Cox proportional hazards regression

returns
a `b`

= coxphfit(`X`

,`T`

)*p*-by-1 vector, `b`

, of coefficient
estimates for a Cox proportional hazards
regression of the observed responses `T`

on
the predictors `X`

, where `T`

is
either an *n*-by-1 vector or an *n*-by-2
matrix, and `X`

is an *n*-by-*p* matrix.

The model does not include a constant term, and `X`

cannot
contain a column of 1s.

returns
a vector of coefficient estimates, with additional options specified
by one or more `b`

= coxphfit(`X`

,`T`

,`Name,Value`

)`Name,Value`

pair arguments.

`[`

also returns the loglikelihood, `b`

,`logl`

,`H`

,`stats`

]
= coxphfit(___)`logl`

,
a structure, `stats`

, that contains additional statistics,
and a two-column matrix, `H`

, that contains the `T`

values
in the first column and the estimated baseline cumulative hazard,
in the second column. You can use any of the input arguments in the
previous syntaxes.

Load the sample data.

load(fullfile(matlabroot,'examples','stats','lightbulb.mat'));

The first column of the light bulb data has the lifetime (in hours) of two different types of bulbs. The second column has the binary variable indicating whether the bulb is fluorescent or incandescent. 0 indicates that the bulb is incandescent, and 1 indicates that it is fluorescent. The third column contains the censorship information, where 0 indicates the bulb was observed until failure, and 1 indicates the bulb was censored.

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

b = coxphfit(lightbulb(:,2),lightbulb(:,1), ... 'Censoring',lightbulb(:,3))

b = 4.7262

The estimate of the hazard ratio is $${e}^{b}$$ = 112.8646. This means that the hazard for the incandescent bulbs is 112.86 times the hazard for the fluorescent bulbs.

Load the sample data.

load(fullfile(matlabroot,'examples','stats','lightbulb.mat'));

The first column of the data has the lifetime (in hours) of two types of bulbs. The second column has the binary variable indicating whether the bulb is fluorescent or incandescent. 1 indicates that the bulb is fluorescent and 0 indicates that it is incandescent. The third column contains the censorship information, where 0 indicates the bulb is observed until failure, and 1 indicates the item (bulb) is censored.

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

b = coxphfit(lightbulb(:,2),lightbulb(:,1),... 'Censoring',lightbulb(:,3))

b = 4.7262

Display the default control parameters for the algorithm `coxphfit`

uses to estimate the coefficients.

`statset('coxphfit')`

`ans = `*struct with fields:*
Display: 'off'
MaxFunEvals: 200
MaxIter: 100
TolBnd: 1.0000e-06
TolFun: 1.0000e-08
TolTypeFun: []
TolX: 1.0000e-08
TolTypeX: []
GradObj: []
Jacobian: []
DerivStep: []
FunValCheck: []
Robust: []
RobustWgtFun: []
WgtFun: []
Tune: []
UseParallel: []
UseSubstreams: []
Streams: {}
OutputFcn: []

Save the options under a different name and change how the results will be displayed and the maximum number of iterations, `Display`

and `MaxIter`

.

coxphopt = statset('coxphfit'); coxphopt.Display = 'final'; coxphopt.MaxIter = 50;

Run `coxphfit`

with the new algorithm parameters.

b = coxphfit(lightbulb(:,2),lightbulb(:,1),... 'Censoring',lightbulb(:,3),'Options',coxphopt)

Successful convergence: Norm of gradient less than OPTIONS.TolFun

b = 4.7262

`coxphfit`

displays a report on the final iteration. Changing the maximum number of iterations did not affect the coefficient estimate.

Generate Weibull data depending on predictor `X`

.

rng('default') % for reproducibility X = 4*rand(100,1); A = 50*exp(-0.5*X); B = 2; y = wblrnd(A,B);

The response values are generated from a Weibull distribution with a shape parameter depending on the predictor variable `X`

and a scale parameter of 2.

Fit a Cox proportional hazards model.

[b,logL,H,stats] = coxphfit(X,y); [b logL]

`ans = `*1×2*
0.9409 -331.1479

The coefficient estimate is 0.9409 and the log likelihood value is –331.1479.

Request the model statistics.

stats

`stats = `*struct with fields:*
covb: 0.0158
beta: 0.9409
se: 0.1256
z: 7.4889
p: 6.9462e-14
csres: [100x1 double]
devres: [100x1 double]
martres: [100x1 double]
schres: [100x1 double]
sschres: [100x1 double]
scores: [100x1 double]
sscores: [100x1 double]

The covariance matrix of the coefficient estimates, `covb`

, contains only one value, which is equal to the variance of the coefficient estimate in this example. The coefficient estimate, `beta`

, is the same as `b`

and is equal to 0.9409. The standard error of the coefficient estimate, `se`

, is 0.1256, which is the square root of the variance 0.0158. The $$z$$-statistic, `z`

, is `beta/se`

= 0.9409/0.1256 = 7.4880. The p-value, `p`

, indicates that the effect of `X`

is significant.

Plot the Cox estimate of the baseline survivor function together with the known Weibull function.

stairs(H(:,1),exp(-H(:,2)),'LineWidth',2) xx = linspace(0,100); line(xx,1-wblcdf(xx,50*exp(-0.5*mean(X)),B),'color','r','LineWidth',2) xlim([0,50]) legend('Estimated Survivor Function','Weibull Survivor Function')

The fitted model gives a close estimate to the survivor function of the actual distribution.

`X`

— Observations on predictor variablesmatrix

Observations on predictor variables, specified as an *n*-by-*p* matrix
of *p* predictors for each of *n* observations.

The model does not include a constant term, thus `X`

cannot
contain a column of 1s.

If `X`

, `T`

, or the value
of `'Frequency'`

or `'Strata'`

contain `NaN`

values,
then `coxphfit`

removes rows with `NaN`

values
from all data when fitting a Cox model.

**Data Types: **`double`

`T`

— Time-to-event datavector | two-column matrix

Time-to-event data, specified as an *n*-by-1
vector or a two-column matrix.

When T is an

*n*-by-1 vector, it represents the event time of right-censored time-to-event data.When T is an

*n*-by-2 matrix, each row represents the risk interval (start,stop] in the counting process format for time-dependent covariates. The first column is the start time and the second column is the stop time. For an example, see Cox Proportional Hazards Model with Time-Dependent Covariates.

If `X`

, `T`

, or the value
of `'Frequency'`

or `'Strata'`

contain `NaN`

values,
then `coxphfit`

removes rows with `NaN`

values
from all data when fitting a Cox model.

**Data Types: **`single`

| `double`

Specify optional
comma-separated pairs of `Name,Value`

arguments. `Name`

is
the argument name and `Value`

is the corresponding value.
`Name`

must appear inside quotes. You can specify several name and value
pair arguments in any order as
`Name1,Value1,...,NameN,ValueN`

.

`'Baseline',0,'Censoring',censoreddata,'Frequency',freq`

specifies
that `coxphfit`

calculates the baseline hazard rate
relative to 0, considering the censoring information in the vector `censoreddata`

,
and the frequency of observations on `T`

and `X`

given
in the vector `freq`

. `'B0'`

— Coefficient initial values`0.01/std(X)`

(default) | numeric vectorCoefficient initial values, specified as the comma-separated
value consisting of `'B0'`

and a numeric vector.

**Data Types: **`double`

`'Baseline'`

— `X`

values at which to compute the baseline hazard`mean(X)`

(default) | scalar value`X`

values at which to compute the baseline
hazard, specified as the comma-separated pair consisting of `'Baseline'`

and
a scalar value.

The default is `mean(X)`

, 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 coefficient estimates, but
the hazard ratio changes.

**Example: **`'Baseline',0`

**Data Types: **`double`

`'Censoring'`

— Indicator for censoringarray of 0s (default) | array of 0s and 1s

Indicator for censoring, specified as the comma-separated pair
consisting of `'Censoring'`

and a Boolean array of
the same size as `T`

. Use 1 for observations that
are right censored and 0 for observations that are fully observed.
The default is all observations are fully observed. For an example,
see Cox Proportional Hazards Model for Censored Data.

**Example: **`'Censoring',cens`

**Data Types: **`logical`

`'Frequency'`

— Frequency or weights of observationsarray of 1s (default) | vector of nonnegative scalar values

Frequency or weights of observations, specified as the comma-separated
pair consisting of `'Frequency'`

and an array that
is the same size as `T`

containing nonnegative
scalar values. The array can contain integer values corresponding
to frequencies of observations or nonnegative values corresponding
to observation weights.

If `X`

, `T`

, or the value
of `'Frequency'`

or `'Strata'`

contain `NaN`

values,
then `coxphfit`

removes rows with `NaN`

values
from all data when fitting a Cox model.

The default is 1 per row of `X`

and `T`

.

**Example: **`'Frequency',w`

**Data Types: **`double`

`'Strata'`

— Stratification variables`[]`

(default) | matrix of real valuesStratification variables, specified as the comma-separated pair
consisting of a matrix of real values. The matrix must have the same
number of rows as `T`

, with each row corresponding
to an observation.

`X`

, `T`

, or the value
of `'Frequency'`

or `'Strata'`

contain `NaN`

values,
then `coxphfit`

removes rows with `NaN`

values
from all data when fitting a Cox model.

The default, `[]`

, is no stratification variable.

**Example: **`'Strata',Gender`

**Data Types: **`single`

| `double`

`'Ties'`

— Method to handle tied failure times`'breslow'`

(default) | `'efron'`

Method to handle tied failure times, specified as the comma-separated pair consisting of
`'Ties'`

and either `'breslow'`

(Breslow's method) or `'efron'`

(Efron's
method).

**Example: **`'Ties','efron'`

`'Options'`

— Algorithm control parametersstructure

Algorithm control parameters for the iterative algorithm used
to estimate `b`

, specified as the comma-separated
pair consisting of `'Options'`

and a structure. A
call to `statset`

creates this argument. For parameter
names and default values, type `statset('coxphfit')`

.
You can set the options under a new name and use that in the name-value
pair argument.

**Example: **`'Options',statset('coxphfit')`

`b`

— Coefficient estimatesvector

Coefficient estimates for a Cox proportional hazards
regression, returned as a *p*-by-1 vector.

`logl`

— Loglikelihoodscalar

Loglikelihood of the fitted model, returned as a scalar.

You can use log likelihood values to compare different models and assess the significance of effects of terms in the model.

`H`

— Estimated baseline cumulative hazardtwo-column matrix | (2+

Estimated baseline cumulative hazard rate evaluated at `T`

values,
returned as one of the following.

If the model is unstratified, then

`H`

is a two-column matrix. The first column of the matrix contains`T`

values, and the second column contains cumulative hazard rate estimates.If the model is stratified, then

`H`

is a (2+*k*) column matrix, where the last*k*columns correspond to the stratification variables using the`Strata`

name-value pair argument.

`stats`

— Coefficient statisticsstructure

Coefficient statistics, returned as a structure that contains the following fields.

`beta` | Coefficient estimates (same as `b` ) |

`se` | Standard errors of coefficient estimates, `b` |

`z` | z-statistics for `b` (that
is, `b` divided by standard error) |

`p` | p-values for `b` |

`covb` | Estimated covariance matrix for |

`csres` | Cox-Snell residuals |

`devres` | Deviance residuals |

`martres` | Martingale residuals |

`schres` | Schoenfeld residuals |

`sschres` | Scaled Schoenfeld residuals |

`scores` | Score residuals |

`sscores` | Scaled score residuals |

`coxphfit`

returns the Cox-Snell, martingale,
and deviance residuals as a column vector with one row per observation.
It returns the Schoenfeld, scaled Schoenfeld, score, and scaled score
residuals as matrices of the same size as X. Schoenfeld and scaled
Schoenfeld residuals of censored data are `NaN`

s.

Cox proportional hazards regression is a semiparametric
method for adjusting survival rate estimates to remove the effect
of confounding variables and to quantify the effect of predictor variables.
The method represents the effects of explanatory and confounding variables
as a multiplier of a common baseline hazard function, *h*_{0}(*t*).

For a baseline relative to 0, this model corresponds to

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

where $${X}_{i}=({x}_{i1},{x}_{i2},\cdots ,{x}_{ip})$$ is
the predictor variable for the *i*th subject, *h*(*X*_{i},*t*)
is the hazard rate at time *t* for *X*_{i},
and *h*_{0}(*t*)
is the baseline hazard rate function. The baseline hazard function
is the nonparametric part of the Cox proportional hazards regression
function, whereas the impact of the predictor variables is a loglinear
regression. The assumption is that the baseline hazard function depends
on time, *t*, but the predictor variables do not
depend on time. See Cox Proportional Hazards Model for details, including
the extensions for stratification and time-dependent variables, tied
events, and observation weights.

[1] Cox, D.R., and D. Oakes. *Analysis
of Survival Data*. London: Chapman & Hall, 1984.

[2] Lawless, J. F. *Statistical
Models and Methods for Lifetime Data*. Hoboken, NJ: Wiley-Interscience,
2002.

[3] Kleinbaum, D. G., and M. Klein. *Survival Analysis*.
Statistics for Biology and Health. 2nd edition. Springer, 2005.

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

`X`

can be a single- or double-precision matrix and can be variable-size.The value of the

`'Ties'`

name-value pair argument must be a compile-time constant. For example, to use Efron's method to handle tied failure times, include`{coder.Constant('Ties'),coder.Constant('efron')}`

in the`-args`

value of`codegen`

.Names in name-value pair arguments must be compile-time constants.

For more information on code generation, see Introduction to Code Generation and General Code Generation Workflow.

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