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Matched Pair Design by Example
Goal:
To compare IQ's of 1st borns (FB) and 2nd borns (SB).
Population of N pairs of siblings:
Pop'n of FB's Pop'n of SB's
Pop'n of diff's
pop'n mean
pop'n var
In particular, we ask
Q: Are first borns more intelligent than their second born siblings?
Answering the question via computing a 100(1- )% C.I. for
:
A sample of n pairs of siblings are selected from the population of N pairs:
where
Now, define
http://www.stat.sfu.ca/~sgchiu/Grace/S330/handouts/paired/
3/26/2011
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Di = Xi - Yi
so that Di> 0 is some indication that the first born of the ith selected pair is more intelligent.
Note that due to common characteristics shared by siblings within each ith pair, Xi and Yi are dependent
variables, even though one pair is independent of another pair because of random selection. In this case, Xi and
Yi are likely to be positively correlated (increase or decrease together). Then
Note that
estimates
:
http://www.stat.sfu.ca/~sgchiu/Grace/S330/handouts/paired/
3/26/2011
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From here on out, we don't worry about the original Xi's and Yi's! All we need are the Di's. (Of course, for
the sake of computing
sD2 if
, simply take
. However, the Di's are still needed for EDA and for computing
is unknown.)
So, a 100(1-
)% C.I. for
is
depending on whether
is known?
is normal? (CLT?)
Di's are normal?
Q: How to check normality in practice???
FACT:
http://www.stat.sfu.ca/~sgchiu/Grace/S330/handouts/paired/
3/26/2011
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Making conclusions:
If the 100(1different (i.e.
)% C.I. for
does not contain 0, then we are quite confident that the siblings' IQ's are
0).
Depending on wether 0 is to the left or right of the C.I., we get a sense of which sibling is more intelligent. A
formal one-sided hypothesis test can be conducted to determine if the first borns have higher IQ's than their
second born siblings.
Grace Chiu
2000-08-17
http://www.stat.sfu.ca/~sgchiu/Grace/S330/handouts/paired/
3/26/2011