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
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
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.
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
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.
Here’s an informal explanation of this t
test. Consider the following schematic representation of the two-by-two crossover trial.
|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
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
), 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
Briefly summarize your findings from this trial. Explain whether the new
treatment appears promising in a 500 words in APA format supported by
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.