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Northern Ireland group meeting: A decade of covariance modelling

Written by Gilbert MacKenzie on . Posted in Sections and local group meeting reports

The Northern Ireland group held a meeting at 4pm on Thursday the 11th of April, 2013 in the David Bates Building in the Queen's University of Belfast. The speaker was Professor Gilbert MacKenzie of the Centre of Biostatistics, The University of Limerick, Ireland.

Professor MacKenzie traced the history of the development of data driven joint mean-covariance modelling from Pourahmadi's (Biometrika, 1999) seminal paper.

The conventional approach to modelling longitudinal RCT data places considerable emphasis on estimation of the mean structure and much less on the covariance structure, between repeated measurements on the same subject. Gilbert described an approach where the mean and covariance parameters were placed on an equal footing and treated symmetrically.

The technique relies on a modified Cholesky decomposition of the marginal covariance matrix which yields transformed parameters which are unique and have the interpretation of autoregressive coefficients and innovation variances. The autoregressive parameters and log-innovation variances are unbounded and can be modelled by two separate linear regression equations. Thus, the whole of the repeated measures inference problem can be reduced to three joint linear regressions in time: one for the mean and two for the covariance matrix.

He discussed optimal joint model selection in the case of a univariate longitudinal Gaussian response with m=11 repeated measures in which the regressions were polynomials of time. Here joint mean-covariance model selection amounts to selecting a joint model with the lowest BIC ('best') in a hyper-cube of dimension m3. While a direct search is always possible, Professor MacKenzie discussed a pairwise BIC search method which linearised the search.

He went on to describe various extensions to this basic univariate case: (a) allowing additional baseline covariates such as the treatment indicator in the two covariance models thereby making possible tests of homogeneity of covariance structure, (b) extending to Linear Mixed Models by applying the Modified Cholesky decomposition to the within subject errors (c) taking a GEE approach and (d) having a constrained mean approach.

The final part of his talk dealt with the non trivial extension of the method to bivariate Gaussian longitudinal response data, Xu and MacKenze (Biometrika, 2012). This retained many properties of the univariate approach but now the main technical statistical challenge was that of modelling innovation matrices. The problem was solved using matrix-logarithm methods and Gilbert illustrated the use of the new methods by analysing bivariate data from the MRC's RCT of teletherapy in age-related macular degeneration. The new approach extends naturally to multivariate longitudinal data and allows many of the previous extensions to be incorporated.

The talk led to a lively and challenging discussion.

 

covariance Northern Ireland local group

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