9/10/2012

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 on p. 87
  • P:\data\math\hartlaub\nonparametrics\1SampSgn.doc
  • P:\data\math\hartlaub\nonparametrics\1SampSgn.MPJ

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 68 and 69 on p. 75

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