Solution file for Problem 8.3 (GO)
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Data: unspecified outcome and 2 factors (denoted generically r and c, 
for row and column) in a 4*3 factorial with 24 units, balanced with 2 
replications per group (not stated but visible in the residual plots, 
although not of importance for the analysis requested).


Model: 2-way ANOVA with replications,

y_ijk = mu + alpha_i + beta_j + (alpha beta)_ij + eps_ijk,

where the errors (eps_ijk) are i.i.d. and N(0,sigma^2).


Model validation: Plots of studentized residuals against fitted values
(denoted Yhat) for analysis on both original and logarithmic scale are
given. The plot on original scale shows a very strong cone (or fan)
shape and thus indicates a gross violation of model assumptions. Except
for one pair of points at the very right (around 3000), the spread among
the two replications (the vertically symmetrical pair of points)
increases with the fitted values. Also, the fitted values seem to span 
several magnitudes, with the lowest values close to zero and the highest
values at around 3000. 

The residual plot obtained from analysis on (natural) log scale looks
much nicer with a fairly even spread across the range of fitted values,
except for the highest fitted value at which the points in the pair are
very close. We conclude that the log-scale analysis is preferable.

Note that it does not matter for our assessment of the plots whether 
the residuals are internally studentized (GO terminology for 
standardised residuals) or externally studentized (deletion residuals). 


Analysis: The DF, SS and MS values from the ANOVA table allow us to
compute the F-test for interaction as the ratio between MS(r.c) and MSE.
We get the P-value for the F-test from an F-distribution (6,12).

Original data     F=7.858/1.4544=5.40      P=0.006
Log data          F=1.2577/1.7308=0.73     P=0.63

The log-scale results are the relevant ones here (but it is interesting
to note the difference in conclusion, as an illustration of the scale-
dependence of interactions), and we conclude that there is no evidence
of additive interaction on the log-scale, and hence that the effects of
the two factors are multiplicative on original scale.
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Addendum: software code to compute P-value from F(6,12) distribution

Minitab:
--------
MTB > CDF 5.40;
SUBC>   F 6 12.
Cumulative Distribution Function 

F distribution with 6 DF in numerator and 12 DF in denominator

  x  P( X <= x )
5.4     0.993545

Note that the P-value (the tail probability) equals 1 minus the
cumulative probability.

Stata:
------
. display Ftail(6,12,5.40)
.0064547

R:
--
pf(5.40,6,12,lower.tail=FALSE)
[1] 0.0064547