math60604a statisticalmodelling §5d ......léo belzile math60604a 5d-compoundsymmetrymodel 14/15...
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MATH 60604AStatistical modelling
§ 5d - Compound symmetry model
HEC MontréalDepartment of Decision Sciences
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Covariance structure of the compound symmetry model
• Assume that the observations within a group are interchangeable.That is, assume that the correlation (conditional on the explanatoryvariables) between two Y observations within a group is always thesame, and that the conditional variance of Y is constant.
• In this case, if there are five observations within group i, theassociated within-group covariance matrix is
Σi =
σ2 + τ τ τ τ τ
τ σ2 + τ τ τ ττ τ σ2 + τ τ ττ τ τ σ2 + τ ττ τ τ τ σ2 + τ
.
• Note here is that the conditional covariance between twoobservations in the same group is τ , and that the conditionalvariance of each observation is σ2 + τ .
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Correlation structure of the compound symmetry model
The corresponding correlation matrix for the compound symmetrycovariance model is
Ri =
1 ρ ρ ρ ρρ 1 ρ ρ ρρ ρ 1 ρ ρρ ρ ρ 1 ρρ ρ ρ ρ 1
,
where ρ = τ/(σ2 + τ).
• The conditional correlation between two observations within agroup is always ρ.
• This covariance structure is called “compound symmetry” and hastwo parameters, σ2 and τ .
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mixed procedure to fit models to correlated data
SAS code
/* Copy t */data revenge;set statmod.revenge;tcat=t;run;
proc mixed data=revenge method=reml;class id tcat;model revenge = sex age vc wom t / solution;repeated tcat / subject=id type=cs r=1 rcorr=1;run;
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Declaring the dependence structure within proc mixed
The command repeated allows us to define the dependencestructure.
• The first argument of the repeated function specifies what orderthe observations are within each group. This variable must be acategorical variable (created via class).
• The option subject specifies the variable which identifies thegroups.
• The option type specifies the model for the within-groupcorrelation.
• The option r=1 (rcorr=1) adds the estimated covariance(correlation) matrix for individual 1 in the output.
We will also use the variable t as a continuous variable in the model, which is why wealso created a copy of the variable t (tcat here), in order to use it as an argument forrepeated.
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Technical aside
• The first argument tcat in the repeated command is ignored here,as the compound symmetry covariance structure does not use theorder of the observations within a group.
• However, the order must be specified for other types of structures.It’s good to specify the “repeated” argument, even when it’s notnecessary.
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Model specification
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Covariance and correlation matrices for individual 1
Since we specified a compound symmetry structure for the covariance,the correlation is the same for all observations within subject 1.
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Parameters for the covariance structure
• The compound symmetry covariance structure is• Var (Yij) = σ2 + τ ;• Cov (Yij, Yij′) = τ .
• The estimate of the conditional covariance between observationsfor the same person is τ = 0.137.
• The estimated conditional variance of an observation isτ + σ2 = 0.386.
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Correlation structure
• The estimate of the conditional correlation between twoobservations from the same person (within-person correlation) is
ρ =τ
τ + σ2=
0.1370.137+ 0.248 = 0.356.
• We can recover these values in the covariance/correlation matricesgiven for the first individual.
• You need to know how to retrieve the correlation based on output(hence the formulae.)
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Likelihood ratio test for covariance parameter
• We can testH0 : τ = 0 againstH1 : τ = 0 using the likelihoodratio test.
• The above table gives the likelihood ratio test forH0 : τ = 0, whichcorresponds to the covariance model of the classic regressionmodel with covariance σ2I (reducedmodel), but ajusted using REML.
• We conclude that the reduced model without a correlationstructure is not an adequate simplification of the more complexmodel with the compound symmetry correlation structure.
• The likelihood ratio test reported by SAS always perform thecomparison with the homoscedastic linear model withoutcorrelation.
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Likelihood ratio test, by hand
• We could obtain the value of the test statistic manually bycomparing the restricted maximum likelihood estimates of the twomodels, here−2ℓreml(θ0) = 776.7 and−2ℓreml(θ) = 709.4, so thelikelihood ratio test statistic is 67.3.• This is the value reported on the previous slide, modulo rounding.
• The null distribution of the likelihood ratio test is χ21 (why?).
• We can compare the value of the test to the 95% quantile of the χ21 ,
3.84. Since the value of the statistic is larger than 3.84, we rejectH0 at level α = 0.05.
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Mean parameter estimates
Desire for revenge seems to decrease in time, after accounting for theother variables.
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Coefficient estimates
• The fitted model is always a linear regression model,
revenge = − 0.169+ 0.136sex+ 0.0459age+ 0.523vc+ 0.399wom − 0.568t.
• It turns out that the estimates β are exactly the same as we saw inthe ordinary linear regression model.
• This is a special case (compound symmetry correlation, and samenumber of observations in each group) and will not always be truefor other models.
• However, these estimates will usually be close to those comingfrom ordinary linear regression.
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Model comparison for coefficients
• The precision of our estimates β changes (left is independence,right is equicorrelation model).
• The standard errors are greater in the model with non-zerocorrelation. The conclusions did not change for any of the predictorvariables, except for sex. It is no longer significant.
• In fact, the correlations make within-person observations redundantto an extent. We actually have less information than we would forindependent observations, so parameter estimates are less precise.
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