The TTEST Procedure

Paired Design

Define the following notation:

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

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.