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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
Introduction
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
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.
Introduction

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
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
Introduction

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
0
1
2
3

0
1
2
3
DI_1: focusing                                                                                                              DI_1: focusing
attention                                                                                                                   attention

DI_2: thinking                                                                                                             DI_2: thinking
disorganized                                                                                                               disorganized
Introduction

DI_3: altered                                                                                                               DI_3: altered
level of                                                                                                                    level of
consciousness                                                                                                               consciousness
DI_4:                                                                                                                       DI_4:
disorientation                                                                                                               disorientation
DI_5: memory                                                                                                               DI_5: memory
problem                                                                                                                    problem

No other symptoms.
DI_6:perceptual                                                                                                                DI_6:perceptual
disturbances                                                                                                                   disturbances
DI_7.1:

disorganized thinking and high level
DI_7.1:
hyperactivity                                                                                                           hyperactivity

of disorientation and memory problems
State 1 Low level of memory problems.

DI_7.2:                                                                                                                DI_7.2:

State 3
State 1

hypoactivity                                                                                                           hypoactivity

State 3 Medium levels of focusing attention,
0
1
2
3

0
1
2
3

DI_1: focusing                                                                                                                DI_1: focusing
attention                                                                                                                     attention

DI_2: thinking                                                                                                                DI_2: thinking
disorganized                                                                                                                  disorganized
DI_3: altered                                                                                                                 DI_3: altered
level of                                                                                                                      level of
consciousness                                                                                                                 consciousness
DI_4:                                                                                                                         DI_4:
disorientation                                                                                                                disorientation
low level of hypoactivity
DI_5: memory                                                                                                                 DI_5: memory
No other symptoms.

problem                                                                                                                      problem
DI_6:perceptual                                                                                                                 DI_6:perceptual
disturbances
4 states of Delirium

disturbances
DI_7.1:                                                                                                                    DI_7.1:
hyperactivity                                                                                                              hyperactivity
State 2 Low level of disorientation and
medium level memory problems.

of altered levels of consciousness and

DI_7.2:
DI_7.2:
disorganized thinking and medium level

hypoactivity
State 2

State 4

hypoactivity
State 4 High level of focusing attention and
Introduction

Clinical course of delirium and
Transitions observed in our data

The   DI is routinely assessed at several points in time, in order to follow
the clinical course of a patient
at admission          2 months later         6 months later

20%                   100%                   100%                   39%
state 1                state 1                state 1

45%                   87%                    95%                    37%
state 2                state 2                state 2
42%
24%                                          79%                    16%
state 3      46%       state 3                state 3
35%
11%                     21%                  100%                   8%
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
Introduction

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
Basic Concepts
Latent Class and Manifest variables

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
state 2
state 3           variables is particularly simple
state 4           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!
Basic Concepts

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
   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

at admission               2 months later              6 months later

   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.
Basic Concepts

Hidden Markov Chains

at admission              2 months later           6 months later

…
…                         …                         …

    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
Model and Estimation
Statistical model 1:
simplified HMC model
at admission             2 months later           6 months later

…
…                          …                        …

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

Statistical model 2: Model that takes
into account death and missingness
at admission        2 months later          6 months later
DI1   …    DI8   DI1   …    DI8    DI1    …     DI8

T0                T1                    T2

Assumptions:                               D1               D2

- Stationarity of transition probabilities
- Homogeneity of the relationship          Mis1             Mis2

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)
Model and Estimation

Statistical model 3:
Latent trajectory model
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
Model and Estimation

Likelihood maximization

   Likelihood maximization is based on the EM algorithm.
The log-likelihood is ‘completed’ by assigning values to
the hidden variables
From Bayes Theorem:

P( S t  j | DI t( 1 )  i ( 1 ) ,..., DI t( 7.2 )  i ( 7.2 ) ) 

P( S t  j )P( DI t( 1 )  i ( 1 ) | S t  j )...P( DI t( 7.2 )  i ( 7.2 ) | S t  j )
   4

 P( S
j 1
t    j )P( DI t( 1 )  i ( 1 ) | S t  j )...P( DI t( 7.2 )  i ( 7.2 ) | S t  j )
Results
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

at admission             2 months later     6 months later

20%                  100%                   100%               39%
state 1                  state 1            state 1

45%                  87%                    95%                37%
state 2                  state 2            state 2
42%
24%                  46%                    79%                16%
state 3                  state 3            state 3
35%
21%
11%                                         100%                   8%
state 4     38%          state 4            state 4
Results

DI distribution conditional on 4
Latent Classes
DI_1: focusing attention              DI_2: thinking disorganized                       DI_3: level of consciousness                   DI_4: disorientation
1.00             0.01    0.01             1.00                        0.04               1.00              0.01      0.05             1.00
0.11                                           0.06              0.15             0.09    0.18
0.32                                               0.08                                                     0.15
0.80                                      0.80                                           0.80                                         0.80    0.17
0.42                                                   0.23
0.57    0.65                                                                                                                                 0.63
0.60                                      0.60                                           0.60                                         0.60            0.38
0.93                                           0.88                                                                                  0.91
0.98                                          0.97                      0.21
0.93
0.40                                      0.40              0.81      0.31               0.40                        0.80             0.40
0.68                                                                                                                                 0.73
0.30
0.20             0.42    0.27             0.20                                           0.20                                         0.20                    0.27
0.23      0.07
0.07    0.07                                           0.05                                                                  0.13    0.09    0.08
0.00                                      0.00                                           0.00                                         0.00
Class_1 Class_2 Class_3 Class_4          Class_1 Class_2 Class_3 Class_4                 Class_1 Class_2 Class_3 Class_4             Class_1 Class_2 Class_3 Class_4

DI_5: memory problem                DI_6:perceptual disturbances                            DI_7: hyperactivity                        DI_8: hypoactivity
1.00                                       1.00                                            1.00     0.04                               1.00                    0.06
0.07     0.10    0.08             0.09                    0.15
0.19                                                                                                                                          0.18
0.80                                                                                       0.80                               0.14     0.80                    0.33    0.15
0.24    0.59                                 0.11   0.11      0.11      0.11
0.60                      0.85                                                             0.60                                        0.60
0.95                                                                                                                                 0.39
0.28                                                                                       0.96     0.93                              0.91
0.40                                                  0.01   0.01                          0.40                       0.87             0.40            0.80
0.76
0.24                                                                                                                                          0.61
0.20                                                  0.88   0.88                          0.20                                        0.20
0.29                                                                                                                                                          0.31
0.13     0.12
0.00             0.04             0.05     0.80                                                                                        0.00
0.00
Class_1 Class_2 Class_3 Class_4           Class_1 Class_2 Class_3 Class_4                  Class_1 Class_2   Class_3 Class_4          Class_1 Class_2 Class_3 Class_4

No symptoms                                Low symptom                                 Medium symptom                               High symptom

4

3

2

1
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
Course 1(22%): stable good
state 1
Course 2 (4%) early improvement fair to good
Course 3 (6%): late improvement fair to good

state 2                                     Course 4 (23%): stable fair

Course 5 (4%) early improvement poor to fair
Memory problems=Low
Disorientation=Low
Memory problems=Medium Course 6 (6%): late improvement poor to fair

state 3        Focusing attention= Medium; Disorganized
thinking=Medium
Course 7
Disorientation=High; Memory problems=High(12%) : stable poor

state 4
Course 8 (4%): stable very poor
at admission 2 months 6 months
later    later
Results

Graphical representation of posterior
probabilities of Latent Class

QuickTime™ et un
décompresseur TIFF (non compressé)
sont requis pour visionner cette imag e.
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
Results

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
   Probability to be in state 1 or 2 or 3 or 4 at different time:

state 1           0.00          0.10      0.10
state 2           0.00          0.40      0.42
state 3           0.72          0.39      0.32
state 4           0.28          0.11      0.15

at admission   2 months   6 months
later      later
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.
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
Questions ???

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