MULTIPLE COMPARISONS UNDER HETEROSCEDASTICITY Simultaneous Multiple
Fit:
Confidence
Comparisons
aov(formula
of
Means:
= elevel
Linear
Tukey
Contrasts
data
=
alpha)
level
Hypotheses:
Estimate med - short == 0 0.4342 long - short == 0 1.1888 long - med == 0 0.1546 R>
Intervals
~ alength,
Quantile = 2.31 95% family-wise confidence
117
lwr -0.4158 -0.0452 -0.3314
upr 1.3441 2.4221 1.8406
summary(alpha.mc) Simultaneous
Multiple
Fit:
Comparisons
aov(formula
Linear
Tests of
Means:
= elevel
General Tukey
~ alength,
Linear
Hypotheses
Contrasts
data
=
alpha)
Hypotheses:
Estimate med - short == 0 0.434 long - short == 0 1.189 long - med == 0 0.155 Signif. (Adjusted
for
codes: 0 p values
Std.
'***' 0.001 reported —
Error 0.384 0.520 0.458 '**' 0.01 single-step
t value 1.13 2.28 1.65 '*'
Pr(>\t\) 0.492 0.061 . 0.221 0.05 '.' method)
0.1
' ' 1
F r o m t h i s o u t p u t w e c o n c l u d e t h a t t h e r e is n o significant difference b e t w e e n a n y c o m b i n a t i o n of t h e t h r e e allele lengths. L o o k i n g a t F i g u r e 4.13, however, t h e v a r i a n c e h o m o g e n e i t y a s s u m p t i o n is questionable a n d one m i g h t challenge t h e v a l i d i t y of t h e s e r e s u l t s . O n e m a y argue t h a t a s a n d w i c h e s t i m a t e is m o r e a p p r o p r i a t e i n t h i s s i t u a t i o n . B a s e d o n t h e r e s u l t s f r o m S e c t i o n 3.2, we use the s a n d w i c h f u n c t i o n f r o m t h e s a n d w i c h package (Zeileis 2 0 0 6 ) , w h i c h provides a h e t e r o s c e d a s t i c i t y - c o n s i s t e n t e s t i m a t e of t h e c o v a r i a n c e m a t r i x . T h e v c o v a r g u m e n t of g l h t c a n b e u s e d to specify the alternative estimate,