# Cox

## Description

Create and analyze a `Cox`

model object to calculate
lifetime probability of default (PD) using this workflow:

Use

`fitLifetimePDModel`

to create a`Cox`

model object.Optionally, use

`discardResiduals`

to remove residual information from the`Cox`

model object.Use

`predict`

to predict the conditional PD and`predictLifetime`

to predict the lifetime PD.Use

`modelDiscrimination`

to return AUROC and ROC data. You can plot the results using`modelDiscriminationPlot`

.Use

`modelCalibration`

to return the root mean square error (RMSE) of observed and predicted PD data. You can plot the results using`modelCalibrationPlot`

.

## Creation

### Syntax

### Description

creates a `CoxPDModel`

= fitLifetimePDModel(`data`

,`ModelType`

,`AgeVar`

=agevar_value)`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.

sets optional properties using additional
name-value arguments in addition to the required arguments in the previous
syntax. For example, `CoxPDModel`

= fitLifetimePDModel(___,`Name=Value`

)```
CoxPDModel =
fitLifetimePDModel(data(TrainDataInd,:),"Cox",ModelID="Cox_A",Descripion="Cox_model",AgeVar="YOB",IDVar="ID",LoanVars="ScoreGroup",MacroVars={'GDP','Market'},ResponseVar="Default",TimeInterval=1,TieBreakMethod='Efron',WeightsVar="Weights")
```

creates a `CoxPDModel`

using a `Cox`

model
type. You can specify multiple name-value arguments.

### Input Arguments

`data`

— Data

table

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`

`ModelType`

— Model type

string with value `"Cox"`

| character vector with value `'Cox'`

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.

**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,WeightsVar="Weights")
```

**Required**

`Cox`

Name-Value Argument`AgeVar`

— Age variable indicating which column in `data`

contains loan age information

string | character vector

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`ModelID`

— User-defined model ID

`Cox`

(default) | string | character vector

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`

`Description`

— User-defined description for model

`""`

(default) | string | character vector

User-defined description for model, specified as
`Description`

and a string or character
vector.

**Data Types: **`string`

| `char`

`IDVar`

— ID variable indicating which column in `data`

contains loan or borrower ID

1st column of `data`

(default) | string | character vector

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`

`LoanVars`

— Loan variables indicating which column in `data`

contains loan-specific
information

all columns of `data`

that
are not the first or last column (default) | string array | cell array of character vectors

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`

`MacroVars`

— Macro variables indicating which column in `data`

contains macroeconomic
information

`""`

(default) | string array | cell array of character vectors

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`

`ResponseVar`

— Variable indicating which column in `data`

contains response variable

string | character vector

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`

`WeightsVar`

— Variable indicating which column in data input contains observation weights

`""`

| string array

Variable indicating which column in `data`

contains the observation weights, specified as
`WeightsVar`

and a string array.

**Note**

The default value (`""`

) results in a
weight of `1`

for each row in the
`data`

.

For an example using `WeightsVar`

, see Create Weighted Lifetime PD Model.

**Data Types: **`string`

`TimeInterval`

— Distance between age values in panel `data`

input

set to most common `AgeVar`

increment in the training `data`

(default) | positive numeric

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`

`TieBreakMethod`

— Method to handle tied default times

`"breslow"`

(default) | string with value `"breslow"`

or
`"efron"`

| character vector with value `'breslow'`

or
`'efron'`

*Since R2023a*

Method to handle tied default times, specified as a string or character vector with one of the following tie-break methods:

`breslow`

— Breslow's approximation to the partial likelihood`efron`

— Efron's approximation to the partial likelihood

For credit applications, the time to default comes discretized and
there are many "ties." This means that are multiple borrowers that
may default at the same (discretized) time (such as, in the second
year of their loan). `TieBreakMethod`

supports the
`breslow`

or `efron`

methods
to handle this scenario.

**Data Types: **`string`

| `char`

## Properties

`ModelID`

— User-defined model ID

`Probit`

(default) | string

User-defined model ID, returned as a string.

**Data Types: **`string`

`Description`

— User-defined description

`""`

(default) | string

User-defined description, returned as a string.

**Data Types: **`string`

`IDVar`

— ID variable indicating which column in `data`

contains loan or borrower ID

1st column of `data`

(default) | string

ID variable indicating which column in `data`

contains the loan or borrower ID, returned as a string.

**Data Types: **`string`

`AgeVar`

— Age variable indicating which column in `data`

contains loan age information

string

Age variable indicating which column in `data`

contains the loan age information, returned as a string.

**Data Types: **`string`

`LoanVars`

— Loan variables indicating which column in `data`

contains loan-specific information

all columns of `data`

that are not
the first or last column (default) | string array

Loan variables indicating which column in `data`

contains the loan-specific information, returned as a string
array.

**Data Types: **`string`

`MacroVars`

— Macro variables indicating which column in `data`

contains macroeconomic information

`""`

(default) | string array

Macro variables indicating which column in `data`

contains the macroeconomic information, returned as a string
array.

**Data Types: **`string`

`ResponseVar`

— Variable indicating which column in `data`

contains response variable

string

Variable indicating which column in `data`

contains
the response variable, returned as a string.

**Data Types: **`string`

`WeightsVar`

— Variable indicating which column in data input contains observation weights

`""`

| string array

Variable indicating which column in `data`

contains the
observation weights, returned as a string array.

**Data Types: **`string`

`TimeInterval`

— Distance between age values in panel `data`

input

set to most common `AgeVar`

increment
in the training data (default) | positive numeric

This property is read-only.

Distance between age values in panel `data`

input,
returned as a scalar positive numeric.

**Data Types: **`double`

`ExtrapolationFactor`

— Extrapolation factor

`1`

(default) | positive numeric between `0`

and
`1`

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`

`TieBreakMethod`

— Method to handle tied default times

`"breslow"`

(default) | string with value `"breslow"`

or
`"efron"`

Method to handle tied default times, returned as a string.

**Data Types: **`string`

## Object Functions

`predict` | Compute conditional PD |

`predictLifetime` | Compute cumulative lifetime PD, marginal PD, and survival probability |

`modelDiscrimination` | Compute AUROC and ROC data |

`modelCalibration` | Compute RMSE of predicted and observed PDs on grouped data |

`modelDiscriminationPlot` | Plot ROC curve |

`modelCalibrationPlot` | Plot observed default rates compared to predicted PDs on grouped data |

`discardResiduals` | Discard residual information of underlying Cox model |

## Examples

### Create Cox Lifetime PD Model

This example shows how to use `fitLifetimePDModel`

to create a `Cox`

model using credit and macroeconomic data.

**Load Data**

Load the credit portfolio data.

```
load RetailCreditPanelData.mat
disp(head(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

disp(head(dataMacro))

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); disp(head(data))

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.

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

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

Display the underlying model.

disp(pdModel.UnderlyingModel)

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 `modelCalibration`

to measure the calibration of the predicted PD values. The `modelCalibration`

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.

CalMeasure = modelCalibration(pdModel,data(Ind,:),{'YOB','ScoreGroup'})

`CalMeasure=`*table*
RMSE
_________
Cox, grouped by YOB, ScoreGroup 0.0012471

disp(CalMeasure)

RMSE _________ Cox, grouped by YOB, ScoreGroup 0.0012471

Use `modelCalibrationPlot`

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

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

**Predict Conditional and Lifetime PD**

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).

LifetimePD = predictLifetime(pdModel,data(Ind,:));

### Select Tie-Break Method for Cox Lifetime PD Models

*Since R2023a*

This example shows how to create a `Cox`

model and select the tie-break method while fitting a Cox lifetime PD model.

**Load Data**

Load the credit portfolio data.

```
load RetailCreditPanelData.mat
disp(head(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

disp(head(dataMacro))

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); disp(head(data))

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

**Join the Data**

Join the two data components into a single data set.

data = join(data,dataMacro); disp(head(data))

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 the 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 with Breslow's Method**

Use `fitLifetimePDModel`

to create a `Cox`

model using the training data. Use the name-value argument `TieBreakMethod`

to set tie-break method to `'breslow'`

. This is the default choice for this argument.

pdModel1 = fitLifetimePDModel(data(TrainDataInd,:),"Cox",... ModelID="Cox-Breslow", IDVar="ID", AgeVar="YOB", ... LoanVars="ScoreGroup", MacroVars={'GDP','Market'}, ... ResponseVar="Default",TieBreakMethod='breslow');

Display the underlying model.

disp(pdModel1.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

Use `predict`

to predict the conditional PD.

pd1 = predict(pdModel1,data(TestDataInd,:));

**Create a Cox Lifetime PD Model with Efron's Method**

Use `fitLifetimePDModel`

to create a `Cox`

model using the training data. Use the name-value argument `TieBreakMethod`

to set tie-break method to `'Efron'`

. This is the default choice for this argument.

pdModel2 = fitLifetimePDModel(data(TrainDataInd,:),"Cox",... ModelID="Cox-Efron", IDVar="ID", AgeVar="YOB", ... LoanVars="ScoreGroup", MacroVars={'GDP','Market'}, ... ResponseVar="Default",TieBreakMethod='efron');

Display the underlying model. The coefficients are only slightly different for this data set.

disp(pdModel2.Model)

Cox Proportional Hazards regression model Beta SE zStat pValue __________ _________ _______ __________ ScoreGroup_Medium Risk -0.6844 0.037029 -18.483 2.8461e-76 ScoreGroup_Low Risk -1.2515 0.045243 -27.662 2.006e-168 GDP -0.084985 0.043691 -1.9452 0.051756 Market -0.0085126 0.0032223 -2.6418 0.0082469 Log-likelihood: -41713.445

Use `predict`

to predict the conditional PD for the second `Cox`

model.

pd2 = predict(pdModel2,data(TestDataInd,:));

**Compare Cox Models**

The predictions for the two `Cox`

models are almost the same for this data set.

[pd1(1:10) pd2(1:10)]

`ans = `*10×2*
0.0162 0.0161
0.0091 0.0090
0.0081 0.0081
0.0073 0.0072
0.0064 0.0064
0.0072 0.0072
0.0030 0.0030
0.0016 0.0016
0.0162 0.0161
0.0091 0.0090

For this data set, the model discrimination (`modelDiscrimination`

) does not seem to change with the `TieBreakMethod`

method and the model accuracy (`modelCalibration`

) shows only a negligible difference in RMSE.

modelDiscriminationPlot(pdModel1,data(TestDataInd,:),ReferencePD=pd2,ReferenceID=pdModel2.ModelID)

`modelCalibrationPlot(pdModel1,data(TestDataInd,:),'Year',ReferencePD=pd2,ReferenceID=pdModel2.ModelID)`

## More About

### Cox Proportional Hazards Models

The *Cox proportional hazards* (PH)
model is a survival model and it models the time until an event of interest
occurs.

For probability of default (PD) models, the event of interest is the default
on a credit obligation. `Cox`

models need information on
whether there was a default and when it happened. For other commonly used PD
models, a binary variable indicating whether there was a default is enough.
`Cox`

PD models need that information, plus the age of the
loan at the time of default.

The `Cox`

proportional hazards (PH) model, also known as a
`Cox`

regression model, assumes the hazard rate is of the form

$$h(t;X)={h}_{0}(t)\mathrm{exp}(X\beta )$$

where

*h*_{0}(*t*) is the baseline hazard rate.*X*is the predictor data.β is a vector of coefficients of the predictors.

exp(

*X*Î²) is the hazard ratio.

The baseline hazard rate is a reference hazard level, common to all
observations, and it does not depend on the predictor values. The hazard ratio
is the factor that scales the baseline hazard value up or down, depending on the
predictor values. For lower risk observations, the hazard ratio is less than
`1`

and this reduces the hazard rate. For higher risk
observations, the hazard ratio increases the hazard rate.

In the hazard rate formula, the predictor values in *X* are
fixed, or *independent of time*. This is the
basic version of the `Cox`

PH model. For PD models, the basic
version of the `Cox`

PH model includes predictors that have
constant values, such as the origination score, or whether a property is for
residential or commercial purposes.

The *time-dependent *
`Cox`

PH model allows predictor values to change over time. For
example, the loan-to-value (LTV) ratio changes over the life of a loan, and the
macroeconomic variables change from period to period. Therefore, the following
hazard rate formula for time-dependent models includes predictor values that can
be a function of time:

$$h(t;X)={h}_{0}(t)\mathrm{exp}(X(t)\beta )$$

The `data`

input for `fitLifetimePDModel`

must be in panel data form. For each ID
(`IDVar`

), there are multiple rows of data. The panel
`data`

input is required for both time-dependent and time
-independent models.

For time-independent predictors, the predictor value is constant for each ID.
For example, the score at origination for each customer is constant throughout
the life of the loan, and this value is repeated for each row corresponding to
the same ID in the panel `data`

format.

For time-dependent predictors, the values may change from one row to the next
for the same ID. The assumption is that the predictor values in each row are
valid in the time interval defined by the age value
(`AgeVar`

) in the previous row and the age value in the
current row.

### Time Interval for `Cox`

Models

Time is discretized into intervals, and predictor values in
the training data (`data`

input) are constant for each
interval: *X*_{1} from
*t*_{0} to
*t*_{1};
*X*_{2} from
*t*_{1} to
*t*_{2}; and so forth.

The `data`

input must be in panel data form, with multiple
observations for each ID, with corresponding age information (the
*t*_{k} values, the
`AgeVar`

column) and the corresponding default indicator
values (the `ResponseVar`

column).

Assume that *t*_{k} -
*t*_{k - 1} =
Δ*t* for all *k* and this is the
*time interval*. This time interval is the age increment
for consecutive observations in the age data (`AgeVar`

). The
assumption is that these increments are regular and that the default indicator
(`ResponseVar`

) is defined consistently with this time
interval, in the sense that a `1`

means there was a default in a
time interval of length Δ*t*. The time interval
Δ*t* is also used for the computation of the probability of
default. For more information, see Lifetime Prediction and Time Interval.

### Survival and Probability of Default for `Cox`

Models

The survival function
*S*(*t*) is a function of time, and gives
the probability of surviving longer than a given time
*t*.

$$S(t)=P(T>t)$$

where

*T*is the failure time, the random variable of interest, and in the`Cox`

model case, the time to default.*t*is the specific time of interest, for example, 1 year.

The main relationship between the survival function and the hazard rate is

$$S(t)=\mathrm{exp}\left(-{\displaystyle {\int}_{0}^{t}h(u)du}\right)$$

Higher values of the hazard rate cause the survival probability to drop faster. Conversely, lower values of the hazard rate cause the survival probability to rise faster.

The probability of default (PD) is the conditional probability of defaulting
in a time interval, given that there has been no default prior to that interval.
For example, the probability of default between time *s* and
*t*, with *s* < *t*,
is represented as:

$$\begin{array}{l}PD(s,t)=P(s<T\le t|T>s)\\ \text{=}\frac{S(s)-S(t)}{S(s)}\\ \text{=1-}\frac{S(t)}{S(s)}\end{array}$$

In credit applications, the time interval of interest, Δ*t*,
is consistent with the training data and the definition of default in the response
variable. The PD is a function of a single time variable *t* and
the implicit time interval Δ*t*:

$$PD(t)=1-\frac{S(t)}{S(t-\Delta t)}$$

## 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**

### R2023b: Added `WeightsVar`

name-value argument for `Cox`

model

The `Cox`

model supports a `WeightsVar`

name-value argument for observation weights.

### R2023a: `modelAccuracy`

object function is renamed to `modelCalibration`

function

The `modelAccuracy`

object function is renamed to
`modelCalibration`

function. The use of
`modelAccuracy`

is discouraged, use `modelCalibration`

instead.

### R2023a: `modelAccuracyPlot`

object function is renamed to `modelCalibrationPlot`

function

The `modelAccuracyPlot`

object function is renamed to
`modelCalibrationPlot`

function. The use of
`modelAccuracyPlot`

is discouraged, use `modelCalibrationPlot`

instead.

### R2023a: Added `TieBreakMethod`

name-value argument

The `TieBreakMethod`

name-value argument enables you to specify
the method to handle tied default times.

### R2023a: Added `discardResiduals`

method for Cox model

Use the `discardResiduals`

method to discard residual information of the underlying Cox model.

### R2023a: `Model`

property renamed to `UnderlyingModel`

The `Model`

property is renamed to
`UnderlyingModel`

.

## See Also

### Functions

### Topics

- Basic Lifetime PD Model Validation
- Compare Logistic Model for Lifetime PD to Champion Model
- Compare Lifetime PD Models Using Cross-Validation
- Expected Credit Loss Computation
- Compare Model Discrimination and Model Calibration to Validate of Probability of Default
- Compare Probability of Default Using Through-the-Cycle and Point-in-Time Models
- Modeling Probabilities of Default with Cox Proportional Hazards
- Create Weighted Lifetime PD Model
- Overview of Lifetime Probability of Default Models

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