This method is based on the two-sample U
-statistic [Kowalski J and Tu XM.] (2007)
], a well established, nonparametric measure of effect based on an investigator-determined scoring mechanism. Our development is modeled after the U
-statistic’s implementation to score the occurrence of a combination of two discrete endpoints in a cardiovascular clinical trial [Moyé LA, et al. (1992)
, Moyé LA, (1991)
, Penicka M, et al. (2007)
]. A recent use of this statistic in medical research has been its application to multivariate ordinal data [Wittkowski KM, et al. (2004)].
In its simplest adaptation, the U-statistic “builds itself up” from a prospectively selected scoring procedure. Let there be n observations in the control group. Let each of the n subjects in the control group have a continuous endpoint measure xi, i = 1, 2, 3, ..., n. Similarly, let the primary endpoint measure for each of the m subjects in the active group be indexed by yj, j = 1, 2, 3, …, m.
-statistic requires a simple scoring mechanism, denoted by i,j
. This is the assignment of a score designed in this paper based on comparing the ith
subject in the control group with the jth
subject in the active group. The score may be as simple as i,j
= 1 if xi
= 0 if xi = yj
; or i,j
= −1 if xi
. Since each of the n
control group subjects will be compared to each of the m
active group subjects, there are nm
comparisons. The U
-score statistic, We
is simply the average of these nm
The normalized statistic based on these scores for a test of the null hypothesis (H0
) of no treatment effect versus the alternative hypothesis (Ha
) of a change in the distribution of the yj
’s based on the treatment is
Under mild regulatory conditions and adequate sample size, we assume that (2) follows a standard normal distribution, then we can compute the sample size from (assuming n
is the probability of a type I error, β
is the probability of a type II error, and Zc
is the cth
percentile value from the standard normal distribution, Alternatively, power may be computed from
) is the cumulative distribution function of the standard normal distribution.
However, the adoption of this statistic requires a careful justification of the scoring mechanism required for the response variables (endpoints). The setting for our evaluations is that of a randomized clinical trial with both a control and an active group. We will construct the score statistic in two cases:
- Case 1. A dichotomous right censored measure combined with a single continuous response variable.
- Case 2. A dichotomous right censored measure combined with two continuous response variables to be used in a hierarchy determined by the precision of the two response variables.
The mathematics of Case 1 will be developed in detail, and then applied to the Case 2 scenario, which is the scenario that we face in the CCTRN network cell therapy studies.
Case 1: A dichotomous right censored measure combined with a single continuous response variable
The investigators’ goal is to compare the change in the measure of a single continuous response variable over time in the control group to the change in that same variable in the active group. Left ventricular ejection fraction (LVEF) is an example of a commonly measured continuous endpoint. LVEF is the percent of the blood in the left ventricle ejected at each beat (for subjects without heart disease LVEF is typically larger than 80%.). To assess changes over time in a variable such as LVEF, there should be a measurement at baseline and at the end of the study. However, the investigators recognize that this goal may not be achievable in all subjects because of the occurrence of death or another SCE. We will assume that (as is the case with LVEF) an increase in the response variable over time corresponds to improved health status.
be this continuous endpoint variable. Then, for the ith
subject in the control group, i
= 1, 2, 3, …, n
, let di
) = ri,2
) − ri,1
) be the change in this endpoint variable over the duration of the study. Assume that di
) has mean μΔR
) and variance
. Analogously, let dj
) = rj,2
) − rj,1
) be the change in the endpoint measure for the jth
subject in the active group, which mean μΔR
)and known variance
. Under the null hypothesis of the study, μΔR
) = μΔR
). If we assume that larger values of μΔR
correspond to improved health, then under the alternative hypothesis, the researchers expect that μΔR
) < μΔR
However, the occurrence of a significant event (SCE) (e.g., a death, a recurrent myocardial infarction (MI), can affect the follow-up measurement of the continuous variable. The hallmark of the SCE is that 1) its occurrence during the trial either precludes the follow-up measurement (as in the case of death), or perturbs the measurement to the point that the effect of therapy can be difficult to assess (e.g., the occurrence of an intercurrent heart attack), and 2) the SCE event rates in the randomized groups may themselves be related to the therapy effect. The occurrence of an intervening SCE (itself an underpowered evaluation in a small study) reduces the power of the LVEF measure by decreasing the number of subjects who survive to have the follow-up measurement.
In this case we define the scoring mechanism, i,,j
|i,,j = 1||if both the ith subject if the control group and the jth subject in the active group experience an SCE during the study, and the time to event for the control group subject is less than the time to event for the active group subject.|
|i,j = 1||if the ith subject in the control group experiences an SCE during the study and the jth subject in the active group does not experience an SCE during the study.|
|i,,j = −1||if both the ith subject in the control group and the jth subject in the active group experiences an SCE during the course of the study, but the time to event for the control group subject is greater than the time to event for the active group subject.|
|i,,j = −1||if the ith subject in the control group does not experience an SCE during the study and the jth subject in the active group does experience an SCE during the study.|
|i,,j = c||if neither the ith subject in the control group nor the jth subject in the active group experience an SCE during the study, and the change in the continuous measure r for the control group subject is less than the change in continuous measure for the active group subject.|
|i,,j = c||If neither the ith subject in the control group nor the jth subject in the active group experiences an SCE during the study, and the change in the continuous measure ri for the control group subject is greater than the change in continuous measure for the active group subject, rj.|
|i,,j = 0||otherwise.|
Under this mechanism, the occurrence of an early SCE (e.g., a death) in one group is considered worse than a subject survival or a later occurring SCE in the other treatment group. If both subjects in the comparison have no SCE, then the change in the response variable is compared.
With some additional notation, the assignment of this scoring system permits the computation of the mean and variance of We under the null and alternative hypothesis.
Define CX(i)(E, R) as the endpoint status of the ith subject in the control group, and CY(j)(E, R) as the endpoint status of the jth subject in the active group. We will use this notation to allow us to capture either 1) the time to the occurrence of an SCE if one has occurred during the course of the trial, or 2) the change in the continuous variable if an SCE has not occurred.
If an SCE has occurred for the ith subject in the control group, then CX(i)(E, R) = CX(i)(+, R), and its value is the time to the occurrence of the SCE. Since the SCE has occurred during the course of the study, then 0 ≤ CX(i)(+, R) ≤ T where T is the maximum time a subject is to be followed in the research protocol. If an SCE has not occurred, then CX(i)(E, R) = CX(i)(–, R), and we set CX(i)(–, R) to equal the change in the continuous measure. Identical notation applies to the jth subject in the active group, CY(j)(E, R).
For example, if in a 180 day clinical trial, the 4th subject in the control group died on day 117, then CX(4)(E, R) = CX(4)(+, R) = 117, the positive sign signifying that the SCE event occurred. Alternatively, if the 5th subject in the active group survived the trial and experienced a six unit increase in the continuous response variable, then CY(5)(E, R) = CY(5)(–, R) = 6, the minus sign in CY(5)(–, R) indicating that no SCE occurred during the study.
Using this notation and letting 1XA
be the indicator function that takes the value of 1 when x
is a member of set A and 0 otherwise, we can write the score function i,j
For this function we can compute its expected value under both the null (E
]) and alternative E
] hypotheses (Appendix A