Random Simulation Methods for Computing Power and Sample Size for Tukey's Trend Test

Tukey’s Trend Test is a simple step-down test for identifying the “No Statistical Significance of Trend” (NOSTASOT) dose [Tukey, Ciminera, and Heyse (1985)]. Obtaining exact power of the experiment and sample size requirements is not always feasible or possible when dealing with this type or other related procedures for handling multiple comparisons. Computer simulations provide a viable alternative to estimate the necessary sample size or power. Methods for obtaining approximate estimates of power and sample size for Tukey’s Trend Test are examined. Specifically, the power and sample size to identify the “No Statistical Significance of Trend” (NOSTASOT) dose is approximated using random simulation techniques. A SAS program is developed to carry out these simulations and estimate sample size and power. The techniques presented here are applicable to a variety of step-down tests used to help account for multiplicity. INTRODUCTION Tukey’s Trend Test is a simple step-down test for identifying the “No Statistical Significance of Trend” (NOSTASOT) dose [Tukey, Ciminera, and Heyse (1985)]. This test is appropriate for doseresponse experiments with a minimum of three, but preferably four or more doses. A trend test is performed on all of the doses. If the trend is significant at alpha, the highest dose is declared statistically significant from placebo (or the lowest dose tested if not placebo), and testing continues. The next trend test excludes the highest dose. If the test is significant, then the next-tohighest dose is declared statistically significant from placebo, and testing continues. This goes on until a test produces a nonsignificant result [Westfall et al. (1999)]. The highest dose found to be non-significant is identified as the NOSTASOT dose. In a typical dose-response clinical trial, the power of the experiment is commonly not based upon the actual procedures that will be used to analyze the results. This may be due to such factors as a very limited prior knowledge for an approximate relationship, and also the multitude of approaches used for handling multiplicity. One approach for estimating experimentwise power is to base required sample-size for a trial on the probability of obtaining at least one statistically significant result, while controlling alpha for the number of possible or interesting comparisons. One conservative method of controlling for alpha is the Bonferroni procedure. This and other related conservative methods often lead to unnecessary expenditures of costly resource. The methods examined here provide a way to compare power and sample size requirements for a number of possible scenarios. In this way, results from the more traditional approach can be assessed and evaluated. TUKEY’S TREND TEST Tukey’s trend test is based upon parametric tests for general linear models. For g treatment groups, the following possible hypotheses are tested: (g) H0g: u1 = u2 = ... = ug at level α (g-1) H0g-1: u1 = u2 = ... = ug-1 at level α (g-2) H0g-2: u1 = u2 = ... = ug-2 at level α . . . (2) H02: u1 = u2 at level α Testing continues until a non-significant result is obtained. The highest dose being tested (in that particular test) is declared the NOSTASOT dose. This NOSTATSOT dose is also referred to as the minimum detectable dose (MDD), [Tamhane, Hochberg, and Dunnett (1996)]. DOSE-RESPONSE RELATIONSHIPS There are two major types of response curves. The ‘S’ shaped curve is characterized by little or no response up to a certain dose level, followed by a steep rise in the curve and then a plateauing of the response [Westfall et al. (1999)]. The ‘U’ shaped (or technically the inverted ‘U’ shaped) curve is similar to the ‘S’ shaped curve, however the response rate declines after a certain dose level. The ‘S’ shaped curve is the most common, while the occurrence of the ‘U’ shaped curve is rare. Selection of the doses to be examined in the dose-response experiment determines the observed dose-response relationship. If all active doses are selected on the steep rise of the ‘S’ curve, then the observed dose-response relationship will appear to be linear. If active doses range from the steep rise in the curve to the plateauing level, the dose-response relationship will appear to be log-linear. If active doses range from the initial flat portion of the curve to the steep rise in the curve, then the dose-response relationship will appear to take on a quadratic appearance. There are a number of approaches for the construction of contrast matrices used for evaluating the dose-response relationship. Among the more common assumptions of the doseresponse relationships are that the effect is linear in dose order, linear in actual dose, or linear in the log-ordinal dose. In addition, a contrast that assumes a linear in the log actual dose relationship, can also be constructed [Westfall et al (1999)]. Other approaches include constructing all pairwise contrasts, Helmert contrasts, and Reverse Helmert contrasts [Tamhane, Hochberg, and Dunnett (1996)]. Tukey, Ciminera, and Heyse (1985) recommend the use of the following contrasts, linear in dose order (ordinal), linear in actual dose (arithmetic), and logarithmic dose (with displaced control value). As previously mentioned, the selection of doses impacts the observed dose-response relationship. However, as long as the monotonicity assumption is not grossly violated and the selected doses lie on some portion of the steep rise of the curve, the trend test is reasonably robust. The trend test is not appropriate for the ‘U’ shaped dose response curve.