The TTEST Procedure

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.