Introduction to Crossover Trials

Transcription

1 Introduction to Crossover Trials Stat 6500 Tutorial Project Isaac Blackhurst

2 A crossover trial is a type of randomized control trial. It has advantages over other designed experiments because, under certain conditions, it can more precisely measuring the treatment effect. These types of trials are particularly well suited for comparing treatments for those with chronic conditions. Often, a patient only has one chance to be given a treatment for their condition. For example, after a cancer has gone into remission, there is only opportunity to assign one anti-relapse treatment to that patient. However, if we are not expecting to cure a patient, but instead improve their condition or some other measured outcome, we can give the patient a series of treatments in different time periods, and measure their response to each treatment. If we do this for multiple patients, and randomize the order of assigned treatments, we are conducting a crossover trial. A crossover trial is a type of designed experiment called a Latin square design. For the simplest example comparing only two treatments, group 1 will receive one treatment, call it treatment A, in period 1, and a second treatment, treatment B, in period 2. Group 2 will receive treatment B in period 1 and treatment A in period 2. This is a Latin square because every group sees every treatment over the course of the trial, and in every period, there is a group receiving each treatment. An example of this simple type of Latin square crossover trial design deals with patients suffering from enuresis (bedwetting). These data come from the Matthews text referenced below. Group 1 received treatment A, in this case the treatment of interest, in period 1, and a placebo in period 2, and group 2 received the treatments in the reverse order. The measured outcome for this trial was the number of dry nights observed over the treatment period. In this case the treatment period was two weeks. To see why this type of trial has the potential to more precisely measure the treatment effect, some theory will be presented. The notation comes from the Matthews text. (As does most everything I know about these types of trials.)

3 Before presenting the theory, we should note the problems with assigning every patient to treatment A in the first period, and treatment B in the second period. If we were to do this and we observed that treatment A improve the measured outcome statistically more than treatment B, we cannot conclude that treatment A is better than treatment B. There are two reasons for this. One is that the first treatment, whichever it is, might always do better than the second treatment. This could be considered an order effect, or a trend effect. By randomizing the assignment of each patient to a different order of treatments, we minimize this source of bias in our measurement by having some patients receive the treatments in the reverse order. Another source of bias has to do with what can be called period effects. For example, in one period, all the measurements might be higher, or lower, than another period. Therefore, if treatment A had a statistically different (suppose higher) measured outcome than treatment B, this could be caused by all the measurements in period 1 being higher because of this period effect. To see why randomizing the order gets rid of the period effect, we need to look at the theory below. Also, as one final point before presenting the theory, it should be noted that the periods need to be separated by sufficient time so that in the second time period, we are not measuring residual or left over effects of the treatment given in the previous period. Theory and Notation The notation for the theory is as follows: n 1 and n 2 are the numbers in each group x ij is measured effect in patient i in period j μ is the overall mean

4 π j is the period effect in period j τ A and τ B are the effects of each treatment ξ i is the effect of the ith patient, distributed N(0, σ ξ 2 ) ε ij is the normally distributed N(0, σ ε 2 ) error for patient i in period j τ is τ A τ B or how much different treatment A is than B π is π 1 π 2 or the period effect of period 1 minus period 2 With this notation, a person assigned to the first group has the following equations for the outcomes in each period: x i1 = μ + π 1 + τ A + ξ i + ε i1 for period 1 x i2 = μ + π 2 + τ B + ξ i + ε i2 for period 2 And for the second group, we see very similar equations, with the exception of the treatment effects being reversed: x i1 = μ + π 1 + τ B + ξ i + ε i1 for period 1 x i2 = μ + π 2 + τ A + ξ i + ε i2 for period 2 And the difference for someone in group 1 simplifies into the following equation: d i = x i1 x i2 = π 1 π 2 + τ A τ B + ε i1 ε i2 We have been able to eliminate the overall mean μ this way, and we have also been able to eliminate ξ i. This elimination of ξ i is perhaps one of the biggest advantages of crossover trials, and the reasons why these types of trials have the potential to be more precise in their measurement of the outcome of

5 interest. ξ i can be thought of as all the person specific variables that could be confounding factors in the experiment. For other types of situations, to eliminate the effect we might try to match similar patients with each other, or we could do a regression including possible confounding variables for each individual. Or if it's another designed experiment, we hope that all those confounding factors are eliminated in the randomization to assigned treatments, which they should be. We might, however, be unlucky in our randomization and not eliminate these effects. This approach has the potential to be better, because we are able to control for all those factors and eliminate them by how the experiment was designed. If these were a large source of variation, their elimination could lead to better estimates of the treatment effect. Continuing with the notation, the expectation of the last equation is the following, since the epsilons have mean zero: π + τ The difference for someone in group 2 is, which is the exact same, except for the reverse order of the treatment effects: d i = x i1 x i2 = π 1 π 2 + τ B τ A + ε i1 ε i2 or d i = x i1 x i2 = π 1 π 2 (τ A τ B) + ε i1 ε i2 Which in expectation is:

6 π τ So when you take the expectation of the difference you have: E d 1 d 2 = 2τ This means, for this example, that the average of the difference for group 1 (period 1 outcome minus the period 2 outcome) minus the average of the difference for group 2 (again period 1 outcome minus the period 2 outcome) ends up being twice the difference of the treatment A effect, minus the treatment B effect. This is essentially a two-sample t-test, except that the difference is twice what we would expect. The confidence intervals for a given alpha level are also twice what would be found from a simple two-sample t-test, and so the p-value ends up being the exact same. Continuing with our example about bed wetting, we find that tao equals , and it is significant. We conclude that treatment A improves bedwetting by eliminating on average of two wet nights over a two week period. The code and SAS output for this example are in an appendix. Finally, it makes sense that because this is essentially a two sample t-test, if the assumptions of normality are violated, we should be able to use a non-parametric two sample equivalent, like a Wilcoxon Rank Sum test, making adjustments for the measure of location being twice the treatment difference we are interested in. Also, for more than two groups, we should be able to do something like a one-way ANOVA or Kruskal Wallis non-parametric equivalent. However, I wasn't able to find the necessary time to confirm my guess, and so I can't be sure that I was right in that hypothesis.

7 References Matthews, John, N.S. Introduction to Randomized Controlled Clinical Trials, Second Edition.

8 Appendix of SAS code and output data enuresis; input patient $ period1 period2 difference group $; datalines; a a a a a b b b b b ; data enuresis; set enuresis; length group2 $ 8; group2 = "a"; if group = "a" then group2 = "b"; run; proc ttest data = enuresis; var period1 period2 / crossover = (group group2); run;

9 SAS output Crossover Variable Information Period Response Treatment 1 period1 group 2 period2 group2 Sequence Treatment Period N Mean Std Dev Std Err Minimum Maximum 1 a a b b Diff (1-2) Diff (1-2) Both Diff (1-2) Both Diff (1-2) Sequence Treatment Period Method Mean 95% CL Mean Std Dev 95% CL Std Dev 1 a a b b Diff (1-2) Diff (1-2) Both Diff (1-2) Pooled Both Diff (1-2) Satterthwaite Both Diff (1-2) Pooled Both Diff (1-2) Satterthwaite

10 Treatment Period Method Variances DF t Value Pr > t Diff (1-2) Pooled Equal Diff (1-2) Satterthwaite Unequal Diff (1-2) Pooled Equal Diff (1-2) Satterthwaite Unequal Equality of Variances Method Num DF Den DF F Value Pr > F Folded F

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