511: Statistics Statistical Methods Dr. Levine
Purdue University Fall 2006
Lecture 19: Analysis of Paired Data
Devore: Section 9.3
Aug, 2006
Statistics 511: Statistical Methods Dr. Levine
Purdue University Fall 2006
Analysis of Paired Data
The data consists of n independently selected pairs (X1 , Y1 ), (X2 , Y2 ),..., (Xn , Yn ) with E Xi = 1 and E Yi = 2 . The differences Di = Xi Yi are assumed to be normally 2 distributed with mean value D = 1 2 and variance D . The last requirement is usually the consequence of Xs and Y s being normally distributed themselves
Aug, 2006
Statistics 511: Statistical Methods Dr. Levine
Purdue University Fall 2006
Example
Consider Ex. 9.8 from Devore. Six river locations were selected and the concentration of zinc in mg/L determined for both
surface water and bottom water at each location. Presumably, there is some connection between surface water and bottom water concentrations...
Aug, 2006
Statistics 511: Statistical Methods Dr. Levine
Purdue University Fall 2006
The Paired t test
The test considered is H0 : D = 0 where D = X Y The test statistic is d 0 t= sD / n
where d and sD are the sample mean and standard deviation of di s Note that the old method of computing the variance of the difference does not work anymore since X and Y are NOT
independent
Aug, 2006
Statistics 511: Statistical Methods Dr. Levine
Purdue University Fall 2006
The differences themselves are independent. Thus, hypotheses about D = 1 2 can be tested using a one-sample t-test with Di s as data H0 : D = 0
Test statistic value is
d 0 t= sD / n where d and sD are the sample mean and standard
deviation of the di s.
Aug, 2006
Statistics 511: Statistical Methods Dr. Levine
Purdue University Fall 2006
Possible alternatives are Ha : D > 0 , Ha : D < 0 and Ha : D = 0 . Their respective rejection regions are t t,n1 , t t,n1 and either t t/2,n1 or t t/2,n1 .
Aug, 2006
Statistics 511: Statistical Methods Dr. Levine
Purdue University Fall 2006
Example
Consider Ex. 9.9 in Devore. Note that differences closely
adhere to the straight line in the normal probability plot but the mean appears to be nonzero.
The formal hypothesis is H0 : D = 0 vs. Ha : D = 0. The value of the test is statistic 6.75 d0 = 3.28 = t= sd / n 8.234/ 16 The P-value for a t curve with 15 df is .004 Note that for this sample size the normality assumption is
important
Aug, 2006
Statistics 511: Statistical Methods Dr. Levine
Purdue University Fall 2006
A condence interval for D
The paired t condence interval for D is d t/2,n1 sD / n Note that for large n this interval is valid without any restrictions on the distribution of differences. The same is not true if n is
relatively small
Aug, 2006
Statistics 511: Statistical Methods Dr. Levine
Purdue University Fall 2006
Example
Consider Example 9.10 in Devore. Yet again, the exploratory
data analysis suggests normality of the differences. The sample size is n
= 13 so this is important.
The needed d = 20.5 sec and sD = 11.96 while the number of df is n 1 = 12. The 95% condence interval is sD t/2,n1 = 20.5 (2.179) 11.96 = (13.3, 27, 7) d n 13
Aug, 2006
Statistics 511: Statistical Methods Dr. Levine
Purdue University Fall 2006
Paired Data and Two-Sample t Procedures
The main difference between the paired data t test and the standard t test lies in how we estimate V (X Y ). In the independent case we have V (X Y ) = V (X) + V (Y )
but...
In the paired data case V (XY ) = V 1 n Di
2 2 1 + 2 21 2 V (Di ) = = n n
Aug, 2006
Statistics 511: Statistical Methods Dr. Levine
Purdue University Fall 2006
In the above, = Corr(X, Y ) = Cov(X, Y )/[
general,
V (X) V (Y )]; in
2 2 V (X Y ) = 1 + 2 21 2
= 0 and, therefore, V (X Y ) = V (X) + V (Y ).
In the independence case,
Thus, using a regular t-test in paired data means overestimating the variance of X Y , and consequently, underestimating the
signicance of the data
Aug, 2006
Statistics 511: Statistical Methods Dr. Levine
Purdue University Fall 2006
Pros and Cons of Pairing
For great heterogeneity and large correlation within
experimental units, the loss in degrees of freedom will be compensated for by an increased precision associated with pairing (use pairing).
If the units are relatively homogeneous and the correlation
within pairs is not large, the gain in precision due to pairing will be outweighed by the decrease in degrees of freedom (use independent samples).
Aug, 2006