[R] randomized block design and two-way factorial design

Peter Dalgaard BSA p.dalgaard at biostat.ku.dk
Wed Dec 13 11:03:28 CET 2000

Michael Wang <mwang at mindspring.com> writes:

> I am still a little unclear in the difference between
> randomized block design and two-way factorial design
> after consulting a few books, including John Rice 
> Mathematical Statistics and Data Analysis.
> Both put observations in cells corresponding to two factors
> of many levels. Both use the same computer program to analyze
> data. 
> It seems that randomized block design can have only one observation
> per cell, is this true? And Friedman's can be used for
> randomized block design but not two-way factorial design?
> If this is not a good place to ask generic statistical questions,
> I will not do so in the future. If you know a good place to do this,
> please recommend. Thanks. 

There are newsgroups on Usenet, sci.stat.math, sci.stat.edu, and
sci.stat.consult as well as the Allstat mailing list and a couple of
others. However, a subject-matter question on R-help now and then
probably won't hurt.

According to Cochran+Cox, randomized blocks have one obs. per cell,
but I wouldn't be too surprised if other sources would allow more than
one (and even C+C is ambiguous since it talks about giving the same
treatment more than once within a block). 

A two-way factorial can certainly have more than one obs per cell and
the term also covers several different experimental designs, including
randomized blocks, but also the straight treatment-combination that
Bill Simpson mentioned, and repeated measures designs in which one
factor might be "subject" and the other "time".

In the one obs. per cell case with one factor being "subject" or
similar, I think the operative word is "randomized" i.e. that in the
absence of treatment effects, the treatments enter symmetrically,
and hence that among Normal models for a randomized block design, the
only relevant model for the covariance is that of compund symmetry and
for the Friedman test that all rankings of the treatments within
blocks are equally likely.

For a repeated measures design you could have substantial serial
correlations which would require a more elaborate model and the
assumptions of the null hypothesis of the Friedman test are not met in
that case. On the other hand you can of course also have data where
the model for randomized blocks fits well, and thus the standard
F-test and the Friedman test applies.

   O__  ---- Peter Dalgaard             Blegdamsvej 3  
  c/ /'_ --- Dept. of Biostatistics     2200 Cph. N   
 (*) \(*) -- University of Copenhagen   Denmark      Ph: (+45) 35327918
~~~~~~~~~~ - (p.dalgaard at biostat.ku.dk)             FAX: (+45) 35327907
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