Advanced statistical analysis of epidemiological studies

Ph.d. course in “Advanced statistical
analysis of epidemiological studies”
Multi-state models, competing risks, recurrent events.
Case-cohort studies
www.biostat.ku.dk/~pka/avepi11
25 November 2011
Per Kragh Andersen
1

Multi-state models
Survival data = cohort studies with a single outcome
• time from “zero” to event (death)
• right-censoring, delayed entry
• basic quantity:
– death intensity
– = mortality rate
– = hazard function
– = h(t) ≈ Prob(die before t + ∆ | alive t)/∆
✲
0 t t + ∆
2

Survival data.
The hazard, h(t) provides a local (in time) description of the
development.
Models for h(t):
• Cox regression
• Poisson regression
The survival function:
S(t) = Prob(alive time t)
describes the cumulative development over time.
3

Two-state model for survival data
0
Alive
h(t)
✲
1
Dead
h(t) ≈ Prob(state 1 time t + ∆ | state 0 time t)/∆
S(t) = Prob(state 0 time t).
F(t) = 1 − S(t) = Prob(state 1 time t) is the cumulative probability
(“risk”) of death over the interval from 0 to t.
4

Relation between rates and risks.
We have the classical relation:
S(t) = exp(−
∫ t
0
h(u)du).
This means that whenever we have a (Cox/Poisson/...) model for the
rate, h(t), we also have a model for the risk: F(t) = 1 − S(t).
In particular, if the rate increases/decreases with a covariate, Z, then
also the risk increases/decreases with Z.
5

Generalisations: competing risks
0
Dead, cause 1
h 1 (t)
✑ ✑✑✑✑✑✸
♣
Alive
♣
◗ ◗◗◗◗◗
1
h k (t)
k
Dead, cause k
6

E.g., k = 3 causes:
Competing risks
• cancer
• cardio-vascular diseases
• other causes
Cause-specific intensities (e.g. cause 1)
h 1 (t) ≈ Prob(state 1 time t + ∆ | state 0 time t)/∆
7

Generalisations: illness-death model
0
Disease-free
h 01 (t)
✲
1
Diseased
❙
❙
❙
❙✇
h 02 (t)
2
Dead
✓
✓
✓
✓✴
h 12 (t)
The illness-death or disability model (chronic disease).
8

Transition intensities
Illness-death model
“Disease incidence”:
h 01 (t) ≈ Prob(state 1 time t + ∆ | state 0 time t)/∆
Mortality among disease-free (e.g. standard mortality)
h 02 (t) ≈ Prob(state 2 time t + ∆ | state 0 time t)/∆
Mortality among diseased (“fatality rate”)
h 12 (t) ≈ Prob(state 2 time t + ∆ | state 1 time t)/∆
9

Generalisations: illness-death model
0
Disease-free
❙
❙
❙
❙✇
h 02 (t)
✛
2
h 01 (t)
h 10 (t)
Dead
✲
1
Diseased
✓
✓
✓
✓✴
h 12 (t)
The illness-death or disability model (recurrent disease).
10

Illness-death model, recurrent disease
Transition intensities
As above, but also “Cure rate”:
h 10 (t) ≈ Prob(state 0 time t + ∆ | state 1 time t)/∆
Also: Recurrent events, no death state.
11

States:
Bone marrow transplantations
• Transplanted
• Graft versus host disease
• Relapse
• Death
etc. etc.
12

Models are given by intensities:
Common features
h ij (t) ≈ Prob(state j time t + ∆ | state i time t)/∆
Intensities may be modelled using:
Cox regression, Poisson regression
and analysed in SAS using
PROC PHREG, PROC GENMOD
In order to use PROC PHREG, a data file must be created for each
transition including:
• entry time (some times 0)
• exit time
• exit status (relevant transition or not)
• covariates
13

Common features
In order to use PROC GENMOD, a data file must be created for each
transition and for each combination of covariates, including:
e.g.
• time spent in state
• number of transitions
Age 1 Age 2 Age 3
Exposed T 01 , D 01 T 02 , D 02 T 03 , D 03
Non-exposed T 11 , D 11 T 12 , D 12 T 13 , D 13
This enables us to analyse the intensities (rates) and estimate rate
ratios
14

Probabilities (risks)
In survival analysis: The classical relation between risk and rate
S(t) = exp(−
∫ t
0
h(u)du).
holds when there are no competing risks (e.g., C & H, ch. 4).
In more general multi-state models:
• transition probabilities are more complex functions of the
intensities
• few general computer programs exist
(SAS MACRO for competing risks with “Cox hazards”, R packages
cmprsk, mstate.)
15

2 causes of death:
In the competing risks model:
P 00 (t) = Prob(alive time t)
= exp(−
∫ t
0
(h 1 (u) + h 2 (u))du).
P 01 (t) = Prob(dead from cause 1 before time t) =
∫ t
0
P 00 (u)h 1 (u)du
0 u u + du t
✲
time
16

This means that the cumulative incidence:
P 01 (t) (and similarly P 02 (t))
may be estimated from
h 1 (t) and h 2 (t).
That is, the risk for cause 1 depends on the rates for both causes 1
and 2.
The SE’s may also be estimated: a SAS MACRO and an R function are
available.
17

What does
estimate
In the competing risks model:
1 − exp(−
∫ t
0
h 1 (u)du)
Prob(Dead from cause 1 before t)
IF h 2 (t) = 0!
i.e., if the competing risk does not exist.
This hypothetical probability is rarely of interest. However, it is used
frequently anyhow!
“Relapse survival curve” in clinical cancer studies.
“1-Kaplan-Meier” as risk estimator in epidemiological (and other)
studies.
18

Censoring in survival studies
Note that rates may be estimated by treating other events as
censoring while risks require modeling of all competing events.
When, in survival studies, we draw the Kaplan-Meier estimator only
the death intensity is taken into account - NOT the censoring
intensity. This makes sense if BOTH (I): the population without
censoring makes sense, AND (II) censoring is “independent”.
Example: event = death due to cancer, consider censoring due to
• end of study
• emigration, loss to follow-up
• death due to traffic accidents
• death due to cardiovascular disease
Magnitude of (I) depends on magnitude of competing risk.
19

Competing risks example: bone marrow transplantation.
1715 leukemia patients with BMT:
• 537 ALL, 340 AML, 838 CML
• 1026 early stage, 410 intermediate stage, 279 advanced stage
• 1224 HLA-identical sibling, 383 HLA-matched unrelated donor,
108 HLA-mismatched unrelated donor
Analysis:
• Cox regression models for cause-specific hazards of “relapse” and
“death in remission”
• Estimation of cumulative incidences
20

Cox regression models for cause-specific hazards
Relapse Death
Covariate ˆβ (SD) ˆβ (SD)
HLA-id. sibling 0 - 0 -
HLA-matched donor 0.011 0.15 0.811 0.097
HLA-mismatched donor -0.944 0.36 1.118 0.14
ALL 0 - 0 -
AML -0.271 0.15 -0.195 0.14
CML -0.721 0.16 0.291 0.117
Early stage 0 - 0 -
Intermed. stage 0.640 0.15 0.474 0.10
Advanced stage 1.848 0.15 0.781 0.13
Karnofsky> 90 -0.118 0.14 -0.504 0.11
21

Cumulative incidences
22

Regression analysis of risks.
The cumulative incidences may be related to covariates by fitting
models for cause-specific hazards and plugging into the rate → risk
relation. However, this does not give a simple relationship, e.g. a
covariate may increase the rate of a given cause but decrease the
corresponding risk depending on its effect on the other rates.
Instead: direct modeling the relationship using the “Fine-Gray”
model.
Cox model for survival data: − log(1 − F(t)) = A 0 (t)e β′Z ,
A 0 (t) cumulative baseline hazard.
Fine-Gray model for P 01 (t): − log(1 − P 01 (t)) = A 01 (t)e β′ 1 Z .
This gives a direct relationship between covariates and risk but the
interpretation of the regression parameters, β 1 is not so simple.
23

Fine-Gray regression models for cumulative incidences
Relapse Death
Covariate ˆβ1 (SD) ˆβ1 (SD)
HLA-id. sibling 0 - 0 -
HLA-matched donor -0.32 0.16 0.76 0.10
HLA-mismatched donor -1.37 0.38 1.15 0.13
ALL 0 - 0 -
AML -0.17 0.15 -0.16 0.14
CML -0.75 0.16 0.33 0.12
Early stage 0 - 0 -
Intermed. stage 0.51 0.15 0.41 0.10
Advanced stage 1.51 0.15 0.45 0.13
Karnofsky> 90 0.17 0.15 -0.44 0.11
24

Recurrent events.
Suppose individuals may experience the event recurrently, e.g.
hospital admissions.
Kessing, Olsen and Andersen, Amer. J. Epidemiol. (1999) studied
data from Danish Central Psychiatric Registry, DCPR:
• All psychiatric admissions 1 April 1970-
• Discharge diagnoses
• ICD-8 until 31 december 1993
• ICD-10 from 1 January 1994
and linked to the Danish Civil Registration System, ”CPR” to
obtain information on vital status and emigration.
25

Sample from register data.
• All patients identified in DCPR before 1 January 1994 with a
bipolar (“manic”) diagnosis at first discharge.
• Restrict attention to patients younger than 52 years at diagnosis,
1307 men and 1655 women.
• Followed to 31 December 1999 w.r.t. later psychiatric
admissions, death, diagnosis of schizophrenia or emigration.
Purpose of study: Evaluate theory of “sensitization” (or “kindling”).
According to this theory, mood episodes themselves stress the brain
so that its sensitivity to biologic and psychosocial stressors increases.
This leads to shorter and shorter intervals between successive
episodes, e.g. (Post, 1992, Amer. J. Psych.)
26

Register data.
Kaplan-Meier estimates of the survivor functions
for 1st, 2nd and later waiting times.
Survival Function
0.0 0.2 0.4 0.6 0.8 1.0
Men
1wt
2wt
3wt
4wt
5wt
Survival Function
0.0 0.2 0.4 0.6 0.8 1.0
Women
1wt
2wt
3wt
4wt
5wt
0 1000 2000 3000 4000
Time(days)
0 1000 2000 3000 4000
Time(days)
27

Using Kaplan-Meier curves.
The Kaplan-Meier curves do not properly address the problem of
sensitization due to selection/heterogeneity – those with 2 episodes is
a select subgroup of those with 1 etc.
Another way of stating the problem is via dependent censoring:
e.g., censoring of the second waiting time depends on the first waiting
time and if successive waiting times are correlated then censoring of
the second waiting time depends on the second waiting time:
T i : duration of follow-up time for subject i, W i1 , W i2 : first and
second waiting time for subject i.
C i2 = T i − W i1 censoring time for W i2 . If W i1 and W i2 are dependent
then W i2 and C i2 are also dependent.
In the analysis we need to take this dependence into account.
28

Random effects models.
Random effects models for survival data are known as “frailty
models”.
Hazard function for waiting time no. j for patient no. i:
h ij (t) = U i h 0 (t) exp(α j + β ′ Z i ).
• U i : random “frailty” for patient i following some distribution
with mean 1 and SD σ across the patient population
• h 0 (t): baseline hazard
• α j : log(hazard ratio) for waiting time no. j compared to
reference waiting time (e.g. first waiting time)
• Z i : covariates
29

Results for younger bipolar patients.
Men
Women
Episode (j) Rate ratio 95% c.i. Rate ratio 95% c.i.
1 1.00 ref. 1.00 ref.
2 1.18 1.03-1.34 1.22 1.08-1.37
3 1.46 1.27-1.69 1.47 1.29-1.67
4 1.72 1.49-2.02 1.61 1.40-1.85
5+ 2.35 2.07-2.67 2.19 1.07-2.45
σ 2 0 (no frailty) 0 (no frailty)
1 1.00 ref. 1.00 ref.
2 0.99 0.84-1.15 1.07 0.93-1.22
3 1.10 0.91-1.33 1.16 0.99-1.36
4 1.16 0.93-1.45 1.17 0.98-1.39
5+ 1.30 1.04-1.64 1.25 1.04-1.50
σ 2 0.45 0.26-0.63 0.41 0.28-0.54
30

Study base: population followed from t 0 to t 1 .
t 0 t 1
❛
❛ ❛
♣
❛
31

Nested case-control study
t 0 t 1
❞
❞
❞
❛
❞
❛
❞
❞
❞
❛ ❞ ❛
♣
❞
32

Estimation of rate ratio θ:
(
)
∑ θ (for case)
log ∑
failures
Case-control set θ
Compare: Cox regression in cohort study or matched case-control
study.
PROC PHREG in SAS may be used (but PROC LOGISTIC may be
simpler).
In the simplest case, the controls are a simple random sample from
the risk set.
33

Matching.
Other nested case-control sampling designs.
Example: Lung cancer incidence, smoking possible confounder.
Many smoking cases, perhaps relatively few smoking controls ⇒
random sampling of m − 1 controls will give few controls per smoking
case and more controls per non-smoking case.
Matching on smoking may be efficient.
• Availability of data
• Inability to estimate effect of smoking
34

θ case = exp(β 1 · exposure case + β 2 · smoke case )
θ control = exp(β 1 · exposure control + β 2 · smoke control )
where exposure is 0 or 1 and and where the value of smoke is the
same for case and controls, i.e. exp(β 2 ) cancels out in log partial
likelihood:
(
)
∑ θ (for case)
log ∑
failures
Case-control set θ .
35

Counter-matching.
To do the matched study, the confounder must be known for every
one.
Suppose instead that exposure is known for every one but the
confounder may be costly to obtain. Then:
• Matching on exposure is possible, but disastrous!
• Information on exposure may be used when selecting controls
E.g. in a given risk set: N 1 = 10 exposed, N 0 = 100 non-exposed.
Simple random sampling then leads to uneven (and inefficient)
exposure distribution in sampled case-control set. Instead, let the
case-control set consist of m = 5 + 1 = n 0 + n 1 = 3 + 3
non-exposed/exposed individuals, i.e. if case is exposed then sample
2 exposed + 3 non-exposed controls and if case is non-exposed then
sample 3 exposed + 2 non-exposed controls.
36

The confounder is ascertained for the sampled case-control set.
In the log-likelihood: Members of the case-control sets must be
weighted differently:
(
)
∑ θ (for case)
log ∑
failures
Case-control set w · θ .
Here: w = N 1
n 1
= 10/3 for exposed
w = N 0
n 0
= 100/3 for non-exposed
“Counter-matching”: m − 1 = 1, case and control must have different
exposure status.
Counter-matching on surrogate exposure is also possible.
Analysis: computer program must be able to deal with different
weights: ‘‘OFFSET’’ in SAS PROC PHREG.
37

Example of matched, nested c-c study.
Josefson, Magnusson, Ylitalo, Sørensen, Qwarforth-Tubbin,
Andersen, Melbye, Adami, Gyllensten Lancet, 2000, 355, 2189-93.
• 146889 women screened between 1969 and 1995 in Uppsala
county cervix cancer screening program: (732887 smears taken)
• 478 cases of cervix cancer in situ (CIS) identified through the
Swedish cancer register
• 5 (potential) controls selected per case from the calendar time
risk set, matched on time of entry into cohort (= time of first
smear) and on age. NO matching on number of smears.
• 1 of the 5 controls randomly selected for inclusion. If the selected
control had only one smear then a second control was selected.
(→ 608 controls.)
38

• Exposure, HPV-16 viral load, ascertained from the 2081/1754
available smears.
Why do a nested case-control study
• To avoid making cytological analyses of many smears.
Why match
• on age Standard, age is a confounder.
• on time of first smear To make ”exposure quality” similar for
cases and controls.
39

Results.
Josefsson et al., Lancet, 2000, 355, 2189-93.
Viral load Cases/controls exp(β)
HPV 16 negative 354/578 1
Below 20 percentile 16/15 1.9 (0.8-4.2)
20-40 percentile 23/7 7.2 (2.7-19.1)
40-60 percentile 28/3 22.8 (5.5-95.0)
60-80 percentile 27/4 18.9 (5.5-64.9)
Above 80 percentile 30/1 59.0 (7.5-462.2)
Dose-response effect of viral load on rate of CIS (here: based on first
smear, only).
In this study (and in many other nested c-c studies): possible to
estimate absolute risk.
40

Case cohort study
t 0 t 1
S
❞
❞
❛
❞ ❞ ❞
❞ ❛ ❛
❞ ❞ ❞
♣
❛
41

Assemble:
Case cohort study.
• all cases
• a random sub-cohort (S) followed from t 0 to t 1 (or a more fancy,
e.g. stratified, random sample)
“Pseudo-likelihood”
(
)
∑ θ (for case)
log ∑
failures
Comparison group θ
The comparison group is the case plus what is left of S at the present
failure time.
NB! One must be able to obtain covariate information for these
persons.
42

Computations
• SAS PROC PHREG, but wrong SE’s (may be amended)
• STATA
Advantages
• several case series may share the same sub-cohort
• savings of computing time and covariate ascertainment
• a small portion of the entire cohort is followed systematically
43

Danish adoption registry:
An adoption study.
• All (14427) adoptions to unrelated granted in DK 1924-47
• name, date of birth for ADoptee, Biologic Mother, Biologic
Father, Adoptive Mother, Adoptive Father
• address of AF, AM at time of adoption
• dates of transfer and formal adoption
44

“Old” study.
1003 AD’s born 1924-26 followed until 1982:
Sørensen, Nielsen, Andersen, Teasdale NEJM (1988).
Status 1982 AD BF BM AF AM
Alive in DK 765 114 367 64 163
Emigrated 75 32 27 4 8
Disappeared 1 4 2 1 0
Not followed 0 146 26 39 7
Dead 119 664 538 852 782
Total 960 960 960 960 960
45

“Old” study.
Cox regression model with lifetime of AD as outcome and
information on lifetimes of parents coded as explanatory variables:
Estimated hazard ratios (95% c.i.) for “at least 1 parent dead (from
relevant cause) before age 70”.
Cause B/A HR c.i.
All B 1.85 1.17-2.92
All A 0.80 0.55-1.16
Natural B 1.49 0.92-2.39
Natural A 0.96 0.65-1.41
Infection B 5.00 1.73-14.4
Infection A 1.00 0.34-2.97
Vascular B 1.92 0.78-4.73
Vascular A 1.50 0.65-3.46
Cancer B 0.87 0.26-2.88
Cancer A 1.49 0.56-3.97
46

“New” study.
All AD’s (12301) followed until 1993, also siblings and half-siblings
(both biologic and adoptive).
It is VERY time consuming to find all those individuals in
non-computerised records prior to 1968.
Therefore, case cohort study:
• all 1403 dead AD’s traced (including entire “family”)
• random sub-cohort of 1683 chosen and traced (1480 new)
• similar analyses performed on the case cohort sample
47

“New” case cohort study.
Cox regression model with lifetime of AD as outcome and information
on lifetimes of parents coded as explanatory variables: Estimated
hazard ratios (95% c.i.) for “at least 1 parent dead (from relevant
cause) before age 70”. (Petersen, Andersen, Sørensen, Gen. Epi., 2005).
Cause B/A HR c.i.
All B 1.27 1.08-1.50
All A 0.92 0.80-1.07
Natural B 1.24 1.01-1.52
Natural A 0.88 0.74-1.05
Infection B 1.35 0.80-2.27
Infection A 0.97 0.62-1.51
Vascular B 1.51 1.05-2.17
Vascular A 0.84 0.57-1.23
Cancer B 1.03 0.72-1.49
Cancer A 1.07 0.77-1.48
48