Documentation

coxphfit

Cox proportional hazards regression

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Syntax

b = coxphfit(X,T)
b = coxphfit(X,T,Name,Value)
[b,logl,H,stats] = coxphfit(___)

Description

example

b= coxphfit(X,T)返回一个p-by-1 vector,b, of coefficient estimates for aCox proportional hazards regressionof the observed responsesTon the predictorsX, whereTis either ann-by-1 vector or ann-by-2 matrix, andXis ann-by-pmatrix.

The model does not include a constant term, andXcannot contain a column of 1s.

example

b= coxphfit(X,T,Name,Value)返回一个vector of coefficient estimates, with additional options specified by one or moreName,Valuepair arguments.

example

[b,logl,H,stats] = coxphfit(___)also returns the loglikelihood,logl, a structure,stats, that contains additional statistics, and a two-column matrix,H, that contains theTvalues 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.

Examples

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Open Script

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.

Open Script

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 algorithmcoxphfituses 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,DisplayandMaxIter。

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

Runcoxphfitwith 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显示一个报告最后的迭代。改变the maximum number of iterations did not affect the coefficient estimate.

Open Script

Generate Weibull data depending on predictorX。

rng('default')% for reproducibilityX = 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 variableXand a scale parameter of 2.

Fit a Cox proportional hazards model.

[b,logL,H,stats] = coxphfit(X,y); [b logL]
ans = 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: [100×1 double] devres: [100×1 double] martres: [100×1 double] schres: [100×1 double] sschres: [100×1 double] scores: [100×1 double] sscores: [100×1 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 asband 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, isbeta/se= 0.9409/0.1256 = 7.4880. The p-value,p, indicates that the effect ofXis 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.

Input Arguments

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Observations on predictor variables, specified as ann-by-pmatrix ofppredictors for each ofnobservations.

The model does not include a constant term, thusXcannot contain a column of 1s.

IfX,T, or the value of'Frequency'or'Strata'containNaNvalues, thencoxphfitremoves rows withNaNvalues from all data when fitting a Cox model.

Data Types:double

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

  • When T is ann-by-1 vector, it represents the event time of right-censored time-to-event data.

  • When T is ann-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, seeCox Proportional Hazards Model with Time-Dependent Covariates。

IfX,T, or the value of'Frequency'or'Strata'containNaNvalues, thencoxphfitremoves rows withNaNvalues from all data when fitting a Cox model.

Data Types:single|double

Name-Value Pair Arguments

Specify optional comma-separated pairs ofName,Valuearguments.Nameis the argument name andValueis the corresponding value.Namemust appear inside single quotes (' '). You can specify several name and value pair arguments in any order asName1,Value1,...,NameN,ValueN。

Example:'Baseline',0,'Censoring',censoreddata,'Frequency',freqspecifies thatcoxphfitcalculates the baseline hazard rate relative to 0, considering the censoring information in the vectorcensoreddata, and the frequency of observations onTandXgiven in the vectorfreq。

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Coefficient initial values, specified as the comma-separated value consisting of'B0'and a numeric vector.

Data Types:double

Xvalues at which to compute the baseline hazard, specified as the comma-separated pair consisting of'Baseline'and a scalar value.

The default ismean(X), so the hazard rate atXish(t)*exp((X-mean(X))*b)。Enter0to compute the baseline relative to 0, so the hazard rate atXish(t)*exp(X*b)。改变the baseline does not affect the coefficient estimates, but the hazard ratio changes.

Example:'Baseline',0

Data Types:double

Indicator for censoring, specified as the comma-separated pair consisting of'Censoring'and a Boolean array of the same size asT。使用1的观察是正确的审查nd 0 for observations that are fully observed. The default is all observations are fully observed. For an example, seeCox Proportional Hazards Model for Censored Data。

Example:'Censoring',cens

Data Types:logical

Frequency or weights of observations, specified as the comma-separated pair consisting of'Frequency'and an array that is the same size asTcontaining nonnegative scalar values. The array can contain integer values corresponding to frequencies of observations or nonnegative values corresponding to observation weights.

IfX,T, or the value of'Frequency'or'Strata'containNaNvalues, thencoxphfitremoves rows withNaNvalues from all data when fitting a Cox model.

The default is 1 per row ofXandT。

Example:'Frequency',w

Data Types:double

Stratification variables, specified as the comma-separated pair consisting of a matrix of real values. The matrix must have the same number of rows asT, with each row corresponding to an observation.

IfX,T, or the value of'Frequency'or'Strata'containNaNvalues, thencoxphfitremoves rows withNaNvalues from all data when fitting a Cox model.

The default,[], is no stratification variable.

Example:'Strata',Gender

Data Types:single|double

Method to handle tied failure times, specified as the comma-separated pair consisting of'Ties'and either'breslow'or'efron'。

Example:'Ties','efron'

Data Types:char

Algorithm control parameters for the iterative algorithm used to estimateb, specified as the comma-separated pair consisting of'Options'and a structure. A call tostatsetcreates this argument. For parameter names and default values, typestatset('coxphfit')。You can set the options under a new name and use that in the name-value pair argument.

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

Data Types:char

Output Arguments

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Coefficient estimates for aCox proportional hazards regression, returned as ap-by-1 vector.

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.

Estimated baseline cumulative hazard rate evaluated atTvalues, returned as one of the following.

  • If the model is unstratified, thenHis a two-column matrix. The first column of the matrix containsTvalues, and the second column contains cumulative hazard rate estimates.

  • If the model is stratified, thenHis a (2+k) column matrix, where the lastkcolumns correspond to the stratification variables using theStrataname-value pair argument.

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

beta Coefficient estimates (same asb)
se Standard errors of coefficient estimates,b
z z-statistics forb(that is,bdivided by standard error)
p p-values forb
covb

Estimated covariance matrix forb
csres

Cox-Snell residuals
devres Deviance residuals
martres Martingale residuals
schres Schoenfeld residuals
sschres Scaled Schoenfeld residuals
scores Score residuals
sscores Scaled score residuals

coxphfitreturns 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 areNaNs.

More About

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Cox Proportional Hazards Regression

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,h0(t).

For a baseline relative to 0, this model corresponds to

h ( X i , t ) = h 0 ( t ) exp [ ∑ j = 1 p x i j b j ] ,

where X i = ( x i 1 , x i 2 , ⋯ , x i p ) is the predictor variable for theith subject,h(Xi,t) is the hazard rate at timetforXi, andh0(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. SeeCox Proportional Hazards Modelfor details, including the extensions for stratification and time-dependent variables, tied events, and observation weights.

References

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

[2] Lawless, J. F.中央集权ical Models and Methods for Lifetime Data。Hoboken, NJ: Wiley-Interscience, 2002.

[3] Kleinbaum, D. G., and M. Klein.Survival Analysis。中央集权ics for Biology and Health. 2nd edition. Springer, 2005.

See Also

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Topics

Introduced before R2006a

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