Testing the Trend of a Response Curve to an Increasing Sequence of Doses: A SAS Macro to Automate the Analysis

In animal toxicity studies, treatments are composed of a series of increasing doses including a vehicle control. The Tukey, Ciminera and Heyse (1985) trend test, referred to as Tukey’s trend test, is often performed to (1) test for linear trend in the response curve corresponding to three dose scales and possibly different dose ranges by deleting the higher doses, (2) identify the NOSTASOT (NO STAtistical Significance Of Trend) dose: the maximum dose concentration which is not significantly different from the vehicle group. This is an iterative and tedious process. A SAS macro was developed to automate Tukey’s trend test. The key to automate this process is to convert the sequential use of Tukey’s trend test in identifying NOSTASOT dose into a set of contrast tests where each contrast corresponds to a different dose scale and dose range combination. In this macro, we utilized and implemented an algorithm for contrast coefficient calculation that works for both balanced and unbalanced studies with equal or unequal dose intervals and equal or unequal sample sizes for one-way designs. The output contrast coefficients can be entered in a set of contrast statements of SAS PROC GLM or PROC MIXED to perform trend test. The macro requires minimum manual intervention to input information about doses and sample sizes. INTRODUCTION Tukey’s trend test is used frequently in toxicity studies for two purposes (1) to detect a linear trend in the response over the dose range. A nonzero (increasing or decreasing) trend is an evidence of treatment affecting the response of interest, thus an indication of toxicity; (2) to identify the NOSTASOT dose. The procedure for (2) involves a sequence of trend tests, each applied to different range of doses. This iterative procedure can be described as follows. Suppose 1 d , , n d are the doses to be examined in the study with 0 1 d being control. First, we apply trend test through the entire dose range 1 d n d . If the trend is not statistically significant (P>0.05), then the NOSTASOT dose is defined to be the highest dose level in the range. If the trend is statistically significant, we delete the highest dose n d and reapply trend test through the range 1 d 1 n d . Repeat trend analysis in this fashion until P>0.05 is observed. The NOSTASOT dose is defined to be the largest dose that yields P 0.05 and no additional P-values are computed. If the lowest active dose group yields P 0.05, the NOSTASOT dose is specified only as below the lowest dose in the experiment. There are three sets of “carriers” n i i X 1 } { frequently used as candidate dose scalings. They span a fairly broad range of possible dose-response relationships: for n i ,..., 1 (i) arithmetic i i d x (ii) ordinal (equally-spaced) i xi (iii) arithmetic-logarithmic i i d x log for 1 i , and ) log (log log 2 3 2 3 1 2 2 1 d d d d d d d x The statistical assessment of Tukey’s trend is performed on each the three candidate scales separately. The minimum p-value is reported. This procedure may result in an inflated type I error rate. This can be justified in toxicity studies since the primary concern is the safety of the treatment. For efficacy studies, Cappizzi et al (1992) proposed an adjusted trend test by adjusting the p-value using the joint distribution, a trivariate t-distribution, of the three test statistics for the three dose scalings. Similar to Tukey’s trend test, this adjusted trend test is applicable for oneway designs or balanced two-way designs. Further extension to unbalanced two-way designs can be found in Quan and Cappizzi (1999). As described above, the step-down analyses of Tukey’s trend test is a tedious process. Thus, it is highly desirable to simplify this process and to automate the analysis. This is the focus of this paper. Statistically, Tukey’s trend test is a sequence of regression analyses to assess the significance of the slopes of regression lines corresponding to different combinations or dose scales and dose ranges when deleting the higher dose(s). It should be pointed out that when we delete a dose, we delete it from the calculation of contrasts, but not from the model fit (ie the pooled within group variance are always estimated from data in all dose groups). Testing the slope of a regression line is equivalent to testing a contrast among the treatment means where the contrast coefficients are properly constructed (Mehta et al 1984). This analysis is equivalent to the orthogonal polynomial analysis of dose response curve when we want to assess the significance of the linear components of the curve. There is an algorithm available (Roger E. Kirk) to convert the orthogonal polynomial analysis to contrast analysis. In this paper, we utilize the approach of converting Tukey’s trend test to contrast analysis to simplify the recursive trend test process, and then apply the algorithm (Roger E. Kirk) for orthogonal linear component analysis to construct the contrast coefficients. The algorithm, SAS macro source code, and examples are provided. ALGORITHM Assuming there are n i ,..., 1 different dose levels, we use the following notations: i m : sample size at the i th dose level. i X : i th dose level, in one of the three scales. i i c m : contrast coefficient corresponding to i X . k : number of doses being included in the step-down Tukey’s trend test, 3 ,..., 1 , n n k .