survival analysis i (chl5209h) · analysis of failure time data, second edition. introductory,...

Survival AnalysisI (CHL5209H)

Olli Saarela

Motivation:Poissonregression

Basic concepts

31-1


Olli Saarela

Dalla Lana School of Public HealthUniversity of Toronto

[email protected]

January 7, 2015


Olli Saarela


Basic concepts

31-2

Literature

I Clayton D & Hills M (1993): Statistical Models inEpidemiology. Not really useful as a reference text butinteresting pedagogical approach.

I Kalbfleisch JD & Prentice RL (2002): The StatisticalAnalysis of Failure Time Data, Second Edition.Introductory, serves as a reference text.

I Klein JP & Moeschberger ML (2003): Survival Analysis -Techniques for Censored and Truncated Data, SecondEdition. Introductory, serves as a reference text.

I Aalen OO, Borgan Ø, Gjessing H (2008): Survival andEvent History Analysis - A Process Point of View. Forthose looking for something more theoretical.


Olli Saarela


Basic concepts

31-3

Motivation: Poisson regression


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Basic concepts

31-4

Survival analysis?

I Survival analysis focuses on a single event per individual(say, first marriage, graduation, diagnosis of a disease,death). Analysis of multiple events would be referred to asevent history analysis.

I Despite the name, a course in survival analysis is mostlynot about survival.

I Rather, most of the time, we concentrate on estimating ormodeling a more fundamental quantity, the rate parameterλ, or the time-dependent version, the hazard functionλ(t).

I Survival probability in turn is determined by the hazardfunction.

I The rates can be for example mortality or incidence rates.


Olli Saarela


Basic concepts

31-5

More about rates

I Rate parameter is the parameter of the Poissondistribution, characterizing the rate of occurrence of theevents of interest.

I The expected number of events µ in a total of Y years offollow-up time and λ are connected by

µ = λY .

I The observed number of events D in Y years of follow-uptime is distributed as D ∼ Poisson(λY ).

I How to estimate the rate parameter λ?


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Basic concepts

31-6

An estimator for λ

I A possible estimator is suggested by

µ = λY ⇔ λ =µ

Y.

I It would seem reasonable to replace here the expectednumber of events µ with the observed number of events Dand take

λ̂ =D

Y.

I This is in fact the maximum likelihood estimator of λ.


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Basic concepts

31-7

Maximum likelihood criterion

I Maximum likelihood estimate is the parameter value thatmaximizes the probability of observing the data D.

I The probability of the observed data is given by thestatistical model, which is now

D ∼ Poisson(λY ).

I Probabilities under the Poisson distribution are given by

(λY )D

D!e−λY .

I We consider this probability as a function of λ, and call itthe likelihood of λ.


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Basic concepts

31-8

Maximizing the likelihood

I We may ignore any multiplicative terms not depending onthe parameter, and instead maximize the expression

λDe−λY .

I Or, for mathematical convenience, its logarithm

D log λ− λY .

I How to find the argument value which maximizes afunction?

I Set the first derivative to zero and solve w.r.t. λ:

D

λ− Y = 0 ⇔ λ =

D

Y.

I Check that the second derivative is negative:

− D

λ2< 0.


Olli Saarela


Basic concepts

31-9

Follow-up data

I Clayton & Hills (1993, p. 41):


Olli Saarela


Basic concepts

31-10

Estimate

I For the 7 subjects (individuals) there is a total of 36 timeunits of follow-up time/person-time, and 2 outcomeevents (for individuals 2 and 6).

I The follow-up of the other individuals was terminated bycensoring (e.g. by events other than the outcome event ofinterest).

I In fact, censoring is the reason why we need to approachthe estimation problem through rates, rather than lookingat the event times directly.

I We arrive at

λ̂ =D

Y=

2

36≈ 0.056.

I To recap:I Estimand/parameter/object of inference: λI Estimator: D

YI Estimate: 0.056.


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Basic concepts

31-11

Interpretation?

I We will define the rate parameter more precisely shortly,but let us first consider the interpretation.

I Unlike the risk parameter, the probability of an eventoccurring within a specific time period, the rate parameterdoes not correspond to a follow-up period of a fixedlength.

I Rather, it characterizes the instantaneous occurrence ofthe outcome event at any given time.

I The rate parameter is not a probability, but it can becharacterized in terms of the risk parameter when thefollow-up period is very short.


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Basic concepts

31-12

Time unit

I Suppose that each of the N = 36 time bins here is oflength h = 0.05 years:

I In total there is Y = Nh = 36× 0.05 = 1.8 years offollow-up.


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Basic concepts

31-13

From risk to rate

I The empirical rate is given by λ̂ = 21.8 = 1.11 per

person-year, or, say, 1110 per 1000 person-years.

I Per person-year, the empirical rate would be the same,had we instead split the person-time into 180 bins oflength 0.01 years.

I Suppose that we have made the time bins short enough sothat at most one event can occur in each bin.

I Whether an event occurred in a particular bin of length his now a Bernoulli-distributed variable, with the expectednumber of events equal to the risk π.

I Thus, because rate is the expected count divided byperson-time, when h is small, we have

λ =π

h⇔ π = λh.

I This connection is important in understanding how rate isrelated to survival probability.


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Basic concepts

31-14

Survival probabilityI One of the particular properties of the natural logarithm

and its inverse is that when x is close to zero, ex ≈ 1 + x ,and conversely, log(1 + x) ≈ x .

I Suppose that we are interested in the probability ofsurviving T years. By splitting the timescale so thatN = T

h , T = Nh.I The probability of surviving through a single time bin of

length h, conditional on surviving until the start of thisinterval, is 1− π = 1− λh.

I By the multiplicative rule, the T year survival probabilityis thus

(1− λh)N .

I This motivates the well-known Kaplan-Meier estimator, tobe encountered later.

I In turn, the logarithm of this is

N log(1− λh) ≈ −Nλh = −λT .


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Basic concepts

31-15

Survival and cumulative hazard

I The quantity λT is known as the cumulative hazard.

I We have (approximately, without calculus) obtained afundamental relationship of survival analysis, namely thatthe T year survival probability is

(1− λh)N ≈ e−λT .

I Let us test whether this approximation actually works.Now λ̂ = 1.11.

I If T = 1 and h = 0.05, N = 20 and we get(1− 1.11× 0.05)20 ≈ 0.319.

I The exact one year survival probability ise−1.11×1 ≈ 0.330.

I We should get a better approximation through a finer splitof the time scale.

I If h = 0.01, N = 100 and (1− 1.11× 0.01)100 ≈ 0.328.


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Basic concepts

31-16

Regression modelsI Recall the relationship µ = λY . If γ = log λ, uquivalently

we have a log-linear form

logµ = γ + log Y ,

where we call log Y an offset term.I γ is an unknown parameter, which we could estimate in an

obvious way. (How?)I Such a one-parameter model is not very interesting, but

serves as a starting point to regression modeling.I Consider now the expected number of events µ1 in Y1

years of exposed person-time and the expected number ofevents µ0 in Y0 years of unexposed person-time.

I The corresponding probability models are now

D1 ∼ Poisson(µ1) and D0 ∼ Poisson(µ0),

where µ1 = λ1Y1 and µ0 = λ0Y0.I Here µ1 + µ0 = µ and D1 + D0 = D.


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Basic concepts

31-17

Combining the two modelsI We may now parametrize the two log-rates in terms of an

intercept term α and a regression coefficient β aslog λ0 = α and log λ1 = α + β.

I By introducing an exposure variable Z , with Z = 1(Z = 0) indicating the exposed (unexposed) person-time,we can express these definitions as a regression equation

log λZ = α + βZ ⇔ λZ = eα+βZ .

I This results in a single statistical model, namely

DZ ∼ Poisson(

YZeα+βZ).

I What is the interpretation of the regression coefficient?I Now we have

λ1

λ0=

eα+β

eα=

eαeβ

eα= eβ,

or β = log(λ1λ0

), that is, the log rate ratio.


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Basic concepts

31-18

Reparametrizing ratesI The model can be easily extended to accommodate more

than one determinant.I For example, unadjusted comparisons of rates are

susceptible to confounding; we can move on to considerconfounder-adjusted rate ratios.

I Consider the following dataset:

I Introduce an exposure variable taking values Z = 1(energy intake < 2750 kcals) and Z = 0 (≥ 2750 kcals),and an age group indicator taking values X = 0 (40− 49),X = 1 (50− 59) and X = 2 (60− 69).


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Basic concepts

31-19

The original parameters

I There are now six rate parameters λZX , corresponding toeach exposure-age combination:

Z = 0 Z = 1X = 0 λ00 λ10

X = 1 λ01 λ11

X = 2 λ02 λ12

I The corresponding statistical distributions are

Z = 0 Z = 1X = 0 D00 ∼ Poisson(Y00λ00) D10 ∼ Poisson(Y10λ10)X = 1 D01 ∼ Poisson(Y01λ01) D11 ∼ Poisson(Y11λ11)X = 2 D02 ∼ Poisson(Y02λ02) D12 ∼ Poisson(Y12λ12)


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Basic concepts

31-20

Transformed parameters

I Now, we are not primarily interested in estimating sixrates; rather, we are interested in the rate ratio betweenthe exposure categories, adjusting for age.

I We could parametrize the rates w.r.t. the baseline, orreference, rate λ00 which is then modified by the exposureand age (cf. Clayton & Hills 1993, p. 220).

I DefineZ = 0 Z = 1

X = 0 λ00 = λ00 λ10 = λ00θX = 1 λ01 = λ00φ1 λ11 = λ00θφ1

X = 2 λ02 = λ00φ2 λ12 = λ00θφ2

I Now θ is the rate ratio within each age group (verify).


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Basic concepts

31-21

Regression parameters

I As before, we can specify the reparametrization in termsof a link function and a linear predictor as

log λZX = α + βZ + γ11{X=1} + γ21{X=2}.

I SinceλZX = eα+βZ+γ11{X=1}+γ21{X=2} ,

we have that λ00 = eα, θ = eβ, φ1 = eγ1 and φ2 = eγ2 .

I The rates are now given by the regression equation as

Z = 0 Z = 1X = 0 λ00 = eα λ10 = eα+β

X = 1 λ01 = eα+γ1 λ11 = eα+β+γ1

X = 2 λ02 = eα+γ2 λ12 = eα+β+γ2

I The number of parameters has been reduced from six tofour.


Olli Saarela


Basic concepts

31-22

Specification in terms of expected counts

I A Poisson model is always specified in terms of theexpected event count: DZX ∼ Poisson(µZX ).

I The regression model for the expected count is specified by

µZX = YZXλZX = YZX eα+βZ+γ11{X=1}+γ21{X=2}

= eα+βZ+γ11{X=1}+γ21{X=2}+log YZX .

I We have obtained the model

DXZ ∼ Poisson(

eα+βZ+γ11{X=1}+γ21{X=2}+log YZX

).

I When fitting the model, log-person years has to beincluded in the linear predictor as an offset variable.


Olli Saarela


Basic concepts

31-23

Fitting the model

I The data as frequency records are entered into R as:

d <- c(4,5,8,2,12,14)

y <- c(607.9,1272.1,888.9,311.9,878.1,667.5)

z <- c(0,0,0,1,1,1)

x <- c(0,1,2,0,1,2)

I The model is specified as

model <- glm(d ~ z + as.factor(x) +

offset(log(y)),

family=poisson(link="log"))

I The as.factor(x) term specifies that we want toestimate separate age group effects (rather than assumethat the X -variable modifies the log-rate additively).


Olli Saarela


Basic concepts

31-24

Results

Call:

glm(formula = d ~ z + as.factor(x) + offset(log(y)),

family = poisson(link = "log"))

Deviance Residuals:

1 2 3 4 5 6

0.73940 -0.58410 0.04255 -0.77385 0.42800 -0.03191

Coefficients:

Estimate Std. Error z value Pr(>|z|)

(Intercept) -5.4177 0.4421 -12.256 < 2e-16 ***

z 0.8697 0.3080 2.823 0.00476 **

as.factor(x)1 0.1290 0.4754 0.271 0.78609

as.factor(x)2 0.6920 0.4614 1.500 0.13366

---

Signif. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’ 0.1 ’ ’ 1

(Dispersion parameter for poisson family taken to be 1)

Null deviance: 14.5780 on 5 degrees of freedom

Residual deviance: 1.6727 on 2 degrees of freedom

AIC: 31.796

Number of Fisher Scoring iterations: 4


Olli Saarela


Basic concepts

31-25

Proportional hazards

I From the model output, we may calculate estimates forthe original rate parameters (per 1000 person-years) as

Z = 0 Z = 1

X = 0 λ̂00 = 4.44 λ̂10 = 10.59

X = 1 λ̂01 = 5.05 λ̂11 = 12.05

X = 2 λ̂02 = 8.86 λ̂12 = 21.20

I Note that the rate ratio stays constant across the agegroups. This is forced by the earlier model specification.

I This is a modeling assumption, namely the proportionalhazards assumption.

I Compare these estimates to the corresponding sixempirical rates. Is assuming proportionality of the hazardrates justified? How could one test this? Or relax thisassumption?


Olli Saarela


Basic concepts

31-26

Basic concepts


Olli Saarela


Basic concepts

31-27

Counting process

I In survival analysis, the outcome is generally a pair ofrandom variables (Ti ,Ei ), where Ti represents the eventtime, and Ei the type of the event that occurred at Ti .

I Usually, we have to consider at least two types of events,namely the outcome event of interest (say, Ei = 1), andcensoring (say, Ei = 0), that is, termination of thefollow-up due to some other reason than the outcomeevent of interest.

I Equivalently, we can characterize the outcomes in terms ofthe cause-specific counting processes

Nij(t) = 1{Ti≤t,Ei=j}, j = 0, 1.

(What is a process?)

I We can also introduce an overall counting processNi (t) =

∑1j=0 Nij(t) for any type of event. If there is no

censoring, this is all we need.


Olli Saarela


Basic concepts

31-28

History

I The observed histories of such counting processes justbefore time t are denoted asFt− = σ({Nij(u) : i = 1, . . . , n; j = 0, 1; 0 ≤ u < t}).

I The mathematical characterization of this as a generatedσ-algebra is not important at this point, we only need tounderstand Ft− as all observed information before t.

I Consider now the conditional probability for an event (ofany type) occurring at t, that is,

P(dNi (t) = 1 | Ft−),

where dNi (t) ≡ Ni (t + dt)− Ni (t).


Olli Saarela


Basic concepts

31-29

Hazard function

I Alternatively, this may be characterized as

P(dNi (t) = 1 | Ft−) = E [dNi (t) | Ft−]

= P(t ≤ Ti < t + dt | Ft−)

≡ λ(t) dt,

or

λ(t) =P(t ≤ Ti < t + dt | Ft−)

dt.

I When Ti ≥ t and the outcome of individual i does notdepend on the outcomes of other individuals, this isequivalent to the familiar definition

λ(t) = lim∆t→0

P(t ≤ Ti < t + ∆t | Ti ≥ t)

∆t.


Olli Saarela


Basic concepts

31-30

Connection to the survival function

I Now

P(t ≤ Ti < t + dt | Ti ≥ t) =P(t ≤ Ti < t + dt)

P(Ti ≥ t)

⇔ λ(t) =f (t)

S(t),

where f (t) is the density function and S(t) the survivalfunction of the event time distribution.

I Note that S(t) = 1− F (t) and f (t) = dF (t)dt .

I Further, d[log F (t)]dt = f (t)

F (t) and −d[log S(t)]dt = f (t)

S(t) .

I This gives us again the fundamental relationship

S(t) = exp

{−∫ t

0λ(u) du

},

where∫ t

0 λ(u) du ≡ Λ(t) is the cumulative hazard.


Olli Saarela


Basic concepts

31-31

Cause-specific hazards

I In the presence of censoring, we are not interested in theoverall hazard rate, but rather, the cause-specific hazardrate

P(dNij(t) = 1 | Ft−) = E [dNij(t) | Ft−]

= P(t ≤ Ti < t + dt,Ei = j | Ft−)

≡ λj(t) dt

for event type j .

I It is easy to see that these are related to the overall hazardby λ(t) =

∑1j=0 λj(t).

I The sub-density function corresponding to event type j isgiven by

fj(t) ≡ P(t ≤ Ti < t + dt,Ei = j)

dt= λj(t)S(t),

where S(t) is the overall survival function.

survival analysis i (chl5209h) · analysis of failure time data, second edition. introductory,...

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