In this representation, Treatment A is the direct effect of treatment A on each patient’s response (AUC value) and similarly for Treatment B; Period One is the effect of period one on each patient’s response and similarly for Period Two. (We are assuming there are no carryover effects.)
Here’s an informal explanation of this t test. Consider the following schematic representation of the two-by-two crossover trial.
Now, consider first the individuals in group one. During Period One, their responses, (i.e., AUC_period1 values), are estimating effects due to treatment A and period one. During Period Two, their responses (i.e., AUC_period2 values) are estimating effects due to treatment B and period two. So when we take the average of the group one AUC_period1 – AUC_period2 values, (let’s call this average x̄), we have a combined estimate of the effects (Treatment A – Treatment B) + (Period 1 – Period 2).
Next, consider the individuals in group two. When we take the average of the group two AUC_period1 – AUC_period2 values (let’s call this average y), we have a combined estimate of the effects (Treatment B – Treatment A) + (Period 1 – Period 2).
Lastly, consider the random variable Z = x̄ – y. This random variable estimates solely the quantity (Treatment A – Treatment B); the period effects (Period 1 – Period 2) cancel out. Under the null hypothesis of no treatment effects, (Treatment A – Treatment B) = 0, so the mean of Z should be zero. The two sample t test for treatment effects outlined above is equivalent to the t test of whether the mean of Z equals zero. Note that since we have equal numbers of patients in group one and group two, there was no need to take sample means when we constructed our t test; but in general, with unequal sample sizes, you should work with sample means when performing the t tests.
Briefly summarize your findings from this trial. Explain whether the new treatment appears promising in a 500 words in APA format supported by scholarly sources.
BONUS. Graphical representations of the findings can be quite illuminating. As a bonus, you are asked to prepare graphical representation(s) of the data. For example, you might prepare a simple plot of mean responses (mean AUC values) for each treatment arm and for each period. Or, you could give patient profile plots of individual AUC values by period and treatment. Describe whether histograms, boxplots, or scatter plots would work with these data. If you assume that there are no significant carryovers or period effects in this trial, explain how you would display the treatment effects in a 250 words in APA format supported by scholarly sources.
| Group | Period One | Period Two |
| T1. AB Sequence | Treatment A + Period One | Treatment B + Period Two |
| 2. BA Sequence | Treatment B + Period One | Treatment A + Period Two |
Now, consider first the individuals in group one. During Period One, their responses, (i.e., AUC_period1 values), are estimating effects due to treatment A and period one. During Period Two, their responses (i.e., AUC_period2 values) are estimating effects due to treatment B and period two. So when we take the average of the group one AUC_period1 – AUC_period2 values, (let’s call this average x̄), we have a combined estimate of the effects (Treatment A – Treatment B) + (Period 1 – Period 2).
Next, consider the individuals in group two. When we take the average of the group two AUC_period1 – AUC_period2 values (let’s call this average y), we have a combined estimate of the effects (Treatment B – Treatment A) + (Period 1 – Period 2).
Lastly, consider the random variable Z = x̄ – y. This random variable estimates solely the quantity (Treatment A – Treatment B); the period effects (Period 1 – Period 2) cancel out. Under the null hypothesis of no treatment effects, (Treatment A – Treatment B) = 0, so the mean of Z should be zero. The two sample t test for treatment effects outlined above is equivalent to the t test of whether the mean of Z equals zero. Note that since we have equal numbers of patients in group one and group two, there was no need to take sample means when we constructed our t test; but in general, with unequal sample sizes, you should work with sample means when performing the t tests.
Briefly summarize your findings from this trial. Explain whether the new treatment appears promising in a 500 words in APA format supported by scholarly sources.
BONUS. Graphical representations of the findings can be quite illuminating. As a bonus, you are asked to prepare graphical representation(s) of the data. For example, you might prepare a simple plot of mean responses (mean AUC values) for each treatment arm and for each period. Or, you could give patient profile plots of individual AUC values by period and treatment. Describe whether histograms, boxplots, or scatter plots would work with these data. If you assume that there are no significant carryovers or period effects in this trial, explain how you would display the treatment effects in a 250 words in APA format supported by scholarly sources.
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