Dynamic analysis of binary longitudinal data

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Dynamic analysis of binary longitudinal data
Ørnulf Borgan Department of Mathematics University
of Oslo Based on joint work with Rosemeire L.
Fiaccone, Robin Henderson and Mauricio L.
Barreto
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Outline
- An example of binary longitudinal data The
Blue Bay project
- Modelling missingness for longitudinal binary
data (including the relation to independent
censoring in event history analysis)
- An additive model for longitudinal binary data
- Dynamic covariates
- Martingale residual processes
- Concluding comments
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Blue Bay project Bahia State, Brazil (size of
France) State capital Salvador (pop 2.5
mill.)
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Public works and education in the areas of
sanitation and environment executed by the Bahia
State Government since 1997 Cost more than 1
billion
Belgica 2002
Belgica 1996
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Data Daily data on diarrhoea for almost a
thousand children (one per family) Collected at
home visits Oct 2000 to Jan 2002 Children less
than 3 years of age at entry Diarrhoea three or
more fluid motions a day Episode of diarrhoea
sequence of days with diarrhea until at least two
consecutive clear days
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The reduced prevalence/incidence over time may
reflect improved health over the study period, or
may be an artefact due to ageing of the cohort
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Social, demographic and economic characteristics
collected at entry to the study
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Follow-up information on 10 children
Under observation
New episode
X
Ongoing episode
X
Drop-out
O
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Pattern of missing observations for all 926
children
Non-available data collector
Police strike
Carnival
Christmas Day
St. John's day
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Three types of missingness - Late entries (16
of children) - Drop-outs (21 of children) -
Intermittent missingness (20 of observations)
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Features of the data Longitudinal binary
data Four time scales calendar, age, study,
episode Calendar time used as basic time scale
Aims Study factors of importance for incidence
and prevalence of diarrhoea and how diarrhoea
incidence and prevalence vary over calendar time
Ignored (for this talk) Spatial associations
Other non-independence
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Conditions on the missingness are defined for
this model
Modelling missingness
Joint model for binary data and missingness
Model for binary data without missingness
Model for observed data
Parameters of interest are defined for this model
Statistical methods are derived and studied for
this model
We need to relate the models for the three
situations (starting with models for one
individual)
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Model without missingness
Observations for child i is a binary time series
Here if the child starts a new
episode of diarrhea at day t (has diarrhoea at
day t)
Let be the s-algebra generated by the
fixed and external time-varying covariates for
child i

is the information that had been available on
child i by day t had there been no missingness
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Introduce the conditional probabilities
The aim for our analysis is to study how the vary
over time and how they depend on covariates,
including dynamic covariates that are functions
of for s lt t
This differs from the common approach in
longitudinal data analysis, where the focus is on
the marginal probabilities
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Joint model for binary longitudinal data and
missingness
Introduce the observation process for individual i
We need to consider the larger filtration
where is generated by and
external aspects of the observation process for
child i
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We make two assumption on the missingness

• These assumption correspond to
• sequential MAR in longitudinal data analysis
• independent censoring in event history analysis

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Modelling the observable data
Binary observations for individual i
Observed filtration
(Note that we for convenience have included
in the definition of )
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Then
We will assume that is
predictable, implying that the
time-dependent dynamic covariates used for
regression modelling depend only on observables

Thus
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Intoduce
The are martingale differences
is a discrete time martingale
Predictable variation process
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Modelling the relation between individuals
Denote by Ft the information available to the
researcher on all children by day t
We impose the following assumptions
(i)
(ii)
The assumptions are weaker than independence
Nevertheless they are debatable (i) in
particular for the diarrhoea data
Note that (ii) implies that the martingales
and are orthogonal
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An additive model for longitudinal binary data
Have the decomposition
Let xi1t ,, xipt be predictable covariates for
child i at day t
Consider the model
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Conditional on "the past" Ft-1 we at day t have
i.e. a linear regression model
We may estimate the by ordinary least
squares at each day t (quick!)
The estimates for each day will be quite
unstable, but they may be accumulated over time
to get stable estimates for the cumulative
regression coefficients
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Some estimated cumulative regression coefficients
for a model for incidence with fixed covariates
(may be interpreted as expected numbers)
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We have (using "obvious" matrix notation)
martingale transformation
Properties may be derived using martingale
methods as for Aalen's additive hazards model for
time-continuous event history data.
In particular is approximately multivariate
normal with a covariance matrix that may be
estimated by
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Dynamic covariates
How can past episodes of diarrhoea be used to
predict future episodes?
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Consider dynamic covariates of the form
with Yis incidence (prevalence) of diarrhoea
Use t 30 days and r 0.01 below
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A dynamic covariate may be on the causal pathway
between a fixed covariate and the event process
The inclusion of dynamic covariates in the
analysis may distort the estimation of the
effects of the fixed covariates
To avoid such distortion we at each time t
regress the dynamic covariates on the fixed
covariates and use the residuals from these fits
as new covariates
This procedure keeps the effect of the fixed
covariates the same as in the model without the
dynamic covariates
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Cumulative regression coefficients for incidence
Average number of diarrhoea episodes
Average number of days with diarrhoea
Also male, 3 or more per bedroom, contaminated
water source, open sewerage, rain affected
accommodation, young mother
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Martingale residual processes
martingale transformation
Examples of standardized martingale residual
processes (standardized by model based SDs)
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Empirical standard deviations of the martingale
residual processes
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Cumulative regression coefficients for
prevalence
Average number of days with diarrhoea
Diarrhoea previous day (lag 1)
Baseline
Lag 2 (residual effect)
Lag 3 (residual effect)
Lag 4 (residual effect)
Also male, age, 3 or more per bedroom, poor
street, contaminated water storage and source,
standing water, open sewerage, rain affected
accommodation, young mother
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Prevalence empirical standard deviations of the
martingale residual processes
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Not Markovian!
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Concluding comments
A dynamic additive model provides a flexible
framework for analyzing longitudinal binary data
The method illustrate how ideas and approaches
from event history analysis may be useful for
analysis of longitudinal data
Advantage method is computationally very quick
Drawback incidence and prevalence are not
restricted to the range 0 to 1
Methodological work is needed, in particular on
methods for model selection and goodness-of-fit