The use of graphical models in multi-dimensional longitudinal data

1. 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

2. 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.htm#diabetes ** 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

3. 99kg 3-monthly consultations • We must control your weight! • Next time we meet you’ll have: • …kept current weight. • …lost x kg. • ...ah, forget about it. • Let’s set our next appointment in about 3 months… Lose 2 kg

4. 3-monthly consultations next consultation How do we decide? 97kg • Very fine, you’ve lost 2 kg! • Next time we meet you’ll have: • …kept current weight. • …lost x kg. • ...ah, forget about it. • Let’s set our next appointment in about 3 months… ?

5. The effect of weight control 55kg This strategy has to be evaluated to the degree in which certain long-term goals have been fulfilled Not the effect of a single goal, but the effect of a sequence of goals, a goal setting strategy, has to be evaluated

6. Markov dynamics Wt-2 Wt Wt-1 or: Wt = f(Wt-1,Wt-2,Gt-1,Gt-2) Gt-2 Gt-1 In principle a mere simulation engine, but for inference purposes a (graphical) model of some sort is assumed.

7. 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.

8. Causality continued Wt-2 Wt Wt-1 ? A variable should be included in the model if: Gt-2 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

9. 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.

10. 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.

11. Assessing a strategy continued • A long-term yield is derived from the simulated weight series. examples include: • normality: BMI<25 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.

12. 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 Set as current strategy Evaluate neighbouring strategies Choose best of these 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.

13. 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.

14. 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

15. Men with 30<BMI<35 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 (BMI<25) after 5 years (20 sessions)

16. Men with 30<BMI<35 at first two post-diagnosis sessions, without heart condition, good HbA1c levels and kidney functioning. Weight control continued The effect of brute force: B&C null 0.0098 full 0.1504 min 0.0000 max 0.1999 300 iterations of a simulated annealing instance. Starting from generic strategy A&B

17. Markov dynamics baseline covariates C Wt-2 Wt Wt-1 Gt-2 Gt-1

18. Markov dynamics time Wt-2 Wt Wt-1 dt Gt-2 Gt-1

19. Markov dynamics time framework t Wt-2 Wt Wt-1 dt Gt-2 Gt-1

20. Markov dynamics multivariate outcome Ht-2 Ht Ht-1 Wt-2 Wt Wt-1 Gt-2 Gt-1

21. Markov dynamics a chain graph model Disease markers Baseline covariates Disease markers time t-1 Disease markers t dt Disease markers t t-1 t-2 t-k Treatment indicators Treatment indicators Treatment indicators t-2 t-1 t-k

22. 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 you’ll 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

23. 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.