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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
         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

           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
     admission as in previous example
    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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