Hi Wolfgang,

One approach is to use latent growth model (SEM). The following sample code
fits a model with AR(1) (labeled "c1" in the model). You may add a
multiple-group analysis for groups A and B. Two models can be then
fitted--one with the same AR(1) by using the same label, i.e., "c1", and
another model with different AR(1) by using a different label, say "c2". A
likelihood-ratio test may be used to compare these two models.

Hope it helps.

my.df <-
structure(list(anti0 = c(2L, 1L, 0L, 1L, 2L, 3L, 5L, 0L, 0L,
4L, 2L, 2L, 4L, 1L, 0L, 3L, 1L, 0L, 1L, 1L), anti1 = c(5L, 2L,
1L, 0L, 3L, 2L, 0L, 0L, 0L, 2L, 2L, 3L, 3L, 1L, 3L, 5L, 0L, 0L,
0L, 3L), anti2 = c(1L, 1L, 1L, 2L, 2L, 3L, 5L, 0L, 0L, 2L, 0L,
6L, 3L, 0L, 4L, 0L, 3L, 2L, 1L, 3L), anti3 = c(5L, 1L, 2L, 4L,
2L, 4L, 3L, 0L, 1L, 2L, 0L, 5L, 2L, 1L, 1L, 0L, 0L, 0L, 0L, 2L
)), .Names = c("anti0", "anti1", "anti2", "anti3"), row.names = c(134L,
76L, 145L, 83L, 53L, 12L, 21L, 177L, 151L, 152L, 96L, 91L, 132L,
198L, 38L, 160L, 101L, 37L, 197L, 31L), class = "data.frame")

library("lavaan")

my.model1 <- 'i =~ 1*anti0 + 1*anti1 + 1*anti2 + 1*anti3
              s =~ 0*anti0 + 1*anti1 + 2*anti2 + 3*anti3
              ## Equality on error variances
              anti0 ~~ sigma*anti0
              anti1 ~~ sigma*anti1
              anti2 ~~ sigma*anti2
              anti3 ~~ sigma*anti3
              ## AR(1): Note that c1 is covariance, not correlation
              anti0 ~~ c1*anti1
              anti1 ~~ c1*anti2
              anti2 ~~ c1*anti3'

my.fit1 <- growth(my.model1, data=my.df)
summary(my.fit1, fit.measures=TRUE)

Cheers,
Mike

-- 
---------------------------------------------------------------------
 Mike W.L. Cheung               Phone: (65) 6516-3702
 Department of Psychology       Fax:   (65) 6773-1843
 National University of Singapore
 http://courses.nus.edu.sg/course/psycwlm/internet/
---------------------------------------------------------------------

On Wed, Dec 12, 2012 at 10:51 PM, Viechtbauer Wolfgang (STAT) <
wolfgang.viechtbauer@maastrichtuniversity.nl> wrote:

> Dear All,
>
> I am currently working with some mixed-effects models for longitudinal
> data where I want to test hypotheses about the difference in the strength
> of the autocorrelation between two groups. In particular, let's say I have
> data of the form:
>
> person  time  group  y
> ----------------------------
> 1       0     A      <value>
> 1       1     A      <value>
> 1       2     A      <value>
> ...     ...   ...    ...
> 2       0     A      <value>
> 2       1     A      <value>
> 2       2     A      <value>
> ...     ...   ...    ...
>
> and I want to fit the model:
>
> y_ij = beta_0 + beta_1 time_ij + u_0i + u_1i time + e_ij,
>
> where i is the index for person, j is the index for time, u_0i ~ N(0,
> tau^2_0) is a random effect for the intercepts, u_1i ~ N(0, tau^2_1) is a
> random effect for the slopes, cov(u_0i, u_1i) = tau_01 is the covariance
> between these two random effects, and now I want to fit an AR(1) structure
> for the errors e_ij, where the autocorelation is rho_A for people from
> group A and rho_B for people from group B. In principle, I may also want to
> let the error variance to differ for the two groups (i.e., sigma^2_A and
> sigma^2_B).
>
> In nlme, I can fit models with a common autocorrelation structure for the
> two groups, with:
>
> lme(y ~ time, random = ~ time | person, correlation = corAR1(form = ~ time
> | person))
>
> but as far as I can tell, it is not possible to allow the autocorrelation
> parameter to differ between the two groups.
>
> Does anybody know of a package/function that would allow fitting such a
> model?
>
> Moreover, the groups are actually created by dichotomizing a continuous
> measure. Ideally, I would like to fit a model, where rho depends on the
> continuous measure directly. Or, to avoid issues related to rho having to
> fall between -1 and 1, it would probably make more sense to assume that
> f(rho) depends on that continuous measure, where f() is the arctanh
> (inverse hyperbolic function or Fisher's r-to-z) transformation. Again, any
> ideas on packages/functions that may handle such a model?
>
> Thanks in advance for any suggestions!
>
> Best,
> Wolfgang
>
> --
> Wolfgang Viechtbauer, Ph.D., Statistician
> Department of Psychiatry and Psychology
> School for Mental Health and Neuroscience
> Faculty of Health, Medicine, and Life Sciences
> Maastricht University, P.O. Box 616 (VIJV1)
> 6200 MD Maastricht, The Netherlands
> +31 (43) 388-4170 | http://www.wvbauer.com
>
> _______________________________________________
> R-sig-mixed-models@r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
>

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