# RE: [R] Anova - interpretation of the interaction term

From: Liaw, Andy <andy_liaw_at_merck.com>
Date: Sun 24 Apr 2005 - 01:55:38 EST

> From: Ted.Harding@nessie.mcc.ac.uk
>
> On 23-Apr-05 Bill.Venables@csiro.au wrote:
[snip]

> [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*.

gm is fitted with aov(), so wouldn't it make sense to look at ?summary.aov? It says:

split an optional named list, with names corresponding

```      to terms in the model. Each component is itself a
list with integer components giving contrasts
whose contributions are to be summed.

```

and even has an example showing how it's used.

Andy

> 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.
>
>
> --------------------------------------------------------------------
> E-Mail: (Ted Harding) <Ted.Harding@nessie.mcc.ac.uk>
> Fax-to-email: +44 (0)870 094 0861
> Date: 23-Apr-05 Time: 14:33:00
> ------------------------------ XFMail ------------------------------
>
> ______________________________________________
> R-help@stat.math.ethz.ch mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help