Anova - interpretation of the interaction term

From: Ted.Harding at nessie.mcc.ac.uk

On 23-Apr-05 Bill.Venables at csiro.au wrote:

[technical setup snipped]

contrasts(dat$vac_inf) <- ginv(m)
gm <- aov(y ~ vac_inf, dat)
summary(gm)

            Df  Sum Sq Mean Sq F value  Pr(>F)
vac_inf      3 12.1294  4.0431   7.348 0.04190
Residuals    4  2.2009  0.5502

This doesn't tell us too much other than there are differences,
probably.  Now to specify the partition:
                

summary(gm, 

      split = list(vac_inf = list("- vs +|N" = 1, 
                                  "- vs +|Y" = 2)))
                    Df  Sum Sq Mean Sq F value  Pr(>F)
vac_inf              3 12.1294  4.0431  7.3480 0.04190
  vac_inf: - vs +|N  1  7.9928  7.9928 14.5262 0.01892
  vac_inf: - vs +|Y  1  3.7863  3.7863  6.8813 0.05860
Residuals            4  2.2009  0.5502

Wow, Bill! Dazzling. This is like watching a rabbit hop
into a hat, and fly out as a dove. I must study this syntax.
But where can I find out about the "split" argument to "summary"?
I've found the *function* split, whose effect is similar, but
I've wandered around the "summary", "summary.lm" etc. forest
for a while without locating the *argument*.

My naive ("cop-out") approach would have been on the lines
of (without setting up the contrast matrix):

  summary(aov(y~vac*inf,data=dat))
              Df  Sum Sq Mean Sq F value  Pr(>F)  
  vac          1  0.3502  0.3502  0.6364 0.46968  
  inf          1 11.3908 11.3908 20.7017 0.01042 *
  vac:inf      1  0.3884  0.3884  0.7058 0.44812  
  Residuals    4  2.2009  0.5502                  

so we get the 2.2009 on 4 df SS for redisuals with mean SS 0.5502.

Then I would do:

  mNp<-mean(y[(vac=="N")&(inf=="+")])
  mNm<-mean(y[(vac=="N")&(inf=="-")])
  mYp<-mean(y[(vac=="Y")&(inf=="+")])
  mYm<-mean(y[(vac=="Y")&(inf=="-")])

  c(     mYp,       mYm,       mNp,      mNm       )
  ##[1]  2.4990492  0.5532018  2.5212655 -0.3058972

  c(    mYp-mYm,   mNp-mNm )
  ##[1] 1.945847   2.827163


after which:

  1-pt(((mYp-mYm)/sqrt(0.5502)),4)
  ##[1] 0.02929801

  1-pt(((mNp-mNm)/sqrt(0.5502)),4)
  ##[1] 0.009458266

give you 1-sided t-tests, and

  1-pf(((mYp-mYm)/sqrt(0.5502))^2,1,4)
  ##[1] 0.05859602

  1-pf(((mNp-mNm)/sqrt(0.5502))^2,1,4)
  ##[1] 0.01891653

give you F-tests (equivalent to 2-sided t-tests) which agree
with Bill's results above.

And, in this case, presenting the results as mean differences
shows that the effect of infection appears to differ between
vaccinated and unvaccinated groups; but a simple test shows
this not to be significant:

  1-pf( (sqrt(1/2)*((mYp-mYm)-(mNp-mNm))/sqrt(0.5502))^2, 1,4)
  ##[1] 0.4481097

As I said above, this would be my naive approach to this
particular case (and to any like it), and I would expect
to be able to explain all this in simple terms to a colleague
who was asking the sort of questions you have reported.
Or is it the case that offering an anova table is needed,
in order to evoke consent to the results by virtue of the
familiar format?

As expected, infection changes the mean for both vaccinated and
unvaccinated, as we arranged when we generated the data.

The challenge to generalise is interesting. However, as implied
above, I'm already impressed by Bill's footwork in R for this
simple case, and it might be some time before I'm fluent
enough in that sort of thing to deal with more complicated
cases, let alone the general one.

For users like myself, a syntax whose terms are closer to
ordinary language would be more approachable. Something
on the lines of

  summary(aov(y ~ vac*inf), by=inf, within=vac)

which would present a table similar to Bill's above (by inf
within the different levels of vac).

Intriguing. The challenge is tempting ... !

Best wishes,
Ted.


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Date: 23-Apr-05                                       Time: 14:33:00
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