[R-sig-ME] Split-plot Design

[Andy Fugard]

Quoting John Maindonald <John.Maindonald at anu.edu.au>:

> A response to your final paragraph.  There's no reason in principle
> why aov() should not allow for random slopes.  In fact, one can
> put continuous variables into the Error() term, but without taking
> time to study the output with some care, it is not obvious to me what
> the results mean.

Aha, thanks.

>
> [For reasons that are not clear to me, you mentioned Type III
> "errors".  The controversy over Type III versus Type I sums of
> squares relates to models with fixed effect interactions.

Interference effect; I meant to say "sums of squares", not "errors".   
I brought this up as I wanted to flag awareness that it's not always  
appropriate (with respect to one's guiding hypotheses/questions) to  
compare nested models where terms are added sequentialy in some  
arbitrary order.

Thanks,

Andy

>
> 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 12:51 AM, Andy Fugard wrote:
>
>> John Maindonald 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.
>>
>> This is something that has been bothering me for a while.  Often   
>> it's argued that ANOVA is just regression; clearly this is not true  
>>  when it's a repeated measures ANOVA, unless "regression" is   
>> interpreted broadly. I think Andrew Gelman argues this somewhere.    
>> I don't see how to get aov to give me a formula, and lm doesn't fit  
>>  stuff with an Error() term, but if it could, logically I would   
>> expect the formula to resemble closely the sort of thing you get   
>> with a mixed effects models.
>>
>> The closest analogy I can find is that doing this...
>>
>>
>>> aov1 = aov(yield ~  N+P+K + Error(block), npk)
>>> summary(aov1)
>>
>> Error: block
>>         Df Sum Sq Mean Sq F value Pr(>F)
>> Residuals  5    343      69
>>
>> Error: Within
>>         Df Sum Sq Mean Sq F value Pr(>F)
>> N          1  189.3   189.3   11.82 0.0037 **
>> P          1    8.4     8.4    0.52 0.4800
>> K          1   95.2    95.2    5.95 0.0277 *
>> Residuals 15  240.2    16.0
>> ---
>> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
>>
>>
>> [I believe the F's and p's here come from a linear construction of   
>> nested models (i.e., N versus intercept only; N+P versus N; plus   
>> N+P+K versus N+P).]
>>
>> ... is a bit like doing this with lmer...
>>
>>
>>> lmer0 = lmer(yield ~  1 + (1|block), npk)
>>> lmer1 = lmer(yield ~  N + (1|block), npk)
>>> lmer2 = lmer(yield ~  N+P + (1|block), npk)
>>> lmer3 = lmer(yield ~  N+P+K + (1|block), npk)
>>> anova(lmer0,lmer1)
>> Data: npk
>> Models:
>> lmer0: yield ~ 1 + (1 | block)
>> lmer1: yield ~ N + (1 | block)
>>     Df   AIC   BIC logLik Chisq Chi Df Pr(>Chisq)
>> lmer0  2 157.5 159.8  -76.7
>> lmer1  3 151.5 155.1  -72.8  7.93      1     0.0049 **
>> ---
>> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
>>> anova(lmer1,lmer2)
>> Data: npk
>> Models:
>> lmer1: yield ~ N + (1 | block)
>> lmer2: yield ~ N + P + (1 | block)
>>     Df   AIC   BIC logLik Chisq Chi Df Pr(>Chisq)
>> lmer1  3 151.5 155.1  -72.8
>> lmer2  4 153.0 157.8  -72.5  0.47      1       0.49
>>> anova(lmer2,lmer3)
>> Data: npk
>> Models:
>> lmer2: yield ~ N + P + (1 | block)
>> lmer3: yield ~ N + P + K + (1 | block)
>>     Df   AIC   BIC logLik Chisq Chi Df Pr(>Chisq)
>> lmer2  4 153.0 157.8  -72.5
>> lmer3  5 149.0 154.9  -69.5  6.02      1      0.014 *
>> ---
>> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
>>
>>
>> Namely, fitting the models, and making comparisons using likelihood  
>>  ratio tests.  Okay, the LLR is asymptotically chisq distributed   
>> with df the difference in parameters whereas... well F is different  
>>  - I always get confused with the df and what bits of the residuals  
>>  plug in where.
>>
>> But, is this vaguely right?  (I'm aware that order matters when the  
>>  design isn't balanced, and have read much about the type I errors   
>> versus type III errors disagreements.)
>>
>> Actually this does lead to one question: following through the   
>> analogy, are repeated measures ANOVAs (those fitted with aov)   
>> always random intercept models, i.e. with no random slopes?
>>
>>
>> Cheers,
>>
>> Andy
>>
>>
>> -- 
>> Andy Fugard, Postgraduate Research Student
>> Psychology (Room F3), The University of Edinburgh,
>> 7 George Square, Edinburgh EH8 9JZ, UK
>> Mobile: +44 (0)78 123 87190   http://www.possibly.me.uk



-- 
Andy Fugard, Postgraduate Research Student
Psychology (Room F3), The University of Edinburgh,
   7 George Square, Edinburgh EH8 9JZ, UK
Mobile: +44 (0)78 123 87190   http://www.possibly.me.uk


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