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Cox proportional hazards model

A Cox proportional hazards model relates to lifetime or failure time data. The
basic Cox model includes a hazard function
*h*_{0}(*t*) and model coefficients
*b* such that, for predictor `X`

, the hazard rate at time
*t* is

$$h\left({X}_{i},t\right)={h}_{0}(t)\mathrm{exp}\left[{\displaystyle \sum _{j=1}^{p}{x}_{ij}{b}_{j}}\right],$$

where the *b* coefficients do not depend on time. The creation function
`fitcox`

infers both the model coefficients *b* and the
hazard rate *h*_{0}(*t*), and stores
them as properties in the resulting `CoxModel`

object.

The full Cox model includes extensions to the basic model, such as hazards with respect to different baselines or the inclusion of stratification variables. See Extension of Cox Proportional Hazards Model.

Create a `CoxModel`

object using `fitcox`

.

`Baseline`

— Baseline hazard`mean(X)`

(default) | real scalarBaseline hazard specified when model was fitted, specified as a real scalar. The Cox
model is a relative hazard model, so it requires a baseline at which to compare hazards
of given data, relative to the baseline. The default is `mean(X)`

(and
the mean within each stratification for stratified models), so the hazard rate at
`X`

is `h(t)*exp((X-mean(X))*b)`

. Enter
`0`

to compute the baseline relative to `0`

, so the
hazard rate at `X`

is `h(t)*exp(X*b)`

. Changing the
baseline does not affect the coefficients.

**Data Types: **`double`

`CoefficientCovariance`

— Covariance matrix for coefficient estimatessquare matrix

Covariance matrix for coefficient estimates, specified as a square matrix with the number of rows equal to the number of predictors.

**Data Types: **`double`

`Coefficients`

— Coefficients and related statisticstable

Coefficients and related statistics, specified as a table with four columns:

`Beta`

— Coefficient estimate`SE`

— Standard error of the coefficient estimate`zStat`

— z statistic`pValue`

—*p*-value for the coefficient (compared to a zero Beta)

Each row of the table corresponds to one predictor.

To obtain any of these columns as a vector, index into the property using dot
notation. For example, in the `coxMdl`

object, the estimated
coefficient vector is

beta = coxMdl.Coefficients.Beta

To perform other tests on the coefficients, use `linhyptest`

.

**Data Types: **`table`

`Formula`

— Representation of the model used in fitformula in Wilkinson notation

Representation of the model used in the fit, specified as a formula in Wilkinson notation. See Wilkinson Notation. For example, to include several predictors, use

`'X ~ a + b + … + c'`

where each of the variables `a`

, `b`

,
`c`

represents one predictor. These variables are column names for
the table `X`

.

`Hazard`

— Estimated baseline cumulative hazardmatrix double

Estimated baseline cumulative hazard, specified as a matrix double. The cumulative hazard is evaluated at time points defined in training.

`Hazard`

has at least two columns. The first column contains the
time values, and the rest of the columns contain the cumulative hazard at each listed
time.

For nonstratified models,

`Hazard`

has two columns.For stratified models,

`Hazard`

has an additional column for each unique combination of the stratification levels. Distinct time values in`Hazard(:,1)`

for each stratification are separated by`0`

entries in`Hazard(:,2)`

. A stratified model is a model trained using the`'Stratification'`

name-value argument.

Theoretically, the cumulative hazard at time *t* is –log(1 –
cdf(*t*)). The empirical cumulative hazard is

$$\widehat{{H}_{0}}(t)={\displaystyle \sum _{{t}_{i}\le t}\widehat{{h}_{0}}}\left({t}_{i}\right)={\displaystyle \sum _{{t}_{i}\le t}\frac{1}{{\displaystyle \sum _{j\in {R}_{i}}\mathrm{exp}}\left(\beta \xb7{x}_{j}\right)},}$$

where *R _{i}* is the risk set at time

**Data Types: **`double`

`IsStratified`

— Indication that model is trained with stratified datalogical

Indication that the model is trained with stratified data, specified as a logical
value (`true`

or `false`

).

**Data Types: **`logical`

`LikelihoodRatioTestPValue`

— real scalar

*P*-value indicating if the model is significant relative to the
null model, specified as a real scalar.

This property contains the *p*-value of performing the likelihood
ratio test against the null model, that is, a model with all coefficients equal to 0.

The likelihood ratio test compares the likelihood function of the data at the
coefficient estimates, and at all the coefficients being 0. The comparison yields a test
statistic that can be used to determine if the trained model is significant, relative to
a model with all coefficients equal to 0. The null hypothesis is that there is no
difference between the null model and the trained model, so a significant
*p*-value implies the trained model is significant.

**Data Types: **`double`

`LogLikelihood`

— Log of likelihood function at coefficient estimatesreal scalar

Log of the likelihood function at the coefficient estimates, specified as a real scalar.

**Data Types: **`double`

`NumPredictors`

— Number of predictorspositive integer

Number of predictors (coefficients) in the model, specified as a positive integer.

**Data Types: **`double`

`PredictorNames`

— Names of predictorscell array of character vectors

Names of the predictors used to fit the model, specified as a cell array of
character vectors. If the model is trained on data in a table, the predictor names are
the names of the table columns. Otherwise, the predictor names are
`X1`

, `X2`

, and so on.

**Data Types: **`cell`

`ProportionalHazardsPValue`

— real vector

*P*-value indicating if each covariate satisfies the proportional
hazards assumption, specified as a real vector, with one entry for each
predictor.

The Cox model relies on the assumption of proportional hazards, that is, for any two data points X1 and X2, hazard(X1)/hazard(X2) is constant. This assumption might be violated if the predictors depend on time. For example, if a predictor corresponds to age, it generally becomes more hazardous as age increases.

The test of this assumption uses the scaled Schoenfeld residuals and was derived by Grambsch and Therneau in [1].

The null hypothesis is that each coefficient satisfies the proportional hazards
assumption. A significant *p*-value implies that a specific coefficient
violates the proportional hazards assumption. The test is performed on each coefficient,
so this property is a vector with as many elements as the number of coefficients.

**Data Types: **`double`

`ProportionalHazardsPValueGlobal`

— real scalar

*P*-value indicating if the whole model satisfies the proportional
hazards assumption, specified as a real scalar.

The Cox model relies on the assumption of proportional hazards, that is, for any two data points X1 and X2, hazard(X1)/hazard(X2) is constant. This assumption might be violated if the predictors depend on time. For example, if a predictor corresponds to age, it generally becomes more hazardous as age increases.

The test of this assumption uses the scaled Schoenfeld residuals and was derived by Grambsch and Therneau in [1].

The null hypothesis is that the model, as a whole, satisfies the proportional
hazards assumption. A significant *p*-value implies the whole model
does not satisfy the proportional hazards assumption.

**Data Types: **`double`

`Residuals`

— Residuals of various typestable

Residuals of various types, specified as a table with seven columns, one for each residual:

`'CoxSnell'`

— The Cox-Snell residuals for an observation`X(i)`

are defined as the cumulative hazard at time`i`

(`cumHazard(i)`

) multiplied by the hazard of`X(i)`

:`csres(i)`

=`cumHazard(i)`

*`exp(X(i) * Beta)`

. Beta is the fitted Beta vector stored in`Coefficients`

.`'Deviance'`

— The deviance residual is defined using the martingale residual as follows:`D(i) = sign(M(i))*sqrt(–2*[M(i) + delta(i)*log(delta(i)–M(i)))]`

, where`D(i)`

is the`i`

th deviance residual,`M(i)`

is the`i`

th martingale residual, and`delta(i)`

indicates if the data point`i`

died or not.`'Martingale'`

— The martingale residual for a point`X(i)`

is`delta(i) – CoxSnell(i)`

, where`delta(i)`

indicates if`X(i)`

died, and`CoxSnell(i)`

is the Cox-Snell residual at`i`

. The martingale residual can be viewed as the difference between the true number of deaths for`X(i)`

minus the expected number of deaths based on the model.`'Schoenfeld'`

— The Schoenfeld residuals are defined as:`scres(i,j) = X(i,j) – M(Beta,i,j)`

, where`X(i,j)`

is the`j`

th element of observation`i`

, and`M(Beta,i,j)`

is the expected value of`X(i,j)`

, given the number of living observations left at time`i`

. The Schoenfeld residuals can be viewed as the difference between a true dead observation at time`i`

and how the model expects a dead observation to look at time`i`

, given the remaining living observations. The residuals are calculated for each covariate, so they have as many columns as the number of learned parameters. The residuals are valid only for times and observations at which there were deaths. For any censored observations, the corresponding residual is`NaN`

.`'ScaledSchoenfeld'`

— The scaled Schoenfeld residuals are the Schoenfeld residuals scaled by the variance of the learned coefficients. Like the Schoenfeld residuals, the scaled residuals are not defined for observations and times at which there were no deaths; a residual at such a point is`NaN`

.`'Score'`

— The score residuals are defined as:`scores(i,t) = integral_{0}^{t}(X(i,u) – Xbar(u)) dScres(i,u)`

, where`Schres(i)`

is the Schoenfeld residual at observation`i`

, and`Xbar`

is the mean of the observations still alive at time`u`

. The residuals are calculated for each covariate, so they have as many columns as the number of learned parameters.`'ScaledScore'`

— The scaled score residuals are the score residuals scaled by the covariance of the fitted coefficients.

`Residuals`

has the same number of rows as the training
data.

**Data Types: **`table`

`ResponseName`

— Response variable namecharacter vector

Response variable name, specified as a character vector. For models where the
response value is in a table, the response variable name is the name of the relevant
table column. Otherwise, `ResponseName`

is
`'T'`

.

**Data Types: **`char`

`StandardError`

— Standard errors of coefficient estimatesreal vector

Standard errors of coefficient estimates, specified as a real vector.
`StandardError`

is the square root of the diagonal of the
`CoefficientCovariance`

matrix.

**Data Types: **`double`

`Stratification`

— Array of unique combinations of input stratificationnumeric array | string array | cell array of strings | categorical array | cell array

Array of unique combinations of input stratification during training, specified as one of the following data types.

Numeric array — All stratification variables are numeric.

String array — All variables are strings.

Cell array of strings — All variables are cell strings.

Categorical array — All variables are categorical.

Cell array — Variables are mixed types.

Given some data `X`

and `T`

, the following table
shows examples of what `Stratification`

contains.

Input Data | Example | Resulting Stratification |
---|---|---|

Double | ```
mdl = fitcox(X,T,'Stratification',[1 2; 1 2; 2 2; 2 2]);
``` | `[1 2; 2 2]` |

String | ```
mdl =
fitcox(X,T,'Stratification',["cat";"dog";"dog";"bird"]);
``` | `["cat"; "dog"; "bird"]` |

Cell string | ```
mdl =
fitcox(X,T,'Stratification',{'cat';'dog';'dog';'bird'});
``` | `{'cat';'dog';'bird'}` |

Categorical | ```
mdl =
fitcox(X,T,'Stratification',categorical([1;2;3;4]));
``` | `categorical([1;2;3;4])` |

Mixed types |
| `{1 'cat'; 2 'cat'; 1 'dog'; 3 'dog'}` |

**Data Types: **`double`

| `char`

| `string`

| `cell`

| `categorical`

`VariableInfo`

— Information about fitting datatable

Information about fitting data, specified as a table with four columns:

`Class`

— The class of the predictor.`Range`

— The minimum and maximum of the predictor if it is not categorical, or the list of all the categories if the predictor is categorical.`InModel`

— A logical indicating if the predictor is used in the model. The response variable is not in the model. Predictor variables used for training are in the model.`IsCategorical`

— A logical indicating if the predictor was treated as categorical during training.

If the model has no categorical predictors, and no formula was used to fit the
model, the number of rows of `VariableInfo`

is the number of model
predictors. Otherwise, the number of rows is the same as the number of elements in
`PredictorNames`

.

**Data Types: **`table`

`coefci` | Confidence interval for Cox proportional hazards model coefficients |

`hazardratio` | Estimate Cox model hazard relative to baseline |

`linhyptest` | Linear hypothesis tests on Cox model coefficients |

`plotSurvival` | Plot survival function of Cox proportional hazards model |

`survival` | Calculate survival of Cox proportional hazards model |

Weibull random variables with the same shape parameter have proportional hazard rates; see Weibull Distribution. The hazard rate with scale parameter $$a$$ and shape parameter $$b$$ at time $$t$$ is

$$\frac{b}{{a}^{b}}{t}^{b-1}$$.

Generate pseudorandom samples from the Weibull distribution with scale parameters 1, 5, and 1/3, and with the same shape parameter `B`

.

rng default % For reproducibility B = 2; A = ones(100,1); data1 = wblrnd(A,B); A2 = 5*A; data2 = wblrnd(A2,B); A3 = A/3; data3 = wblrnd(A3,B);

Create a table of data. The predictors are the three variable types, 1, 2, or 3.

predictors = categorical([A;2*A;3*A]); data = table(predictors,[data1;data2;data3],'VariableNames',["Predictors" "Times"]);

Fit a Cox regression to the data.

`mdl = fitcox(data,"Times")`

mdl = Cox Proportional Hazards regression model: Beta SE zStat pValue _______ _______ _______ __________ Predictors_2 -3.5834 0.33187 -10.798 3.5299e-27 Predictors_3 2.1668 0.20802 10.416 2.0899e-25

rates = exp(mdl.Coefficients.Beta)

`rates = `*2×1*
0.0278
8.7301

Perform a Cox proportional hazards regression on the `lightbulb`

data set, which contains simulated lifetimes of light bulbs. The first column of the light bulb data contains the lifetime (in hours) of two different types of bulbs. The second column contains a binary variable indicating whether the bulb is fluorescent or incandescent; 0 indicates the bulb is fluorescent, and 1 indicates it is incandescent. The third column contains the censoring information, where 0 indicates the bulb was observed until failure, and 1 indicates the observation was censored.

Load the `lightbulb`

data set.

`load lightbulb`

Fit a Cox proportional hazards model for the lifetime of the light bulbs, accounting for censoring. The predictor variable is the type of bulb.

coxMdl = fitcox(lightbulb(:,2),lightbulb(:,1), ... 'Censoring',lightbulb(:,3))

coxMdl = Cox Proportional Hazards regression model: Beta SE zStat pValue ______ ______ ______ __________ X1 4.7262 1.0372 4.5568 5.1936e-06

Find the hazard rate of incandescent bulbs compared to fluorescent bulbs by evaluating $$\mathrm{exp}(Beta)$$.

hr = exp(coxMdl.Coefficients.Beta)

hr = 112.8646

The estimate of the hazard ratio is $${e}^{Beta}$$ = 112.8646, which means that the estimated hazard for the incandescent bulbs is 112.86 times the hazard for the fluorescent bulbs. The small value of `coxMdl.Coefficients.pValue`

indicates there is a negligible chance that the two types of light bulbs have identical hazard rates, which would mean `Beta`

= 0.

[1] Grambsch, Patricia M., and Terry
M. Therneau. *Proportional Hazards Tests and Diagnostics Based on Weighted
Residuals.* Biometrika, vol. 81, no. 3, 1994, pp. 515–526. JSTOR, `https://www.jstor.org/stable/2337123`

.

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