[R-sig-ME] Split-plot Design

[John Maindonald]

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.

[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.

With models that have a single error term, drop1() will example
the effect of single term deletions, often in unbalanced designs
more relevant than the sequential information in the anova table.
Importantly, drop1() respects hierarchy, whereas Type III sums of
squares do not.]

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