**Background:** Randomized controlled trials are the gold standard for clinical research. Biostatisticians are heavily involved in such trials, from the planning stage (e.g., sample size and power considerations) through the analysis of findings (e.g., estimation of treatment effects). In this assignment, we will examine treatment outcomes in a two treatment, two period (two-by-two) crossover design.

In the two-by-two crossover design, subjects are randomly assigned to one of two groups. The first group initially receives treatment A in the first period of the trial followed by treatment B in the second period of the trial, and the other group initially receives treatment B in the first period of the trial followed by treatment A in the second period. The response, or primary endpoint of the trial, is measured at least twice in each patient, at the end of the first period and again at the end of the second period. Each patient is his or her own control for comparison of treatment A and treatment B.

Crossover designs are used when the treatments alleviate a condition, rather than effect a cure. After the response to the treatment administered in the first period is measured, there is a washout period in which any lingering effect of the treatment administered in the first period dissipates, and then the response to the second treatment is measured.

An advantage of a crossover design is increased precision afforded by comparison of both treatments on the same subject, compared to a parallel group clinical trial (in which patients are randomized onto different treatment arms). Disadvantages of crossover trials are complex statistical analyses of findings (typically, by complex analyses of variance), potential difficulties in separating the treatment effects from the time effect (patients may respond differently in the first period and the second period), and the carryover effect (the effect of the treatment given in the first period may not totally wash out, but may carry over onto the second period).

We will give a simple example of a two-by-two crossover trial, and undertake analyses of the trial results via *t* tests. The trial was meant to assess the efficacy of a new experimental therapy for interstitial cystitis (IC). Interstitial cystitis is a chronic bladder condition affecting primarily women; symptoms include bladder pressure and pain, urgency, and occasionally pelvic pain. The new experimental therapy was meant to reduce pain and urgency relative to standard therapy. A total of 24 patients were enrolled in the trial; trial results are given in the Excel workbook titled MHA610_Week 4_Assignment_Crossover_Trial_Data.xls.

Open the workbook, and examine the worksheet. The first row contains column headings, and the next 24 rows represent the 24 patients entered into the trial. The group one patients received experimental therapy in the first period of the trial followed by standard therapy in the second period of the trial. The group two patients received standard therapy in the first period of the trial followed by experimental therapy in the second period.

The primary outcome of the trial was an area under the curve (AUC) calculation of relative pain and urgency the patient experienced following therapy: the smaller the AUC, the less severe the patient’s pain and urgency. AUC_period1 denotes each patient’s AUC during the first period of the trial, and AUC_period2 denotes the patient’s AUC during the second period of the trial. The column headed Rx denotes the treatment each patient received during the first period of the trial.

- We will first test for carryover effects.
- The
*t*test formulation for the test for carryover proceeds as follows: calculate the total (sum) of the AUC_period1 and AUC_period2 values for each patient in group one (12 patients) and separately for each patient in group two (12 patients). - The test for carryover is the two sample
*t*test for assessing whether these AUC totals differ significantly between group one and group two under the assumption that the variances of the AUC totals in the two groups are identical. - Calculate the sample means and standard deviations for the AUC totals for each group, and perform the two sample
*t*tests. Analyze whether there is a significant carryover effect in this clinical trial.

- The
- We will next test for treatment effects.
- The
*t*test formulation for assessing treatment effects proceeds as follows:- Calculate the difference of the AUC values for each patient in group one, that is, the 12 individual AUC_period1 – AUC_period2 values, and similarly calculate for each patient in group two.
- If there is no treatment effect, one would expect the AUC_period1 and AUC period 2 values to be similar, except perhaps for an offset due to period effects; we need to account for potential period effects when we compare the group one and group two AUC differences.
- It turns out that the
*t*test for a treatment effect is the two sample*t*test for assessing whether these AUC_period1 – AUC_period2 differences differ significantly between group one and group two, under the assumption that the variances of the AUC differences are the same in the two groups.

- Calculate the sample means and standard deviations for the AUC differences as defined above in each group, and perform the two sample
*t*test. Analyze whether there a significant treatment effect in this clinical trial.

- The

Here’s an informal explanation of this *t* test. Consider the following schematic representation of the two-by-two crossover trial.

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 |

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.)

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.

Carefully review the Grading Rubric for the criteria that will be used to evaluate your assignment.

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