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:
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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.dun...@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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