Analysis of ordinal repeated categorical response data by using marginal model (Maximum likelihood approach)

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Analysis of ordinal repeated categorical
response data by using marginal model (Maximum
likelihood approach) by Abdul Salam Instructor
K.C. Carriere Stat 562
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Contents

• Introduction
• Background of data
• Objective of the study
• Basic theory
• Marginal model
• Model fitting using ML
• SAS Codes
• Results
• Conclusion

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Introduction

• Definition
• Categorical data
• Repeated categorical data
• Advantages and Disadvantages of repeated
  Measurements Designs

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Definition

• Categorical data
• Categorical data fits into a small number of
  discrete categories (as opposed to continuous).
  Categorical data is either non-ordered (nominal)
  such as gender or city, or ordered (ordinal) such
  as high, medium, or low temperatures.

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Definition (cont-)

• Repeated categorical data
• The term repeated measurements refers broadly
  to data in which the response of each
  experimental unit or subject is observed on
  multiple occasions or under multiple conditions.
  When the response is categorical then it is
  called repeated categorical data.

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Definition (cont-)

• Application of Repeated categorical data
• Repeated categorical response data occur commonly
  in health-related application, especially in
  longitudinal studies. For example, a physician
  might evaluate patients at weekly intervals
  regarding whether a new drug treatment is
  successful. In some cases explanatory variable
  also vary over time.

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Advantages of Repeated Measurements Designs

• Individual patterns of change.
• Provide more efficient estimates of relevant
  parameters than cross-sectional designs with the
  same number and pattern of measurement.
• Between subjects sources of variability can be
  excluded from the experimental error.

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Disadvantages of Repeated Measurements Designs

• Analysis of repeated data is complicated by the
  dependence among the repeated observations made
  on the same experimental unit.
• Often investigator cannot control the
  circumstances for obtaining measurements, so that
  the data may be unbalanced or partially
  incomplete.

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Background of Insomnia data

• A randomized, double blind clinical trail has
  been performed for comparing an active hypnotic
  drug with a placebo in patients who have insomnia
  problems. The outcome variable which is patients
  response to the question, How quickly did you
  fall asleep after going to bed? measured using
  categories (lt20 minutes, 20-30 minutes, 30-60
  minutes, and gt60 minutes). Patients were asked
  this question before and following a two-week
  treatment period.

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Background of Insomnia data

• Patients were randomly assigned to one of the two
  treatments active and placebo. The two
  treatments, active and placebo, form a binary
  explanatory variable. Patients receiving the two
  treatments were independent samples.

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Table1 Time to falling Asleep, by Treatment and
Occasion.(n239).
Time to Falling Asleep Time to Falling Asleep Time to Falling Asleep Time to Falling Asleep Time to Falling Asleep
Follow Up Follow Up Follow Up Follow Up
Treatment Initial lt20 min 20 30 min 30 60 min gt 60 min
Active lt20 7 4 1 0
20 30 11 5 2 2
30 60 13 23 3 1
60 9 17 13 8
Placebo lt20 7 4 2 1
20 30 14 5 1 0
30 60 6 9 18 2
gt 60 4 11 14 22
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Objectives

• To study the effect of time on the response.
• To study the effect of treatment on the response.
  Is the time to fall asleep is quicker for active
  treatment than placebo?
• Is there any interaction between treatment and
  time? How does the treatment affect the time to
  fall asleep over time?

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Pharmaceutical Company Interest
Company hope that patients with a Active
treatment have a significantly higher rate of
improvement than patients with placebo.
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Generalized linear model to the analysis of
Repeated Measurements Designs

• Marginal Models
• Random Effect Models
• Transition models.

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Basic Theory
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GLMs for ordinal response.

• Extensions of generalized linear model
  methodology for the analysis of repeated
  measurements accommodate discrete or continuous,
  time-independent or dependent covariates. GLMs
  have three components A random component, which
  identify the response variable Y and its
  probability distribution a systematic component
  specify explanatory variables used in a linear
  predictor function a link function specifies the
  functional relationship between the systematic
  component and the E(Y)..

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Random Component.

• Since the response is ordinal, so it is often
  advantageous to construct logits that account for
  categorical ordering and are less affected by the
  number of choice of categories of the response,
  which is known as cumulative response
  probabilities, from which the cumulative logits
  are defined. For ordinal response with c 1
  ordered categories labeled as 0,1, 2,.,C for
  each individuals or experimental unit. The
  cumulative response probabilities are

j 0,1,.c Thus

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Systematic component.

• The systematic component of the generalized
  linear model specifies the explanatory variables.
  The linear combination of these explanatory
  variables is called the linear predictor denoted
  by

The vector ß characterizes how the
cross-sectional response distribution depends on
the explanatory variables.
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Link Function.

• The link function explain the relation ship
  between random and systematic component, that how

relates to the explanatory variables in the
linear predictor. For ordinal response having c1
categories, one might use the cumulative logit.
Logitj logit P(Y j),
j1,..c
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Link Function.
where
GLM is simplified to proportional odds model,
then ßj may simplify to ß indicating the same
effect for each logit. The proportional odds
model is
for j 1,.c,
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Link Function.
For individuals with covariate vector x and x,
the odds ratio for the response below category j
is
The odds ratio does not depend on response
category j. The regression coefficient can be
calculated by taking log, which indicate the
difference in logit (log odds) of response
variable per unit change in the x.
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Maximum Likelihood Method (ML).

• The standard approach to maximum likelihood (ML)
  fitting of marginal models involves solving the
  score equations using the Newton-Raphson method,
  Fisher scoring, or some other iterative
  reweighted least squares algorithm. ML fitting of
  marginal logit models is awkward. For T
  observations on an I-category response, at each
  setting of predictors the likelihood refers to IT
  multinomial joint probabilities, but the model
  applies to T sets of marginal multinomial
  parameters, and assume that marginal multinomial
  variates are independent.

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ML Model Speciofication.

• Let consider T categorical responses, where the
  tth variable has It categories. The responses are
  ordinal observed for P covariate patterns,
  defined by a set of explanatory variables. Let r
  

denote the number of response profiles for each
covariate pattern. The vector of counts for
covariate pattern p is denoted by Yp. The Yp are
assumed to be independent multinomial random
vectors,
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ML Model Speciofication.

• Where is a vector of positive probabilities
  and 1rT is a r-dimensional vector of 1s. Since
  the model applies to T sets of marginal
  multinomial parameters, the marginal models can
  be written as a generalized linear model with the
  link function,

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ML Fitting of marginal Models
Lang and Agresti (1994) considered the likelihood
as a function of rather then. The likelihood
function for a marginal logit model is the
product of the multinomial mass functions from
the various predictors setting. One approach for
ML fitting views the model as a set of
constraints and uses methods for maximizing a
function subject to constraints
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ML Fitting of marginal Models
Let be a vector having elements and the
lagrange multipliers . The Lagrangian
likelihood equations have form
where
is a vector with terms involving the contents in
marginal logits that the model specifies
constraints as well as log-likelihood derivative.
The Newton-Raphson iterative scheme is
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ML Fitting of marginal Models
After obtaining the fitted values on convergence
of the algorithm, they calculate model parameter
estimates using
This maximum likelihood fitting method makes no
assumption about the model that describes the
joint distribution. Thus, when the marginal
model holds, the ML estimate are consistent
regardless of the dependence structure for that
distribution.
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Inference

• Hypothesis testing for parameters
• After obtaining model parameter estimates and
  estimated covariance matrix, one can apply
  standard methods of inference, for instance Wald
  chi-squared test for marginal homogeneity.
• Goodness of Fit test
• To assess model goodness of fit, one can compare
  observed and fitted cell counts using the
  likelihood-ratio statistics G2 or the Pearson
  Chi-square statistics. For nonsparse tables,
  assuming that the model holds, these statistics
  have approximate chi-squared distributions with
  degree of freedom equal to the number of
  constraints implied by

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Limitations of ML

• The number of multinomial probabilities increases
  dramatically as the number of predictors
  increases.
• ML approaches are not practical when T is large
  or there are many predictors, especially when
  some are continuous.
• It does not make any assumption about the model
  that describes the joint distribution .

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Results
Time to Falling Asleep Time to Falling Asleep Time to Falling Asleep Time to Falling Asleep Time to Falling Asleep
Treatment Occasion lt20 min 20 30 min 30 60 min gt 60 min
Active Initial 0.101 0.168 0.336 0.395
Follow up 0.336 0.412 0.160 0.092
Placebo Initial 0.117 0.167 0.292 0.425
Follow up 0.258 0.242 0.292 0.208
Table2 Sample Marginal Proportions for Insomnia
Data.
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Figure 1 Sample Marginal Proportions Insomnia
data.
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Marginal Proportion

• sample proportion of time to falling asleep in
  lt20 minutes for subject who received Active
  treatment at initial occasion is
• (7410) / (741011138)
  12/1190.1008
• Similarly the sample proportion of time to
  falling asleep in gt60 minutes for subject
  received placebo at follow up is
• (10222) / (7421..1422)
  25/1200.20833
• And so on.

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What did you get from Marginal Proportion table?

• From initial to follow up occasion, time to
  falling asleep seems to shift downward for both
  treatments.
• The degree of shift seems greater for the active
  treatment than placebo, indicating possible
  interaction. Or we could say that effect of
  treatment on the response is different at
  different occasion.

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Fitted Marginal Model

• Let x represent the treatment, with x1 for an
  Active treatment and x0 for
• the placebo. Let t denote the occasion
  measurement , with t0 for initial and
• t1 for follow up. Let (Yt) represent the outcome
  variable which is patients
• response at time t to the question, How quickly
  did you fall asleep after
• going to bed? with j0 for lt20 minutes, j1 for
  20-30 minutes, j2 for 30-60
• minutes, and j3 for gt60 minutes). The marginal
  model with cumulative link
• can be written for our data set as

logit P(Y j)
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SAS code
data isomnia input treatment initial
follow count _at__at_ If count0 then
count1E-8 datalines active lt20 lt20
7 active lt20 20-30 4 active
lt20 30-60 1 active lt20 gt60
0 active 20-30 lt20 11
active 20-30 20-30 5 active 20-30
30-60 2 active 20-30 gt60
2 active 30-60 lt20 13 active
30-60 20-30 23 active 30-60 30-60
3 active 30-60 gt60 1 active
gt60 lt20 9 active gt60
20-30 17 active gt60 30-60 13
active gt60 gt60 8 placbo lt20
lt20 7 placbo lt20 20-30
4 placbo lt20 30-60 2 placbo lt20
gt60 1 placbo 20-30 lt20
14 placbo 20-30 20-30 5 placbo
20-30 30-60 1 placbo 20-30 gt60
0 placbo 30-60 lt20 6
placbo 30-60 20-30 9 placbo 30-60
30-60 18 placbo 30-60 gt60 2
placbo gt60 lt20 4 placbo gt60
20-30 11 placbo gt60 30-60 14
placbo gt60 gt60 22
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SAS code
proc catmod orderdata dataisomnia weight
count population Treatment response
clogit model initialfollow(1 0 0 1 1 1,
a 1 ß1 ß2 ß3 active follow, j1
0 1 0 1 1 1, a
2 ß1 ß2 ß3 active follow, j2
0 0 1 1 1 1, a 3
ß1 ß2 ß3 active follow, j3
1 0 0 1 0 0, a 1 ß1
active initial, j1
0 1 0 1 0 0, a 2 ß1 active initial ,
j2 0 0 1 1
0 0, a 3 ß1 active initial, j3
1 0 0 0 1 0, a 1
ß2 placebo follow, j1
0 1 0 0 1 0, a 2 ß2
placebo follow, j2
0 0 1 0 1 0, a 3 ß2 placebo
follow, j3 1 0 0 0 0 0, a 1
placebo initial, j1
0 1 0 0 0 0, a 2 placebo initial,
j2 0 0 1 0 0
0) a 3 placebo initial, j3 (1 2 3
'Cutpoint', 4'Treatment', 5'TIme effect',
6'TimeTreatment effect') /
freq quit
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Fitted Marginal Model

• After fitting the marginal model using maximum
  likelihood
• method to the above marginal distribution gave
  the following
• results
• Logit P (Y J) -1.16 0.10 1.371.074
  (Occasion)
• 0.046 (Treatment)
• 0.662 (Occasion
  Treatment)

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Hypothesis testing for estimators

• For Occasion
• ß1 1.074 S.E (ß1) 0.162 p-valuelt0.0001
• For Treatment
• ß2 0.046 S.E (ß2) 0.236 p-value 0.84
• For interaction (Occasion time)
• ß3 0.662 S.E (ß3) 0.244 p-value 0.00665

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Model Goodness of fit test

• The Likelihood ratio test (G2) has been used for
  Goodness of fit
• test. ML model fitting, comparing the observed to
  fitted cell
• counts in modeling the 12 marginal logits using
  these six
• parameters with df6 gives G2 8.0 and p-value
  0.238,
• indicating that the model fit the given data set
  well

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Interpretation of Parameters

• Effect of Treatment (Active vs Placebo)
• 1. At initial observation
• The estimated odds that the time to falling
  asleep for the active treatment is below any
  fixed equal Exp 0.0461.04 times the estimated
  odds for the placebo treatment.
• 2. At Follow up observation
• The estimated odds that the time to falling
  asleep for the active treatment is below any
  fixed equal Exp0.0460.662 2.03 times the
  estimated odds for the placebo treatment.

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Interpretation of Parameters (cont.)

• For the Active treatment the slope is ß3 0.662
  (SE0.244) higher than for the placebo, giving
  strong evidence of faster improvement. In other
  words, initially the two treatments had similar
  effect, but at the follow up those patients with
  the active treatment tended to fall asleep more
  quickly.

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Conclusion

• Using the maximum likelihood methods for the
  marginal distribution for the above given
  Insomnia data set, we have sufficient evidence to
  conclude that treatment and time have substantial
  effects on the response (time to fall asleep).

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Thank You For Your Attention