Hello hotmark,
Here are the answers to your questions.
Question a.
As you can see from the following page, test-retest reliability
coefficient can be estimated by computing the correlation coefficient
between the results of both tests. That is, we have to find the
correlation coeffcient between sets of data (2,4,3,5,6) and
(3,4,2,4,7). I used Microsoft Excel to compute this, finding that the
correlation coefficient is 0.84. So, 0.84 is the test-retest
reliability coefficient.
Test reliability
http://www.psych.ualberta.ca/~chrisw/L7Reliability/L7Reliability.pdf
Question b.
We're assuming here that the test was taken only once (to many
people), so we're splitting in half the results of them. Test 1 would
represent the scores of 5 people, while test 2 would represent the
ones of the other 5.
So, first we compute the reliability coefficient for this test from
the Spearman-Brown formula. The Split-Half reliability, of course,
will be the same as the one we calculated in question a, 0.84. Thus,
plugging this in the S-B formula (look it up in the same page as
above), we get that the reliability of the test is:
2*0.84/(1+0.84) = 0.91
Now, the formula for computing the effect of changing the length of
the test can be found at the following page
On-Line Classes
http://www.ed.sc.edu/edpyrmfn/johnson/meetn09.htm
(search for the "methods for estabilishing reliability" section in
this page)
The formula is
Kr
-------
1+(K-1)r
where r is the original reliability coefficient, and K means that the
test is K times longer than the original one.
So, you want to know what happens to this when K varies. We compute
the first derivative of the formula with respect to K, to find that
the derivative is:
r(1+(K-1)r)-Krr
-----------------
((1+(K-1)r)^2
r+Krr-rr-Krr
= -----------------
((1+(K-1)r)^2
r-rr
= -----------------
((1+(K-1)r)^2
Now r > rr because r lies between 0 and 1, so the numerator is
positive. The denominator is also positive, because it's a squared
number. Therefore, the first derivate of the formula with respect to K
is positive. Thus, when K rises (when the test is longer) the
reliability coefficient rises, so the test becomes more reliable.
Google search terms used:
"test-retest" reliability
spearman-brown
I hope this was clear enough. If you have any doubts, please request a
clarification. Otherwise, I await your comments and rating.
Best luck!
elmarto |
Clarification of Answer by
elmarto-ga
on
22 Jun 2003 18:56 PDT
I'm very sorry, but I don't understand what do you mean by
"calculations". I'd really like to help you with this but I don't
understand what is missing from my answer.
Again, in question (b), the calculations are there, because the
derivation is done step by step. Do you want me to post the
substractions, additions and multiplications that are needed in order
to arrive to the 0.84 result from question (a)? If that is what you're
looking for, here it is. I will be following the numerical example in
the following page adapted to your data.
Numerical Example
http://davidmlane.com/hyperstat/A56626.html
X = 2 4 3 5 6
Y = 3 4 2 4 7
sum(X*Y) = 2*3 + 4*4 + 3*2 + 5*4 + 6*7 = 90
sum(X) = 2+4+3+5+6 = 20
sum(X^2) = 4 + 16 + 9 + 25 + 36 = 90
sum(Y) = 3 + 4 + 2 + 4 + 7 = 20
sum(Y^2) = 9 + 16 + 4 + 16 + 49 = 94
N = 5
sum(X*Y) - (sum(X)*sum(Y)) /N = 90-(20*20)/5 = 10
sum(X^2)-((sum(X))^2)/N = 90 - (20^2)/5 = 10
sum(Y^2)-((sum(Y))^2)/N = 94 - (20^2)/5 = 14
r = 10 / sqrt(14*10) = 0.84
where sqrt means "square root of"
Please, if this is not what you're looking for, be more specific on
what calculations do you need on another clarification request so I
can further assist you.
Best regards,
elmarto
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