Post on 20-Jan-2016
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1STA 617 – Chp11 STA 617 – Chp11 Models for repeated dataModels for repeated data
Analyzing Repeated CategoricalResponse Data
Repeated categorical responses may come from repeated measurements over time on each
individual or from a set of measurements that are related
because they belong to the same group or cluster (e.g., measurements made on siblings from the same family, measurements made on a set of teeth from the same mouth).
Observations within a cluster are not usually independent of each other, as the response from one child of a family, say, may influence the response from another child, because the two grew up together.
Matched-pairs are the special case of each cluster having two members.
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Using repeated measures within a cluster can be an efficient way to estimate the mean response at each measurement time without estimating between-cluster variability.
Many times, one is interested in the marginal distribution of the response at each measurement time, and not substantially interested in the correlation between responses across times.
Estimation methods for marginal modeling include maximum likelihood estimation and generalized estimating equations (GEE).
Maximum likelihood estimation is difficult because the likelihood is written in terms of the IT multinomial joint probabilities for T responses with I categories each, but the model applies to the marginal probabilities.
Lang and Agresti give a method for maximum likelihood fitting of marginal models in Section 11.2.5. Modeling a repeated multinomial response or repeated ordinal response is handled in the same way.
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Topics
In Section 11.1 we compare marginal distributions in T-way tables. The remaining sections extend models to include explanatory variables.
In Section 11.2 we use ML methods for fitting marginal models.
In Section 11.3 we use generalized estimating equations (GEE), a multivariate version of quasi-likelihood that is computationally simpler than ML.
Section 11.4 covers technical details about the GEE approach.
In the final section we introduce a transitional approach that models observations in terms of previous outcomes.
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11.1 COMPARING MARGINAL DISTRIBUTIONS: MULTIPLE RESPONSES Please review 10.1-10.3. Example: in treating a chronic condition with some
treatment, the primary goal might be to study whether the probability of success increases over the T weeks of a treatment period.
The T success probabilities refer to the T first-order marginal distributions
We want to compare marginal distributions.
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11.1.1 Binary Marginal Models and Marginal Homogeneity
T binary responses Marginal logit model
with
All possible outcomes where
Let
the joint distribution of is Mult (n, (1, 2, …, 2^T))
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Marginal homogeneity
Likelihood The likelihood-ratio test of marginal homogeneity
where sample proportions and is maximized likelihood estimate assuming marginal homogeneity.
asymptotic null chi-squared distribution with DF=T-1
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11.1.2 Crossover Drug Comparison Example
each subject used each of three drugs for treatment of a chronic condition at three times.
The response measured the reaction as favorable or unfavorable. (binary)
assume that the drugs have no carryover effects and that the severity of the condition remained stable for each subject throughout the experiment.
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Test marginal homogeneity
Sample proportions favorable (n=46)[(6+2+16+4)/46=0.61, 28/46=0.61, 16/46=0.35]for drug A, B, C
Clearly, from the sample proportion, A and B are similar, and better than C
The likelihood-ratio test statistic is 5.95 (DF=2). P-value=0.05.
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SAS
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simultaneous confidence intervals
The confidence interval for the true difference is (0.00133, 0.520) between B and C
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CATMOD
Suppose the dependent variable A has three levels and is the only response-effect in the MODEL statement.
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Design Matrix
p_A=alpha+beta1+beta2 P_B=alpha+beta1 P_C=alpha
Alpha=intercept Beta1=p_B-p_C Beta2=p_A-p_B
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Design Matrix
p_A=parameter1 P_B=parameter2 P_C=parameter3
Analysis of Weighted Least Squares Estimates
Effect Parameter Estimate StandardError
Chi-Square
Pr > ChiSq
Model 1 0.6087 0.0720 71.56 <.0001
2 0.6087 0.0720 71.56 <.0001
3 0.3478 0.0702 24.53 <.0001
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11.1.3 Modeling Margins of a Multicategory Response Saturated model
marginal homogeneity
Test
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Ordinal response
marginal homogeneity
Test
Model fitting 11.2.5
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11.1.4 Wald and Generalized CMH Score Tests of Marginal Homogeneity Similar with paired data in Chapter 10
SAS
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11.2 MARGINAL MODELING: MAXIMUM LIKELIHOOD APPROACH
compared marginal distributions, but accounting for explanatory variables.
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Longitudinal Mental Depression Example
comparing a new drug with a standard drug Outcome: mental depression (normal, abnormal) Stratified randomization by severity of depression (was
mild or severe). Four arms n=80, 70, 100, 90 Follow up 1 week, 2 weeks, and 4 weeks
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explanatory variables: treatment type and severity of initial diagnosis
T=3 12 marginal distributions result from three repeated
observations for each of the four groups. Let s denote the severity of the initial diagnosis, with
s=1 for severe and s=0 for mild. Let d denote the drug, with d=1 for new and d=0 for
standard. Let t denote the time of measurement. Use score (0, 1,
2), the logs to base 2 of the week (1, 2, 4).
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Descriptive statistics (sample proportions)
the sample proportion of normal responses after week 1 for subjects with mild initial diagnosis using the standard drug was
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data depress;input case diagnose treat time outcome ;
* outcome=1 is normal;
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proc sort; by diagnose treat time;
proc means n mean std; class diagnose treat time; var outcome; run;
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The sample proportion of normal responses increased over time for each group; increased at a faster rate for the new drug than the
standard, for each fixed initial diagnosis; and was higher for the mild than the severe initial
diagnosis, for each treatment at each occasion. The company would hope to show that patients have a
significantly higher rate of improvement with the new drug.
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Modeling
The marginal logit model 1 (main effects model)
Time (t) is continuous The natural sampling assumption is multinomial for the
eight cells in the 23 cross-classification of the three responses
A check of model fit compares the 32 cell counts in Table 11.2 to their ML fitted values. Since the model describes 12 marginal logits using four parameters, residual df=8. The deviance G2=34.6.
Lack of fit, since model assumes a common rate of improvement (should be higher for new drug)
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Model 2
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For each drug-time combination, the estimated odds of a normal response when the initial diagnosis was severe equal exp(-1.29)=0.27 times the estimated odds when the initial diagnosis was mild.
The estimate indicates an insignificant difference between the drugs after 1 week.
At time t, the estimated odds of normal response with the new drug are exp(-0.06+1.01 t) times the estimated odds for the standard drug, for each initial diagnosis level.
Conclusion: severity of initial diagnosis, drug treatment, and time all have substantial effects on the probability of a normal response.
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11.2.2 Modeling a Repeated Multinomial Response
At observation t, the marginal response distribution has I-1 logits.
nominal responses, baseline-category logit models describe the odds of each outcome relative to a baseline.
For ordinal responses, one might use cumulative logit models.
checking for interaction is crucial.
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11.2.3 Insomnia Example randomized, double-blind clinical trial comparing an
active hypnotic drug with a placebo in patients who have insomnia problems.
response is the patient’s reported time in minutes to fall asleep after going to bed.
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Proportional odds model
Sample marginal distributions
proc sort; by treat time;
proc freq; tables treat*time*outcome /nocol NOFREQ NOPERCENT; run;
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ML model fitting
G2=8.0 (df=6)
shows evidence of interaction At the initial observation, the estimated odds that time
to falling asleep for the active treatment is below any fixed level equal exp(0.046)=1.04 times the estimated odds for the placebo treatment;
at the follow-up observation, the effect is exp(0.046+0.662)=2.03.
In other words, initially the two groups had similar distributions, but at the follow-up those with the active treatment tended to fall asleep more quickly.
Follow-up with placebo or treatment, both tended to fall sleep more quickly (exp(1.07)=2.9)
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11.2.4 Comparisons That Control for Initial Response
Model assumption: the marginal distributions for initial response are identical for the treatment groups.
This is true if random assignment of subjects to the groups (one of the principles in experimental design: randomization, other two: replication, blocking)
If the initial marginal distributions are not identical, however, the difference between follow-up and initial marginal distributions may differ between treatment groups, even though their conditional distributions for follow-up response are identical.
In such cases, although marginal models can be useful, they may not tell the entire story. It may be more informative to construct models that compare the follow-up responses while controlling for the initial response.
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transitional model
Let Y2 denote the follow-up response, for treatment x with initial response y1.
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11.2.5 ML Fitting of Marginal Logit Models*
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
The marginal multinomial variates are not independent. Marginal logit models have the generalized loglinear
model form
where denote the complete set of multinomial joint probabilities for all settings of predictors.
33STA 617 – Chp11 STA 617 – Chp11 Models for repeated dataModels for repeated data
Example, model (11.1)
the model of marginal homogeneity (T=2)
34STA 617 – Chp11 STA 617 – Chp11 Models for repeated dataModels for repeated data
likelihood
The likelihood function for a marginal logit model is the product of the multinomial mass functions from the various predictor settings.
Usually, no continuous predictor is allowed
if U denote a full column rank matrix such that the space spanned by the columns of U is the orthogonal complement of the space spanned by the columns of X.
maximizing the likelihood incorporates these model constraints as well as identifiability constraints
35STA 617 – Chp11 STA 617 – Chp11 Models for repeated dataModels for repeated data
ML
Joseph Lang ( jblang@stat.uiowa.edu) has R and S-Plus functions for ML fitting of marginal models through the generalized loglinear model (11.8), using the constraint approach with Lagrange multipliers.http://www.stat.uiowa.edu/~jblang/mph.fitting/mph.fit.documentation.2.0.htm
The program MAREG (Kastner et al. 1997) provides GEE fitting and ML fitting of marginal models with the Fitzmaurice and Laird (1993) approach, allowing multicategory responses.
36STA 617 – Chp11 STA 617 – Chp11 Models for repeated dataModels for repeated data
Generalized Estimating Equation (GEE)