[R] Manually calculating values from aov() result

[John Fox]

Dear Brian,

As Duncan mentioned, the terms type-I, II, and III sums of squares 
originated in SAS. The type-II and III SSs computed by the Anova() 
function in the car package take a different computational approach than 
in SAS, but in almost all cases produce the same results. (I slightly 
regret using the "type-*" terminology for car::Anova() because of the 
lack of exact correspondence to SAS.) The standard R anova() function 
computes type-I (sequential) SSs.

The focus, however, shouldn't be on the SSs, or how they're computed, 
but on the hypotheses that are tested. Briefly, the hypotheses for 
type-I tests assume that all terms later in the sequence are 0 in the 
population; type-II tests assume that interactions to which main effects 
are marginal (and higher-order interactions to which lower-order 
interactions are marginal) are 0. Type-III tests don't, e.g., assume 
that interactions to which a main effect are marginal are 0 in testing 
the main effect, which represents an average over levels of the 
factor(s) with which the factor in the main effect interact. The 
description of the hypotheses for type-III tests is even more complex if 
there are covariates. In my opinion, researchers are usually interested 
in the hypotheses for type-II tests.

These matters are described in detail, for example, in my applied 
regression text <https://www.john-fox.ca/AppliedRegression/index.html>.

I hope this helps,
  John

-- 
John Fox, Professor Emeritus
McMaster University
Hamilton, Ontario, Canada
web: https://www.john-fox.ca/
--
On 2024-08-07 8:27 a.m., Brian Smith wrote:
> [You don't often get email from briansmith199312 using gmail.com. Learn why this is important at https://aka.ms/LearnAboutSenderIdentification ]
> 
> Caution: External email.
> 
> 
> Hi,
> 
> Thanks for this information. Is there any way to force R to use Type-1
> SS? I think most textbooks use this only.
> 
> Thanks and regards,
> 
> On Wed, 7 Aug 2024 at 17:00, Duncan Murdoch <murdoch.duncan using gmail.com> wrote:
>>
>> On 2024-08-07 6:06 a.m., Brian Smith wrote:
>>> Hi,
>>>
>>> I have performed ANOVA as below
>>>
>>> dat = data.frame(
>>> 'A' = c(-0.3960025, -0.3492880, -1.5893792, -1.4579074, -4.9214873,
>>> -0.8575018, -2.5551363, -0.9366557, -1.4307489, -0.3943704),
>>> 'B' = c(2,1,2,2,1,2,2,2,2,2),
>>> 'C' = c(0,1,1,1,1,1,1,0,1,1))
>>>
>>> summary(aov(A ~ B * C, dat))
>>>
>>> However now I also tried to calculate SSE for factor C
>>>
>>> Mean = sapply(split(dat, dat$C), function(x) mean(x$A))
>>> N = sapply(split(dat, dat$C), function(x) dim(x)[1])
>>>
>>> N[1] * (Mean[1] - mean(dat$A))^2 + N[2] * (Mean[2] - mean(dat$A))^2
>>> #1.691
>>>
>>> But in ANOVA table the sum-square for C is reported as 0.77.
>>>
>>> Could you please help how exactly this C = 0.77 is obtained from aov()
>>
>> Your design isn't balanced, so there are several ways to calculate the
>> SS for C.  What you have calculated looks like the "Type I SS" in SAS
>> notation, if I remember correctly, assuming that C enters the model
>> before B.  That's not what R uses; I think it is Type II SS.
>>
>> For some details about this, see
>> https://mcfromnz.wordpress.com/2011/03/02/anova-type-iiiiii-ss-explained/
>>
> 
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