PMML 4.0 - General Regression

Model XSD and Tag Description

  <xs:element name="GeneralRegressionModel">
    <xs:complexType>
      <xs:sequence>
        <xs:element minOccurs="0" maxOccurs="unbounded" ref="Extension" />
        <xs:element ref="MiningSchema" />
        <xs:element minOccurs="0" ref="Output" />
        <xs:element minOccurs="0" ref="ModelStats" />
        <xs:element ref="ModelExplanation" minOccurs="0"/>
        <xs:element minOccurs="0" ref="Targets" />
        <xs:element minOccurs="0" ref="LocalTransformations" />
        <xs:element ref="ParameterList" />
        <xs:element minOccurs="0" ref="FactorList" />
        <xs:element minOccurs="0" ref="CovariateList" />
        <xs:element ref="PPMatrix" />
        <xs:element minOccurs="0" ref="PCovMatrix" />
        <xs:element ref="ParamMatrix" />
        <xs:element minOccurs="0" ref="EventValues" />
        <xs:element minOccurs="0" ref="BaseCumHazardTables" />
        <xs:element ref="ModelVerification" minOccurs="0"/>
        <xs:element ref="Extension" minOccurs="0" maxOccurs="unbounded"/>
      </xs:sequence>
      <xs:attribute name="targetVariableName" type="FIELD-NAME" />

      <xs:attribute name="modelType" use="required">
        <xs:simpleType>
          <xs:restriction base="xs:string">
            <xs:enumeration value="regression" />
            <xs:enumeration value="generalLinear" />
            <xs:enumeration value="multinomialLogistic" />
            <xs:enumeration value="ordinalMultinomial" />
            <xs:enumeration value="generalizedLinear" />
            <xs:enumeration value="CoxRegression"/>
         </xs:restriction>
        </xs:simpleType>
      </xs:attribute>
      <xs:attribute name="modelName" type="xs:string" />
      <xs:attribute name="functionName" type="MINING-FUNCTION" use="required" />
      <xs:attribute name="algorithmName" type="xs:string" />
      <xs:attribute name="targetReferenceCategory" type="xs:string" />
      <xs:attribute name="cumulativeLink" type="CUMULATIVE-LINK-FUNCTION" />
      <xs:attribute name="linkFunction" type="LINK-FUNCTION" />
      <xs:attribute name="linkParameter" type="REAL-NUMBER" />   
      <xs:attribute name="trialsVariable" type="FIELD-NAME" />
      <xs:attribute name="trialsValue" type="INT-NUMBER" />
      <xs:attribute name="distribution" >
      <xs:simpleType >
          <xs:restriction base="xs:string">
            <xs:enumeration value="binomial"/>
            <xs:enumeration value="gamma"/>
            <xs:enumeration value="igauss"/>
            <xs:enumeration value="negbin"/>
            <xs:enumeration value="normal"/>
            <xs:enumeration value="poisson"/>
            <xs:enumeration value="tweedie"/>
          </xs:restriction >
        </xs:simpleType >
      </xs:attribute >
      <xs:attribute name="distParameter" type="REAL-NUMBER" />   
      <xs:attribute name="offsetVariable" type="FIELD-NAME" />
      <xs:attribute name="offsetValue" type="REAL-NUMBER" />
      <xs:attribute name="modelDF" type="REAL-NUMBER" />
      <xs:attribute name="endTimeVariable" type="FIELD-NAME" />
      <xs:attribute name="startTimeVariable" type="FIELD-NAME" />
      <xs:attribute name="subjectIDVariable" type="FIELD-NAME" />
      <xs:attribute name="statusVariable" type="FIELD-NAME" />
      <xs:attribute name="baselineStrataVariable" type="FIELD-NAME" />
    </xs:complexType>
  </xs:element>

  <xs:element name="ParameterList">
    <xs:complexType>
      <xs:sequence>
        <xs:element ref="Extension" minOccurs="0" maxOccurs="unbounded"/>
        <xs:element ref="Parameter" minOccurs="0" maxOccurs="unbounded"/>
      </xs:sequence>
    </xs:complexType>
  </xs:element>

  <xs:element name="Parameter">
    <xs:complexType>
      <xs:sequence>
        <xs:element ref="Extension" minOccurs="0" maxOccurs="unbounded"/>
      </xs:sequence>
      <xs:attribute name="name" type="xs:string" use="required" />
      <xs:attribute name="label" type="xs:string" />
      <xs:attribute name="referencePoint" type="REAL-NUMBER" default="0" />
    </xs:complexType>
  </xs:element>

  <xs:element name="FactorList">
    <xs:complexType>
      <xs:sequence>
        <xs:element ref="Extension" minOccurs="0" maxOccurs="unbounded"/>
        <xs:element minOccurs="0" maxOccurs="unbounded" ref="Predictor" />
      </xs:sequence>
    </xs:complexType>
  </xs:element>

  <xs:element name="CovariateList">
    <xs:complexType>
      <xs:sequence>
        <xs:element ref="Extension" minOccurs="0" maxOccurs="unbounded"/>
        <xs:element minOccurs="0" maxOccurs="unbounded" ref="Predictor" />
      </xs:sequence>
    </xs:complexType>
  </xs:element>

  <xs:element name="Predictor">
    <xs:complexType>
      <xs:sequence>
        <xs:element ref="Extension" minOccurs="0" maxOccurs="unbounded"/>
        <xs:element ref="Matrix" minOccurs="0" />
      </xs:sequence>
      <xs:attribute name="name" type="FIELD-NAME" use="required" />
      <xs:attribute name="contrastMatrixType" type="xs:string" />
    </xs:complexType>
  </xs:element>

  <xs:element name="PPMatrix">
    <xs:complexType>
      <xs:sequence>
        <xs:element ref="Extension" minOccurs="0" maxOccurs="unbounded"/>
        <xs:element ref="PPCell" minOccurs="0" maxOccurs="unbounded"/>
      </xs:sequence>
    </xs:complexType>
  </xs:element>

  <xs:element name="PPCell">
    <xs:complexType>
      <xs:sequence>
        <xs:element ref="Extension" minOccurs="0" maxOccurs="unbounded"/>
      </xs:sequence>
      <xs:attribute name="value" type="xs:string" use="required" />
      <xs:attribute name="predictorName" type="FIELD-NAME" use="required" />
      <xs:attribute name="parameterName" type="xs:string" use="required" />
      <xs:attribute name="targetCategory" type="xs:string" />
    </xs:complexType>
  </xs:element>

  <xs:element name="PCovMatrix">
    <xs:complexType>
      <xs:sequence>
        <xs:element ref="Extension" minOccurs="0" maxOccurs="unbounded"/>
        <xs:element maxOccurs="unbounded" ref="PCovCell" />
      </xs:sequence>
      <xs:attribute name="type">
        <xs:simpleType>
          <xs:restriction base="xs:string">
            <xs:enumeration value="model" />
            <xs:enumeration value="robust" />
          </xs:restriction>
        </xs:simpleType>
      </xs:attribute>
    </xs:complexType>
  </xs:element>

  <xs:element name="PCovCell">
    <xs:complexType>
      <xs:sequence>
        <xs:element ref="Extension" minOccurs="0" maxOccurs="unbounded"/>
      </xs:sequence>
      <xs:attribute name="pRow" type="xs:string" use="required" />
      <xs:attribute name="pCol" type="xs:string" use="required" />
      <xs:attribute name="tRow" type="xs:string" />
      <xs:attribute name="tCol" type="xs:string" />
      <xs:attribute name="value" type="REAL-NUMBER" use="required" />
      <xs:attribute name="targetCategory" type="xs:string" />
    </xs:complexType>
  </xs:element>

  <xs:element name="ParamMatrix">
    <xs:complexType>
      <xs:sequence>
        <xs:element ref="Extension" minOccurs="0" maxOccurs="unbounded"/>
        <xs:element ref="PCell" minOccurs="0" maxOccurs="unbounded"/>
      </xs:sequence>
    </xs:complexType>
  </xs:element>

  <xs:element name="PCell">
    <xs:complexType>
      <xs:sequence>
        <xs:element ref="Extension" minOccurs="0" maxOccurs="unbounded"/>
      </xs:sequence>
      <xs:attribute name="targetCategory" type="xs:string" />
      <xs:attribute name="parameterName" type="xs:string" use="required" />
      <xs:attribute name="beta" type="REAL-NUMBER" use="required" />
      <xs:attribute name="df" type="INT-NUMBER" />
    </xs:complexType>
  </xs:element>

  <xs:element name="BaseCumHazardTables">
    <xs:complexType>
      <xs:sequence>
        <xs:element ref="Extension" minOccurs="0" maxOccurs="unbounded"/>
        <xs:choice> 
          <xs:element maxOccurs="unbounded" ref="BaselineStratum" />
          <xs:element maxOccurs="unbounded" ref="BaselineCell" />
        </xs:choice> 
      </xs:sequence>
      <xs:attribute name="maxTime" type="REAL-NUMBER" use="optional" />
    </xs:complexType>
  </xs:element>

  <xs:element name="BaselineStratum">
    <xs:complexType>
      <xs:sequence>
        <xs:element ref="Extension" minOccurs="0" maxOccurs="unbounded"/>
        <xs:element minOccurs="0" maxOccurs="unbounded" ref="BaselineCell" />
      </xs:sequence>
      <xs:attribute name="value" type="xs:string" use="required" />
      <xs:attribute name="label" type="xs:string" />
      <xs:attribute name="maxTime" type="REAL-NUMBER" use="required" />
    </xs:complexType>
  </xs:element>

  <xs:element name="BaselineCell">
    <xs:complexType>
      <xs:sequence>
        <xs:element ref="Extension" minOccurs="0" maxOccurs="unbounded"/>
      </xs:sequence>
      <xs:attribute name="time" type="REAL-NUMBER" use="required" />
      <xs:attribute name="cumHazard" type="REAL-NUMBER" use="required" />
    </xs:complexType>
  </xs:element> 

  <xs:element name="EventValues">
    <xs:complexType>
      <xs:sequence>
        <xs:element ref="Extension" minOccurs="0" maxOccurs="unbounded"/>
        <xs:element minOccurs="0" maxOccurs="unbounded" ref="Value" />
        <xs:element minOccurs="0" maxOccurs="unbounded" ref="Interval" />
      </xs:sequence>
    </xs:complexType>
  </xs:element> 

GeneralRegressionModel: marks the beginning of a general regression model. As the name says it, this is intended to support a multitude of regression models.

ParameterList: lists all Parameters. Each Parameter contains a required name and optional label. Parameter names should be unique within the model and as brief as possible (since Parameter names appear frequently in the document). The label, if present, is meant to give a hint on a Parameter's correlation with the Predictors. The optional attribute referencePoint is used in Cox regression models only and has a default value of 0. ParameterList can be empty only for CoxRegression models, for other models at least one Parameter should be present.

FactorList: list of factor (categorical predictor) names. Not present if this particular regression flavor does not support factors (ex. linear regression). If present, the list may or may not be empty. Each name in the list must match a DataField name or a DerivedField name. The factors must be categorical variables. Predictor elements corresponding to categorical predictors can optionally contain a contrast matrix. Such matrix describes the codings of categorical variable. If the categorical variable has n values, there will be n rows and n-1 columns in the matrix. The rows and columns correspond to the categories of the factor in the order listed in DataField or DerivedField. This approach helps to reduce the total number of parameters in the model. The use of such a matrix during scoring is described below.

CovariateList: list of covariate names. Will not be present when there is no covariate. Each name in the list must match a DataField name or a DerivedField name. The covariates will be treated as continuous variables.

targetVariableName: name of the target variable (also called response variable). Must match a DataField name. It should also match the name of the predicted variable in MiningSchema.

modelType: specifies the type of regression model in use. This information will be used to select the appropriate mathematical formulas during scoring. The supported regression algorithms are listed.

modelName and algorithmName can have arbitrary strings describing the specific model.

functionName can only be classification or regression.

targetReferenceCategory can be used for specifying the reference category of the target variable in a multinomial classification model. Normally the reference category is the one from DataDictionary that does not appear in the ParamMatrix, but when several models are combined in one PMML file an explicit specification is needed.

cumulativeLink: specifies the type of cumulative link function to use when ordinalMultinomial model type is specified.

CUMULATIVE-LINK-FUNCTION data type

The definition

<xs:simpleType name="CUMULATIVE-LINK-FUNCTION"> <xs:restriction base="xs:string"> <xs:enumeration value="logit"/> <xs:enumeration value="probit"/> <xs:enumeration value="cloglog"/> <xs:enumeration value="loglog"/> <xs:enumeration value="cauchit"/> </xs:restriction> </xs:simpleType>

is used for specifying a cumulative link function used in ordinalMultinomial model. Specific formulas are listed below in the scoring example.

linkFunction: specifies the type of link function to use when generalizedLinear model type is specified.

LINK-FUNCTION data type

The definition

<xs:simpleType name="LINK-FUNCTION"> <xs:restriction base="xs:string"> <xs:enumeration value="cloglog"/> <xs:enumeration value="identity"/> <xs:enumeration value="log"/> <xs:enumeration value="logc"/> <xs:enumeration value="logit"/> <xs:enumeration value="loglog"/> <xs:enumeration value="negbin"/> <xs:enumeration value="oddspower"/> <xs:enumeration value="power"/> <xs:enumeration value="probit"/> </xs:restriction> </xs:simpleType>

is used for specifying a link function used in generalizedLinear model. Specific formulas are listed below in the scoring example.

linkParameter: specifies an additional number the following link functions need: oddspower and power.

trialsVariable: specifies an additional variable used during scoring some generalizedLinear models (see the description of scoring procedure below). This attribute must refer to a DataField or a DerivedField.

trialsValue: a positive integer used during scoring some generalizedLinear models (see the description of scoring procedure below). At most one of the attributes trialsVariable and trialsValue can be present in a model.

distribution: the probability distribution of the dependent variable for generalizedLinear model may be specified as normal, binomial, gamma, inverse Gaussian, negative binomial, or Poisson.

distParameter: specifies an ancillary parameter value for the negative binomial distribution.

offsetVariable: if present, this variable is used during scoring generalizedLinear, ordinalMultinomial, or multinomialLogistic models (see the description of scoring procedures below). This attribute must refer to a DataField or a DerivedField.

offsetValue: if present, this value is used during scoring generalizedLinear, ordinalMultinomial, or multinomialLogistic models. It works like a user-specified intercept (see the description of the scoring procedures below). At most one of the attributes offsetVariable and offsetValue can be present in a model.

modelDF: the value of degrees of freedom for the model. This value is needed for computing confidence intervals for predicted values.

endTimeVariable: if modelType is CoxRegression, this variable is required during scoring (see the description of scoring procedures below). This attribute must refer to a DataField or a DerivedField containing a continuous variable.

startTimeVariable: if modelType is CoxRegression, this variable is optional, it is not used during scoring but is an important piece of information about model building. This attribute must refer to a DataField or a DerivedField containing a continuous variable.

subjectIDVariable: if modelType is CoxRegression, this variable is optional, it is not used during scoring but is an important piece of information about model building. This attribute must refer to a DataField or a DerivedField. Explicitly listing all categories of this variable is not recommended.

statusVariable: if modelType is CoxRegression, this variable is required during scoring (see the description of scoring procedures below). This attribute must refer to a DataField or a DerivedField.

baselineStrataVariable: if modelType is CoxRegression, this variable is optional, if present it is used during scoring (see the description of scoring procedures below). This attribute must refer to a DataField or a DerivedField containing a categorical variable.

PPMatrix: Predictor-to-Parameter correlation matrix. It is a rectangular matrix having a column for each Predictor (factor or covariate) and a row for each Parameter. The matrix is represented as a sequence of cells, each cell containing a number representing the correlation between the Predictor and the Parameter. The cell values are computed as follows:

• For each Predictor variable v and each Parameter p, the corresponding cell value is missing (empty) if there is no correlation between v and p.
• If there is a correlation between a covariate Predictor and the Parameter, the cell value is set to the exponent that the covariate is raised to in the dependency expression. Example: assuming variable jobcat is a factor and work is a covariate, the Parameter [jobcat=professional] * work * work is correlated to the covariate work, and the number that should be entered in the cell is 2 because work is present at second power in the expression.
• If there is a correlation between a factor variable and the Parameter, the cell value is set to the Predictor value that determines the correlation. Example: Assuming the categories of the factor variable jobcat are professional, clerical, skilled, unskilled, the cell in the matrix that corresponds to (jobcat, jobcat=skilled) has a value of skilled.

The empty cells are not required to be present in the exported model file. All cells determined to be missing from the xml file at model parsing will be assumed to be empty. Since empty cells make up a large chunk of the matrix, this will reduce the size of the exported model. Note that PPMatrix can be empty if a model is intercept-only or Cox regression without parameters.

Note the implied targetCategory attribute. This is permitted in order to allow usage of different PPMatrices for different response values in classification models. For multinomialLogistic model if any PPCell contains this attribute, the expectation is that for that particular response level, a full PPMatrix can be reconstructed from the PMML document. It is that matrix which will be used during scoring in order to get the probability (and other statistics) for the response level. By default, all target categories share the PPMatrix.

targetCategory attribute can thus be used to override the default for some or all target categories.

PPCell: cell in the PPMatrix. Knows its row name, column name, and information as described above.

PCovMatrix: matrix of Parameter estimate covariances. Made up of PCovCells, each of them being located via row information for Parameter name (pRow), row information for target variable value (tRow), column information for Parameter name (pCol) and column information for target variable value (tCol). Note that the matrix is symmetric with respect to the main diagonal (interchanging tRow and tCol together with pRow and pCol will not change the value). Therefore it is sufficient that only half of the matrix be exported. Attributes tRow and tCol are optional since they are not needed for linear regression models. This element has an optional attribute type that can take values model and robust. This attribute describes the way the covariance matrix was computed in generalizedLinear model. The robust option is also known as Huber-White or sandwich or HCCM.

ParamMatrix: Parameter matrix. A table containing the Parameter values along with associated statistics (degrees of freedom). One dimension has the target variable's categories, the other has the Parameter names. The table is represented by specifying each cell. There is no requirement for Parameter names other than that each name should uniquely identify one Parameter.

PCell: cell in the ParamMatrix. The optional targetCategory and required parameterName attributes determine the cell's location in the Parameter matrix. The information contained is: beta (actual Parameter value, required), and df (degrees of freedom, optional). For ordinalMultinomial model ParamMatrix specifies different values for the intercept parameter: one for each target category except one. Values for all other parameters are constant across all target variable values.

EventValues contains a list of Value and/or Interval elements that describe values of the status variable in Cox Regression model corresponding to the "Event". Please see example of Cox Regression below for explanation.

BaseCumHazardTables: Values of baseline cumulative hazard for Cox regression. In the presence of baseline strata variable there is a separate table for each baseline stratum value, otherwise only one table is needed. There is a value for maximum time for which data was available, and a set of pairs of time and cumulative hazard values, in BaselineCell elements.

BaselineCell: cell in the BaseCumHazardTables. The required time and cumHazard attributes contain all needed information.

BaselineStratum contains a set of BaselineCells plus the maximum time for one value of baseline strata variable. The optional label attribute makes it more human-readable.

General Regression Samples: Multinomial Logistic Example

Here is the information about the variables:

The Parameter estimates are displayed as follows:

The PPMatrix is:

This Predictor-to-Parameter combinations mapping is the same for each target variable category. The corresponding XML model is:

  <?xml version="1.0" ?>
  <PMML version="4.0" xmlns="https://www.dmg.org/PMML-4_0" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
    <Header copyright="dmg.org"/>
    <DataDictionary numberOfFields="5">
      <DataField name="jobcat" optype="continuous" dataType="double"/>
      <DataField name="minority" optype="continuous" dataType="double"/>
      <DataField name="sex" optype="continuous" dataType="double"/>
      <DataField name="age" optype="continuous" dataType="double"/>
      <DataField name="work" optype="continuous" dataType="double"/>
    </DataDictionary>

    <GeneralRegressionModel
          targetVariableName="jobcat"
          modelType="multinomialLogistic"
          functionName="classification">

      <MiningSchema>
        <MiningField name="jobcat" usageType="predicted"/>
        <MiningField name="minority" usageType="active"/>
        <MiningField name="sex" usageType="active"/>
        <MiningField name="age" usageType="active"/>
        <MiningField name="work" usageType="active"/>
      </MiningSchema>

      <ParameterList>
        <Parameter name="p0" label="Intercept"/>
        <Parameter name="p1" label="[SEX=0]"/>
        <Parameter name="p2" label="[SEX=1]"/>
        <Parameter name="p3" label="[MINORITY=0]([SEX=0])"/>
        <Parameter name="p4" label="[MINORITY=1]([SEX=0])"/>
        <Parameter name="p5" label="[MINORITY=0]([SEX=1])"/>
        <Parameter name="p6" label="[MINORITY=1]([SEX=1])"/>
        <Parameter name="p7" label="age"/>
        <Parameter name="p8" label="work"/>
      </ParameterList>

      <FactorList>
        <Predictor name="sex" />
        <Predictor name="minority" />
      </FactorList>

      <CovariateList>
        <Predictor name="age" />
        <Predictor name="work" />
      </CovariateList>

      <PPMatrix>
        <PPCell value="0" predictorName="sex" parameterName="p1"/>
        <PPCell value="1" predictorName="sex" parameterName="p2"/>
        <PPCell value="0" predictorName="sex" parameterName="p3"/>
        <PPCell value="0" predictorName="sex" parameterName="p4"/>
        <PPCell value="1" predictorName="sex" parameterName="p5"/>
        <PPCell value="1" predictorName="sex" parameterName="p6"/>
        <PPCell value="0" predictorName="minority" parameterName="p3"/>
        <PPCell value="1" predictorName="minority" parameterName="p4"/>
        <PPCell value="0" predictorName="minority" parameterName="p5"/>
        <PPCell value="1" predictorName="minority" parameterName="p6"/>
        <PPCell value="1" predictorName="age" parameterName="p7"/>
        <PPCell value="1" predictorName="work" parameterName="p8"/>
      </PPMatrix>

      <ParamMatrix>
        <PCell targetCategory="1" parameterName="p0" beta="26.836" df="1"/>
        <PCell targetCategory="1" parameterName="p1" beta="-.719" df="1"/>
        <PCell targetCategory="1" parameterName="p3" beta="-19.214" df="1"/>
        <PCell targetCategory="1" parameterName="p5" beta="-.114" df="1"/>
        <PCell targetCategory="1" parameterName="p7" beta="-.133" df="1"/>
        <PCell targetCategory="1" parameterName="p8" beta="7.885E-02" df="1"/>
        <PCell targetCategory="2" parameterName="p0" beta="31.077" df="1"/>
        <PCell targetCategory="2" parameterName="p1" beta="-.869" df="1"/>
        <PCell targetCategory="2" parameterName="p3" beta="-18.99" df="1"/>
        <PCell targetCategory="2" parameterName="p5" beta="1.01" df="1"/>
        <PCell targetCategory="2" parameterName="p7" beta="-.3" df="1"/>
        <PCell targetCategory="2" parameterName="p8" beta=".152" df="1"/>
        <PCell targetCategory="3" parameterName="p0" beta="6.836" df="1"/>
        <PCell targetCategory="3" parameterName="p1" beta="16.305" df="1"/>
        <PCell targetCategory="3" parameterName="p3" beta="-20.041" df="1"/>
        <PCell targetCategory="3" parameterName="p5" beta="-.73" df="1"/>
        <PCell targetCategory="3" parameterName="p7" beta="-.156" df="1"/>
        <PCell targetCategory="3" parameterName="p8" beta=".267" df="1"/>
        <PCell targetCategory="4" parameterName="p0" beta="8.816" df="1"/>
        <PCell targetCategory="4" parameterName="p1" beta="15.264" df="1"/>
        <PCell targetCategory="4" parameterName="p3" beta="-16.799" df="1"/>
        <PCell targetCategory="4" parameterName="p5" beta="16.48" df="1"/>
        <PCell targetCategory="4" parameterName="p7" beta="-.133" df="1"/>
        <PCell targetCategory="4" parameterName="p8" beta="-.16" df="1"/>
        <PCell targetCategory="5" parameterName="p0" beta="5.862" df="1"/>
        <PCell targetCategory="5" parameterName="p1" beta="16.437" df="1"/>
        <PCell targetCategory="5" parameterName="p3" beta="-17.309" df="1"/>
        <PCell targetCategory="5" parameterName="p5" beta="15.888" df="1"/>
        <PCell targetCategory="5" parameterName="p7" beta="-.105" df="1"/>
        <PCell targetCategory="5" parameterName="p8" beta="6.914E-02" df="1"/>
        <PCell targetCategory="6" parameterName="p0" beta="6.495" df="1"/>
        <PCell targetCategory="6" parameterName="p1" beta="17.297" df="1"/>
        <PCell targetCategory="6" parameterName="p3" beta="-19.098" df="1"/>
        <PCell targetCategory="6" parameterName="p5" beta="16.841" df="1"/>
        <PCell targetCategory="6" parameterName="p7" beta="-.141" df="1"/>
        <PCell targetCategory="6" parameterName="p8" beta="-5.058E-02" df="1"/>
      </ParamMatrix>

    </GeneralRegressionModel>

  </PMML>

Scoring Algorithm

We will use the above example to illustrate the steps that should be followed in the scoring process. Say the following case (observation) must be scored:

         obs = (sex=1 minority=0 age=25 work=4)

1. Do model file parsing. Reconstruct the PPMatrix and the Parameter matrix.
2. To score a case, construct the vector x (of length equal to the number of Parameters in the model) as follows.
   • If row i of the PP correlation matrix is empty, that means the i-th parameter is an intercept, set x_i = 1.
   • If row of the PP correlation matrix is nonempty and corresponds to a factor value or set of factor values, set x_i to 1 if the case being scored matches this row, 0 if it does not.
   • If row i of the PP correlation matrix is nonempty and corresponds to a covariate c, with the entry r, then r is the multiplicity of the covariate c in the parameter, so set x_i=c^r using the value of c in the record.
   • If row i of the PP correlation matrix is nonempty and corresponds to a number of covariates, then set x_i to be the product of covariate values (from the record) using their corresponding multiplicities found in the PP matrix row.
   • Finally, if row i of the PP correlation matrix is nonempty and corresponds to a combination of factors and covariates, then set x_i to be the product of covariate values (from the record) using their corresponding multiplicities found in the PP matrix if the factor values in the record match those in the PP matrix row and 0 otherwise.
3. Now for each response category (value of the target variable) j, let β_j be the vector of Parameter estimates for that response category. (If k is the last response category, remember that by convention β _k= 0.) Set r _j= <x,β_j > and s _j= exp r_i. The probability that our case falls into category j is then p _j= s_j/ (s₁ + ... + s _k).
4. If you just want to assign each case to the category into which it has the highest probability of falling, it is not necessary to compute anything after r_j; the category whose r_j value is highest is the one you want. If you want to compute the actual probabilities (for instance, in order to know whether you are assigning a case to a 51% good or a 99% good category), we use a little dodge to avoid overflow. Namely, p_j is the reciprocal of exp (r₁-r_j ) +... + exp (r_k-r_j). If r_i-r_j> 700 for any i, then the exponential will overflow; but in this case P_j is so small that we can set it to zero. Underflow in the denominator can be ignored since the term exp (r_j-r_j) ensures the denominator is at least 1.

General Regression Samples: General Linear Example

The information about the variables is the same as in the previous example, but now the target variable JOBCAT is considered to be continuous.

The Predictor-to-Parameter combinations mapping is the same as above. The corresponding XML model is:

  <?xml version="1.0" ?>
  <PMML version="4.0" xmlns="https://www.dmg.org/PMML-4_0" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
    <Header copyright="dmg.org"/>
    <DataDictionary numberOfFields="5">
      <DataField name="jobcat" optype="continuous" dataType="double"/>
      <DataField name="minority" optype="continuous" dataType="double"/>
      <DataField name="sex" optype="continuous" dataType="double"/>
      <DataField name="age" optype="continuous" dataType="double"/>
      <DataField name="work" optype="continuous" dataType="double"/>
    </DataDictionary>

    <GeneralRegressionModel
          targetVariableName="jobcat"
          modelType="generalLinear"
          functionName="regression">

      <MiningSchema>
        <MiningField name="jobcat" usageType="predicted"/>
        <MiningField name="minority" usageType="active"/>
        <MiningField name="sex" usageType="active"/>
        <MiningField name="age" usageType="active"/>
        <MiningField name="work" usageType="active"/>
      </MiningSchema>

      <ParameterList>
        <Parameter name="p0" label="Intercept"/>
        <Parameter name="p1" label="[SEX=0]"/>
        <Parameter name="p2" label="[SEX=1]"/>
        <Parameter name="p3" label="[MINORITY=0]([SEX=0])"/>
        <Parameter name="p4" label="[MINORITY=1]([SEX=0])"/>
        <Parameter name="p5" label="[MINORITY=0]([SEX=1])"/>
        <Parameter name="p6" label="[MINORITY=1]([SEX=1])"/>
        <Parameter name="p7" label="age"/>
        <Parameter name="p8" label="work"/>
      </ParameterList>

      <FactorList>
        <Predictor name="sex" />
        <Predictor name="minority" />
      </FactorList>

      <CovariateList>
        <Predictor name="age" />
        <Predictor name="work" />
      </CovariateList>

      <PPMatrix>
        <PPCell value="0" predictorName="sex" parameterName="p1"/>
        <PPCell value="1" predictorName="sex" parameterName="p2"/>
        <PPCell value="0" predictorName="sex" parameterName="p3"/>
        <PPCell value="0" predictorName="sex" parameterName="p4"/>
        <PPCell value="1" predictorName="sex" parameterName="p5"/>
        <PPCell value="1" predictorName="sex" parameterName="p6"/>
        <PPCell value="0" predictorName="minority" parameterName="p3"/>
        <PPCell value="1" predictorName="minority" parameterName="p4"/>
        <PPCell value="0" predictorName="minority" parameterName="p5"/>
        <PPCell value="1" predictorName="minority" parameterName="p6"/>
        <PPCell value="1" predictorName="age" parameterName="p7"/>
        <PPCell value="1" predictorName="work" parameterName="p8"/>
      </PPMatrix>

      <ParamMatrix>
        <PCell  parameterName="p0" beta="1.602" df="1"/>
        <PCell  parameterName="p1" beta="0.580" df="1"/>
        <PCell  parameterName="p3" beta="0.831" df="1"/>
        <PCell  parameterName="p5" beta="0.429" df="1"/>
        <PCell  parameterName="p7" beta="-0.012" df="1"/>
        <PCell  parameterName="p8" beta="0.010"  df="1"/>
      </ParamMatrix>

    </GeneralRegressionModel>

  </PMML>

Scoring Algorithm

For this example the steps that should be followed in the scoring process are similar to the previous one but fewer. Say the following case (observation) must be scored:

         obs = (sex=1 minority=0 age=25 work=4)

1. Do model file parsing. Reconstruct the PPMatrix and the Parameter matrix.
2. To score a case, construct the vector x (of length equal to the number of Parameters in the model) as follows.
   • If row i of the PP correlation matrix is empty, set x _i = 1.

   • If row of the PP correlation matrix is nonempty and corresponds to a factor value or set of factor values, set x _i to 1 if the case being scored matches this row, 0 if it does not.
   • If row i of the PP correlation matrix is nonempty and corresponds to a covariate c, with the entry r, then r is the multiplicity of the covariate c in the parameter, so set x_i=c^r using the value of c in the record.
   • If row i of the PP correlation matrix is nonempty and corresponds to a number of covariates, then set x_i to be the product of covariate values (from the record) using their corresponding multiplicities found in the PP matrix row.
   • Finally, if row i of the PP correlation matrix is nonempty and corresponds to a combination of factors and covariates, then set x_i to be the product of covariate values (from the record) using their corresponding multiplicities found in the PP matrix if the factor values in the record match those in the PP matrix row and 0 otherwise.
3. Now let β be the vector of Parameter estimates. The inner product r = <x,β> is the predicted value for the considered case.

General Regression Samples: Ordinal Multinomial Example

The information about the variables is the same as in the previous examples, but now the target variable JOBCAT is considered to be ordinal.

The Predictor-to-Parameter combinations mapping is the same as above. The corresponding XML model is:

  <?xml version="1.0" ?>
  <PMML version="4.0" xmlns="https://www.dmg.org/PMML-4_0" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
    <Header copyright="dmg.org"/>
    <DataDictionary numberOfFields="5">
      <DataField name="jobcat" optype="continuous" dataType="double"/>
      <DataField name="minority" optype="continuous" dataType="double"/>
      <DataField name="sex" optype="continuous" dataType="double"/>
      <DataField name="age" optype="continuous" dataType="double"/>
      <DataField name="work" optype="continuous" dataType="double"/>
    </DataDictionary>

    <GeneralRegressionModel
          targetVariableName="jobcat"
          modelType="ordinalMultinomial"
          functionName="classification"
          cumulativeLink="logit">

      <MiningSchema>
        <MiningField name="jobcat" usageType="predicted"/>
        <MiningField name="minority" usageType="active"/>
        <MiningField name="sex" usageType="active"/>
        <MiningField name="age" usageType="active"/>
        <MiningField name="work" usageType="active"/>
      </MiningSchema>

      <ParameterList>
        <Parameter name="p0" label="Intercept"/>
        <Parameter name="p1" label="[SEX=0]"/>
        <Parameter name="p2" label="[SEX=1]"/>
        <Parameter name="p3" label="[MINORITY=0]([SEX=0])"/>
        <Parameter name="p4" label="[MINORITY=1]([SEX=0])"/>
        <Parameter name="p5" label="[MINORITY=0]([SEX=1])"/>
        <Parameter name="p6" label="[MINORITY=1]([SEX=1])"/>
        <Parameter name="p7" label="age"/>
        <Parameter name="p8" label="work"/>
      </ParameterList>

      <FactorList>
        <Predictor name="sex" />
        <Predictor name="minority" />
      </FactorList>

      <CovariateList>
        <Predictor name="age" />
        <Predictor name="work" />
      </CovariateList>

      <PPMatrix>
        <PPCell value="0" predictorName="sex" parameterName="p1"/>
        <PPCell value="1" predictorName="sex" parameterName="p2"/>
        <PPCell value="0" predictorName="sex" parameterName="p3"/>
        <PPCell value="0" predictorName="sex" parameterName="p4"/>
        <PPCell value="1" predictorName="sex" parameterName="p5"/>
        <PPCell value="1" predictorName="sex" parameterName="p6"/>
        <PPCell value="0" predictorName="minority" parameterName="p3"/>
        <PPCell value="1" predictorName="minority" parameterName="p4"/>
        <PPCell value="0" predictorName="minority" parameterName="p5"/>
        <PPCell value="1" predictorName="minority" parameterName="p6"/>
        <PPCell value="1" predictorName="age" parameterName="p7"/>
        <PPCell value="1" predictorName="work" parameterName="p8"/>
      </PPMatrix>

      <ParamMatrix>
        <PCell targetCategory="1" parameterName="p0" beta="-0.683" df="1"/>
        <PCell targetCategory="2" parameterName="p0" beta="0.723" df="1"/>
        <PCell targetCategory="3" parameterName="p0" beta="1.104" df="1"/>
        <PCell targetCategory="4" parameterName="p0" beta="1.922" df="1"/>
        <PCell targetCategory="5" parameterName="p0" beta="3.386" df="1"/>
        <PCell targetCategory="6" parameterName="p0" beta="4.006" df="1"/>
        <PCell  parameterName="p1" beta="1.096" df="1"/>
        <PCell  parameterName="p3" beta="0.957" df="1"/>
        <PCell  parameterName="p5" beta="1.149" df="1"/>
        <PCell  parameterName="p7" beta="-0.067" df="1"/>
        <PCell  parameterName="p8" beta="0.060"  df="1"/>
      </ParamMatrix>

    </GeneralRegressionModel>

  </PMML>

Scoring Algorithm

For this example the steps that should be followed in the scoring process are somewhat similar to the first example but also the link function is used. Say the following case (observation) must be scored:

         obs = (sex=1 minority=0 age=25 work=4)

1. Do model file parsing. Reconstruct the PPMatrix and the Parameter matrix.
2. To score a case, construct the vector x (of length equal to the number of Parameters in the model) as follows.
   • If row i of the PP correlation matrix is empty, set x _i= 1.
   • If row of the PP correlation matrix is nonempty and corresponds to a factor value or set of factor values, set x _i to 1 if the case being scored matches this row, 0 if it does not.
   • If row i of the PP correlation matrix is nonempty and corresponds to a covariate c, with the entry r, then r is the multiplicity of the covariate c in the parameter, so set x_i=c^r using the value of c in the record.
   • If row i of the PP correlation matrix is nonempty and corresponds to a number of covariates, then set x_i to be the product of covariate values (from the record) using their corresponding multiplicities found in the PP matrix row.
   • Finally, if row i of the PP correlation matrix is nonempty and corresponds to a combination of factors and covariates, then set x_i to be the product of covariate values (from the record) using their corresponding multiplicities found in the PP matrix if the factor values in the record match those in the PP matrix row and 0 otherwise.
3. Obtain the values of the offsetVariable or offsetValue a.
   Set
   • a = value from the observation information if offsetVariable is used
   • a = offsetValue from the XML file if offsetValue is used
   • a = 0 otherwise.
4. When the target variable has only two categories, the inverse of link function transforms the value predicted by the regression equation into the corresponding probability of the first target category. If target variable is ordinal with more than two categories, a different intercept parameter value is specified by the model for each target category except the last. Inverse of the link function transforms value predicted by the regression equation with specified intercept value into the corresponding cumulative probability for the given category.

How to compute p_j := probability of target=Value_j

For each response category (value of the target variable) j, let β_j be the vector of Parameter estimates for that response category. (If k is the last response category, β_k is not specified.) For the given case let <x,β_j> be the result of evaluating the inner product just like in the multinomialLogistic model and y_j = <x,β_j> + a. Predicted probability for each category is then computed according to the following formulas:

p₁ = F(y₁)
p_j = F(y_j) - F(y_j-1) , for 2 ≤ j < k
p_k = 1 - F(y_k-1)

Function F is an inverse of the specified link function:

logit, ordinal

inverse of logit function: F(y)= 1/(1+exp(-y)).

probit, ordinal

inverse of probit function: F(y)= integral(from -∞ to y)(1/sqrt(2*π))exp(-0.5*u*u)du.

cloglog, ordinal

inverse of cloglog function: F(y)= 1 - exp( -exp(y) ).

loglog, ordinal

inverse of loglog function: F(y)= exp( -exp(-y) ).

cauchit, ordinal

inverse of cauchit function: F(y)= 0.5 + (1/π) arctan(y).

General Regression Samples: Simple Regression Example

Only two continuous predictors are used in this example, and the target variable JOBCAT is considered to be continuous.

The Predictor-to-Parameter combinations mapping is trivial. The corresponding XML model is:

  <?xml version="1.0" ?>
  <PMML version="4.0" xmlns="https://www.dmg.org/PMML-4_0" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
    <Header copyright="dmg.org"/>
    <DataDictionary numberOfFields="5">
      <DataField name="jobcat" optype="continuous" dataType="double"/>
      <DataField name="minority" optype="continuous" dataType="double"/>
      <DataField name="sex" optype="continuous" dataType="double"/>
      <DataField name="age" optype="continuous" dataType="double"/>
      <DataField name="work" optype="continuous" dataType="double"/>
    </DataDictionary>

    <GeneralRegressionModel
          targetVariableName="jobcat"
          modelType="regression"
          functionName="regression">

      <MiningSchema>
        <MiningField name="jobcat" usageType="predicted"/>
        <MiningField name="age" usageType="active"/>
        <MiningField name="work" usageType="active"/>
      </MiningSchema>

      <ParameterList>
        <Parameter name="p0" label="Intercept"/>
        <Parameter name="p1" label="age"/>
        <Parameter name="p2" label="work"/>
      </ParameterList>

      <CovariateList>
        <Predictor name="age" />
        <Predictor name="work" />
      </CovariateList>

      <PPMatrix>
        <PPCell value="1" predictorName="age" parameterName="p1"/>
        <PPCell value="1" predictorName="work" parameterName="p2"/>
      </PPMatrix>

      <ParamMatrix>
        <PCell  parameterName="p0" beta="2.922" df="1"/>
        <PCell  parameterName="p1" beta="-0.031" df="1"/>
        <PCell  parameterName="p2" beta="0.034" df="1"/>
      </ParamMatrix>

    </GeneralRegressionModel>

  </PMML>

Scoring Algorithm

For this example the steps that should be followed in the scoring process are somewhat similar to the general linear example but are even simpler. Say the following case (observation) must be scored:

         obs = (age=25 work=4)

1. Do model file parsing. Reconstruct the PPMatrix and the Parameter matrix.
2. To score a case, construct the vector x (of length equal to the number of Parameters in the model) as follows.
   • If row i of the PP correlation matrix is empty, set x _i = 1.
   • If row i of the PP correlation matrix is nonempty and corresponds to a covariate c, the row should contain exactly one nonzero entry, in the column corresponding to the independent variable c. The value of this entry should be 1 since it is a linear model, so set x_i=x (using the value of c which appears in this case).
3. Now let β be the vector of Parameter estimates. The inner product r = <x,β> is the predicted value for the considered case.

General Regression Samples: Generalized Linear Model Example

The information about the variables is the same as in the previous examples, but now the target variable JOBCAT is considered to be continuous.

The Predictor-to-Parameter combinations mapping is the same as above. The corresponding XML model is:

  <?xml version="1.0" ?>
  <PMML version="4.0" xmlns="https://www.dmg.org/PMML-4_0" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
    <Header copyright="dmg.org"/>
    <DataDictionary numberOfFields="5">
      <DataField name="jobcat" optype="continuous" dataType="double"/>
      <DataField name="minority" optype="continuous" dataType="double"/>
      <DataField name="sex" optype="continuous" dataType="double"/>
      <DataField name="age" optype="continuous" dataType="double"/>
      <DataField name="work" optype="continuous" dataType="double"/>
    </DataDictionary>

    <GeneralRegressionModel
          targetVariableName="jobcat"
          modelType="generalLinear"
          modelName="GZLM"
          functionName="regression"
          distribution="gamma"
          linkFunction="power"
          linkParameter="-1"
          offsetValue="3">

      <MiningSchema>
        <MiningField name="jobcat" usageType="predicted"/>
        <MiningField name="minority" usageType="active"/>
        <MiningField name="sex" usageType="active"/>
        <MiningField name="age" usageType="active"/>
        <MiningField name="work" usageType="active"/>
      </MiningSchema>

      <ParameterList>
        <Parameter name="p0" label="Intercept"/>
        <Parameter name="p1" label="[SEX=0]"/>
        <Parameter name="p2" label="[SEX=1]"/>
        <Parameter name="p3" label="[MINORITY=0]([SEX=0])"/>
        <Parameter name="p4" label="[MINORITY=1]([SEX=0])"/>
        <Parameter name="p5" label="[MINORITY=0]([SEX=1])"/>
        <Parameter name="p6" label="[MINORITY=1]([SEX=1])"/>
        <Parameter name="p7" label="age"/>
        <Parameter name="p8" label="work"/>
      </ParameterList>

      <FactorList>
        <Predictor name="sex" />
        <Predictor name="minority" />
      </FactorList>

      <CovariateList>
        <Predictor name="age" />
        <Predictor name="work" />
      </CovariateList>

      <PPMatrix>
        <PPCell value="0" predictorName="sex" parameterName="p1"/>
        <PPCell value="1" predictorName="sex" parameterName="p2"/>
        <PPCell value="0" predictorName="sex" parameterName="p3"/>
        <PPCell value="0" predictorName="sex" parameterName="p4"/>
        <PPCell value="1" predictorName="sex" parameterName="p5"/>
        <PPCell value="1" predictorName="sex" parameterName="p6"/>
        <PPCell value="0" predictorName="minority" parameterName="p3"/>
        <PPCell value="1" predictorName="minority" parameterName="p4"/>
        <PPCell value="0" predictorName="minority" parameterName="p5"/>
        <PPCell value="1" predictorName="minority" parameterName="p6"/>
        <PPCell value="1" predictorName="age" parameterName="p7"/>
        <PPCell value="1" predictorName="work" parameterName="p8"/>
      </PPMatrix>

      <ParamMatrix>
        <PCell  parameterName="p0" beta="-2.30824444845005" df="1"/>
        <PCell  parameterName="p1" beta="-0.268177596945098" df="1"/>
        <PCell  parameterName="p3" beta="-0.169104566719988" df="1"/>
        <PCell  parameterName="p5" beta="-0.219215962160056" df="1"/>
        <PCell  parameterName="p7" beta="0.00427629446211706" df="1"/>
        <PCell  parameterName="p8" beta="-0.00397117497757107"  df="1"/>
      </ParamMatrix>

    </GeneralRegressionModel>

  </PMML>

Scoring Algorithm

For this example the steps that should be followed in the scoring process are somewhat similar to the second example but also the link function is used. Say the following case (observation) must be scored:

         obs = (sex=1 minority=0 age=25 work=4)

1. Do model file parsing. Reconstruct the PPMatrix and the Parameter matrix.
2. To score a case, construct the vector x (of length equal to the number of Parameters in the model) as follows.
   • If row i of the PP correlation matrix is empty, set x _i = 1.
   • If row of the PP correlation matrix is nonempty and corresponds to a factor value or set of factor values, set x _i to 1 if the case being scored matches this row, 0 if it does not.
   • If row i of the PP correlation matrix is nonempty and corresponds to a covariate c, with the entry r, then r is the multiplicity of the covariate c in the parameter, so set x_i=c^r using the value of c in the record.
   • If row i of the PP correlation matrix is nonempty and corresponds to a number of covariates, then set x_i to be the product of covariate values (from the record) using their corresponding multiplicities found in the PP matrix row.
   • Finally, if row i of the PP correlation matrix is nonempty and corresponds to a combination of factors and covariates, then set x_i to be the product of covariate values (from the record) using their corresponding multiplicities found in the PP matrix if the factor values in the record match those in the PP matrix row and 0 otherwise.
3. Obtain the values of the offsetVariable or offsetValue (a), TrialsVariable or TrialsValue (b), distParameter (c) and linkParameter (d).
   Set
   • a = value from the observation information if offsetVariable is used
   • a = offsetValue from the XML file if offsetValue is used
   • a = 0 otherwise
   Set
   • b = value from the observation information if TrialsVariable is used
   • b = trialsValue from the XML file if TrialsValue is used
   • b = 1 otherwise
   Set
   • c = distParameter from the XML file if link = negbin and distribution = negbin
   Set
   • d = linkParameter from the XML file if link = oddspower or power.
4. Let β be the vector of Parameter estimates and <x,β> be the inner product of two vectors x and β. The predicted value for the considered case is F(<x,β> + a)*b,

   where function F is an inverse of the specified link function:

   cloglog

   inverse of cloglog function: F(y) = 1 - exp( -exp(y) ).
   
   

   identity

   inverse of identity function: F(y) = y.
   
   

   log

   inverse of log function: F(y) = exp(y).
   
   

   logc

   inverse of logc function: F(y) = 1 - exp(y).
   
   

   logit

   inverse of logit function: F(y)= 1/(1 + exp(-y)).
   
   

   loglog

   inverse of loglog function: F(y) = exp( -exp(-y) ).
   
   

   negbin(c)

   inverse of negbin(c) function: F(y) = 1/(c(exp(-y) - 1)).
   
   

   oddspower(d)

   inverse of oddspower(d) function:

   F(y) = 1/(1 + (1 + d*y)^-1/d) if d!=0;
   

   F(y) = 1/(1 + exp(-y)) if d=0.
   
   

   power(d)

   inverse of power(d) function:

   F(y) = y^1/d if d!=0;
   

   F(y) = exp(y) if d=0.
   
   

   probit

   inverse of probit function: F(y) = integral(from -∞ to y)(1/sqrt(2*π))exp(-0.5*u*u)du.
   
   

   General Regression Samples: Example of a model with contrast matrices

   The following example illustrates the use of contrast matrices in a regression model. Here salCat is a target variable with two categories, "Low" and "High". There are two factors in the model and two covariates. The factor gender has two categories and uses "Simple" contrast matrix, while jobcat has three categories and "Helmert" contrast matrix. The model uses main effects and some interaction effects as indicated in PPMatrix and in parameter labels.

   <GeneralRegressionModel targetVariableName="salCat" modelType="multinomialLogistic" modelName="contrastLogistic" functionName="classification" targetReferenceCategory="High"> <MiningSchema> <MiningField name="salCat" usageType="predicted" /> <MiningField name="gender" usageType="active" missingValueTreatment="asIs" /> <MiningField name="educ" usageType="active" missingValueTreatment="asIs" /> <MiningField name="jobcat" usageType="active" missingValueTreatment="asIs" /> <MiningField name="salbegin" usageType="active" missingValueTreatment="asIs" /> </MiningSchema> <ParameterList> <Parameter name="P0000001" label="Constant" /> <Parameter name="P0000002" label="gender(1)" /> <Parameter name="P0000003" label="educ" /> <Parameter name="P0000004" label="jobcat(1)" /> <Parameter name="P0000005" label="jobcat(2)" /> <Parameter name="P0000006" label="gender(1) by jobcat(1)" /> <Parameter name="P0000007" label="gender(1) by jobcat(2)" /> <Parameter name="P0000008" label="educ by gender(1) by salbegin" /> </ParameterList> <FactorList> <Predictor name="gender" contrastMatrixType="Simple" > <Matrix nbRows="2" nbCols="1"> <Array type="real" n="1">.5</Array> <Array type="real" n="1">-.5</Array> </Matrix> </Predictor> <Predictor name="jobcat" contrastMatrixType="Helmert"> <Matrix nbRows="3" nbCols="2"> <Array type="real" n="2">.666666666667 0</Array> <Array type="real" n="2">-.333333333333 .5</Array> <Array type="real" n="2">-.333333333333 -.5</Array> </Matrix> </Predictor> </FactorList> <CovariateList> <Predictor name="educ" /> <Predictor name="salbegin" /> </CovariateList> <PPMatrix> <PPCell value="f" predictorName="gender" parameterName="P0000002" /> <PPCell value="1" predictorName="educ" parameterName="P0000003" /> <PPCell value="1" predictorName="jobcat" parameterName="P0000004" /> <PPCell value="2" predictorName="jobcat" parameterName="P0000005" /> <PPCell value="f" predictorName="gender" parameterName="P0000006" /> <PPCell value="1" predictorName="jobcat" parameterName="P0000006" /> <PPCell value="f" predictorName="gender" parameterName="P0000007" /> <PPCell value="2" predictorName="jobcat" parameterName="P0000007" /> <PPCell value="1" predictorName="educ" parameterName="P0000008" /> <PPCell value="f" predictorName="gender" parameterName="P0000008" /> <PPCell value="1" predictorName="salbegin" parameterName="P0000008" /> </PPMatrix> <ParamMatrix> <PCell targetCategory="Low" parameterName="P0000001" beta="17.0599111512836" df="1" /> <PCell targetCategory="Low" parameterName="P0000002" beta="-2.79578119817189" df="1" /> <PCell targetCategory="Low" parameterName="P0000003" beta="-0.625739483585618" df="1" /> <PCell targetCategory="Low" parameterName="P0000004" beta="-5.76523337984277" df="1" /> <PCell targetCategory="Low" parameterName="P0000005" beta="17.743574615114" df="1" /> <PCell targetCategory="Low" parameterName="P0000006" beta="0.421913613872923" df="1" /> <PCell targetCategory="Low" parameterName="P0000007" beta="0" df="0" /> <PCell targetCategory="Low" parameterName="P0000008" beta="1.1136356754678E-005" df="1" /> </ParamMatrix> </GeneralRegressionModel>

   Scoring Algorithm

   For this example the following steps are needed to score the case

            obs = ( gender="f"  educ=19  jobcat=3  salbegin=45000 )
   

   Note that the categories for factor gender have the following indices: "f" is 1, "m" is 2. For jobcat the categories are "1", "2", "3".
   1. Do model file parsing. Reconstruct the PPMatrix, the Parameter matrix, contrast matrices C_gender and C_jobcat for the factors.
   2. To score the above case, construct the vector x (of length equal to the number of Parameters in the model, 8) as follows.
      • If row i of the PP correlation matrix is empty, set x _i = 1. Here

                x1=1.
        

      • If row of the PP correlation matrix corresponds to exactly one factor value, set x _i to the entry of the contrast matrix for this factor with row index defined by the factor category in the record and column index defined by the category in the PP matrix. For our example we get:

                x2=Cgender(1,1)=0.5
                x4=Cjobcat(3,1)=-0.333333333333
                x5=Cjobcat(3,2)=-0.5 
        

      • If row of the PP correlation matrix corresponds to a set of factor values, set x_i to the product of the entries of contrast matrices as described above. In our example we get:

                x6=Cgender(1,1)*Cjobcat(3,1)=0.5*(-0.333333333333)=-0.16666666666666
                x7=Cgender(1,1)*Cjobcat(3,2)=0.5*(-0.5)=-0.25 
        

      • If row i of the PP correlation matrix corresponds to a covariate c, with the entry r, then r is the multiplicity of the covariate c in the parameter, so set x_i=c^r using the value of c in the record. In our example we get:

                x3=educ=19
        

      • If row i of the PP correlation matrix corresponds to a number of covariates, then set x_i to be the product of covariate values (from the record) using their corresponding multiplicities found in the PP matrix row.
      • Finally, if row i of the PP correlation matrix corresponds to a combination of factors and covariates, then set x_i to be the product of covariate values (from the record) using their corresponding multiplicities found in the PP matrix times the product of contrast matrix entries as described above. In the example we get:

                x8=educ*Cgender(1,1)*salbegin=19*0.5*45000=427500
        

   3. Now let β be the vector of Parameter estimates (for a specified target category, if applicable). The inner product r = <x,β> can be used as described before for various types of regression models. In our example the probability of target category "Low" will be computed as p("Low") = exp( r )/(1 + exp( r ) ).

   General Regression Samples: Cox Regression Model Example

   Cox proportional hazards model of survival is often used in real-life research studies in various industries including pharmaceutical and telecommunications. The idea is as follows: the data must contain an end time variable and a status variable, in addition to any number of predictor variables, and optionally a baseline strata variable, a start time variable, and a subject ID variable. Usually the status variable has certain values or intervals of values that are considered "an event", such as the death of a patient or a telephone customer switching to a competing carrier. The event is happenning or not at the time indicated by the end time variable. Survival is the probability of the event not happening. Cumulative hazard is defined as the negative log of survival. The main assumption is that cumulative hazard for a case with predictors x at time t is computed as

      H( t | x ) = H0( t ) exp( x' * β ),
   

   where H₀(t) is the baseline cumulative hazard at time t, vector β has regression parameter estimates. The probability of survival is

      S( t | x ) = exp( -H( t | x ) ).
   

   The start time and subject ID variables provide an opportunity to represent time-dependent predictors that often appear in survival models. The baseline strata variable, if present, divides all data into several strata based on its categories, with separate baseline hazard values for each stratum. The same regression coefficients β are used in all strata.

   In the following examples variable childs is used as the end time variable, variable life is the status variable with value 1 corresponding to the "event", happy and educ are a factor and a covariate, respectively, and the model is using their main effects and their interaction term. The first example does not have a baseline strata variable, while the second one uses region for that.

   <GeneralRegressionModel targetVariableName="life" modelType="CoxRegression" modelName="CSCox" functionName="regression" endTimeVariable="childs" statusVariable="life" > <MiningSchema> <MiningField name="childs" usageType="active" missingValueTreatment="asIs" /> <MiningField name="happy" usageType="active" missingValueTreatment="asIs" /> <MiningField name="educ" usageType="active" missingValueTreatment="asIs" /> <MiningField name="life" usageType="predicted"/> </MiningSchema> <ParameterList> <Parameter name="P0000001" label="[happy=1]" referencePoint="0"/> <Parameter name="P0000002" label="[happy=2]" referencePoint="0"/> <Parameter name="P0000003" label="[happy=3]" referencePoint="0"/> <Parameter name="P0000004" label="educ" referencePoint="12.85536159601"/> <Parameter name="P0000005" label="[happy=1] * educ" referencePoint="0"/> <Parameter name="P0000006" label="[happy=2] * educ" referencePoint="0"/> <Parameter name="P0000007" label="[happy=3] * educ" referencePoint="0"/> </ParameterList> <FactorList> <Predictor name="happy" /> </FactorList> <CovariateList> <Predictor name="educ" /> </CovariateList> <PPMatrix> <PPCell value="1" predictorName="happy" parameterName="P0000001" /> <PPCell value="2" predictorName="happy" parameterName="P0000002" /> <PPCell value="3" predictorName="happy" parameterName="P0000003" /> <PPCell value="1" predictorName="educ" parameterName="P0000004" /> <PPCell value="1" predictorName="happy" parameterName="P0000005" /> <PPCell value="1" predictorName="educ" parameterName="P0000005" /> <PPCell value="2" predictorName="happy" parameterName="P0000006" /> <PPCell value="1" predictorName="educ" parameterName="P0000006" /> <PPCell value="3" predictorName="happy" parameterName="P0000007" /> <PPCell value="1" predictorName="educ" parameterName="P0000007" /> </PPMatrix> <ParamMatrix> <PCell parameterName="P0000001" beta="2.19176500383392" df="1" /> <PCell parameterName="P0000002" beta="0.839584538765938" df="1" /> <PCell parameterName="P0000003" beta="0" df="0" /> <PCell parameterName="P0000004" beta="0.207006511267958" df="1" /> <PCell parameterName="P0000005" beta="-0.124788379173099" df="1" /> <PCell parameterName="P0000006" beta="-0.0652692443310469" df="1" /> <PCell parameterName="P0000007" beta="0" df="0" /> </ParamMatrix> <EventValues> <Value value="1"/> </EventValues> <BaseCumHazardTables maxTime="8"> <BaselineCell time="1" cumHazard="0.0805149154781295" /> <BaselineCell time="2" cumHazard="0.208621561646413" /> <BaselineCell time="3" cumHazard="0.367889107749672" /> <BaselineCell time="4" cumHazard="0.610515527436034" /> <BaselineCell time="5" cumHazard="0.782436645962723" /> <BaselineCell time="6" cumHazard="0.898256334351415" /> <BaselineCell time="7" cumHazard="1.34645277785058" /> <BaselineCell time="8" cumHazard="1.92644296943848" /> </BaseCumHazardTables> </GeneralRegressionModel> <GeneralRegressionModel targetVariableName="life" modelType="CoxRegression" modelName="CSCox" functionName="regression" endTimeVariable="childs" statusVariable="life" baselineStrataVariable="region"> <MiningSchema> <MiningField name="childs" usageType="active" missingValueTreatment="asIs" /> <MiningField name="happy" usageType="active" missingValueTreatment="asIs" /> <MiningField name="educ" usageType="active" missingValueTreatment="asIs" /> <MiningField name="region" usageType="active" /> <MiningField name="life" usageType="predicted"/> </MiningSchema> <ParameterList> <Parameter name="P0000001" label="[happy=1]" referencePoint="0"/> <Parameter name="P0000002" label="[happy=2]" referencePoint="0"/> <Parameter name="P0000003" label="[happy=3]" referencePoint="0"/> <Parameter name="P0000004" label="educ" referencePoint="12.85536159601"/> <Parameter name="P0000005" label="[happy=1] * educ" referencePoint="0"/> <Parameter name="P0000006" label="[happy=2] * educ" referencePoint="0"/> <Parameter name="P0000007" label="[happy=3] * educ" referencePoint="0"/> </ParameterList> <FactorList> <Predictor name="happy" /> </FactorList> <CovariateList> <Predictor name="educ" /> </CovariateList> <PPMatrix> <PPCell value="1" predictorName="happy" parameterName="P0000001" /> <PPCell value="2" predictorName="happy" parameterName="P0000002" /> <PPCell value="3" predictorName="happy" parameterName="P0000003" /> <PPCell value="1" predictorName="educ" parameterName="P0000004" /> <PPCell value="1" predictorName="happy" parameterName="P0000005" /> <PPCell value="1" predictorName="educ" parameterName="P0000005" /> <PPCell value="2" predictorName="happy" parameterName="P0000006" /> <PPCell value="1" predictorName="educ" parameterName="P0000006" /> <PPCell value="3" predictorName="happy" parameterName="P0000007" /> <PPCell value="1" predictorName="educ" parameterName="P0000007" /> </PPMatrix> <ParamMatrix> <PCell parameterName="P0000001" beta="1.96429877799117" df="1" /> <PCell parameterName="P0000002" beta="0.487952271605177" df="1" /> <PCell parameterName="P0000003" beta="0" df="0" /> <PCell parameterName="P0000004" beta="0.186388616742954" df="1" /> <PCell parameterName="P0000005" beta="-0.0964727062694649" df="1" /> <PCell parameterName="P0000006" beta="-0.0257167272021955" df="1" /> <PCell parameterName="P0000007" beta="0" df="0" /> </ParamMatrix> <EventValues> <Value value="1"/> </EventValues> <BaseCumHazardTables> <BaselineStratum value="1" label="[region=North East]" maxTime="7"> <BaselineCell time="1" cumHazard="0.0480764996657994" /> <BaselineCell time="2" cumHazard="0.213530888447458" /> <BaselineCell time="3" cumHazard="0.347177590555568" /> <BaselineCell time="4" cumHazard="0.700088580976311" /> <BaselineCell time="5" cumHazard="0.756857216338272" /> <BaselineCell time="6" cumHazard="0.880125294006154" /> <BaselineCell time="7" cumHazard="1.79261158114014" /> <BaselineStratum> <BaselineStratum value="2" label="[region=South East]" maxTime="7"> <BaselineCell time="1" cumHazard="0.104783416911293" /> <BaselineCell time="2" cumHazard="0.149899368179306" /> <BaselineCell time="3" cumHazard="0.344676164146026" /> <BaselineCell time="4" cumHazard="0.447807317242553" /> <BaselineCell time="5" cumHazard="0.602148704727296" /> <BaselineCell time="6" cumHazard="0.996057753780737" /> </BaselineStratum> <BaselineStratum value="3" label="[region=West]" maxTime="8"> <BaselineCell time="1" cumHazard="0.0798136487904092" /> <BaselineCell time="2" cumHazard="0.148350388305914" /> <BaselineCell time="3" cumHazard="0.252784132000578" /> <BaselineCell time="4" cumHazard="0.366288821244008" /> <BaselineCell time="5" cumHazard="0.562653812085775" /> <BaselineCell time="6" cumHazard="0.61271473319101" /> <BaselineCell time="7" cumHazard="0.81698327174713" /> <BaselineCell time="8" cumHazard="1.28475458929774" /> </BaselineStratum> </BaseCumHazardTables> </GeneralRegressionModel>

   Scoring Algorithm

   For this example some steps that should be followed in the scoring process are somewhat similar to the general linear example but also additional work is done to compute survival and cumulative hazard values using the equations presented above.

   1. Check if baseline strata variable is present. If it is, get its value from the case and check if there is a BaselineStratum element for that value. If not, return missing values as the result, else get maxTime from the BaselineStratum's attribute. In the absence of strata variable get maxTime from the BaseCumHazardTables element.
   2. Get the value of the end time variable from the case. If it is less than the minimum time in a BaselineCell in the previously chosen BaselineStratum or BaseCumHazardTables, then predicted survival is 1 and cumulative hazard is 0. If the time value is greater than maxTime, return missing value. Otherwise find the BaselineCell that has the largest time attribute value that is not greater than the time from the case. Extract baseline cumulative hazard value H₀(t) from its attribute cumHazard.
   3. Compute the inner product r = <x,β> as described above for regression models.
   4. Compute the inner product of the reference point vector x₀ (its values are located in Parameter elements) and the parameter estimates β: s = <x₀,β>
   5. Finally, compute the cumulative hazard and survival:

         H( t | x ) = H0( t ) exp( r - s ),
         S( t | x ) = exp( -H( t | x ) ).
      

   Note that Cox Regression model can be valid even when there are no parameters at all. In that case r=0, s=0, so the cumulative hazard is the same as baseline hazard, and survival is still computed by the formula presented above.