Patterns of Delirium: Latent Classes and HiddenMarkov Chains as Modeling Tools

1
Patterns of Delirium Latent Classes and
HiddenMarkov Chains as Modeling Tools

• Antonio CIAMPI, Alina DYACHENKO,
• Martin COLE, Jane McCUSKER
• McGill University

BIRS, 11-16 December 2011
2
Outline

• Introduction
• Basic Concepts
• Model and Estimation
• Results
• Conclusion

3
State and course of a disease
Introduction

• A patient with a particular illness presents a
  number of symptoms and signs. The underlying
  clinical concept is that of disease state
• As the illness evolves in time, the presentation
  may change. The underlying clinical concept is
  that of disease course
• These concepts may be operationalized by
  measuring clinical indices. An example would be a
  one-dimensional severity index, usually measured
  on a continuous scale
• More generally, one could use a multivariate
  index, reflecting a potential multidimensionality
  of the disease
• In either case, a patient may be represented by a
  vector describing a curve in time y(t)
• Can statistical learning method help discover
  patterns in this type of data?

4
Introduction
Example Delirium

• Delirium is a disorder prevalent in hospitalized
  elderly populations characterized by acute,
  fluctuating and potentially reversible
  disturbances in consciousness, orientation,
  memory, thought, perception and behavior.
• The Delirium Index (DI) is a clinical instrument
  which is used
• to measure the severity of delirium
• to classify patients with delirium into clinical
  states
• It consists of eight 4-level ordinal subscales
  assessing symptoms and sign of Delirium.

5
Delirium Index subscales
Introduction

• DI_1 Focusing attention
• DI_2 Disorganized thinking
• DI_3 Altered level of consciousness
• DI_4 Disorientation
• DI_5 Memory problem
• DI_6 Perceptual disturbances
• DI_7.1 Hyperactivity
• DI_7.2 Hypoactivity

6
Introduction
Note

• In this presentation we work with the
  multivariate DI only
• The univariate DI, defined as a sum of the
  subscales, represents the state of a patient as a
  continuous value. It is best modelled as a
  mixture of mixed regression models (for
  longitudinal data)
• Though less informative, this approach is more
  flexible, as it allows for continuous time, hence
  measuring times varying from patient to patient

7
Clinical states
Introduction

• Anticipating our results, we show here a graph
  representing 4 clinical states
• These were empirically defined from a data
  analysis of 413 elderly patients at risk of
  developing delirium, some with some without
  delirium at admission
• 225 of 413 patients (46) have missing values
• The analysis does not use the diagnosis, but only
  the subscales of DI
• Delirium Index was measured at diagnosis, and at
  2 and 6 months from diagnosis

8
4 states of Delirium
Introduction
State 1
State 2
State 2 Low level of disorientation and
medium level memory problems. No
other symptoms.
State 1 Low level of memory problems.
No other symptoms.
State 4
State 3
State 3 Medium levels of focusing attention,
disorganized thinking and high level of
disorientation and memory problems
State 4 High level of focusing attention and
disorganized thinking and medium level of
altered levels of consciousness and low level of
hypoactivity
9
Clinical course of delirium and Transitions
observed in our data
Introduction

• The DI is routinely assessed at several points in
  time, in order to follow the clinical course of a
  patient

6 months later
2 months later
at admission
100
20
39
100
state 1
state 1
state 1
37
95
state 2
state 2
state 2
87
45
42
16
79
24
state 3
state 3
state 3
46
35
21
8
100
11
state 4
state 4
state 4
38

• By clinical course we mean the sequence of
  transitions from one state to an other over time.
  Each patient has his or her own clinical course
  however, we speak of typical clinical courses,
  meaning typical or common sequences of transitions

10
Defining clinical course the statistical
approach
Introduction

• Defining the clinical course of a disease is a
  very general problem in medicine and
  Epidemiology. Usually clinicians solve it on the
  basis of their experience
• HOWEVER, appropriate statistical methods exist to
  help define clinical course directly from data
• These statistical methods are latent class
  analysis especially in the more modern versions
  which include hidden Markov chains and other
  dynamical models
• The rest of this presentation is devoted to
  explaining these notions in as an intuitive
  manner as possible

11
Latent Class and Manifest variables
Basic Concepts
DI 1
DI 2
DI 7.1
DI 7.2
Manifest variables Delirium Index


Latent classes Delirium states
Latent variable
state 1

• If we knew the latent class, the description of
  the manifest variables is particularly simple
• In the most classical definition of latent class,
  given the latent class, the manifest variables
  are assumed to be independent
• We only need the univariate probability
  distributions to entirely describe the data, a
  major simplification!

state 2
state 3
state 4
12
Example
Basic Concepts

• Consider a patient in clinical state (latent
  class) 1. Then we can calculate from the data
  that the probability of observing a low level of
  Disorientation is about 0.16
• Consider a patient in clinical state 2. Then the
  probability of observing a low level of
  Disorientation and a medium level of Memory
  problem are respectively 0.28 and 0.30. The
  probability of observing both is 0.280.30
  0.084
• Conversely, consider a patient with a high level
  of Disorientation and Memory problems but no
  other symptoms, then the probabilities that the
  patient is in states 1 to 4 are respectively
  0.003, 0.944, 0.053, 0.00
• Notice that these values are extracted from the
  data through latent class analysis.

13
Markov Chains
Basic Concepts

• A patient is examined at different points in
  time. At each point in time he is in one of a
  number of possible states. For instance one of
  the states of delirium described above.
• A Markov Chain (MC) is a description of the
  evolution of a patient over time. It consists of
  a series of states and of a set of transition
  probabilities from one time point to the next.
• In a MC, the probability of a transition in the
  time interval (t1, t2) is only influenced by the
  state of the patient at time t1.
• A MC is stationary if the transition
  probabilities do not depend on time.

14
Hidden Markov Chains
Basic Concepts

• In our case we do not have access to the state of
  the patient but only to the manifest variables
  from which we can extract the probability of the
  states. Thus our model will have to be of the
  form above. This is called a Hidden Markov Chain

• Our analytic tools allow us to extract from the
  level of the manifest variables information,
  concerning the hidden level, e.g.
• Probability to belong to a particular state at
  time t0
• Transition probabilities
• We can also test stationarity of the transition
  probabilities

15
Statistical model 1 simplified HMC model
Model and Estimation

• Properties
• Each manifest variable depends only on the
  corresponding latent variable
• Conditionally on the latent variables the
  manifest variables are independent (classical
  latent class definition)
• Conditionally on the latent variables the
  manifest variables are independent (classical
  latent class definition)
• Transition structure for the latent variables has
  the form of a first-order Markov chain

16
Statistical model 2 Model that takes into
account death and missingness
Model and Estimation
at admission
2 months later
6 months later



DI1
DI8

• DI1

• DI8

• DI1

• DI8

T1
T2
T0

• D1

• D2

• Assumptions
• Stationarity of transition probabilities
• Homogeneity of the relationship
• between manifest and latent variables across
  times
• Linearity in the latent variables
• Additional assumptions of independence or
  dependence between latent variables and other
  indicator variables (ex., Death and Missingness)

• Mis1

• Mis2

17
Statistical model 3 Latent trajectory model
Model and Estimation
at admission
2 months later
6 months later

• Graph has two layers of latent classes
• Lower level consists of one latent variable its
  laten classes can be directly interpretable as
  distinct courses of the disorder

18
Likelihood maximization
Model and Estimation

• Likelihood maximization is based on the EM
  algorithm. The log-likelihood is completed by
  assigning values to the hidden variables

From Bayes Theorem
19
Latent classes from Manifest variables with Death
and Missingness information
Results

• Model selection strategy
• determine the number of latent classes using
  statistical criteria like AIC and BIC (in our
  case we have 4 latent classes)
• test the models assumption on missingness and
  death indicator mutually independence and
  independence of all other variable in the model
• test the model assumption of stationarity,
  homogeneity and linearity
• examine more complex models

20
Dynamics through Hidden Markov Chain
Results
6 months later
2 months later
at admission
state 1
state 1
state 1
100
100
20
39
state 2
state 2
state 2
87
95
45
37
42
state 3
state 3
state 3
46
79
24
16
35
21
state 4
state 4
state 4
100
11
8
38
21
DI distribution conditional on 4 Latent Classes
Results
22
List of most probable courses with the a priori
probability
Results
23
Graphical representation of posterior
probabilities of Latent Class
Results
24
Example 1 Conditional Probability of Clinical
Course given Clinical State at admission

• Patient is in State 1 at admission

Course 1 stable good
0.97

• Patient is in State 4 at admission

Course 4 early improvement
0.30 Course 6 early very poor to poor
0.15 Course 7 late very poor to
poor 0.08 Course 8 stable very
poor 0.29
25
Example 2 Predicting clinical course from
manifest variables
Results

• Example 2 a patient has the following manifest
  variables at admission
• Focusing attention Disorganized
  thinking Medium
• Disorientation Memory problem
  High
• Hypoactivity
  Low
• Probability of each of the most probable course.
• Course 3 early light
  improvement 0.26
•   Course 4 early improvement
  0.08
• Course 5 stable poor
  0.23
• Course 6 early very poor to
  poor 0.04
• Course 7 late very poor to poor
  0.02
• Course 8 stable very poor
  0.08

26
Example 3 Predicting clinical states from
manifest variables
Results

• Example 3 a patient has the same manifest
  variables at admission as in previous example
• Probability to be in state 1 or 2 or 3 or 4 at
  different time

0.00
0.10
0.10
state 1
state 2
0.00
0.40
0.42
0.72
0.39
0.32
state 3
0.28
0.11
0.15
state 4
6 months later
2 months later
at admission
27
Conclusion
Conclusion

• We have shown that latent class analysis is a
  useful tool to extract information from clinical
  data
• It provides means to obtain directly from data
  the key concepts of clinical state and clinical
  course of a disease
• It counts for realistic features of clinical
  studies eg Death and Missingness.
• We have shown how this applies in the case of
  Delirium
• See A. Ciampi, A. Dyachenko, M. Cole, J.
  McCusker (2011). Delirium superimposed on
  dementia Defining disease states and course from
  longitudinal measurements of a multivariate index
  using latent class analysis and hidden Markov
  chains. International Psychogeriatrics.

28
Future research
Conclusion

• Inclusion of patients characteristics
  (covariates)
• Improve tests of model fit
• Develop non-stationary models
• Develop mixtures of Hidden Markov chains
  (addition of another level of latent classes)
• Develop latent trait models

29
Questions ???