Transcript Layouts and Models.pptx

Two Sample t-test vs. Paired t-test
The Layout of an Experiment
 The Layout for an Experiment is a graphical display
indicating elements of the structure of an Experiment
and the data which go along with it.
 It indicates how many factors there are and any
structural relationship among factors (important for
more complicated designs).
 It also indicates the numbers of observations (data
points) and where they are located.
Layout for simple comparative experiment
Associated Model
The layout suggest two independent samples and so we
can use the simple linear model:
Yij=µi+εij
This suggests an independent sample t-test assuming
that:
εij ~ Normal(0,σ2)
Suppose there is more structure to the data
 Suppose that the individual data points have more
“structure”.
 Suppose that each data point is a Pre or Post
Treatment score for subjects/experimental units.
 This needs to reflected in the Layout.
Corrected Layout
Correct Model
 Model including Subject for Paired data:
Yij= µi+Sj+ εij
 Note that in the second model we have partitioned the
experimental error into two terms, because Subject is now
included in the Model. This is a fundamental idea in ANOVA
where the variation in the data is partitioned into its
components. That is why it is referred to as Analysis of Variance
(ANOVA).
 The important implication is that σ for the second model (paired
data model) is smaller than σ for the first model (independent
sample model). This is because our estimate of σ in the second
model does not include Subject to subject variation.
How they differ
 For testing, our estimates of the means for each group
are the same, but our estimates of experimental error
variation are quite different.
 In terms of the signal to noise ratio, our estimate of the
signal is the same, but the estimate of “noise” is
different, since the subject to subject variation is
controlled for.
 This does not affect the hypotheses, but it can affect
the likelihood of finding significance since the second
model has more Power.
This can be simplified to a Paired t-test
 We can simplify the model by simply computing the
differences between Pre and Post values.
 Then instead of testing
H0:µ1= µ2
Ha: µ1≠ µ2,
we are testing
H0: µd=0
Ha: µd≠0
where µd= µ1-µ2, with a one sample t-test.