPLIMMOMENT_(k, u)

returns the empirical limited moment of order k evaluated at the current value of a data set variable `u` that represents the upper limit of the limited moment. The order k can also be a data set variable. Internally, PROC SEVERITY computes the moment using the SVRTUTIL_EMPLIMMOMENT function as described in the section Predefined Utility Functions. The EDF estimate that is required by the SVRTUTIL_EMPLIMMOMENT function is computed by using the response variable values in the current BY group or in the entire input data set if you do not specify the BY statement.

_LIMMOMENT_(k, u)

returns the limited moment of order k evaluated at the current value of a data set variable `u` that represents the upper limit of the limited moment. The order k can be a data set variable or a constant. Internally, for each candidate distribution, PROC SEVERITY computes the moment using the LIMMOMENT function as described in the section Predefined Utility Functions.

All the preceding functions are right-hand side functions. They act as placeholders for distribution-specific functions, with the exception of _EDF_ and _EMPLIMMOMENT_ functions.

As an example, let the data set `Work.Test` contain a response variable `Y` and a left-truncation threshold variable `T`. The following statements use the values in this data set to fit a model with distribution `D` such that the parameters of the model minimize the value of the objective function symbol `MYOBJ`:

```options cmplib=(work.mydist);
proc severity data=work.test objective=myobj;
loss y / lt=t;

myobj = -_LOGPDF_(y);
if (not(missing(t))) then
myobj = myobj + log(1-_CDF_(t));

dist d;
run;
```

The symbol `MYOBJ` is designated as an objective function symbol by using the OBJECTIVE= option in the PROC SEVERITY statement. The response variable `Y` and left-truncation variable `T` are specified in the LOSS statement. The distribution `D` is specified in the DIST statement. The remaining statements constitute a program that computes the value of the `MYOBJ` symbol.

Let the distribution `D` have parameters `P1` and `P2`. In order to estimate the model for this distribution, PROC SEVERITY internally converts the generic program to the following program specific to distribution `D`:

```myobj = -D_LOGPDF(y, p1, p2);
if (not(missing(t))) then
myobj = myobj + log(1-D_CDF(t, p1, p2));
```

Note that the generic keyword functions _LOGPDF_ and _CDF_ have been replaced with distribution-specific functions D_LOGPDF and D_CDF, respectively, with appropriate distribution parameters. The D_LOGPDF and D_CDF functions must have been defined previously and are assumed to be available in the `Work.Mydist` library that you specify in the CMPLIB= option.

The program is executed for each observation in `Work.Test` to compute the value of `MYOBJ` by using the values of variables `Y` and `T` in that observation and internally computed values of the model parameters `P1` and `P2`. The values of `MYOBJ` are then added over all the observations of the data set or over all the observations of the current BY group if you specify the BY statement. The resulting aggregate value is the value of the objective function, and it is supplied to the optimizer. If the optimizer requires derivatives of the objective function, then PROC SEVERITY automatically differentiates `MYOBJ` with respect to the parameters `P1` and `P2`. The optimizer iterates over various combinations of the values of parameters `P1` and `P2`, each time computing a new value of the objective function and the needed derivatives of it, until it finds a combination that minimizes the objective function.

Note the following points when you define your own program to compute the custom objective function:

• The value of the objective function is always minimized by PROC SEVERITY. If you want to maximize the value of a certain objective, then add a statement that assigns the negated value of the maximization objective to the objective function symbol that you specify in the OBJECTIVE= option. Minimization of the negated objective is equivalent to the maximization of the original objective.

• The contributions of individual observations are always added to compute the overall objective function in a given iteration of the optimizer. If you specify the WEIGHT statement, then the contribution of each observation is weighted by multiplying it with the normalized value of the weight variable for that observation.

• If you are fitting multiple distributions in one PROC SEVERITY step and use any of the keyword functions in your program, then it is recommended that you do not explicitly use the parameters of any of the specified distributions in your programming statements.

• If you use a specific keyword function in your programming statements, then the corresponding distribution functions must be defined in a library that you specify in the CMPLIB= system option or in `Sashelp.Svrtdist`, the predefined functions library. In the preceding example, it is assumed that the functions D_LOGPDF and D_CDF are defined in the `Work.Mydist` library that is specified in the CMPLIB= option.

• You can use most DATA step statements and functions in your program. The DATA step file and the data set I/O statements (for example, INPUT, FILE, SET, and MERGE) are not available. However, some functionality of the PUT statement is supported. See the section "PROC FCMP and DATA Step Differences" in Base SAS Procedures Guide for more information. In addition to the differences listed in that section, the following differences exist:

• Only numeric-valued variables can be used in PROC SEVERITY programming statements. This restriction also implies that you cannot use SAS functions or call routines that require character-valued arguments, unless you pass those arguments as constant (literal) strings or characters.

• You cannot use functions that create lagged versions of a variable in PROC SEVERITY programming statements. If you need lagged versions, then you can use a DATA step prior to the PROC SEVERITY step to add those versions to the input data set.

• When coding your programming statements, avoid defining variables that begin with an underscore (_), because they might conflict with internal variables created by PROC SEVERITY.

Custom Objective Functions and Regression Effects

If you specify regression effects by using the SCALEMODEL statement, then PROC SEVERITY automatically adds a statement prior to your programming statements to compute the value of the scale parameter or the log-transformed scale parameter of the distribution using the values of the regression variables and internally created regression parameters. For example, if your specification of the SCALEMODEL statement results in three regression effects `x1`, `x2`, and `x3`, then for a model that contains the distribution `D` with scale parameter `S`, PROC SEVERITY prepends your program with a statement that is equivalent to the following statement:

```    S = _SEVTHETA0 * exp(_SEVBETA1 * x1 + _SEVBETA2 * x2 + _SEVBETA3 * x3);
```

If a model contains a distribution `D1` with a log-transformed scale parameter `M`, PROC SEVERITY prepends your program with a statement that is equivalent to the following statement:

```    M = _SEVTHETA0 + _SEVBETA1 * x1 + _SEVBETA2 * x2 + _SEVBETA3 * x3;
```

The _SEVTHETA0, _SEVBETA1, _SEVBETA2, and _SEVBETA3 are the internal regression parameters associated with the intercept and the regression effects `x1`, `x2`, and `x3`, respectively.

Since the names of the internal regression parameters start with a prefix _SEV, if you use a variable in your program with a name that begins with _SEV, then PROC SEVERITY writes an error message to the SAS log and stops processing.