The Xxxxxxx Model Sample Clauses

The Xxxxxxx Model.Β Analyzing a crossover trial is more complex compared to parallel-groups designs. Using a simple pre/post comparison or paired t test without examining the carry-over effects beforehand will result in errors and provide false estimates. Xxxxxxx (1965) developed a model that has the ability to test the effects of treatment, period and carry-over simultaneously for a simple AB/BA crossover trial: π‘¦π‘–π‘—π‘˜ = πœ‡ + 𝑏𝑖𝑗 + πœ‹π‘˜ + πœ™π‘™ + πœ†π‘™ + πœ–π‘–π‘—π‘˜ 𝑖 = 1,2; 𝑗 = 1,2, … , 𝑛𝑖; π‘˜ = 1,2; 𝑙 = 1,2; where πœ‡ is the overall mean, 𝑏𝑖𝑗 is the random effect of the jth subject within the ith sequence, πœ‹π‘˜ is the kth period effect, πœ™π‘™ is the direct effect of the lth treatment, πœ†π‘™ is the carry-over effect of the lth treatment, and πœ–π‘–π‘—π‘˜ is the random error. (Xxxxxxx, 1965). In order to perform the hypothesis testing for direct treatment effects, carry-over effects and period effects, Xxxxxxx asserted that 𝑏𝑖𝑗 and πœ–π‘–π‘—π‘˜ must be assumed to be independent and normally distributed with mean 0 and variance 𝜎2 and 𝜎2, respectively. Therefore, observations from different subjects are 𝑖𝑗 𝑒 considered independent. In an AB/BA trial, the hypothesis of absent carry-over effects (π»π‘œ: πœ†1 = πœ†2) can be tested by using the sum of the observations in the two periods on the same subject: 𝑦𝑖𝑗1 + 𝑦𝑖𝑗2 = 2(πœ‡ + 𝑏𝑖𝑗) + (πœ‹1 + πœ‹2) + (πœ™1 + πœ™2) + πœ†π‘™ + πœ–οΏ½οΏ½π‘—1 + πœ–π‘–π‘—2 where πœ†π‘™ reflects the carry-over effect of the lth treatment following the AB treatment sequence. (Xxxx & Xxxx, 2004). If no carry-over effect is present, then the hypothesis of no direct treatment effects (π»π‘œ: πœ™1 = πœ™2) can be tested by using the difference between the observations in the two periods on the same subject, denoted as follows: 𝑦𝑖𝑗1 βˆ’ 𝑦𝑖𝑗2 = (πœ‹1 βˆ’ πœ‹2) + (βˆ’1)𝑖+1(πœ™1 βˆ’ πœ™2) + (πœ–π‘–π‘—1 βˆ’ πœ–π‘–π‘—2) where (βˆ’1)𝑖+1 = 1 if the subject follows an AB treatment sequence and (βˆ’1)𝑖+1 = βˆ’1 if follows a BA treatment sequence. (Xxxx & Xxxx, 2004). However, if the carry-over effects cannot be ignored or eliminated, only data from the first period can be used and the analysis will be the same as the one for a conventional parallel design. If there are significant clinically important carry-over effects, it is possible that the subject’s disease status or response is affected permanently, in which case the crossover design is less efficient than a conventional independent-groups trial. (Piatadosi, 1997). Nonetheless, some researchers consider these designs still pertinent unde...
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