[R-sig-ME] Split-plot Design

John Maindonald John.Maindonald at anu.edu.au
Sat Mar 22 07:40:17 CET 2008

I like this - a good robust defence!  Let the debate go on.

Designs that are in some sense balanced are, as far as I can judge,  
still the staple of agricultural trials.  The sorts of issues that  
arise in time series modeling must also be increasingly important.   
The thinking behind those designs is important, and ought to penetrate  
more widely.

That general style of design is not however limited to agricultural  
experimentation.  It is used widely, with different wrinkles, in  
psychology in industrial experimentation, in medicine, in laboratory  
experimentation, and with internet-based experimentation a new major  
area of application.  It ought to be used much more widely outside of  
agriculture.  Agriculture has been somewhat unique in having  
professional statisticians working alongside agricultural scientists  
for many decades now.

The wrinkles can be important.  Fisher famously commented on the  
frequent need to ask Nature more than one question at a time:
"No aphorism is more frequently repeated in connection with field  
trials, than that we must ask Nature few questions, or, ideally, one  
question, at a time. The writer is convinced that this view is wholly  
mistaken. Nature, he suggests, will best respond to a logical and  
carefully thought out questionnaire"

For each fixed effect and interaction, it is however necessary to be  
clear what is the relevant error term.  Some time ago, someone posted  
a question about the analysis of a psychology experiment where the  
number of possible error terms was very large, where some pooling was  
desirable, and where it was not at all clear what terms in the anova  
table to pool, and where d.f, were so small that the mean squares did  
not provide much clue.  (I looked for the email exchange, but could  
not immediately find it.) That is not a good design.  There may be  
much more potential for this sort of difficultly in psychology as  
compared to agriculture.  In agriculture the sites/blocks/plots/ 
subplots hierarchy is the order of the day, albeit often crossed with  
seasons to ensure that the analysis is not totally obvious.

The data sets are often larger than formerly. (But not always, again  
note some psychology experiments. Or they may not be large in the  
sense of allowing many degrees of freedom for error) Large datasets  
readily arise when, as for the internet-based experimentation,  
randomization can be done and data collected automatically.  (Some  
advertisers are randomizing their pop-up ads, to see which gives the  
best response.) With largish degrees of freedom for error, it is no  
longer necessary to worry about exact balance.  I consider Doug that  
you are too tough on the textbooks.  Maybe they ought to be branching  
out from agricultural experimentation more than they do; that is as  
much as I'd want to say.  There are any number of examples from  
psychology, some of them very interesting, in the psychology books on  
experimental design.  (Data from published papers ought nowadays as a  
matter of course go into web-based archives - aside from other  
considerations this would provide useful teaching resources. In some  
areas, this is already happening.)

Degrees of freedom make sense in crossed designs also; it is the F- 
statistics for SEs for fixed effects that can be problematic.  It may  
happen that one or two sources of error (maybe treatments by years)  
will dominate to such an extent that other sources can pretty much be  
ignored.  The conceptual simplification is worth having; its utility  
may not be evident from a general multi-level modeling perspective.

Maybe one does not want statisticians to be too dyed in the wool  
agricultural.  Still, a bit of that thinking goes a long way, not  
least among social scientists.  The arguments in Rosenbaum's  
"Observational Data" are much easier to follow if one comes to them  
with some knowledge and practical experience of old-fashioned  
experimental design.  That seems to me a good indication of the quite  
fundamental role of those ideas, even if one will never do a  
randomized experiment.

I'd have every novice statistician do apprenticeship's that include  
experience in horticultural science (they do not have a long tradition  
of working alongside professional statisticians), medicine and  
epidemiology, business statistics, and somewhere between health social  
science and psychology.  It is unfortunate that I did not myself have  
this broad training, which may go some way to explaining why I am not  
in complete agreement with Doug's sentiments!!

This is not to defend, in any large variety of places and  
circumstances, inference that relies on degrees of freedom.  Not that  
my point relate directly to degrees of freedom.  On degrees of  
freedom, I judge them to have a somewhat wider usefulness than Doug  
will allow.  It would be nice if lmer() were to provide Kenward &  
Rogers style degrees of freedom ( as the commercial ASReml-R software  
does), but Doug is right to press the dangers of giving information  
that can for very unbalanced designs be misleading.  Given a choice  
between mcmcsamp() and degrees of freedom, maybe I would choose  

There's enlightening discussion of the opportunities that internet- 
based business offers for automated data collection and for  
experimentation on website visitors, in Ian Ayres "SuperCrunchers. Why  
thinking by numbers is the new way to be smart" (Bantam).   
Fortunately, the hype of the title pretty much goes once on gets into  
the text.

John Maindonald             email: john.maindonald at anu.edu.au
phone : +61 2 (6125)3473    fax  : +61 2(6125)5549
Centre for Mathematics & Its Applications, Room 1194,
John Dedman Mathematical Sciences Building (Building 27)
Australian National University, Canberra ACT 0200.

On 22 Mar 2008, at 1:31 AM, Douglas Bates wrote:

> On Thu, Mar 20, 2008 at 7:04 PM, John Maindonald
> <john.maindonald at anu.edu.au> wrote:
>> I do not think it quite true that the aov model that has an
>> Error() term is a fixed effects model.  The use of the word
>> "stratum" implies that a mixed effects model is lurking
>> somewhere.  The F-tests surely assume such a model.
>> Some little time ago, Doug Bates invested me, along with
>> Peter Dalgaard, a member of the degrees of freedom police.
>> Problem is, I am unsure of the responsibilities, but maybe
>> they include commenting on a case such as this.
> Indeed, I should have been more explicit regarding the duties of a
> member of the degrees of freedom police when I appointed you :-)  They
> are exactly what you are doing - to weigh in on discussions of degrees
> of freedom.
> Perhaps I can expand a bit on why I think that the degrees of freedom
> issue is not a good basis for approaching mixed models in general.
> Duncan Temple Lang used to have a quote about "Language shapes the way
> we think." in his email signature and I think similar ideas apply to
> how we think about models.  Certainly the concept of error strata and
> the associated degrees of freedom are one way of thinking about
> mixed-effects models and they are effective in the case of completely
> balanced designs.  If one has a completely balanced design and the
> number of units at various strata is small then decomposition of the
> variability into error strata is possible and the calculation of
> degrees of freedom is important (because the degrees of freedom are
> small - exact values of degrees of freedom are unimportant when they
> are large).  Such data occur frequently - in text books.  In fact,
> they are the only type of data that occur in most text books and hence
> they have shaped the way we think about the models.  Generations of
> statisticians have looked at techniques for small, balanced data sets
> and decided that these define "the" approach to the model.  So when
> confronted with large observational (and, hence, unbalanced or
> "messy") data sets they approach the modeling with this mind set.
> Certainly such data did occur in some agricultural trials (I'm not
> sure if this is the state of the art in current agricultural trials)
> and this form of analysis was important but it depends strongly on
> balanced data and balance is a very fragile characteristic in most
> data.  The one exception is data in text books - most often data for
> examples in older texts were added as an afterthought and, for space
> reasons and to illustrate the calculations, were small data sets which
> just happened to be completely balanced so that all the calculations
> worked out.   Fortunately this is no longer the case in books like
> your book with Braun where the data sets are real data and available
> separately from the book in R packages but it will take many years to
> flush the way of thinking about models induced by those small,
> balanced data sets from statistical "knowledge".  (There is an
> enormous amount of inertia built into the system.  Most introductory
> statistics textbooks published today, and probably those published for
> the next couple of decades, will include "statistical tables" as
> appendices because, as everyone knows, all of the statistical analysis
> we do in practice involves doing a few calculations on a hand
> calculator and looking up tabulated values of distributions.)
> Getting back to the "language shapes the way we think" issue, I would
> compare it to strategy versus tactics.  I spent a sabbatical in
> Australia visiting Bill Venables.  I needed to adjust to driving on
> the left hand side of the road and was able to do that after a short
> initial period of confusion and discomfort.  If I think of places in
> Adelaide now it is natural for me to think of myself sitting on the
> right hand side of the front seat and driving on the left hand side of
> the road.  That's tactics.  However, even after living in Adelaide for
> several months I remember leaving the parking lot of a shopping mall
> and choosing an indirect route out the back instead of the direct
> route ahead because the direct route would involve a left hand turn
> onto a busy road.  Instead I chose to make two right hand turns to get
> to an intersection with traffic lights where I could make the left
> turn.  At the level of strategy I still "knew" that left hand turns
> are difficult and right hand turns are easy - exactly the wrong
> approach in Australia.
> This semester I am attending a seminar on hierarchical linear models
> organized in our social sciences school by a statistician whose
> background is in agricultural statistics.  I make myself unpopular
> because I keep challenging the ideas of the social scientists and of
> the organizer.  The typical kinds of studies we discuss are
> longitudinal studies where the "observational units" (i.e. people) are
> grouped within some social contexts.  These are usually large, highly
> unbalanced data  sets. Because they are longitudinal they inevitably
> involve some migration of people from one group to another.
> The migration means that random effects for person and for the various
> types of groups are not nested.  The models are not hierarchical or at
> least they should not be.  Trying to jam the analysis of such data
> into the hierarchical/multilevel framework of software like HLM or
> MLwin doesn't work well but that certainly won't stop people from
> trying.  The hierarchical language in which they learned the model
> affects their strategy.
> The agricultural statistician, by contrast, doesn't worry about the
> nesting etc.,  For him the sole issue of important is to determine the
> number of degrees of freedom associated with the various error strata.
> These studies involve tens of thousands of subjects and are nowhere
> close to being balanced but his strategy is still based on having a
> certain number of blocks and plots within the blocks and subplots
> within the plots and everything is exactly balanced.
> So I am not opposed to approaches based on error strata and computing
> degrees of freedom where appropriate and where important.  But try not
> to let the particular case of completely balanced small data sets
> determine the strategy of the approach to all models involving random
> effects.  It's fragile and does not generalize well, just as assuming
> a strict hierarchy of factors associated with random effects is
> fragile.  Balance is the special case.  Nesting is the special case.
> It is a bad idea to base strategy on what happens in very special
> cases.
> they are always balanced and small data sets
> Another very fragile
>> lme() makes a stab at an appropriate choice of degrees
>> of freedom, but does not always get it right, to the extent
>> that there is a right answer.  [lmer() has for the time being
>> given up on giving degrees of freedom and p-values for
>> fixed effects estimates.]  This part of the output from lme()
>> should, accordingly, be used with discretion.  In case of
>> doubt, check against a likelihood ratio test.  In a simple
>> enough experimental design, users who understand how
>> to calculate degrees of freedom will reason them out for
>> themselves.
>> John Maindonald.
>> John Maindonald             email: john.maindonald at anu.edu.au
>> phone : +61 2 (6125)3473    fax  : +61 2(6125)5549
>> Centre for Mathematics & Its Applications, Room 1194,
>> John Dedman Mathematical Sciences Building (Building 27)
>> Australian National University, Canberra ACT 0200.
>> On 21 Mar 2008, at 9:23 AM, Kevin Wright wrote:
>>> Your question is not very clear, but if you are trying to match the
>>> results in Kuehl, you need a fixed-effects model:
>>> dat <- read.table("expl14-1.txt", header=TRUE)
>>> dat$block <- factor(dat$block)
>>> dat$nitro <- factor(dat$nitro)
>>> dat$thatch <- factor(dat$thatch)
>>> colnames(dat) <- c("block","nitro","thatch","chlor")
>>> m1 <- aov(chlor~nitro*thatch+Error(block/nitro), data=dat)
>>> summary(m1)
>>> Mixed-effects models and degrees of freedom have been discussed many
>>> times on this list....search the archives.
>>> K Wright
>>> On Thu, Mar 20, 2008 at 12:39 PM,  <marcioestat at pop.com.br> wrote:
>>>> Hi listers,
>>>> I've been studying anova and at the book of Kuehl at the chapter
>>>> about split-plot there is a experiment with the results... I am
>>>> trying to
>>>> understand the experiments and make the code in order to obtain the
>>>> results... But there is something that I didn't understand yet...
>>>> I have a split-plot design (2 blocks) with two facteurs, one
>>>> facteur has 4 treatments and the other facteur is a measure
>>>> taken in three years...
>>>> I organize my data set as:
>>>> Nitro Bloc Year Measure
>>>> a
>>>> x
>>>> 1         3.8
>>>> a
>>>> x
>>>> 2         3.9
>>>> a         x         3         2.0
>>>> a         y         1         3.7
>>>> a         y         2
>>>> 2.4
>>>> a         y         3
>>>> 1.2
>>>> b         x
>>>> 1         4.0
>>>> b         x
>>>> 2         2.5
>>>> and so on...
>>>> So, I am trying this code, because I want to test each factor and  
>>>> the
>>>> interaction...
>>>> lme=lme(measure ~ bloc + nitro + bloc*nitro, random= ~ 1|year,
>>>> data=lme)
>>>> summary(lme)
>>>> The results that I am obtaining are not correct, because
>>>> I calculated the degrees of fredom and they are not
>>>> correct... According to this design I will get two errors one for  
>>>> the
>>>> whole plot and other for the subplot....
>>>> Well, as I told you, I am still learning... Any suggestions...
>>>> Thanks in advance,
>>>> Ribeiro
>>>>      [[alternative HTML version deleted]]
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