WEEK SIX ASSIGNMENT MHA 610

A Crossover Clinical Trial

Alban Evans

MHA 610 Introductions to Biostatistics (NDF1506A)

March 2, 2015

Instructor: Glenn Cummings

Carry over effects

Rx	group	AUC_period1	AUC_period2	(p1-mean)^2	(p2-mean)^2
standard	2	0.597	0.641818182	0.037143246	0.293646024
standard	2	0.808888889	0.405	0.000367223	0.334855703
standard	2	1.198333333	0.655	0.166960092	0.169805236
standard	2	0.115111111	0.3	0.455105019	0.015358492
standard	2	0.584285714	0.590666667	0.042205641	0.288185117
standard	2	0.512	0.39	0.077131637	0.251906385
standard	2	1.14	0.96	0.122691994	0.208248392
standard	2	0.843333333	0.343333333	0.002873764	0.331961079
standard	2	0.731666667	0.531111111	0.003370866	0.331388505
standard	2	0.989090909	0.850769231	0.039746435	0.290831511
standard	2	0.612	0.405714286	0.031586471	0.299699236
standard	2	1.345	0.875	0.308329404	0.073281195
sum		9.476709957	6.948412809	1.287511791	2.889166877
mean		0.78972583	0.579034401
variance				0.117046526	0.262651534
standard deviation				0.342120631	0.512495399

Rx	group	AUC_period1	AUC_period2	(p1-mean)^2	(p2-mean)^2
experimental	1	0.4475	0.442222222	0.034883984	0.305072842
experimental	1	0.66	1.16	0.000661902	0.027371645
experimental	1	0.815	1.081428571	0.032662412	0.007546806
experimental	1	0.948	0.914285714	0.098424915	0.006443354
experimental	1	0.854285714	0.7	0.048405794	0.086763365
experimental	1	0.408	1.473333333	0.051199266	0.229227527
experimental	1	0.666666667	1.473333333	0.001049379	0.229227527
experimental	1	0.261818182	0.508571429	0.138722254	0.236181215
experimental	1	0.667142857	0.774	0.001080457	0.048645045
experimental	1	0.717142857	0.8075	0.006867488	0.034990028
experimental	1	0.582857143	1.286666667	0.002643544	0.085328515
experimental	1	0.582857143	1.313333333	0.002643544	0.10161885
sum		7.611270563	11.9346746	0.419244939	1.398416718
mean		0.634272547	0.994556217
variance				0.038113176	0.127128793
standard deviation				0.195225962	0.356551248

There is no significant carryover effect in this clinical trial because the AUC totals differ significantly. AUC totals for group one is7.6 and 11.9 for period one and two respectively whereas for group two the AUC totals are 9.4 and 6.94 for period one and two respectively. The variances of the AUC totals are not identical.

Treatment effect

Rx	group	AUC_period1	AUC_period2	AUC Difference	(AUC Difference-mean)^2
standard	2	0.597	0.641818182	-0.044818182	0.065285161
standard	2	0.808888889	0.405	0.403888889	0.037325259
standard	2	1.198333333	0.655	0.543333333	0.110650637
standard	2	0.115111111	0.3	-0.184888889	0.156483788
standard	2	0.584285714	0.590666667	-0.006380952	0.047120419
standard	2	0.512	0.39	0.122	0.00786617
standard	2	1.14	0.96	0.18	0.000941964
standard	2	0.843333333	0.343333333	0.5	0.083699449
standard	2	0.731666667	0.531111111	0.200555556	0.000102736
standard	2	0.989090909	0.850769231	0.138321678	0.005237381
standard	2	0.612	0.405714286	0.206285714	1.94103E-05
standard	2	1.345	0.875	0.47	0.067240935
sum				2.528297147	0.581973308
mean				0.210691429
variance					0.052906664
standard deviation					0.230014487

Rx	group	AUC_period1	AUC_period2	AUC Difference	(AUC Difference-mean)^2
experimental	1	0.4475	0.442222222	0.005277778	0.133635172
experimental	1	0.66	1.16	-0.5	0.019520653
experimental	1	0.815	1.081428571	-0.266428571	0.00880878
experimental	1	0.948	0.914285714	0.033714286	0.155234389
experimental	1	0.854285714	0.7	0.154285714	0.264781651
experimental	1	0.408	1.473333333	-1.065333333	0.497095028
experimental	1	0.666666667	1.473333333	-0.806666667	0.19925778
experimental	1	0.261818182	0.508571429	-0.246753247	0.012889157
experimental	1	0.667142857	0.774	-0.106857143	0.064225005
experimental	1	0.717142857	0.8075	-0.090357143	0.07286033
experimental	1	0.582857143	1.286666667	-0.703809524	0.118010012
experimental	1	0.582857143	1.313333333	-0.73047619	0.137042502
sum				-4.32340404	1.683360458
mean				-0.36028367
variance					0.153032769
standard deviation					0.39119403

Comparing the group one and two differences, we see that there is a significant difference between the group one and group two AUC differences. Looking at the sums and means of the two groups, group one has a sum of -4.3 and a mean of -3.6 whereas group two has a sum of 2.5 and a mean of 0.2.

Two-by-two crossover trial

Rx	group	AUC_period1	AUC_period2	difference
experimental	1	0.4475	0.442222222	0.005278
experimental	1	0.66	1.16	-0.5
experimental	1	0.815	1.081428571	-0.26643
experimental	1	0.948	0.914285714	0.033714
experimental	1	0.854285714	0.7	0.154286
experimental	1	0.408	1.473333333	-1.06533
experimental	1	0.666666667	1.473333333	-0.80667
experimental	1	0.261818182	0.508571429	-0.24675
experimental	1	0.667142857	0.774	-0.10686
experimental	1	0.717142857	0.8075	-0.09036
experimental	1	0.582857143	1.286666667	-0.70381
experimental	1	0.582857143	1.313333333	-0.73048
average x̄				-0.36028

Rx	group	AUC_period1	AUC_period2	difference
standard	2	0.597	0.641818182	-0.044818
standard	2	0.808888889	0.405	0.4038889
standard	2	1.198333333	0.655	0.5433333
standard	2	0.115111111	0.3	-0.184889
standard	2	0.584285714	0.590666667	-0.006381
standard	2	0.512	0.39	0.122
standard	2	1.14	0.96	0.18
standard	2	0.843333333	0.343333333	0.5
standard	2	0.731666667	0.531111111	0.2005556
standard	2	0.989090909	0.850769231	0.1383217
standard	2	0.612	0.405714286	0.2062857
standard	2	1.345	0.875	0.47
average y				0.2106914

Considering the random variable Z = x̄ - y from the calculations we find that Z= -0.1496. Since the value is not equal to zero we find out that there was a treatment effect when treatment A and B were administered.

It is conventional for data to be pretested for evidence of carry over in the cross over trials analysis. The outcome of a given treatment always varies according to the position it has in the sequence treatments. This approach relies on the questionable assumption that all carry over are absent when the statically tests fails to find any. For instance, Chisholm et al. (1996) in the study of hypercholesterolemia concluded that there was no carry over when an analysis of variance found no statistically significant interaction between treatment sequence and outcome (Chisholm et al., 1996). However such tests had limited power and could rule out a type II error (wrong conclusion that there is no carry over effect) (Senn, 1993).

If carry over was detected convention suggests this may be dealt with in the analysis in one of two ways. The usual approach was to treat the study as though it were a parallel group trial and confine analysis to the first period alone. The advantages of the crossover were lost, with the wasted expense of discarding the data from the second period. More importantly, the significance test of comparing the first periods may be invalid (Freeman, 1989). A second approach that is applicable only to studies with at least three treatment periods (ABB/BAA) is to model the carry over effect and using it to adjust the estimate treatment. Such approaches, while statistically elegant, were based on assumptions which can rarely be justified in practice (Senn, 1993).

The best advice is therefore to avoid the usage of a crossover design if there is any good reason to suppose that carry over effects are likely to occur. A readable approach to the problems of designing and analyzing crossover trials are provided by Senn (1993).

Bonus

Histograms, box plots, or scatter plots would not work with these data because it is a comparison between two modes of treatment administered to one patient.

If the preliminary test for differential carryover were not significant, then the data from both periods were analyzed in the usual manner. Recent work, however, had a revelation that this 2-stage analysis performed poorly because the unconditional Type I error rate operated at a much higher level than desired. We cannot go into the specific details here, but part of the reason for this is that, the test for differential carryover and the test for treatment differences in the first period are highly correlated and they do not act independently.

Even worse, this two-stage approach can lead to losing one-half of the data. If differential carryover effects are of concern, then a best approach would be the use of a study design that can account for them. Before the development of a general statistical model and investigations into its implications for, we more definitions are required (Piantadosi, 2005).

References

Chisholm, A., Mann, J., Sutherland, W., Duncan, A., Skeaff, M., & Frampton, C. (January 01, 1996). Effect on lipoprotein profile of replacing butter with margarine in a low fat diet: randomized crossover study with hypercholesterolemia subjects. Bmj (clinical Research Ed.), 312, 7036, 931-4.

Freeman, P. R. (January 01, 1989). The performance of the two-stage analysis of two-treatment, two-period crossover trials. Statistics in Medicine, 8, 12, 1421-32.

Piantadosi, S. (2005). Clinical trials: A methodologic perspective. Hoboken, N.J: Wiley-Interscience.

Senn SJ.Cross-over trials in clinical research. Chichester: John Wiley, 1993.