Paired Design :: SAS/STAT(R) 12.3 User's Guide

Paired Design

Define the following notation:

$\displaystyle n^\star$	$\displaystyle = \mbox{number of observations in data set}$
$\displaystyle y_{1i}$	$\displaystyle = \mbox{value of \Mathtext{i}th observation for first PAIRED variable,} \; \; i \in \{ 1, \ldots , n^\star \}$
$\displaystyle y_{2i}$	$\displaystyle = \mbox{value of \Mathtext{i}th observation for second PAIRED variable,} \; \; i \in \{ 1, \ldots , n^\star \}$
$\displaystyle f_ i$	$\displaystyle = \mbox{frequency of \Mathtext{i}th observation,} \; \; i \in \{ 1, \ldots , n^\star \}$
$\displaystyle w_ i$	$\displaystyle = \mbox{weight of \Mathtext{i}th observation,} \; \; i \in \{ 1, \ldots , n^\star \}$
$\displaystyle n$	$\displaystyle = \mbox{sample size} = \sum _ i^{n^\star } f_ i$

Normal Difference (DIST=NORMAL TEST=DIFF)

The analysis is the same as the analysis for the one-sample design in the section Normal Data (DIST=NORMAL) based on the differences

$d_ i = y_{1i} - y_{2i} \; \; , \; \; i \in \{ 1, \ldots , n^\star \}$

Lognormal Ratio (DIST=LOGNORMAL TEST=RATIO)

The analysis is the same as the analysis for the one-sample design in the section Lognormal Data (DIST=LOGNORMAL) based on the ratios

$r_ i = y_{1i} / y_{2i} \; \; , \; \; i \in \{ 1, \ldots , n^\star \}$

Normal Ratio (DIST=NORMAL TEST=RATIO)

The hypothesis $H_0\colon \mu _1 / \mu _2 = \mu _0$ , where $\mu _1$ and $\mu _2$ are the means of the first and second PAIRED variables, respectively, can be rewritten as $H_0\colon \mu _1 - \mu _0\mu _2 = 0$ . The t value and p-value are computed in the same way as in the one-sample design in the section Normal Data (DIST=NORMAL) based on the transformed values

$z_ i = y_{1i} - \mu _0 y_{2i} \; \; , \; \; i \in \{ 1, \ldots , n^\star \}$

Estimates and confidence limits are not computed for this situation.

The TTEST Procedure

Paired Design

Normal Difference (DIST=NORMAL TEST=DIFF)

Lognormal Ratio (DIST=LOGNORMAL TEST=RATIO)

Normal Ratio (DIST=NORMAL TEST=RATIO)