Survival
Analysis:
Survival time is a common response for studying the effects of treatments on
life time. The properties of Survival times are usually characterized by (1)
survival function (also known as reliability function, or cumulative survival
rate) and (2) hazard function (aslo known as failure
rate function). A unique characteristic is that the survival time often is
censored. It may be left-censored, right censored or both. A hazard
function of survival time T is the conditional failure rate defined as the
probability of failure during a small time interval
given the individual has survived. Survival analysis usually studies the
survival time based on some treatment effects and the covariates. SPSS provides
four techniques for studying survival time.
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Life Tables: The first step of constructing a lift time table is to
subdivide the period of observation into smaller time intervals. For each small time interval, the subjects which are observed at
least that long are used to calculate the probability of a terminal event
occurring in that interval. The probabilities estimated from each of the
intervals are then used to estimate the overall probability of the event
occurring at different time points. See SPSS Help Menu for additional
information. |
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Kaplan-Meier
Model:
Kaplan-Meier method is a nonparametric technique for estimating the survival
rates with the presence of censored cases. The basic idea is to first compute
the conditional probabilities at each time point when an event occurs and
then, compute the product limit of those probabilities to estimate the
survival rate at each point in time. It is also called Product-Limit method.
This technique is often used for comparing the effects of treatments on the
survival time. |
The data set used for this demonstration is Rat Tumor data set. See the Data Set page for details. The data is taken from the study by King et al. (1979). They studied the tumor-free time (in days) of 90 rats on three different diets. The Fat level is the treatment level of three diets: 1 is low fat diet, 2 is saturated fat diet and 3 is unsaturated fat diet. The variable, TumorFreeTime, is the time until the rat developed tumors. Censored is the indicator of uncensored data, coded as 0, and censored data coded as 1.
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Cox
Regression Model:
Cox regression model is a common technique used for comparing the survival
time among treatment levels and taking into account
the covariate effects with the presence of censored cases. This is also known
as a proportional hazard model. Proportional hazard model assumes that the
covariate effects on a hazard function is the same for different factor
levels for all time points. That is, the ratio of the hazard functions for
two individuals with values of covariate vectors x1 and x2
does not vary with time t. This implies that the hazard function of T
given covariates x can be written as h(t|x) = h0(t)g(x). h0(t)
is the baseline hazard function where g(x) = 1. |
The data set used for this demonstration is the Lung Cancer data set taken from Prentice (1973), which consists of 40 patients’ survival time due to lung cancer using two different treatments. See the Data Set page for details. The dependent variable is the survival time. The treatment has two levels-1: standard treatment, and 2: experimental treatment. There are also several covariates including medical condition, patient's age, tumor type, and time between diagnosis and treatment.
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Cox Regression with Time-dependent Covariates: The Cox regression
assumes that the covariate effects on a hazard function is the same for
different factor levels for all time points. This assumption may not be
satisfied. The Cox regression with time-dependent covariates is a technique
for modeling survival time with time-dependent covariates. See SPSS Help Menu
for additional information. |