The TTEST Procedure

AB/BA Crossover Design

Let "A" and "B" denote the two treatment values. Define the following notation:

$\begin{align*} n^\star _1 & = \mbox{number of observations with treatment sequence AB} \\ n^\star _2 & = \mbox{number of observations with treatment sequence BA} \\ y_{11i} & = \mbox{response value of }i\mbox{th observation in sequence AB during period 1,} \; \; i \in \{ 1, \ldots , n^\star _1\} \\ y_{12i} & = \mbox{response value of }i\mbox{th observation in sequence AB during period 2,} \; \; i \in \{ 1, \ldots , n^\star _1\} \\ y_{21i} & = \mbox{response value of }i\mbox{th observation in sequence BA during period 1,} \; \; i \in \{ 1, \ldots , n^\star _2\} \\ y_{22i} & = \mbox{response value of }i\mbox{th observation in sequence BA during period 2,} \; \; i \in \{ 1, \ldots , n^\star _2\} \end{align*}$

So $\{ y_{11i}, \ldots , y_{11{n^\star _1}} \}$ and $\{ y_{22i}, \ldots , y_{22{n^\star _2}} \}$ are all observed at treatment level A, and $\{ y_{12i}, \ldots , y_{12{n^\star _2}} \}$ and $\{ y_{21i}, \ldots , y_{21{n^\star _1}} \}$ are all observed at treatment level B.

Define the period difference for an observation as the difference between period 1 and period 2 response values:

$\mr{pd}_{kji} = y_{k1i} - y_{k2i}$

for $\; k \in \{ 1, 2\} \;$ and $\; i \in \{ 1, \ldots , n^\star _ k\} \;$ . Similarly, the period ratio is the ratio between period 1 and period 2 response values:

$\mr{pr}_{kji} = y_{k1i} / y_{k2i}$

The crossover difference for an observation is the difference between treatment A and treatment B response values:

$\mr{cd}_{kji} = \left\{ \begin{array}{ll} y_{k1i} - y_{k2i} \; \; , & k=1 \\ y_{k2i} - y_{k1i} \; \; , & k=2 \\ \end{array} \right.$

Similarly, the crossover ratio is the ratio between treatment A and treatment B response values:

$\mr{cr}_{kji} = \left\{ \begin{array}{ll} y_{k1i} / y_{k2i} \; \; , & k=1 \\ y_{k2i} / y_{k1i} \; \; , & k=2 \\ \end{array} \right.$

In the absence of the IGNOREPERIOD option in the PROC TTEST statement, the data are split into two groups according to treatment sequence and analyzed as a two-independent-sample design. If DIST= NORMAL, then the analysis of the treatment effect is based on the half period differences $\{ \mr{pd}_{kji} / 2\}$ , and the analysis for the period effect is based on the half crossover differences $\{ \mr{cd}_{kji} / 2\}$ . The computations for the normal difference analysis are the same as in the section Normal Difference (DIST=NORMAL TEST=DIFF) for the two-independent-sample design. The normal ratio analysis without the IGNOREPERIOD option is not supported for the AB/BA crossover design. If DIST= LOGNORMAL, then the analysis of the treatment effect is based on the square root of the period ratios $\{ \sqrt {\mr{pr}_{kji}} \}$ , and the analysis for the period effect is based on the square root of the crossover ratios $\{ \sqrt {\mr{cr}_{kji}} \}$ . The computations are the same as in the section Lognormal Ratio (DIST=LOGNORMAL TEST=RATIO) for the two-independent-sample design.

If the IGNOREPERIOD option is specified, then the treatment effect is analyzed as a paired analysis on the (treatment A, treatment B) response value pairs, regardless of treatment sequence. So the set of pairs is taken to be the concatenation of $\{ (y_{111}, y_{121}), \ldots , (y_{11{n^\star _1}}, y_{12{n^\star _1}}) \}$ and $\{ (y_{221}, y_{211}), \ldots , (y_{22{n^\star _2}}, y_{22{n^\star _2}}) \}$ . The computations are the same as in the section Paired Design.

See Senn (2002, Chapter 3) for a more detailed discussion of the AB/BA crossover design.