by Timothy Bilash MD

June 2004

based on:

Review of Basic & Clinical Biostatistics by

Beth Dawson, Robert Trapp (2001) CH9

- Survival analysis adds a complexity to interpretation of events and rates (TDB)

- patients do not enter study at same time

- time is mixed in with event occurence and contibute indistinguishably for a given outcome
- time and event are mixed and equated
- compound variable (person-years)
- change in time-to-event (earlier time) can have same effect as additional occurance (number)
- cannot distinguish statistically with survival analysis

- exposure and event are not time distinct (sum of given amounts of both)
- event determines amount of exposure, not given exposure leads to event
- time dimension feeds event occurance back into exposure
- event can occur prior to or during beginnings of exposure yet be unrelated to it, even tho in the exposure group

- length of time can be compared, but important to note
__exposure and event are in____different____time__**periods**

- event determines amount of exposure, not given exposure leads to event
- withdrawal
__for event__mixed with withdrawal from__attrition__or__non-compliance__- drop out for event not the same as drop out for compliance

- startpoint problem at beginning of period (partial exposure amount to risk)
- limits exposure for early event group

- endpoint problem at end of period (partial time obsrvation for outcome to occur)
- cut of inclusion at same given time
- limits events for later exposure group

- time and event are mixed and equated
- survival analysis is different from experiment
- for survival must analyze data before all events have occurred, as opposed to after all events to be counted have occurred
- events cumulative over time
- always have a truncation
- alters interpetation of the statistics (conditional probabilities)

- Survival parameters that affect risk and outcome
- exposure to
**risk**- onset delay
- length of exposure (integrated exposure)
- dose
- total
- maximum

- confounders/modifiers
- before
- during
- after

- possibility of
**outcome**- incubation time to event (delay of time to event after exposure has occurred)
- time to symptom
- time to diagnosis (test+)
- threshold for detection
- "accuracy" of test (PPV, NPV)
- gating
- censoring

- exposure to
*Censored observations*- Time of Entry
*Simultaneously Censored*(entry time simultaneous, like experiment)- [Fig9-1 p 212]

*Progressively Censored*(entry time not simultaneous)- [Fig 9-2 p212]

- [Fig 9-2 p212]

- Time of Entry

- patients do not enter study at same time
- SURVIVAL CURVES (Characterizing one Group)

**Life Table**(Actuarial) vs**Kaplan-Meier**Methods- both calculate proportion surviving thru an interval
- # of events in given time interval
- 1 event in that time interval

- different (weighted) averages to give survival proportion
- Kaplan-Meier is exact
- Life Table is approximation (averages over interval)

- equivalent if constant rate of events over each interval and over study time

- both calculate proportion surviving thru an interval
**Life Table**(**Actuarial Table)**Analysis

*also called*

**Cutler-Edurer Method**

- different ways to collect life table data
life table*Current*- cross sectional data
- different people at one point in time

- used by insurers

- cross sectional data
life table*Cohort*- (longitudinal data)
- same people over a period of time

- most medical studies

- (longitudinal data)
- not the same statistic

- assumptions for Life Table method
- intervals
**fixed**- # of events in given time interval

- allows
**mild censoring** - assumes events average out to
**midpoint of intervals**- equivalent to a
**random withdrawal**during the period - compares to
__constant__rate of withdrawl [?]

- equivalent to a
- assumes survival in one period not dependent on survival in other periods
- time interval duration used in the analysis is somewhat arbitrary but should be selected so that the number of censored observations in any interval is small

- intervals
- uses conditional probability, must not have had that event all the periods before that interval
**survival function**- assumes probability of survival in a given period does not depend on survival in any other period
- "probably violated in much medical research but does not appear to cause major concern to biostatisticians"

- count patients in the study
**at the beginning of each interval who are not left by the end of the interval (had an event)**- used for the
**numerator**

- used for the
- count patients who were left
**at the beginning of each interval**- used for
**denominator** - some not in the study that long (stopped)
- some lost to followup
- denominator reduced by half the number of patients withdrawn for other reasons during the period
- assumes patients withdraw randomly throughout the interval
- on average patient withdrawals for an interval occur on average at the midpoint of the interval
- so subtract 1/2 of the number who withdraw during that period instead of all
- less of a concern if time intervals are short

__varying event# for fixed time interval__- n events for that time interval

- used for
**Survival Function****per intervals**= proportion surviving thru the__ith interval__=

**S(i) = p(i) p(i-1) ... p(1)**[D&T p 216]**i= ith**__time____interval__**(***fixed*time in denominator of rate)**p(i)= 1-q(i)= percent survival (with no event) for ith time interval****q(i)= d(i)/[n(i)-w(i)/2]= event rate**for ith time interval**n(i)**=**n(i-1) - w(i-1)**=**#pts**at beginning of interval- affects
__denominator__not numerator - withdrawals in previous interval(i-1) affects number in currnt interval (i)

- affects
**d(i)**=**#termination events**in the interval**w(i)**=**#withdrawals**in interval (censored for other reasons than event)

- uses
**Greenwood's formula for the standard error**SE [D&T p 215]- formula for confidence interval (CI)
- assumes
__mild censoring__ - assumes the proportion surviving in an interval is approximately
__normally distributed__ - assumes sufficiently large sample sizes
- assumes a normal distribution for proportion surviving

- assumes
- SE for S(i)=
**S(i) * Sqrt[Sum( q(i) / (n(i) - d(i) - w(i)/2) )]**

- formula for confidence interval (CI)

- typically as the interval from entry into the study gets longer, the number of patients remaining for the next interval gets smaller.
- this means that the uncertainty (standard deviation) of the proportion surviving gets larger (
__statistics get worse with time__) - 95% confidence intervals get wider (often drawn on the graph as a band on either side of the curve)

- this means that the uncertainty (standard deviation) of the proportion surviving gets larger (
- considerable
**bias**can occur [p217]- if the intervals are large
__if many withdrawals occur__- if the
**withdrawals not midway in the interval**- Kaplan-Meier removes this problem

- Kaplan-Meier removes this problem

**Kaplan-Meier Product Limit**method [D&T p217]

- actuarial-type method, with
__analysis at each event occurance__(time since entry divided intoby each event)**unequal intervals**- event proportions in group estimated at the variable event moment, rather than at fixed intervals
- good for studies even involving small numbers of patients
- gives
__exact survival proportions__because it uses exact survival times __fixed event# (1) per varying time__- events per interval time= 1

**Survival Function per event**= proportion surviving__thru jth event__=**S(j)=p(j)p(j-1)...p(1)****j= jth**__event__**(***varying*time in denominator of rate)**p(j)=1-q(j)= percent survival (with no event) for time interval of the jth event****q(j)= d(j)/n(j)= event rate**for jth event interval**d(j)= jth event (1)****n(j)= #pts**at jth event

- [SE for S(j)]
**= S(j) * Sqrt[Sum( (d(j) / (n(j) * (n(j) - d(j)) )]**[see D&T p218] - note that
*withdrawls are ignored*- patients who are
**lost to follow-up**and those who**drop out**before an event time merely**drop out of the calculations**by no longer being considered. **[ISLT]**effects of censoring- if no censored observations occur, the Wilcoxon rank sum test is appropriate for comparing the ranks of survival times
- according to D&T, Kaplan-Meier
**removes the problem of withdrawals not occurring midway in the interval**, however: over the time of study (that is*K-M gives valid rates only if withdrawals and lost-to-follow-ups occur at a**constant**rate**,*or as an approximation at a random rate**withdrawals are uncorrelated**)- other problems from many withdrawals would still remain however [?]
- no way to account if the patients who have been censored (withdrawals and lost to followup) would be more or less likely to have an event in a given group, if the withdrawals were related to treatment medication or other events related to treatment group or perceptions
*for example, if a patient drops out of an estrogen study because of bleeding (from higher unopposed estrogen levels), and does not have an event (if estrogen lowers the event risk), then the event rate would be artifically increased*- d(j) for that interval would be inflated because does not contain these patients
- so only removes the withdrawal problem if withdrawals are constant over time (or random), withdrawals are random as to patients who are exposed and not to risk, and withdrawals are random as to risk factors for outcome.

- large interval problem would still remain (for low rates) [?]

- patients who are

- actuarial-type method, with
**Summarizing Survival Data****(**__within__**a group)**[p225]

**Hazard**or__Function__**Hazard**__Rate__- (a
__Rate__**is***within*as opposed to__group____Ratio__**is**__between__)__groups__ - also called conditional failure rate (events per total time)
- used to obtain estimate of
*Mean Survival Time*for survival data - useful for comparing two groups at risk
*if assumption of an*__exponential distribution__*is reasonable**(***ie,**__constant event rate__*)*- allows censored observations (makes corrections to
__denominator__for dropouts and exposure). - often used to characterize Kaplan-Meier Curves and in Cox Proportional Model
**H = D / (SumF + SumC)****D**is number of deaths**SumF**is the sum of event times**SumC**is the sum of censored times- assumes exponential distribution for survival curve (constant event rate)
- [see pg 225 for formula]

- (a
__when____assumption of an exponential distribution____is not appropriate__, other forms of the hazard function based on different probability distributions are used [Lee 1992]

**MEASURES OF SIGNIFICANCE (Comparing groups)**

- "Little information is available to guide investigators in deciding which procedure is appropriate in which situation." [p224]
- in addition sometimes cannot determine which procedure used to compare survival distributions
- multiple names
*research on biostatistical methods for analyzing survival data is still underway*

- Independent groups t-test is not appropriate for comparing survival curves directly because survival times are not normally distributed and tend to be positively skewed (p220)

**Comparing Survival Data (**__between__**groups -**data__Uncensored__**)**

**Wilcoxon rank sum**test- assumes
__constant rates__ - compares the ranks of survival time
- if
__no censored observations__occur, appropriate for comparing the ranks of survival times [D&T p221]

- assumes

**Comparing Survival Data (**__between__**groups -**__Censored__data*or*__Uncensored__**)**

**Tests for significance****to compare survival curves with censored observations**- presence of censored observations requires special methods for comparing two or more survival distributions
- conclusions that result are approximations and can be calculated in different ways (also have name confusions for techniques)
__Hazard Ratio__statistic__Logrank__- most commonly reported
- Logrank compares the differences of the (group sum over all periods)

statistic__Mantel-Haenszel Chi-Square__- can be applied to any set of 2x2 tables
- Mantel-Haenszel combines the series of 2x2 tables to estimate an odds ratio

- independent-groups t-test is not appropriate for Kaplan-Meier because survival time (denominator) is not normally distributed, and tends to be positively skewed. [D&T p220]

__Constant Hazard Ratio__*of Hazard Rates between groups,*

__Constant (or Proportional)____Hazard Rate__*in each group*

**Hazard**for*Ratio***comparing two groups**[p221]- ratio of proportions (
**proportion of outcomes**group to__in at risk__**proportion of outcomes**group for bi-valued risk)__in not at risk__**(O**_{1}_{/ }**E**_{1})**/**_{ }**(O**_{2}/**E**_{2})**O**is_{i}**observed****E**_{i}_{ }_{is expected}

- ratio of rates

__constant Hazard Ratio__**of Hazard Rates****between groups**__assumes____constant Hazard Rates__*of events*in each group*throughout the time of study*- or could also have
to make the ratio constant (Cox)__non-constant but proportional Hazard rates__

- allows censored observations
- interpreted as odds ratio between groups if bi-valued (binary) outcome
- HR can be calculated from logrank statistics [p221]
- used in Cox Proportional Model

- ratio of proportions (
**Logrank Statistic**

*also called*

**Mantel logrank statistic**

**Cox-Mantel logrank statistic**(more general use than just survival curves)- assumes
between groups throughout time of study__constant hazard ratio____rate in each group may vary__*, but*__rates stay in constant ratio__- ie,
of non-constant rates rates are at least*ratio**proportional*

- for each interval, the number of events
*observed in each group,*is compared to the number of events*expected in each group (calculated for the group*as a proportion by group number of the total number of events*from rate in all groups combined*who are at risk, as if group membership did not matter), and these are used to calculate a Chi-square statistic test for significance - at
*fixed*intervals (or computer programs can determine at*each event instead)*- determine the
*number at risk*in that*jth*interval- remove those not in study at start of that time interval from the number at risk
- died in previous interval
- censored because of event/outcome in previous interval
- censored because of atrition

- remove those not in study at start of that time interval from the number at risk
- calculate the
*number of**observed events**O*_{ij}*in the**jth interval,*for failures/events/deaths*in each**ith**group* - calculate the
*number of**expected events**E*_{ij}_{ }*in the**jth interval, for*failures/events/deaths*in each**ith**group**divides up the expected events for the interval by the proportion in each group**(as if by chance)***E**= (# of events/deaths in_{ij}in the*ith*group) * (proportion of patients at risk in*jth*interval*ith*__group__in the)*jth*interval

the__total__*numbers of observed events**O*_{ij }*and expected events**E*_{ij}*over all j intervals for each i group to get*__O___{i}*,*__E___{i}- Oi, Ei are sums over all periods (or events) for each group

- determine the
- sums these
*O*_{i}*,*totals in an approximate*E*_{i}**Chi-square test**for significance (if distributions are not the same as expected)- Chi-square statistic for events over all intervals

**X**^{2}**= SUM [(O**_{i}**-E**_{i })^{2}**/****E**_{i}**]**(over all groups)- approximates Chi-square distribution with N-1 degress of freedom (N is # of groups)
**O**is sum of the observed events over all_{i}*j*intervals for the*i*group**E**is sum of the expected events over all_{i}*j*intervals for the*i*groupindicates distributions are*X*^{2}>Nthe same and there is a difference between groups*not***alternate rewrite X**^{2}**= SUM [E**_{i}(1 - O_{i}**/E**_{i})^{2}]- expected value times the square of fractional risk ratio subtracted from 1

- for bi-valued outcome (yes/no, 1 degree of freedom):

**X**^{2}_{yes/no}**= [O**_{yes }- E_{yes}]^{2}**/****E**_{yes }+**[O**_{no}**- E**_{no}]^{2}**/****E**_{no}*X*^{2}*> ~7 indicates observed is different from expected at the 0.01 level (1%)*

- Chi-square statistic for events over all intervals
- allows censored observations
- better with exponential distributions [p224]
- unweighted
**Petro logrank test**- weighted logrank test
- weighted by number of patients at risk
- gives more weight to early events when the number of patients at risk is large (more patients happens to be at beginning of study period, and since time and events are mixed in survival function cannot separate them)

**Mantel-Haenszel test**(**chi-square statistic**) [D&T p223]

__non-constant hazard rates__*OK but assumes a*__constant hazard ratio__- sometimes called logrank test but
the same*not* - test for homogenity of a relationship between 2 factors across changes in a third factor; is the relationship the same across levels of the third factor
- time intervals determined
*by events*- calculate the observed and expected numbers each time an event occurs
- that is, compare percentage in each group caused by that event over time to event interval

- approximate test (compares to expected based on average for the period, a pooled odds ratio) [p222]
- unweighted
- similar results to logrank tests
- can be used to compare any distributions [p223]
- MH Chi Square =

__[SUM(observed numbers)-SUM(expected numbers)]**2__

**[SUM(variances)]**

over every time period ª

- Would be subject to small number (frequency) restrictions of chi-square analysis?

**Cox Proportional Hazard Model**- assumes the
between groups for risk of an event*hazard ratio**is*throughout the study (p214 Dawson and Trapp)*constant*hazard__constant__(__ratios__**rates are proportional**)

- allows
*censored observations* - uses the
*Hazard Function*to evaluate the length of time to event - entry point problem (start point)
- divides periods into intervals
- mismatch endpoint problem for time until event if less than a year, and the analysis being done in yearly segments (like compounding interest yearly instead of daily)
- all patients with an event treated as if event happened at one year mark, whether in first month or last day of period
- for example, a patient dying on day 357 after entering is not counted for a tally of one-year survival with Cox Model since did not make it to the second year
- Kaplan-Meier product limit method gives credit for time up to actual event rather than within interval which corrects this

- see Chapter 10

- assumes the

__Non-constant__(arbitrary)*Hazard ratio**between groups**of Hazard Rates**for each group*

- Methods that are difficult to quantify statistically

**Person-years**[p214 D&T]- patient contributes to the average however long they are in the study
__event marks one in numerator__with__time to event in denominator____number of events divided__by__average time to event of all patients__

- used to compare numbers for a period with a different time period or from another study
- no statistical methods available to compare these numbers however
- mixes time and number
- same number is obtained by observing 1000 patients for 1 year as observing 10 patients for 100 years

- assumes chance of and event is constant throughout the study
- risk of exposure

- risk of exposure

- patient contributes to the average however long they are in the study
**1-Year, 5-Year Survival(Mortality) Rates**- endpoint problem for patients that dont stay in group for that total time period
- lose partial participants (have to be in study at least that long)
- Life Table and Kaplan-Meier product limit methods give credit for the amount of time subjects survive up to the time when data are analyzed.

**Generalized Wilcoxon test**__non-constant hazard ratios__*OK*- other names for this test
**Generalized Kuskal-Wallis**test**Gehan**test**Breslow**test

- extension of Wilcoxon rank sum test allowing for
*censored data* - weights earlier events more

- Methods that are difficult to quantify statistically
**Intention-to-Treat Principle**[p228]

- The results for each patient who entered the trial are included in the analysis of the group to which the patient was randomized, regardless of any subsequent events.
**dropouts**- possible that the patients who dropped out of the treatment group had some characteristics that, independent of treatment, could affect the outcome

**crossovers**- patients cross-over from one treatment group to the other (or have poor commpliance if comparing to placebo group)
- do not know why cross-overs occur

- Comparison of techniques
- analyze the patients by group they were randomized to (intention-to-treat)
- analyze the patients by group they ended up in at end of study
- eliminate all crossovers
*both of these approaches are potentially biased*- advocates of evidence-based medicine recommend intention-to-treat
- [ISLT] these issues are more problematic than is currently addressed

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*page views since Mar2007*

ªcorrection 8.27.2004

- "Little information is available to guide investigators in deciding which procedure is appropriate in which situation." [p224]