# Cox

Create Cox model object for lifetime probability of default

## Description

Create and analyze a Cox model object to calculate lifetime probability of default (PD) using this workflow:

1. Use fitLifetimePDModel to create a Cox model object.

2. Use predict to predict the conditional PD and predictLifetime to predict the lifetime PD.

3. Use modelDiscrimination to return AUROC and ROC data. You can plot the results using modelDiscriminationPlot.

4. Use modelAccuracy to return the root mean square error (RMSE) of observed and predicted PD data. You can plot the results using modelAccuracyPlot.

## Creation

### Description

example

CoxPDModel = fitLifetimePDModel(data,ModelType,AgeVar=agevar_value) creates a Cox PD model object.

If you do not specify variable information for IDVar, LoanVars, MacroVars, and ResponseVar, then:

• IDVar is set to the first column in the data input.

• LoanVars is set to include all columns from the second to the second-to-last columns of the data input.

• ResponseVar is set to the last column in the data input.

example

CoxPDModel = fitLifetimePDModel(___,Name=Value) sets optional properties using additional name-value arguments in addition to the required arguments in the previous syntax. For example, CoxPDModel = fitLifetimePDModel(data(TrainDataInd,:),"Cox",ModelID="Cox_A",Descripion="Cox_model",AgeVar="YOB",IDVar="ID",LoanVars="ScoreGroup",MacroVars={'GDP','Market'},ResponseVar="Default",TimeInterval=1) creates a CoxPDModel using a Cox model type. You can specify multiple name-value arguments.

### Input Arguments

expand all

Data, specified as a table, in panel data form. The data must contain an ID column and an Age column. The response variable must be a binary variable with the value 0 or 1, with 1 indicating default.

Data Types: table

Model type, specified as a string with the value "Cox" or a character vector with the value 'Cox'.

Data Types: char | string

Name-Value Arguments

Specify required and optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Required Cox Name-Value Argument

expand all

Age variable indicating which column in data contains the loan age information, specified as AgeVar and a string or character vector.

Note

The required name-value argument AgeVar is not treated as a predictor in the Cox lifetime PD model. When using a Cox model, you must specify predictor variables using LoanVars or MacroVars. The AgeVar values are the event times for the underlying Cox proportional hazards model.

AgeVar values for each ID should be increasing. If there are nonpositive age increments, fitLifetimePDModel warns when you create a Cox model and removes the IDs with nonpositive age increments. By default, the TimeInterval value is set to the most common age increment in the training data.

Data Types: string | char

Optional Cox Name-Value Arguments

expand all

User-defined model ID, specified as ModelID and a string or character vector. The software uses the ModelID to format outputs and is expected to be short.

Data Types: string | char

User-defined description for model, specified as Description and a string or character vector.

Data Types: string | char

ID variable indicating which column in data contains the loan or borrower ID, specified as IDVar and a string or character vector.

Data Types: string | char

Loan variables indicating which column in data contains the loan-specific information, such as origination score or loan-to-value ratio, specified as LoanVars and a string array or cell array of character vectors.

Data Types: string | cell

Macro variables indicating which column in data contains the macroeconomic information, such as gross domestic product (GDP) growth or unemployment rate, specified as MacroVars and a string array or cell array of character vectors.

Data Types: string | cell

Variable indicating which column in data contains the response variable, specified as ResponseVar and a logical value.

Note

The response variable values in the data must be a binary variable with 0 or 1 values, with 1 indicating default.

In Cox lifetime PD models, the ResponseVar values define the censoring information for the underlying Cox proportional hazards model.

Data Types: string | char

Distance between age values in training data in the panel data input, specified as TimeInterval and a positive numeric scalar.

Use the TimeInterval name-value argument to fit time-dependent models and also as the time interval for the PD computation when you use the predict function. For example, if the age data (AgeVar) is 1, 2, 3, ..., then the TimeInterval is 1; if the age data is 0.25, 0.5, 0.75,..., then the TimeInterval is 0.25. For more information, see Time Interval for Cox Models and Lifetime Prediction and Time Interval.

Note

Unlike Logistic and Probit models, a Cox model requires an AgeVar variable. By default, if you do not specify a TimeInterval when creating a Cox model, the TimeInterval is inferred from the increments in the AgeVar values in the training data.

Data Types: double

## Properties

expand all

User-defined model ID, returned as a string.

Data Types: string

User-defined description, returned as a string.

Data Types: string

Underlying statistical model, returned as a returned as a Cox proportional hazards model object. For more information, see fitcox and CoxModel.

Data Types: CoxModel

ID variable indicating which column in data contains the loan or borrower ID, returned as a string.

Data Types: string

Age variable indicating which column in data contains the loan age information, returned as a string.

Data Types: string

Loan variables indicating which column in data contains the loan-specific information, returned as a string array.

Data Types: string

Macro variables indicating which column in data contains the macroeconomic information, returned as a string array.

Data Types: string

Variable indicating which column in data contains the response variable, returned as a string.

Data Types: string

Distance between age values in panel data input, returned as a scalar positive numeric.

Data Types: double

Extrapolation factor, returned as a positive numeric scalar between 0 and 1.

By default, the ExtrapolationFactor is set to 1. For age values (AgeVar) greater than the maximum age observed in the training data, the conditional PD, computed with predict, uses the maximum age observed in the training data. In particular, the predicted PD value is constant if the predictor values do not change and only the age values change when the ExtrapolationFactor is 1. For more information, see Extrapolation for Cox Models, Extrapolation Factor for Cox Models, and Use Cox Lifetime PD Model to Predict Conditional PD.

Data Types: double

## Object Functions

 predict Compute conditional PD predictLifetime Compute cumulative lifetime PD, marginal PD, and survival probability modelDiscrimination Compute AUROC and ROC data modelAccuracy Compute RMSE of predicted and observed PDs on grouped data modelDiscriminationPlot Plot ROC curve modelAccuracyPlot Plot observed default rates compared to predicted PDs on grouped data

## Examples

collapse all

This example shows how to use fitLifetimePDModel to create a Cox model using credit and macroeconomic data.

ID    ScoreGroup    YOB    Default    Year
__    __________    ___    _______    ____

1      Low Risk      1        0       1997
1      Low Risk      2        0       1998
1      Low Risk      3        0       1999
1      Low Risk      4        0       2000
1      Low Risk      5        0       2001
1      Low Risk      6        0       2002
1      Low Risk      7        0       2003
1      Low Risk      8        0       2004
Year     GDP     Market
____    _____    ______

1997     2.72      7.61
1998     3.57     26.24
1999     2.86      18.1
2000     2.43      3.19
2001     1.26    -10.51
2002    -0.59    -22.95
2003     0.63      2.78
2004     1.85      9.48

Join the two data components into a single data set.

data = join(data,dataMacro);
ID    ScoreGroup    YOB    Default    Year     GDP     Market
__    __________    ___    _______    ____    _____    ______

1      Low Risk      1        0       1997     2.72      7.61
1      Low Risk      2        0       1998     3.57     26.24
1      Low Risk      3        0       1999     2.86      18.1
1      Low Risk      4        0       2000     2.43      3.19
1      Low Risk      5        0       2001     1.26    -10.51
1      Low Risk      6        0       2002    -0.59    -22.95
1      Low Risk      7        0       2003     0.63      2.78
1      Low Risk      8        0       2004     1.85      9.48

Partition Data

Separate the data into training and test partitions.

nIDs = max(data.ID);
uniqueIDs = unique(data.ID);

rng('default'); % For reproducibility
c = cvpartition(nIDs,'HoldOut',0.4);

TrainIDInd = training(c);
TestIDInd = test(c);

TrainDataInd = ismember(data.ID,uniqueIDs(TrainIDInd));
TestDataInd = ismember(data.ID,uniqueIDs(TestIDInd));

Create a Cox Lifetime PD Model

Use fitLifetimePDModel to create a Cox model using the training data.

AgeVar="YOB", ...
IDVar="ID", ...
LoanVars="ScoreGroup", ...
MacroVars={'GDP','Market'}, ...
ResponseVar="Default");
disp(pdModel)
Cox with properties:

TimeInterval: 1
ExtrapolationFactor: 1
ModelID: "Cox"
Description: ""
Model: [1x1 CoxModel]
IDVar: "ID"
AgeVar: "YOB"
LoanVars: "ScoreGroup"
MacroVars: ["GDP"    "Market"]
ResponseVar: "Default"

Display the underlying model.

disp(pdModel.Model)
Cox Proportional Hazards regression model

Beta          SE         zStat       pValue
__________    _________    _______    ___________

ScoreGroup_Medium Risk       -0.6794     0.037029    -18.348     3.4442e-75
ScoreGroup_Low Risk          -1.2442     0.045244    -27.501    1.7116e-166
GDP                        -0.084533     0.043687     -1.935       0.052995
Market                    -0.0084411    0.0032221    -2.6198      0.0087991

Log-likelihood: -41742.871

Validate Model

Use modelDiscrimination to measure the ranking of customers by PD.

DataSetChoice = "Testing";
if DataSetChoice=="Training"
Ind = TrainDataInd;
else
Ind = TestDataInd;
end

DiscMeasure = modelDiscrimination(pdModel,data(Ind,:),SegmentBy="ScoreGroup")
DiscMeasure=3×1 table
AUROC
_______

Cox, ScoreGroup=High Risk      0.64112
Cox, ScoreGroup=Medium Risk    0.61989
Cox, ScoreGroup=Low Risk        0.6314

disp(DiscMeasure)
AUROC
_______

Cox, ScoreGroup=High Risk      0.64112
Cox, ScoreGroup=Medium Risk    0.61989
Cox, ScoreGroup=Low Risk        0.6314

Use modelDiscriminationPlot to visualize the ROC curve.

modelDiscriminationPlot(pdModel,data(Ind,:),SegmentBy="ScoreGroup")

Use modelAccuracy to measure the accuracy (or calibration) of the predicted PD values. The modelAccuracy function requires a grouping variable and compares the accuracy of the observed default rate in the group with the average predicted PD for the group.

AccMeasure = modelAccuracy(pdModel,data(Ind,:),{'YOB','ScoreGroup'})
AccMeasure=table
RMSE
_________

Cox, grouped by YOB, ScoreGroup    0.0012471

disp(AccMeasure)
RMSE
_________

Cox, grouped by YOB, ScoreGroup    0.0012471

Use modelAccuracyPlot to visualize the observed default rates compared to the predicted PD.

modelAccuracyPlot(pdModel,data(Ind,:),{'YOB','ScoreGroup'})

Use the predict function to predict conditional PD values. The prediction is a row-by-row prediction.

%dataCustomer1 = data(1:8,:);
CondPD = predict(pdModel,data(Ind,:));

Use predictLifetime to predict the lifetime cumulative PD values (computing marginal and survival PD values is also supported).

expand all

## References

[1] Baesens, Bart, Daniel Roesch, and Harald Scheule. Credit Risk Analytics: Measurement Techniques, Applications, and Examples in SAS. Wiley, 2016.

[2] Bellini, Tiziano. IFRS 9 and CECL Credit Risk Modelling and Validation: A Practical Guide with Examples Worked in R and SAS. San Diego, CA: Elsevier, 2019.

[3] Breeden, Joseph. Living with CECL: The Modeling Dictionary. Santa Fe, NM: Prescient Models LLC, 2018.

[4] Roesch, Daniel and Harald Scheule. Deep Credit Risk: Machine Learning with Python. Independently published, 2020.

## Version History

Introduced in R2021b