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 non-parametric methods (life-table, Kaplan-Maier, 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 follow-up, 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. Non-random 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 NON-REGRESSION SURVIVAL ANALYSIS

Two non-regression methods are used in survival analysis: the life-table and the Kaplan-Maier methods. The life-table 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 Kaplan-Maier method is best used for small data sets in which the time of event occurrence is measured precisely. It is an improvement on the life-table method in the handling of withdrawals. T. The assumption could therefore create bias or imprecision. The Kaplan-Maier method avoids this complication by not fixing the time intervals in advance.

3.0 REGRESSION METHODS FOR SURVIVAL ANALYSIS

The Proportional hazards, a semi-parametric 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 non-parametric methods for comparing 2 survival distributions are: Gehan’s generalized Wilcoxon test, the Cox-Mantel test, the log-rank test, Peto’s generalized Wilcoxon test, the Mantel-Haenszel test, and Cox’s F test. The parametric tests are the likelihood ratio test and Cox’s F test. The log-rank 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 log-rank test for later failure times. It gives more weight to the earlier part of the survival curve. The Mantel-Haenszel test relies on methods of analyzing incidence density ratios. The log-rank test attaches equal importance to all failure times irrespective of whether they are early or late. A modification of the log-rank test by Peto attaches more importance to earlier failure times. Cox’s regression is a semi-parametric method for studying several covariates simultaneously. The log-linear 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 time-dependent covariates (time-varying 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 follow-up 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 non-medical 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_{Dt ®0} {Pr (t  =<T =< t +  Dt|T >=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 NON-PARAMETRIC 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 non-parametric methods have to be used. Three methods of survival analysis are commonly: the life-table method, the Kaplan-Maeir method, and the Proportional hazards method. Other methods used are: exponential regression, log-normal regression, the competing risk model, and the discrete-time methods. A modification of the life-table method uses the subject-years approach. In both methods a survival curve is constructed showing the probability of the end-point 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. Cross-over 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 Mantel-Haenszel test (b) The log-rank 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 log-rank method by using a single dichotomous covariate or to behave like the K-M 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 Kaplan-Meier method is suitable for small samples with progressive censoring. The lifetable method is better for large samples.

DATA SET-UP FOR SURVIVAL ANALYSIS

All survival programs use a rectangular data file set up as shown below

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 (mdy₂ - mdy₁).                          

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 socio-economic 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 tumor-free time. Survival can be described as relative survival or absolute survival. Relative survival is to 1-year 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 follow-up, withdrawal from the study, study termination when subjects had different dates of enrolment, loss to follow-up, 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 non-random.

Random censoring occurs uniformly throughout the study, is not informative but creates fewer problems than non-random censoring. Non-random 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 non-informative. Informative censoring occurs when dropping out of the study is related to the outcome. In non-informative 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 NON-REGRESSION 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 LIFE-TABLE ( 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 O-W. Column #6 is the probability of dying in the interval. It is computed as P = D / O-W. Column #7 is the probability of surviving to the end of the interval and is computed as Q=1-P. 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 mid-interval. The log-rank 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 life-table 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 mid-interval 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 KAPLAN-MAIER 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 K-M 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 non-events. 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 K-M PROCEDURE

The K-M 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 Kaplan-Maier method is best used for small data sets in which the time of event occurrence is measured precisely. The Kaplan-Maier method is an improvement on the life-table method in the handling of withdrawals. The life-table 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 Kaplan-Maier method avoids this complication by not fixing the time intervals in advance. Intervals are defined in two ways: (a) An interval ends when the end-point 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 non-proportional 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 time-varying 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, h₀(t). It is a more robust semi-parametric 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 time-dependent 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 built-in 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 = h₀t exp (^på_j=1 b_jx_j) where h_t = hazard at time t, h₀t = baseline hazard, b_j = regression coefficient. The hazard ratio is computed as exp {^på_i=1 b_i (x_1i – x_ii)} where x_1i = exposed and x_0i = unexposed. Cox’s model is called non-parametric because h₀(t) is undefined.

The exponent  å b_jx_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 time-dependent part. The model can be written as h{t, x(t)} = h₀ (t) exp {^på_i=1 b₁x_i + ^på_i=1 d₁x_i (t)} where ^på_i=1 b₁x_i is time-independent and ^på_i=1 d₁x_i (t) is time-dependent. 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 {^på_j=1 b_j (x_ij - x_kj)}. The relative hazard ratio does not depend on t because the common factor h₀t 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 p-value, 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 log-lognormal 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 time-dependendent covariate * time is non-significant, 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 c₁² = {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 time-dependent 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 semi-parametric 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 R² 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 non-parametric methods for comparing 2 survival distributions are: Gehan’s generalized Wilcoxon test, the Cox-Mantel test, the log-rank test, Peto’s generalized Wilcoxon test, the Mantel-Haenszel 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 Log-rank (MH) test, Wilcoxon test, and the likelihood test are based on a chisquare statistic and a corresponding p value. The formula for the log-rank chisquare is given as c² = ^rå_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 log-rank test is more sensitive if the assumptions of proportional hazards hold.

The formula for the Wilcoxon chisquare is given as c² = {^rå_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 log-rank test for later failure times. The Peto test is an alternative to the Log-rank test. It gives more weight to the earlier part of the survival curve.

M-H TEST

The Mantel-Haenszel test relies on methods of analyzing incidence density ratios, IDR. IDR = {a/PT₁}/ {b/PT₂} where a = number of remissions in group1. b = number of remissions in group2 and PT = person-time of follow-up. 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 p-value for testing the null hypothesis of no difference between the two survival curves.

LOG-RANK TEST

The log-rank test is a non-parametric 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 f₁ and f₂ and the corresponding risks are r₁ and r₂, the expected numbers of failures are e₁ = (f₁ and f₂). r₁/ (r₁ + r₂) and e₂ = (f₁ and f₂). r₂/ (r₁ + r₂). The expected failure times summed uo over all failure times are E₁ = åe₁ and E₂ = åe₂. The observed failures summed up over all failure times are O₁ = åf₁ and O₂ = åf₂. The chisquare test statistic is then computed with 1 degree of freedom as å (O-E)²/E. The log-rank test can be used in cases with more than 2 curves where the chosquate test statistic is computed with c-1 degrees of freedom where c = number of curves. The log-rank test attaches equal importance to all failure times irrespective of whether they are early or late. A modification of the log-rank 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 semi-parametric method for studying several covariates simultneously. The loglinear exponential and the linear exponential regression methods are parametric approaches to studying prognostic covariates.