Hi Peter, I might be unclear in my description of the data. Each patient was measured for a response variable "y" at 3 time points, there is no drug or other treatment involved. The objective was to examine the repeatability of the measurements of response variable "y". Since this is repeated measure, I thought it should be analyzed by a simple mixed model? When you suggested a MxK (K=3) design, what is M then?

Thanks very much,

John





________________________________
 From: peter dalgaard <pdalgd@gmail.com>

Cc: "r-help@r-project.org" <r-help@r-project.org> 
Sent: Sunday, May 27, 2012 12:09 AM
Subject: Re: [R] a simple mixed model


On May 27, 2012, at 07:12 , array chip wrote:

> Hi, I was reviewing a manuscript where a linear mixed model was used. The data is simple: a response variable "y" was measured for each subject over 3 time points (visit 1, 2 and 3) that were about a week apart between 2 visits. The study is a non-drug study and one of the objectives was to evaluate the repeatability of response variable "y". 
> 
> 
> The author wanted to estimate within-subject variance for that purpose. This is what he wrote "within-subject variance was generated from SAS 'Prog Mixed' procedure with study visit as fixed effect and subject as random effect". I know that the study visit was a factor variable, not a numeric variable. Because each subject has 3 repeated measurements from 3 visits, how can a model including subject as random effect still use visit as fixed factor? If I would do it in R, I would just use a simple model to get within-subject variance:
> 
> obj<-lmer(y~1+(1|subject),data=data)
> 
> What does a model "obj<-lmer(y~visit+(1|subject),data=data)" mean?
> 
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Sounds like a pretty standard two-way ANOVA with random row effects. 

If the design is complete (M x K with K = 3 in this case), you look at the row and column means. An additive model is assumed and the residual (interaction) is used to estimate the error variance. 

The variation of the row means is compared to the residual variance. If tau is the variance between row levels, the variance of the row means is sigma^2/K + tau, and tau can be estimated by subtraction. 

The column averages can be tested for systematic differences between visits with the usual F test. A non-zero effect here indicates that visits 1, 2, 3 have some _systematic_ difference across all individuals. 

For an incomplete design, the model is the same, but the calculations are less simple.

-- 
Peter Dalgaard, Professor,
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
Phone: (+45)38153501
Email: pd.mes@cbs.dk  Priv: PDalgd@gmail.com
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