9/5/2016

Paired Replicates Data

• Two sets of observations, typically referred to as "pre-treatment" (X) and "post-treatment" (Y), are obtained on n experimental units. We are interested in assessing the effect of the treatment.
• Let (theta) refer to the "typical effect" of the treatment.
  • All of the information about (theta) in the pair (Xi, Yi) is contained in the difference Zi = Yi - Xi.
• Model: Let Zi = Yi - Xi, i=1, ..., n and assume that:
  • The n Zi's are mutually independent;
  • Each Zi has a continuous distribution with median (theta).
• We want to make inferences about (theta).
  • Three ways to use the information in the Zi's
    1. Use all of the information (paired t-test)
    2. Use minimal information on whether Y is larger than X (Sections 3.4 - 3.6 and 3.8)
    3. Use the information on whether Y is larger than X and the relative magnitudes of the Zi's. (Sections 3.1 - 3.3 and 3.7)
• Sign Test for paired data

The One Sample Location Problem

• Sign Test (Competitor to the one sample t test)

Example

• Exercise 98 (oxygen consumption) on p. 93
  • P:\data\math\hartlaub\nonparametrics\OxygenComp.csv
  • P:\data\math\hartlaub\nonparametrics\OxygenComp.R

Large Sample Approximation for the sign test

Dealing with ties

• If there are zero Zs (note that these have zero probability), discard the zero values and redefine n to be the number of nonzero Zs. ***{Do NOT use the sign test if you have a lot of zero Zs.}
• An alternative and recommended approach

Point estimators associated with the sign test

• Exercises 63 (blood-level data) and 64 (salary data) on p. 79

Confidence intervals and confidence bounds based on the sign test

Please read Sections 3.1, 3.4, 3.5, 3.6, amd 3.8 for class on Wednesday