patterns of delirium: latent classes and hiddenmarkov chains as modeling tools antonio ciampi, alina...
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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
Outline
Introduction Basic Concepts Model and Estimation Results Conclusion
State and course of a disease
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?
Introduction
Introduction
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.
Example: Delirium
Delirium Index subscales
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
IntroductionIntroduction
Introduction
Note
In this presentation we work with the multivariate DI onlyThe 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
Clinical states
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
IntroductionIntroduction
State 2 Low level of disorientation and medium level memory problems. No other symptoms.
4 states of DeliriumIntroductionIntroduction
0
1
2
3
DI_1: focusingattention
DI_2: thinkingdisorganizedDI_3: alteredlevel of
consciousness
DI_4:
disorientationDI_5: memoryproblem
DI_6:perceptualdisturbances
DI_7.1:hyperactivity
DI_7.2:hypoactivity
0
1
2
3
DI_1: focusingattention
DI_2: thinkingdisorganizedDI_3: alteredlevel of
consciousness
DI_4:
disorientationDI_5: memoryproblem
DI_6:perceptualdisturbances
DI_7.1:hyperactivity
DI_7.2:hypoactivity
0
1
2
3
DI_1: focusingattention
DI_2: thinkingdisorganizedDI_3: alteredlevel of
consciousness
DI_4:
disorientationDI_5: memoryproblem
DI_6:perceptualdisturbances
DI_7.1:hyperactivity
DI_7.2:hypoactivity
0
1
2
3
DI_1: focusingattention
DI_2: thinkingdisorganizedDI_3: alteredlevel of
consciousness
DI_4:
disorientationDI_5: memoryproblem
DI_6:perceptualdisturbances
DI_7.1:hyperactivity
DI_7.2:hypoactivity
State 1
State 1 Low level of memory problems. No other symptoms.
State 2
State 4State 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
Clinical course of delirium and Transitions observed in our data
IntroductionIntroduction
6 months later2 months later
state 1
state 2
state 3
state 4
state 1
state 2
state 3
state 4
state 1
state 2
state 3
state 4
100%
87%
46%
38%
42%
21%35%
100%
95%
79%
100%
20%
45%
24%
11%
39%
37%
16%
8%
at admission
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
The DI is routinely assessed at several points in time, in order to follow the clinical course of a patient
Defining clinical course: the statistical approach
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
IntroductionIntroduction
Latent Class and Manifest variables
DI 1 DI 2 DI 7.1 DI 7.2 Manifest variables
Delirium Index
Latent classes
Delirium states
Basic Concepts
state 1state 2state 3state 4
……
Latent variable
If we knew the latent class, the description of the manifest variables is particularly simpleIn the most classical definition of latent class, given the latent class, the manifest variables are assumed to be independentWe only need the univariate probability distributions to entirely describe the data, a major simplification!
Example
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 0.28*0.30 = 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.
Basic Concepts
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.
6 months later2 months laterat admission
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
6 months later2 months laterat admission
………
…
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
Statistical model 1: simplified HMC model
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
Model and Estimation
6 months later2 months laterat admission
………
…
Statistical model 2: Model that takes into account death and missingness
6 months later2 months laterat admission
…DI1 DI8 DI1
T1
DI8 DI1
T2
DI8
D1 D2
Mis1 Mis2
… …
Model and Estimation
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)
T0
Statistical model 3: Latent trajectory model
6 months later2 months laterat admission
………
…
Model and Estimation
- 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
Likelihood maximization
Likelihood maximization is based on the EM algorithm. The log-likelihood is ‘completed’ by assigning values to the hidden variables
Model and Estimation
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From Bayes Theorem:
Latent classes from Manifest variables with Death and Missingness information
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 model’s 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
Results
Dynamics through Hidden Markov Chain
Results
6 months later2 months later
state 1
state 2
state 3
state 4
state 1
state 2
state 3
state 4
state 1
state 2
state 3
state 4
100%
87%
46%
38%
42%
21%35%
100%
95%
79%
100%
20%
45%
24%
11%
39%
37%
16%
8%
at admission
DI distribution conditional on 4 Latent Classes
DI_4: disorientation
DI_5: memory problem DI_6:perceptual disturbances DI_7: hyperactivity DI_8: hypoactivity
DI_1: focusing attention DI_2: thinking disorganized DI_3: level of consciousness
0.68
0.42
0.07
0.32
0.57
0.27
0.01
0.65
0.07
0.01
0.93
0.00
0.20
0.40
0.60
0.80
1.00
Class_1 Class_2 Class_3 Class_4
0.980.81
0.23
0.08
0.31
0.05
0.11
0.42
0.07
0.04
0.88
0.00
0.20
0.40
0.60
0.80
1.00
Class_1 Class_2 Class_3 Class_4
0.73
0.13
0.17
0.30
0.09
0.09
0.38
0.27
0.08
0.18
0.63
0.91
0.00
0.20
0.40
0.60
0.80
1.00
Class_1 Class_2 Class_3 Class_4
0.29
0.04
0.28
0.13
0.24
0.24
0.120.05
0.19
0.59
0.850.95
0.00
0.20
0.40
0.60
0.80
1.00
Class_1 Class_2 Class_3 Class_4
0.910.80
0.61
0.31
0.090.18
0.33
0.39
0.06
0.15
0.15
0.00
0.20
0.40
0.60
0.80
1.00
Class_1 Class_2 Class_3 Class_4
0.97 0.930.80
0.060.15
0.21
0.01 0.05
0.23
0.15
0.00
0.20
0.40
0.60
0.80
1.00
Class_1 Class_2 Class_3 Class_4
0.88 0.88
0.01 0.01
0.11 0.11 0.11 0.11
0.80
1.00
Class_1 Class_2 Class_3 Class_4
0.96 0.93 0.870.76
0.04 0.07 0.100.14
0.08
0.00
0.20
0.40
0.60
0.80
1.00
Class_1 Class_2 Class_3 Class_4
No symptoms Low symptom Medium symptom High symptom
1
2
3
4
DI_1 DI_2 DI_3 DI_4 DI_5 DI_6 DI_7 DI_8
Results
List of most probable courses with the a priori probability
Focusing attention= Medium; Disorganized thinking=Medium
Disorientation=High; Memory problems=High
Disorientation=Low
Memory problems=Medium
Memory problems=Low
6 months
later
2 months
later
state 2
at admission
state 1
state 3
state 4
Course 1(22%): stable good
Course 4 (23%): stable fair
Course 6 (6%): late improvement poor to fair
Course 5 (4%) early improvement poor to fair
Course 7 (12%) : stable poor
Course 8 (4%): stable very poor
Course 3 (6%): late improvement fair to good
Course 2 (4%) early improvement fair to good
Results
Graphical representation of posterior probabilities of Latent Class
Results
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sont requis pour visionner cette image.
Example 1: Conditional Probability of Clinical Course given Clinical State at admission
Course 1: stable good 0.97
Patient is in State 1 at admission:
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
Example 2: Predicting clinical course from manifest variables
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
Results
Example 3: Predicting clinical states from manifest variables
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:
state 1
state 2
state 3
state 4
6 months
later
2 months
later
at admission
0.00
0.00
0.72
0.28
0.10
0.40
0.39
0.11
0.10
0.42
0.32
0.15
Results
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.
Conclusion
Future research
Inclusion of patient’s 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
Conclusion
Questions ???