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Interval-censored survival analysis

intcensdepvar1depvar2[indepvars] [weight] [ifexp] [inrange] ,distribution(distname)[timeoddseformrobustcluster(varname)cwien(varlist)small(#)level(#)init(matname)maximize_options] Description

intcensfits various distributions to a non-negative outcomeywhich is known to be in the interval [depvar1,depvar2] and which depends onindepvars.yfor each observation is either point data, interval-censored, left-censored or right-censored.depvar1anddepvar2should have the following form:Type of data

depvar1depvar2---------------------------------------------- point dataa= [a,a]aainterval-censored [a,b]a<bleft-censored (0,b].bor0bright-censored [a,inf)a.----------------------------------------------Any

stsettings are ignored.

Options

distribution(distname)is not optional. Available distributions are:Distribution syntax remarks ---------------------------------------- Exponential

expWeibullweibGompertzgompLog-logisticloglorlloLog-normallognorln2 parameter gammagamnote difference fromstregGeneralized gammagen3 parameter gamma Inverse GaussianinvgInverse GaussianwienAlternative parameterisation Time to endpoint ofwienranSee below for parameterisations Wiener process with random drift

timerequests that exponential or Weibull results be reported in accelerated failure form.

oddsrequests that log-logistic estimates be presented as log odds ratios. The default is accelerated failure form.If

eformis specified then exponentiated coefficients are reported. The default is to report log hazard ratios etc.

cwien(varlist)is only valid with distributionswienorwienran. It allows the initial distance from the endpoint of the Wiener process to be modelled as a function of covariates (see below).

robustspecifies that the sandwich estimator of variance is to be used in place of the conventional MLE variance estimator.robustcombined withcluster()further allows observations which are not independent within cluster (although they must be independent between clusters).

cluster(varname)specifies that the observations are independent across groups (clusters) but not necessarily independent within groups.varnamespecifies to which group each observation belongs.cluster()impliesrobust; that is, specifyingrobust cluster()is equivalent to typingcluster()by itself.

small(#)specifies a tolerance within which interval-censored observations are treated as point data. The default is 1E-6.level(#)specifies the confidence level, in percent, for the confidence intervals of the coefficients; see help level.

maximize_optionscontrol the maximization process; see help maximize.

init(matname)supplies a matrix of inital values. They must be in the correct order but the columns do not need to be labelled. Use this option rather than from(), which is one of themaximize_options.

ParameterisationsThe exponential, Weibull, Gompertz, log-logistic, log-normal and generalized gamma distributions are parameterised as in

streg. The log-likelihoods reported for the distributions which are special cases of the generalized gamma are calculated so as to be comparable to the generalized gamma using likelihood ratio tests.The log-logistic results can also be shown as odds ratios, in which case the survivor function is: S(t) = 1/(1+lambda*t^(1/gamma)), lambda=exp(X*B) The other densities are:

Two parameter gamma

gamf(t) = (lambda*alpha)^alpha*t^(alpha-1)*exp(-lambda*alpha*t)/gamma(alpha), lambda = exp(-X*B)Inverse Gaussian

invgf(t) = 1/sqrt(2*pi*phi*t^3)*exp(-(t-eta)^2/(2*eta^2*phi*t)), eta = exp(X*B)Inverse Gaussian

wienf(t) = c/sqrt(2*pi*t^3)*exp(-(mu*t-c)^2/(2*t)), mu = exp(X*B), c = exp(X1*B1)This is the time to first reach a distance c from the starting point for a Wiener process with drift rate mu and variance parameter 1. The two parameterisations are related by eta = c/mu, phi = 1/c^2.

Allowing the drift rate to have a normal distribution between subjects with mean mu and standard deviation tau gives the following density for time to reach c (not a proper density because the event may never happen). There may be convergence problems with this model:

wienranf(t) = c/(sqrt(2*pi)*t*sqrt(t^2*tau^2+t))*exp(-(mu*t-c)^2/(2*(t^2*tau^2+t))), mu = exp(X*B), c = exp(X1*B1)With the distributions

wienorwienran, X isindepvars, the covariates listed before the comma, and X1 is the covariates in the optioncwien( ). X would be factors which affect the rate of movement towards the endpoint, while X1 would be factors which reflect the initial distance from the endpoint.See: Aalen O.O. & Gjessing H.K. (2001) Understanding the shape of the hazard rate: A process point of view. Statistical Science, 16, 1.

Examples

. intcens t0 t1 age trt, dist(weib). intcens t0 t1 age trt, dist(weib) time. intcens t0 t1 age trt, dist(logl) odds. intcens t0 t1 age trt, dist(invg). intcens t0 t1 trt, dist(wien) cwien(age)

Jamie Griffin Infectious Disease Epidemiology Unit London School of Hygiene and Tropical Medicine jamie.griffin@lshtm.ac.uk Updated 11th October 2005AuthorManual:

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