1.0 INTRODUCTION TO SURVIVAL ANALYSIS
Survival analysis is used to study survival duration and the effects of covariates on survival. It uses parametric
methods (Weibull, lognormal, or gamma) or nonparametric methods (lifetable, KaplanMaier, and the Proportional hazards).
Time is measured as time to relapse, length of remission, remission duration, survival after relapse, time to death, or time
to a complication. The best zero time is point of randomization. Other zero times are: enrolment, the first visit, first symptoms,
diagnosis, and start of treatment. Problems of survival analysis are censoring, truncation, and competing causes of death.
Censoring is loss of information due to withdrawal from the study, study termination, loss to followup, or death due to a
competing risk. In left censoring observation ends before a given point in time. In right censoring the subject is last seen
alive at a given time and is not followed up subsequently. Interval censoring, a mixture of left and right censoring, occurs
between two given time given points in time. Right censoring is more common than left censoring. Random censoring occurs uniformly
throughout the study, is not related to outcome, and is not biased. Nonrandom censoring is due to investigator manipulation
and can cause bias. Progressive censoring occurs in studies in which entry and censoring times are different for each subject.
Clinical trials analysis based on the intention to treat is more conservative than censored analysis. In left truncation,
only individual who survive a certain time are included in the sample. In right truncation only individuals who have experienced
the event of interest by a given time are included in the sample. Competing causes of death are one cause of censoring that
bias survival estimates.
2.0 NONREGRESSION SURVIVAL ANALYSIS
Two nonregression methods are used in survival analysis: the lifetable and the KaplanMaier methods. The lifetable
methods better with large data sets and when the time of occurrence of an event cannot be measured precisely. It leads to
bias by assuming that withdrawals occur at the start of the interval when in reality they occur throughout the interval. The
KaplanMaier method is best used for small data sets in which the time of event occurrence is measured precisely. It is an
improvement on the lifetable method in the handling of withdrawals. T. The assumption could therefore create bias or imprecision.
The KaplanMaier method avoids this complication by not fixing the time intervals in advance.
3.0 REGRESSION METHODS FOR SURVIVAL ANALYSIS
The Proportional hazards, a semiparametric method proposed by Sir David Cox in 1972,
is the most popular regression method for survival analysis. It is used on data whose distribution is unknown.
4.0 COMPARING SURVIVAL CURVES
The nonparametric methods for comparing 2 survival distributions are: Gehan’s generalized Wilcoxon test,
the CoxMantel test, the logrank test, Peto’s generalized Wilcoxon test, the MantelHaenszel test, and Cox’s
F test. The parametric tests are the likelihood ratio test and Cox’s F test. The logrank test is more sensitive if
the assumptions of proportional hazards hold. The Wilcoxon test is more sensitive to differences between the curves at the
earlier failure times. It is less sensitive than the logrank test for later failure times. It gives more weight to the earlier
part of the survival curve. The MantelHaenszel test relies on methods of analyzing incidence density ratios. The logrank
test attaches equal importance to all failure times irrespective of whether they are early or late. A modification of the
logrank test by Peto attaches more importance to earlier failure times. Cox’s regression is a semiparametric method
for studying several covariates simultaneously. The loglinear exponential and the linear exponential regression methods are
parametric approaches to studying prognostic covariates. Risk factors for death can be identified using linear discriminant
functions and the linear logistic regression method.
SURVIVAL ANALYSIS – DETAILED NOTES
1.0 INTRODUCTION TO SURVIVAL ANALYSIS
A. SURVIVAL ANALYSIS: DEFINITION, OBJECTIVES, and USES
DEFINITION OF SURVIVAL ANALYSIS
Survival analysis is study of the occurrence and timing of events. Covariates are studied to determine their effect
on survival duration. Although applicable for both retrospective and prospective data, they are best for the latter. Quantitative
changes in variables are treated as events. Two features of survival analysis are not found in conventional statistics: censoring
and timedependent covariates (timevarying explanatory variables).
Censoring is essentially failure to complete observation until the event of interest. Covariates change with time
and this has to be taken into account in a longitudinal analysis.
OBJECTIVES OF SURVIVAL ANALYSIS
The three main objectives of survival analysis are: (a) estimation and interpretation of the hazard function (b)
comparing survival/hazard functions (c) assessment of the relationship of explanatory covariates to survival. Treatment effects
can be confounded.
USES OF SURVIVAL ANALYSIS
Survival analysis is used in followup of patients on treatment by various experimental therapies. It is also used
to evaluate survival after diagnosis with specific diseases. It is also used to summarize and evaluate mortality in different
groups. The methods of survival analysis can be extended to other uses that are nonmedical such as: survival of animals in
drug trials, survival of electric bulbs, survival of machine tools, survival of equipment, survival of friendships, time to
promotion, time to divorce. The techniques of survival analysis are employed in various disciplines for example event history
in sociology, reliability analysis in engineering, failure time analysis in engineering, and duration analysis in economics.
B. MATHEMATICAL FORMULATIONS
Survival functions are probabilistic and can be described in three ways: as survivorship function, probability
density functions, and as hazard functions. The three formulations can be shown to be mathematically equivalent. The survivorship
function is a cumulative distribution function. Survival up to time t is defined as S(t) = 1 – F(t) where F(t) is the
cumulative distribution function of failure up to time t defined by the expression F(t) = Pr(T =<t) where T = survival
time. Note that S(0) = 1. We could alternatively write S(t) = Pr(T>t). A survival curve is exponential. The probability
density function is a derivative of the cumulative distribution function defined as f(t) = d F(t)/dt =  dS(t)/dt. The definition
of f(x) = number dying in the interval / total number of patients x interval width. The hazard function is defined as the
h(t) = lim_{D}_{t }_{®}_{0}
{Pr (t =<T =< t + DtT >=t)}
/Dt. The hazard function can be defined simply as the number dying in the interval that begins at time t / number
surviving at time t. The hazard function is related to the survival and probability density functions in this expression h(t)
= f(t) / S(t).
C. METHODS OF SURVIVAL ANALYSIS
PARAMETRIC AND NONPARAMETRIC METHODS
Survival analysis can use parametric methods (like Weibull, lognormal, or gamma) if the underlying distribution
of the hazard is known. In most practical situations the distribution is unknown and nonparametric methods have to be used.
Three methods of survival analysis are commonly: the lifetable method, the KaplanMaeir method, and the Proportional hazards
method. Other methods used are: exponential regression, lognormal regression, the competing risk model, and the discretetime
methods. A modification of the lifetable method uses the subjectyears approach. In both methods a survival curve is constructed
showing the probability of the endpoint event of interest against time.
SURVIVAL CURVES
Survival curves are used for preliminary examination of data. Median survival can be read off the curves. Visual
inspection can tell us whether there are obvious differences between the 2 groups and whether those differences are increasing
or decreasing. Crossover of the curves is also important because it could invalidate more sophisticated tests. However we
need to apply more sophisticated tests described later to test the null hypothesis that there is no significant difference
between the two curves. Median survival can be read straight off the curve.
COMPARING SURVIVAL CURVES
Usually 2 curves are generated one for each treatment or experimental group. Four statistical tests are generally
used to test for significant differences between the curves: (a) The MantelHaenszel test (b) The logrank test (c) the generalized
Wilcoxon (Gehan) test and (d) The proportional hazards model due to Cox. Several guidelines can be given for choice of analytic
method. Cox regression should be treated as the default method. It can be modified to behave like the logrank method by using
a single dichotomous covariate or to behave like the KM model by fitting a model without covariates. Logist, probit, and
general linear models are used for large data sets with repeated covariate measurements. LIFEREG is used for studying the
shape of the hazard function, computing predicted survival, and for left censored data. Logit models are used where there
are too many ties. The KaplanMeier method is suitable for small samples with progressive censoring. The lifetable method
is better for large samples.
DATA SETUP FOR SURVIVAL ANALYSIS
All survival programs use a rectangular data file set up as shown below
Observation 
Survival 
Status 
Group 
Covariates 
1. 



V1 
V2 
V3 
V4 
V5 
2 








3 








4 








. 








The observations are arranged and are identified by a serial number or a special identification number. Survival
duration is given in units of time either days, weeks, or months. Duration can be computed if the zero time and the failure/censoring
time are given in calendar times by using the routine Duration = mdy (mdy_{2}  mdy_{1}).
D. MEASUREMENT OF TIME
There are several ways of measuring time to the event of interest. Time may be measured as duration for example
time since birth (age), time since a given event, time since the last occurrence of the same event. Time may also be measured
as calendar time although this is less popular in clinical trials.
The following examples illustrate various descriptions of time periods: time to relapse, survival after relapse,
time to death, time to infection or any other complication. In survival analysis our interest is in survival duration which
is usually time measured from zero time until the event of interest: failure/relapse, death, 1^{st} response, or censoring.
Zero time is defined as the point in time when the hazard starts operating, the point of randomization, the time of enrolment
into the study, the date of the first visit, the date of the first symptoms, the date of diagnosis, or the date of starting
treatment. The best zero time is the point if randomization. Use of time at diagnosis or start of treatment may introduce
bias because socioeconomic factors may determine access to diagnosis and treatment facilities.
Survival duration is measured by subtraction of the zero time from time at failure or censoring. Thus we may be
interested in time from start of treatment to the 1^{st} response. Sometimes the interest is in the length of remission,
remission duration. Sometimes the interest is in the tumorfree time. Survival can be described as relative survival or absolute
survival. Relative survival is to 1year survival of trial subjects with the general population. Absolute survival is the
proportion of the trial subjects who live up to 5 years. Absolute survival is more popular in usage.
E. PROBLEMS OF SURVIVAL ANALYSIS
CENSORING
A problem in survival analysis is censoring. Censoring occurs when an individual is not followed up until occurrence
of the event of interest. Censoring leads to loss of information due to incomplete observation. Those not followed up fully
may have a different experience that would lead to bias in the study. Censoring is caused by loss to followup, withdrawal
from the study, study termination when subjects had different dates of enrolment, loss to followup, or death due to a competing
risk. Censored observations contribute to the analysis until the time of censoring. Censored analysis makes the assumption
that if censored subjects had been followed beyond the point in time at which they were censored, they would have had the
same rates of outcomes as those not censored at that time. Existence of similar censoring patterns between different treatment
groups suggests that censoring assumptions are holding.
Left censoring is when observation ends before a given point in time ie T < c. In left censoring it is known
that an individual experienced the event of interest before start of the study. Right censoring is when observation ends beyond
a given point in time, T > c. In right censoring the individual is known to be alive at a certain time before experiencing
the event of interest but was not followed after that. Interval censoring occurs between two given time given points in time
a < T < b. In interval censoring all what is known is that the event occurred in a given time interval. Right censoring
is more common than left censoring. Interval censoring is in essence a combination of left and right censoring.
Censoring can be described as random and nonrandom.
Random censoring occurs uniformly throughout the study, is not informative but creates fewer problems than nonrandom
censoring. Nonrandom censoring is under the control of the investigator and is classified as either type 1 or type 2 censoring.
Type 1 censoring occurs when the censoring time is fixed by the investigator, in other words he decides to terminate the study
at a given point in time. The censoring time may be the same for all subjects or may differ. Type 11 censoring is when censoring
time is determined by a given number of events. When that number of events has occurred observation is terminated. Usual methods
of survival analysis do not distinguish between type 1 and type 2 censoring. Both type 1 and type 11 censoring are types of
right censoring. Random censoring is one not under the control of the investigator and includescensoring due to loss to follow
up, censoring due to death from competing causes of death, or when only the termination time is fixed but entry times are
left to vary randomly.
Censoring can be described as informative or noninformative. Informative censoring occurs when dropping out of
the study is related to the outcome. In noninformative censoring there is no relationship between censoring and outcome.
Singly censored events occur in animal experiments in which all animals are treated at the same time and are observed
for the duration of the study. All surviving animals are sacrificed at the end of the study. Progressive censoring is what
occurs in human clinical trials because patients enter the study at different times and are censored at different times.
Analysis of clinical trials data can be based on intention to treat which will include in the survival series all
those who were censored alive. This is a more conservative approach than censored analysis.
TRUNCATION
There are basically two types of truncation. In left truncation, only individuals who survive a certain time are
included in the sample. In right truncation only individuals who have experienced the event of interest by a given time are
included in the sample.
COMPETING CAUSES OF DEATH
Competing causes of death are one cause of censoring that bias survival estimates.
3.0 NONREGRESSION SURVIVAL ANALYSIS
A. SURVIVAL ANALYSIS USING THE LIFE TABLE METHOD
INTRODUCTION AND DEFINITION
The formula for survival is given as S(t) = Õ (1 – q_{j})
MANUAL CONSTRUCTION OF THE LIFETABLE ( 8 COLUMNS)
Column #1 is the time at the start of the time interval. The first row of the table is assigned time 0. Column
#2 is the number of subjects under observation at the start of the time interval, O. Column #4 is the number who died during
the time interval, D. Column #4 is the number withdrawn during the time interval, W. Withdrawals are considered to occur at
the start of the time interval. We assume that there are no secular trends in risk of death in different calendar periods.
Those who withdraw and those who stay under observation have the same probability of death. Column #5 is the number under
observation during the interval. It is computed as OW. Column #6 is the probability of dying in the interval. It is computed
as P = D / OW. Column #7 is the probability of surviving to the end of the interval and is computed as Q=1P. Column #8 is
the probability of survival from time 0 until the end of the interval. The probability for the first row is 1.0. Subsequent
probabilities are computed by multiplying Q into the survival probability of the prior row. The survival probabilities in
column #8 are plotted against time in column #1 to generate a survival curve. Two or more curves can be generated depending
on the treatment or experimental groups.
COMPUTER PROCEDURE FOR THE LIFETABLE METHOD
The Lifetable procedure can work on either grouped on ungrouped data. If the data is ungrouped, the intervals needed
for the computations can be chosen automatically by the procedure or they can be fixed by the investigator. The procedure
treats a censored observation as if it was censored at the middle of the interval during which it was censored. The inputs
are: survival duration (grouped or ungrouped), status, and group. The outputs are: median survival, survival probabilities
at different points in time. And hazard at midinterval. The logrank test can be used to compare two curves.
ADVANTAGES
The life table methods works well with large data sets and when the time of occurrence of an event can not be measured
precisely. It is an advantage of being able to make a credible analysis without knowing the exact times of censoring or withdrawal.
DISADVANTAGES
The lifetable method is not efficient in handling withdrawals. This could be a source of bias. The choice of the
interval is arbitrary. The method assumes that withdrawal occurs at midinterval which may not be the case.
SENSITIVITY ANALYSIS
The sensitivity of the analysis can be assessed by comparing analysis assuming left censoring at the start of the
interval and left censoring at the end of the interval.
B. SURVIVAL ANALYSIS USING THE KAPLANMAIER METHOD
INTRODUCTION and DEFINITION
The KM involves defining a risk set at each time there is a failure and computation of the instantaneous probability
of death at that time. The formula for survival using the KM method is S(t) = Õ (1 –
d_{j}/n_{j}).
MANUAL CONSTRUCTION OF THE KM TABLE
Column #1 is the time at occurrence of an event, t_{i}. It is an exact time and not a time interval. It
is not fixed in advance but is defined by events of death or withdrawal. Deaths and withdrawals occur at different times.
The notation t refers to any time when death, withdrawal, or censoring of an event occur.
Column #2 is the number of subjects at risk at time, t_{i}. This number decreases progressively down the column
as the number of deaths, the number of withdrawals, and the number of censored observations are subtracted. Column #4 is the
number of deaths at time t. Column #4 is the number of withdrawals at time t. Column #5 is the probability of death at time
t_{i}. It is computed as the number of deaths at time t_{i} (column
#4) divided by the number at risk just before time t_{i} (column #2). Occurrence of withdrawals is recorded in the
table but they are considered nonevents. A withdrawal affects only the number at risk when the next event of death occurs.
Column #6 is the probability of survival at time t_{i} . It is computed as the 1  probability of death at time t_{i…}Column
#7 indicates cumulative survival from time 0 to time t_{i} . It is computed by multiplying the row probability of
survival into the probability of survival of the previous row.
THE COMPUTER KM PROCEDURE
The KM curve can be produced using a computer program. The inputs are: survival duration, status, treatment group,
and covariates. The output are several measures of sirvival duration with 95% confidence intervals: 1^{st} quartile,
median, 3^{rd} quartile, and the mean; the number of failures, and the number censored.
ADVANTAGES and DISADVANTAGES
The KaplanMaier method is best used for small data sets in which the time of event occurrence is measured precisely.
The KaplanMaier method is an improvement on the lifetable method in the handling of withdrawals. The lifetable method considers
withdrawals to occur at the start of the interval but it reality withdrawals occur throughout the interval. The assumption
could therefore create bias or imprecision. The KaplanMaier method avoids this complication by not fixing the time intervals
in advance. Intervals are defined in two ways: (a) An interval ends when the endpoint event of interest occurs. (b) An interval
ends when a withdrawal occurs.
5.1.4 REGRESSION METHODS FOR SURVIVAL ANALYSIS
Three regression methods can be used: LIFEREG, PHREG, and Poisson regression. The Proportional
hazards regression is the most popular. This procedure uses regression methods proposed in 1972 by the British statistician
Sir David Cox in his famous paper ‘Regression Models and Lifetables’ published in the Journal of the Royal Statistical
Society. It became one of the most quoted papers in statistics literature. It has two distinguishing features: use of partial
MLE estimates and use of proportional hazards. It can however be extended to handling nonproportional hazards by use of interaction
or stratification covariates.
Proportional hazards regression PHREG is supplanting LIFEREG which produces regression
estimates using MLE. LIFEREG is parametric regression that accommodates left censoring and tests hypotheses about the shape
of the hazard function but can not handle timevarying covariates. Proportional hazards regression has become very popular
for many reasons. It can estimate the hazard ratio, the hazard at any time and with a given set of covariates, h(t,x), and
the survivor function without having to define the baseline hazard, h_{0}(t). It is a more robust semiparametric
regression that fits the data well and does not require selection or assumption of any particular probability model. It assumes
a constant hazard over time. Its regression is based on partial likelihoods and handles only right censoring. It handles timedependent
covariates. Time dependent covariates alow incorporation of changes in the independent variables and new events during the
course of the study. It also handles both continuous and discrete time periods. The covariates may represent different time
periods and may be measured at regular or irregular time intervals. It can be modified to handle competing risks in which
there is more than one outcome like death due to the disease under investiugation and death due to an accident. The competing
risk removes the individual from the risk set of the event of interest a problem solved by treating the competing risk as
a censoring event. Cox regression has the advantage of controlling for confounding variables. It provies several approaches
for handling tied data. Its disadvantage is lack of builtin graphics capability.
The proportional hazards model due to Cox is a parametric test for comparing survival curves. Its special advantage
is that it enables modeling the data such that incidence or hazard depends only on elapsed time, t but the relative hazard
ratio does not. It also has the advantage of giving instantaneous risk of failure as a function of the risk factors of interest
whereas survival curves estimate the probability of survival up to a certain time. The
Cox model is robust giving results that closely approximate the correct parameter values. It also can be used to model exponential
and Weibull distributions. Using an exponential model is allowed only if we are sure of the distribution. If not sure, the
use of the Cox model is a safe bet since it is not parametric and will still give approximaltely correct parameter estimates.
The likelihood function is a mathematical expression of the joint
probability of obtaining the data actually observed on the subjects in the study as a function of the unknown b parameters.
It is fitted by interation. The maximum likelihood estimates in the Cox model are based on consideration of probabilities
for subjects who fail and not those who are considered censored and that is why it is called a partial likelihood.
There is a close relation between survival, S_{t}, and hazard, H_{t}. Differentiation of –lnS_{t}
yields the corresponding hazard function. The survival curve can be derived from the integrated hazard, H_{t}, by
taking exponents of  H_{t.} The basic proportional hazards model of Cox is h_{t} = h_{0}t exp (^{p}å_{j=1} b_{j}x_{j}) where h_{t} = hazard at time t, h_{0}t
= baseline hazard, b_{j }= regression coefficient. The hazard ratio is computed as exp {^{p}å_{i=1} b_{i} (x_{1i} – x_{ii})} where x_{1i}
= exposed and x_{0i} = unexposed. Cox’s model is called nonparametric because h_{0}(t) is undefined.
The exponent å b_{j}x_{j} contains the
term x_{j }but this does not imply that in the ordinary Cox model is x_{j }is time dependent. If time dependent
covariates are used, the model becomes the extended Cox model. The extended Cox model has two components, a time independent
part and a timedependent part. The model can be written as h{t, x(t)} = h_{0} (t) exp {^{p}å_{i=1} b_{1}x_{i} + ^{p}å_{i=1} d_{1}x_{i}
(t)} where ^{p}å_{i=1} b_{1}x_{i}
is timeindependent and ^{p}å_{i=1}
d_{1}x_{i} (t) is timedependent. The time
dependent portion contains interaction terms that involve a function of time.
A stratified Cox model is one in which the PH model is modified to allow stratification for a variable that is
not included in the model because it does not satisfy PH assumptions. Two or more curves can be drawn side by side for each
of the strata of the stratifying variable.
We can compute the relative hazard ratio at two times as follows h_{it }/_{ }h_{kt} = exp
{^{p}å_{j=1} b_{j }(x_{ij}
 x_{kj})}. The relative hazard ratio does not depend on t because the common factor h_{0}t cancels out. The
hazards are in constant proportion and their ratio does not depend on survival duration. The comparison of survival is effected
by including a covariate for group in the proportional hazards regression model. The input of the PHREG is time at failure
and covariates. The output of a PHREG procedure is: regression coefficient, standard error of the regression coefficient,
the pvalue, and the hazard ratio. The Wald statistic is computed as coefficient / se of coefficient ~ N(0,1) the
p value being looked up in the z or normal table.
The proportionality assumption in proportional hazards analysis is that the hazards for persons with different
profiles of covariates are constant over time. This implies that the regression coefficients obtained from PH regression cover
the whole time period. There are 4 methods of assessing the PH assumptions. The PH assumptions are void if the loglognormal
curves of 2 different treatements cross. If plots of the observed and the expected are close, the PH assumptions are valid.
Godness of fit tests based on the chisquare can be used to test validity of the PH assumptions. If the interaction term of
a timedependendent covariate * time is nonsignificant, the PH assumptions is satisfied.
There are 4 alternatives if the proportionality assumptions do not hold for a particular covariate. The PH analysis
is repeated by stratifying on the covariate in question. Separate PH analyses can be repeated for each time period for example
an analysis for the first year and another one for the second year. Logistic regression can be performed instead of the PH
regression because logistic analysis does not take time into consideration. The analysis can allow for lack of proportionality
by letting the hazard ratio vary by time.
The significance of the regression coefficient of any covariate can be assessed using Wald’s statistic ie
c_{1}^{2} = {b/se(b)}^{2}. Alternatively the likelihood is computed with the variable in
the model and when the variable is not in the model. A chisquare statistic is then constructed using the difference in the
two log likelihoods.
The Poisson regression is used for rare outcomes (<5%) doe which the proportional hazards model is not appropriate.
It models the log transform of incidence at time t as a linear function of covariates. It assumes that incidence does not
depend on the elapsed time, t.
Logistic, probit, and general linear models based on MLE can be used. They are easier
and more intuitive than Cox’s regression. They however treat trime like any other variable and treat covariates as fixed.
The Cox PH regression is used in the analysis of nested case control studies, case cohort studies, and cohort studies
because they all have timedependent covariates. Proportional hazards regression differs from linear and logistic regressions
in several important ways. The outcome in multiple linear regression is measured
as a mean. The outcome in multiple logistic regression is measured as the logit which is the logarithm of the odds of the
outcome variable. In proportional hazards regression, the outcome is measured as a relative
hazard. The means, logits and relative hazards of the outcome variable change linearly with each unit changes in the independent
variables. The distribution of the outcome variable in normal in linear regression, binomial in logistic regression, and has
no specified distribution in the semiparametric proportional hazards regression. Proportional hazards regression can deal
with censored data whereas the other two can not. It has constant hazard over time. Hazard measures are not applicable to
linear and logistic regression. The methods of assessing model fit also vary by type of regression. To assess whether the model accounts for outcome better than chance we use the F test for linear regression,
the likelihood ratio test for multiple logistic regression and the proportional hazards regression. The R^{2} statistic
is used to make a quantitative assessment of how well the model accounts for the outcome of multiple linear and multiple logistic
models. The comparison of observed to estimated values is used to assess the fit of multiple logistic and proportional hazards
models. Various sensitivity, specificity, and accuracy measures are used to assess how well the predictive ability of the
multiple logistic regression model. There are no comparable methods of assessing prediction of multiple linear models and
proportional hazards models. The interpretation of the regression coefficient varies by type of regression. In multiple linear
regression, a positive coefficient is interpreted to mean that the outcome increases as the independent variable increases
and a negative coefficient is interpreted to mean that the outcome decreases as the independent variable increases. In multiple
logistic regression, a positive coefficient is interpreted to mean that the logit increases with increase of the independent
variable and a negative coefficient is interpreted to mean that the logit decreases with increase of the independent variable.
In proportional hazards regression a positive coefficient is interpreted to mean that the logarithm of the relative hazard
increases with increase of the independent variable and a negative coefficient is interpreted to mean that the logarithm of
the relative hazard decreases with decrease of the independent variable.
5.1.5 COMPARING SURVIVAL CURVES
The nonparametric methods for comparing 2 survival distributions are: Gehan’s generalized Wilcoxon test,
the CoxMantel test, the logrank test, Peto’s generalized Wilcoxon test, the MantelHaenszel test, and Cox’s
F test. The parametric tests for comparing survival distributions vary by the type of distribution. The likelihood ratio test
and Cox’s F test are used for exponential and Weibull distributions. The likelihood test assumes exponential distribution
and equal hazards at different times. The F test is used for gamma distributions.
The Logrank (MH) test, Wilcoxon test, and the likelihood test are based on a chisquare statistic and a corresponding
p value. The formula for the logrank chisquare is given as c^{2} = ^{r}å_{j=1}
{(d_{1j}  e_{1j}) / e_{1j} }^{2} where d_{1j} = number of failures in group 1 at
time j and e_{1j} = expected number of failures in group 1 at time j. The
logrank test is more sensitive if the assumptions of proportional hazards hold.
The formula for the Wilcoxon chisquare is given as c^{2} = {^{r}å_{j=1}
n_{j}(d_{1j}  e_{1j})}^{2} where n_{j }= weight. The Wilcoxon test is more sensitive
to differences between the curves at the earlier failure times. It is less sensitive than the logrank test for later failure
times. The Peto test is an alternative to the Logrank test. It gives more weight to the earlier part of the survival curve.
MH TEST
The MantelHaenszel test relies on methods of analyzing incidence density ratios, IDR. IDR = {a/PT_{1}}/
{b/PT_{2}} where a = number of remissions in group1. b = number of remissions in group2 and PT = persontime of followup.
We can compute the standard error of IDR by using its log transform thus se(lnIDR) = {1/a + 1/b}^{1/2}. The z test
statistic is computed as z = ln(IDR) / se(lnIDR) and is used to derive a pvalue for testing the null hypothesis of no difference
between the two survival curves.
LOGRANK TEST
The logrank test is a nonparametric test used to compare survival curves. It is a special MH procedure for survival
curves. By assuming that events at each failure time are independent of one another we can stratify by failure time and use
MH methods for stratified data by forming a stratum at each failure time. If
observed failures at each failure time in the two groups are f_{1} and f_{2} and the corresponding risks are
r_{1} and r_{2}, the expected numbers of failures are e_{1} = (f_{1} and f_{2}). r_{1}/
(r_{1} + r_{2}) and e_{2} = (f_{1} and f_{2}). r_{2}/ (r_{1} + r_{2}).
The expected failure times summed uo over all failure times are E_{1} = åe_{1}
and E_{2} = åe_{2}. The observed failures summed up over all failure times are O_{1}
= åf_{1} and O_{2} = åf_{2}. The chisquare test statistic is then computed with 1 degree of freedom
as å (OE)^{2}/E. The logrank test can be used in cases with more than 2 curves where the chosquate test statistic
is computed with c1 degrees of freedom where c = number of curves. The logrank test attaches equal importance to all failure
times irrespective of whether they are early or late. A modification of the logrank test by Peto attaches more importance
to earlier failure times.
GENERALIZED WILCOXON
The generalized Wilcoxon (Gehan) test
ANALYSIS FOR PROGNOSTIC COVARIATES
A KM curve can be drawn for each level of covariate and visual inspection is used to compare. Cox’s regression
is a semiparametric method for studying several covariates simultneously. The loglinear exponential and the linear exponential
regression methods are parametric approaches to studying prognostic covariates.
Risk factors for death can be identified using linear discriminant
functions and the linear logistic regression method.