Stat 562 course presentation 1 Analysis of ordinal repeated categorical response data by using marginal model (Maximum likelihood approach) by Abdul Salam

Stat 562 course presentation 1

Analysis of ordinal repeated categorical response data by using

marginal model (Maximum likelihood approach)

by Abdul Salam

Instructor: K.C. CarriereStat 562


Contents:• Introduction• Background of data• Objective of the study• Basic theory

– Marginal model– Model fitting using ML

• SAS Codes• Results• Conclusion


Introduction• Definition

– Categorical data

– Repeated categorical data

– Advantages and Disadvantages of repeated Measurements Designs


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.


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.


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.


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.


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.


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

patient’s response to the question, How quickly did you fall

asleep after going to bed?” measured using categories (<20

minutes, 20-30 minutes, 30-60 minutes, and >60 minutes).

Patients were asked this question before and following a two-

week treatment period.


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.


Table#1: Time to falling Asleep, by Treatment and Occasion.(n=239).

Time to Falling Asleep

Follow Up

Treatment Initial <20 min 20 – 30 min 30 – 60 min > 60 min

Active <20 7 4 1 0

20 – 30 11 5 2 2

30 – 60 13 23 3 1

60 9 17 13 8

Placebo <20 7 4 2 1

20 – 30 14 5 1 0

30 – 60 6 9 18 2

> 60 4 11 14 22


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?


Pharmaceutical Company Interest

Company hope that patients with a Active treatment have

a significantly higher rate of improvement than patients

with placebo.


Generalized linear model to the analysis of Repeated

Measurements Designs

• Marginal Models;

• Random Effect Models;

• Transition models.


Basic Theory


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)..


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 rP Y j j = 0,1,…….c

Thus 0 1 1, ......., 1o o c


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

0 1 1 2 2 ........i i i p ipx x x

The vector β characterizes how the cross-sectional response

distribution depends on the explanatory variables.


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 c+1 categories, one might use

the cumulative logit.

Logitj = logit [P(Y ≤ j)], j=1,…………..c

( )E y


Link Function.1

1

log , 1,.......1

jj

j

j c

where j rP Y j

GLM is simplified to proportional odds model, then βj may

simplify to β indicating the same effect for each logit. The

proportional odds model is

j jx x for j =1,……….c,


Link Function.For individuals with covariate vector x* and x, the odds ratio for the response below category j is

*

**

*

*

* *

* *

* *

* *

/

/,

/

/

exp,

exp

, exp

, exp

, exp

, exp .

r

rj

r

r

j

j

j

j j j

j j j

j

j

P Y j x

P Y j xx x

P Y j x

P Y j x

xx x

x

x x x x

x x x x

x x x x

x x x x

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.


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.


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,

T

tt

I

, ;1 1 , 1,......,Tp p p r pY mult n p P


ML: Model Speciofication.

• Where is a vector of positive probabilities and 1rT is a r-

dimensional vector of 1’s. 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,

logC A X

p


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 log( ) 0U C A


ML Fitting of marginal Models:

Let be a vector having elements and the lagrange multipliers . The Lagrangian likelihood equations have form

0h

, , logh h f l f

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


ML Fitting of marginal Models:

1

1 , 1,...............

t

t t th

h t

After obtaining the fitted values on convergence of the algorithm, they calculate model parameter estimates using

^ ^1

logX X X C A

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.


InferenceHypothesis 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

logC A X


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 .


Results:

Table#2: Sample Marginal Proportions for Insomnia Data.

Time to Falling Asleep

Treatment Occasion <20 min 20 – 30 min

30 – 60 min

> 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


Figure# 1: Sample Marginal Proportions Insomnia data.

0

0.2

0.4

0.6

0.8

1

<20 20-30 30-60 >60

Pro

po

rtio

n

active and initial active at follow up

placebo at initial placebo at follow up


Marginal Proportion • sample proportion of time to falling asleep in <20 minutes for

subject who received Active treatment at initial occasion is

= (7+4+1+0) / (7+4+1+0+11+…………+13+8) = 12/119=0.1008

• Similarly the sample proportion of time to falling asleep in >60

minutes for subject received placebo at follow up is

= (1+0+2+22) / (7+4+2+1+………..+14+22) = 25/120=0.20833

And so on.


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.


Fitted Marginal ModelLet ‘x’ represent the treatment, with x=1 for an Active treatment and x=0 for

the placebo. Let t denote the occasion measurement , with t=0 for initial and

t=1 for follow up. Let (Yt) represent the outcome variable which is patient’s

response at time t to the question, “How quickly did you fall asleep after

going to bed?” with j=0 for <20 minutes, j=1 for 20-30 minutes, j=2 for 30-60

minutes, and j=3 for >60 minutes). The marginal model with cumulative link

can be written for our data set as

1 2 3 *j t x x t logit [P(Y ≤ j)] =


SAS codedata isomnia; input treatment $ initial $ follow $ count @@;

If count=0 then count=1E-8;datalines;active <20 <20 7 active <20 20-30 4active <20 30-60 1 active <20 >60 0 active 20-30 <20 11 active 20-30 20-30 5active 20-30 30-60 2 active 20-30 >60 2active 30-60 <20 13 active 30-60 20-30 23active 30-60 30-60 3 active 30-60 >60 1 active >60 <20 9 active >60 20-30 17active >60 30-60 13 active >60 >60 8placbo <20 <20 7 placbo <20 20-30 4placbo <20 30-60 2 placbo <20 >60 1 placbo 20-30 <20 14 placbo 20-30 20-30 5placbo 20-30 30-60 1 placbo 20-30 >60 0placbo 30-60 <20 6 placbo 30-60 20-30 9placbo 30-60 30-60 18 placbo 30-60 >60 2 placbo >60 <20 4 placbo >60 20-30 11placbo >60 30-60 14 placbo >60 >60 22;


SAS codeproc catmod order=data data=isomnia; weight count; population Treatment; response clogit; model initial*follow=(1 0 0 1 1 1, α 1+ β1+ β2 +β3 active + follow, j=1

0 1 0 1 1 1, α 2+ β1+ β2 +β3 active + follow, j=2 0 0 1 1 1 1, α 3+ β1+ β2 +β3 active + follow, j=3 1 0 0 1 0 0, α 1+ β1 active+ initial, j=1

0 1 0 1 0 0, α 2+ β1 active+ initial , j=2 0 0 1 1 0 0, α 3+ β1 active + initial, j=3 1 0 0 0 1 0, α 1 + β2 placebo+ follow, j=1

0 1 0 0 1 0, α 2 + β2 placebo+ follow, j=2 0 0 1 0 1 0, α 3 + β2 placebo+ follow, j=3

1 0 0 0 0 0, α 1 placebo+ initial, j=1

0 1 0 0 0 0, α 2 placebo+ initial, j=2

0 0 1 0 0 0) α 3 placebo+ initial, j=3

(1 2 3 ='Cutpoint', 4='Treatment', 5='TIme effect', 6='Time*Treatment effect') / freq; quit;


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 +1.37+1.074 (Occasion) +

0.046 (Treatment) +

0.662 (Occasion * Treatment)


Hypothesis testing for estimators:• For Occasion

– β1= 1.074 S.E (β1)= 0.162 p-value=<0.0001

• For Treatment – β2= 0.046 S.E (β2)= 0.236 p-value= 0.84

• For interaction (Occasion * time) – β3= 0.662 S.E (β3)= 0.244 p-value= 0.00665


Model Goodness of fit testThe 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 df=6 gives G2 = 8.0 and p-value 0.238,

indicating that the model fit the given data set well


Interpretation of ParametersEffect 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 {0.046}=1.04 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 Exp{0.046+0.662} = 2.03 times

the estimated odds for the placebo treatment.


Interpretation of Parameters (cont.)

• For the Active treatment the slope is β3= 0.662 (SE=0.244)

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.


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).


Thank You For Your Attention

Documents

Stat 562 course presentation 1 Analysis of ordinal repeated categorical response data by using marginal model (Maximum likelihood approach) by Abdul Salam