Covariate information in complex event history data some thoughts arising from a case study

1
Covariate information in complex event history
data -some thoughts arising from a case study

• Elja Arjas
• Department of Mathematics and Statistics,
  University of Helsinki
• and
• National Public Health Institute (KTL)
• Based on ongoing joint work with Olli Saarela and
  Sangita Kulathinal

2
Background and motivation

• Assessment of risk factors of cardiovascular
  diseases (e.g. coronary heart disease, stroke)
• Traditional approach for cohort analysis hazard
  regression model, with covariates (e.g. blood
  pressure, cholesterol level, or body mass index)
  measured only at the baseline
• Adding a genetic component usually candidate
  loci, potentially causative on the basis of the
  available information about their function.

3
Emphasis on causal ideas

• Stressing probabilistic predictions How would
  the probability of the outcome change if a
  covariate would have a different value?
• Association vs. causation the issue of
  confounding (change by intervention,
  do-conditioning, Pearl 2000).

4
Cosidering causal effects

• Compare, e.g., predictive probabilities of future
  response y
• p(ydata, attrib, hist, do(exposure))
• vs.
• p(ydata, attrib, hist, do(exposure))
• for a generic individual (or, for an
  equivalence class of exchangeable individuals)
  characterized by attributes and past history used
  in conditioning (cf. Arjas and Parner 2004).

5
Causal ideas

• Causal mechanisms can involve pathways that are
• direct in the sense that they influence, in the
  postulated model structure, directly the outcome
  variable, or
• indirect in that their effect on the outcome is
  mediated via the levels of the measured risk
  factors.

6
MORGAM study

• Evans et al. (2005)
• Individuals of different ages in a cohort are
  monitored for
• (fatal and non-fatal) occurrences of coronary
  heart disease (CHD) or stroke,
• death from other causes.
• Information on risk factors such as
• smoking status,
• blood pressure (BP),
• body mass index (BMI),
• total cholesterol and HDL cholesterol and
• possible earlier occurrences (yes/no) of CHD or
  stroke
• is collected at cohort baseline.

7
Genetic information

• SNP (single nucleotide polymorphism) level
  genotype data from candidate loci, e.g.
• functionally connected e.g. to blood clotting,
• associated with cardiovascular diseases,
• associated with increased lipid levels.
• Due to the cost involved genotyping is only done
  on
• all known cases of CHD or stroke, and
• individuals belonging to a random subset of the
  original cohort.

8
Information missing

• There is
• no genetic information of any kind available on
  most members of the original cohort, and even for
  those belonging to the case-cohort set, only on
  the chosen candidate loci
• no knowledge of early fatal occurrences of CHD or
  stroke from outside the cohort.

9
Graphical representation
event endpoint
parameters of interest
time (age)
underlying covariate process
candidate gene
measure- ment error variance
covariate measurement
10
Aspects to be considered...

• Time
• BMI, BP and cholesterol level do not remain
  constant over time individually varying
  stochastic processes.
• Even an accurate measurement at a particular time
  cannot be directly related to the endpoints as a
  "cause.
• The interpretation, and value for a causal
  analysis, of covariate measurements made in the
  past will generally depend on how long ago they
  were measured.

11
Further aspects

• Feed back to covariate values from earlier
  events
• Covariate values of individuals who had
  experienced a CHD event or stroke already before
  being recruited to the cohort may have been
  influenced by this event (e.g., the person quits
  smoking, changes diet, or gets medication to
  lower blood pressure).
• Influence of an earlier treatment
• After a first occurrence of non-fatal CHD or
  stroke, the risk for later similar events or
  death is likely to be more strongly influenced by
  the availability and success of the acute medical
  treatment than by the values of the measured risk
  factors/covariates.

12
Further aspects

• Potential confounding issue
• The considered candidate loci can influence both
  the values of the measured covariates and those
  of the outcome variables. If this is not properly
  accounted for in the modelling and analysis of
  data, they become a potential source of
  confounding in an observational study.
• Here also How about the rest of the
  genome, outside the selected candidate loci?

13
Further aspects

• Large dimension of parameter space
• The degree of SNP-based polymorphisms present in
  the data generally exceeds by far numbers for
  which it would be possible, given the amounts of
  data, to reliably estimate risks associated with
  individual genotypes.
• Particularly problematic in this sense
  is the
• MHC/HLA region.

14
Some shortcuts

• Problem 2
• Ignore the current status covariate information
  that may have been influenced by the earlier
  occurrence, only keeping information on
  covariates that do not change in time (age, sex,
  genotype).
• Problem 3
• Consider follow-up data only up to the first
  occurrence of CHD or stroke.
• Problems 1, 4 and 5
• Try something more systematic For problem 5,
  apply a monotonicity postulate and consequent
  partial ordering of risks. For problems 1 and 4,
  treat the missing covariate information in a
  distributional form (using data augmentation and
  MCMC).

15
Problem 5 dimension

• Partial ordering
• The two variants (alleles) of a biallelic SNP are
  labeled as 0 and 1, with 0 for the "common and 1
  for the "rare form
• Within each gene (more generally, linkage group),
  arrange the sequence of SNP genotypes (pairs of
  the form 00, 01, 10 and 11), each determined from
  the same SNP locus, into haplotypes. (Alleles
  belonging to the same - maternal or paternal -
  chromosome form a haplotype.)

16
Problem 5 dimension (2)

• Denote (-,ø,) to indicate less risky,
  neutral and more risky allele, respectively.
• For each pair of alleles, there are three
  possibilities
• allele 0 is less risky than allele 1 (-),
• no effect (øø) and
• allele 1 is less risky than 0 (-).
• Postulate this ordering of alleles is extendible
  to a partial ordering of haplotype risks. For
  example, haplotype h1 is more risky than
  haplotype h2 if all its alleles are either more
  risky or neutral compared to the corresponding
  alleles in h2, and at least one is more risky.
• Haplotypes can then be classified into groups,
  each being represented by a vector with elements
  chosen from -,ø,. Modelling of risks is then
  done via such classes.
• Extend this partial ordering into a partial
  ordering between to haplotype pairs (diplotypes).

17
Problem 5 dimension (3)
18
Problem 5 dimension (4)
19
Problem 5 dimension (5)
event endpoint
genotype
diplotype
restrictions for parameters from the allele
ordering
population haplotype frequencies
ordering of alleles of causal loci
number of causal loci
location of causal loci
20
Problem 1 time

• Regression dilution
• Measuring time dependent and individually
  varying covariates (such as BP, cholesterol level
  and BMI) at a single time point generally leads
  to an under-estimation of the effect size.
• But what should one do if for each individual
  there is only a single covariate measurement in
  the data?

21
Problem 1 time (2)

• Modelling the underlying covariate process
• For dealing with time dependent covariates in an
  explicit form, one needs a generator (stochastic
  intensities) for the covariate process considered
  as a function of pre-t histories, as well as
  corresponding stochastic intensities for the end
  point (TX) itself.
• One possibility is to apply the Marked Point
  Process (MPP) framework. The considered end
  point, with a corresponding description of the
  outcome, can then be imbedded into this process
  in a natural way as a marked point (TX).

22
Problem 1 time (3)

• Measurement error
• If also the covariate measurements involve a
  random error, we need a measurement model. The
  model parameters can be estimated if there are
  additional data available on the progression of
  the covariates.
• Numerical implementation
• Using MCMC and data augmentation methods but
  practical implementation can be difficult.
• Dependence of the covariates on genotype
  information?
• Fortunately, only long time averages of
  covariates are likely to be of importance for the
  considered endpoints. But potential confounding
  problem remains.

23
Problem 4 missing data, confounding

• Genetic factors are potential confounders in
  causal questions. If the relevant genotype
  information is known and its role has been
  properly accounted for in the statistical model,
  this problem can be dealt with by proper
  conditioning on such information.
• But what to do when a majority of the cohort
  members, as in MORGAM, have not been genotyped?
• Usual solution restrict the analysis only to
  those individuals who have been genotyped. But
  then the relevant follow-up and covariate
  information that exists on the other cohort
  members will not be used in the analysis at all.

24
Problem 4 missing data, confounding

• Treat also problem 4 as a missing data problem,
  considering a probability model for the missing
  genotypes and applying "full likelihood and
  Bayesian inference (Kulathinal and Arjas 2006,
  cf. Scheike and Martinussen 2004). This solution
  involves considering the unknown genotypes in a
  distributional form.
• Note, however, that a person's genotype, the
  measured risk factors and phenotype (time to
  event and event type) may all be statistically
  dependent of each other. Therefore the likelihood
  contribution from an individual who has not been
  genotyped involves an integration with respect to
  a (conditional) genotype distribution (which is
  generally different for different individuals).

25
Problem 4 missing data, confounding

• In general, and depending on the information
  available, one can consider different levels of
  conditioning in the predictive probabilities
• p(ydata, attrib, hist,
  do(exposure)).
• Depending on such a level, the interpretation of
  the results from causal analysis will differ,
  with more detailed conditioning taking us closer
  to individual causal effect - which, however,
  can never be achieved by a statistical analysis
  of data.

26
Problem 4 missing data, confounding

• More detailed conditioning is also attractive as
  a recipe against potential confounders (no
  unmeasured confounders postulate).
• Playing with finer level conditioning by using
  latent variable modelling can be attractive, but
  also risky if there is very little data, noisy
  data, or no data at all to support such modelling
  efforts.
• In essence, such finer level predictive
  probabilities are calibrated against data that
  are actually observed.

27
Take home-messages

• Careful consideration of sources of information
  is important.
• Interpretation of results is often facilitated by
  establishing intuitive links to causal what if
  ideas (do-conditioning).
• Less emphasis on inference (particularly
  statistical significance testing) concerning
  individual regression coefficients.

28
Take home-messages (2)

• General modelling approach based on MPPs is
  useful, offering possibilities to consider
  conditioning of probabilities on different levels
  of information.
• Bayesian approach, and applying MCMC for
  numerical computations, provides a flexible
  framework for statistical inference, keeping it
  within the domain of probability.