The use of graphical models in multidimensional longitudinal data

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The use of graphical models in multi-dimensional
longitudinal data

• Volkert Siersma
• Department of Biostatistics
• University of Copenhagen
• IBS Nordic Regional Conference
• Oslo, June 2-4, 2005

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Weight control in type 2 diabetes (T2DM) patients
Diabetes Care in General Practice T2DM is an
increasingly common illness that is linked to
considerable excessive mortality. There are many
indications that treatment () can postpone the
development of diabetic complications. Treatment
of T2DM is primarily done in general practice,
where the results are not satisfactory. RCT
Structured vs. Routine care. 1428 newly
diagnosed T2DM patients included among 600 Danish
GPs. The structured care group is regularly
every third month reviewed for a period of
about 6 years. This observed cohort inspires the
following discussion.
http//www.gpract.ku.dk/Ansatte/olivarius.htmdi
abetes Olivarius, N.d.F., Beck-Nielsen, H.,
Andreassen, A.H., Horder, M. and Pedersen, P.A.
(2001) Randomised controlled trial of structured
personal care of type 2 diabetes mellitus. Ann.
Intern. Med., 323(7319) 970-975
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3-monthly consultations

• We must control your weight!
• Next time we meet youll have
• kept current weight.
• lost x kg.
• ...ah, forget about it.
• Lets set our next appointment in about 3 months

Lose 2 kg
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3-monthly consultations next consultation
How do we decide?
97kg

• Very fine, youve lost 2 kg!
• Next time we meet youll have
• kept current weight.
• lost x kg.
• ...ah, forget about it.
• Lets set our next appointment in about 3 months

?
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The effect of weight control
55kg
Not the effect of a single goal, but the effect
of a sequence of goals, a goal setting strategy,
has to be evaluated
This strategy has to be evaluated to the degree
in which certain long-term goals have been
fulfilled
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Markov dynamics
Wt-2
Wt-1
Wt
Gt-2
or Wt f(Wt-1,Wt-2,Gt-1,Gt-2)
Gt-1
In principle a mere simulation engine, but for
inference purposes a (graphical) model of some
sort is assumed.
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Causality
The model, estimated from the data, can be used
as a simulation engine to simulate weight
development relative to a sequence of goals when
the relationships are causal. Specifically, when
there are no unmeasured confounders to the direct
relationships with Wt. Then the do conditional
probability, the one used when dictating the
goal-setting in a simulation programme, is the
same as the see or observed conditional
probability, the one we estimate from the data.
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Causality continued
Wt-2
Wt-1
Wt
?
Gt-2
A variable should be included in the model if
Gt-1

• It confounds a relation between Wt and another
  variable

• It has a relation with Wt and is might be used
  in a strategy

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Causality continued
Some causality is induced by temporal
relations. Causality of the model can be
constructed if the mechanism is well-known. In
behavioural studies, causality has to be
introduced by adding the potential confounders to
the model. This often leads to large models and
may render the model unstable.
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Assessing a strategy
Within the model there is no information to
determine the goal for the next session beyond
the present weight and the weight and goal at the
previous session. Thus, a strategy is a
(deterministic) function to determine a goal for
the next session from the present weight and the
weight and goal at the previous session. The
model for the dynamics is used to simulate the
next weight, given the goal and the previous
weight. Given start values (W0 and W1, no goal is
set at the first session) a series of weights and
weight goals can be simulated.
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Assessing a strategy continued

• A long-term yield is derived from the simulated
  weight series. examples include
• normality BMIlt25 after 10 sessions
• stability sum of weight differences.
• The process of simulating a weight development
  and calculating the yield is done several times
  to get an empirical estimate of the distribution
  of the yield.
• This distribution can be contrasted to a
  similarly derived distribution of the yield of a
  null strategy, i.e. no goal set or indeed any
  other interesting strategy.

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Optimising a strategy
A strategy can be viewed as a function of weight
and previous goal with several parameters.
Optimising yield w.r.t. the strategy parameters
is a difficult, often high-dimensional,
optimisation problem.
Heuristic search methods
Start with a sensible strategy
Evaluate neighbouring strategies
Choose best of these
Set as current strategy
Repeat until convergence
A collection of generic strategies should be
constructed for fast evaluation of intuitive
strategies, start values for the optimisation,
and base camps for other strategies.
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Optimal strategy scope
The simulated weight series and thus a strategy
is evaluated conditional on the start values of
the process. An optimal strategy is therefore
also only optimal for patients with these start
values.
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Strategy analysis

• Operationalised optimisation could take the form
  of a black box on-line data mining exercise.
• Strategy analysis on a more general level is
  wanted in many cases.
• An overview of the yields of various generic
  strategies
• An overview of the strategy effect of some sort
  for the most usual combinations of start values.
• A description or visualisation of some sort of
  the optimal strategy

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Men with 30ltBMIlt35 at first two post-diagnosis
sessions, without heart condition, good HbA1c
levels and kidney functioning.
Weight control
Estimated (10.000 simulations) probability of
normal body weight (BMIlt25) after 5 years (20
sessions)
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Men with 30ltBMIlt35 at first two post-diagnosis
sessions, without heart condition, good HbA1c
levels and kidney functioning.
Weight control continued
The effect of brute force BC
null 0.0098
full 0.1504
min 0.0000
max 0.1999
300 iterations of a simulated annealing instance.
Starting from generic strategy AB
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Markov dynamics baseline covariates
C
Wt-2
Wt-1
Wt
Gt-2
Gt-1
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Markov dynamics time
Wt-2
Wt
Wt-1
dt
Gt-2
Gt-1
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Markov dynamics time framework
t
Wt-2
Wt
Wt-1
dt
Gt-2
Gt-1
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Markov dynamics multivariate outcome
Ht-2
Ht-1
Ht
Wt-2
Wt-1
Wt
Gt-2
Gt-1
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Markov dynamics a chain graph model
Disease markers
Disease markers time t-1
Baseline covariates
Disease markers
Disease markers
t dt
t
t-1
t-2
t-k
Treatment indicators
Treatment indicators
Treatment indicators
t-2
t-1
t-k
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Chain graph model tools

• Much of the analysis is the investigation of
  large chain graph models. Several types of
  inference are needed.
• Recall our goals this is not an ordinal
  variable. Methods are needed to relate partly
  ordinal variables.
• Finding interactions and including them in the
  model.
• Relate sets of variables to other sets of
  variables.

Next time we meet youll have a) kept current
weight. b) lost x kg. c) ...ah, forget about it.
Sets of ordinal variables can be identified with
partly ordinal variables pseudo gamma
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Using graphical models

• The graphical model serves as a simulation
  engine.
• Inference on the graphical model is used to
  check the causality of the relations that are
  used to simulate the sequences of disease markers
  and treatment.
• Inference on the graphical model can reveal
  factors to be included in or excluded from a
  strategy.
• Examination of interactions can reveal
  influences of passing time and unrealistic goal
  setting.