The article The Ames Salmonell/ Microsome Mutagenicity Assay: Issues of Inference and Validation [ Journal of American Statistical Association (1989) 84: 651 661] discusses the

The article €œ The Ames Salmonell/ Microsome Mutagenicity Assay: Issues of Inference and Validation€ [ Journal of American Statistical Association (1989) 84: 651€“ 661] discusses the importance of chemically induced mutation for human health and the biological basis for the primary in vitro assay for mutagenicity, the Ames Salmonell/ microsome assay. In an Ames test, the ­response obtained from a single sample is the number of visible colonies that result from plating approximately 108 microbes. A common protocol for an Ames test includes multiple samples at a control dose and four or five logarithmically spaced doses of a test compound. The following data are from one such experiment with 20 samples per dose level. The dose levels were mg/ sample.

We want to determine whether there is an increasing trend in the mean number of colonies as the dose level increases. One method of making such a determination is to use a contrast with constants ai determined in the following fashion. ­Suppose the treatment levels are t values of a continuous variable x: x1, x2, . . . , xt. Let ai = xi €“ and = ˆ‘ a. If is significantly different from zero and positive, then we state there is a positive trend in the μis. If is significantly different from zero and negative, then we state there is a negative trend in the μis. In this experiment, the dose levels are the treatments x1 = 0, x2 = .3, x3 = 1.0, x4 = 3.0, and x5 = 10.0 with x = 2.86. Thus, the coefficients for the contrasts are a1 = 0 €“ 2.86 = €“ 2.86, a2 = 0.3 €“ 2.86 = €“ 2.56, a3 = 1.0 €“ 2.86 = €“ 1.86, a4 = 3.0 €“ 2.86 = 1.14, and a5 = 10.0 €“ 2.86 = + 7.14. We therefore need to evaluate the significance of the ­following contrast in the treatment means given by €“2.86c €“ 2.56. 3 €“ 1.861.0 + 0.143.0 + 7.1410.0. If the contrast is significantly different from zero and is positive, we conclude that there is an ­increasing trend in the dose means.
a. Test whether there is an increasing trend in the dose mean. Use a = .05.
b. Do there appear to be any violations in the conditions necessary to conduct the test in part (a)? If there are violations, suggest a method that would enable us to validly test whether the positive trend exists.

Dose Number of Visible Colonies Control 11 13 14 14 15 15 15 15 16 17 17 18 18 19 20 21 22 23 25 27 178 17.5 3 39 39 42 43 44 45 46 50 50 50 51 52 52 52 55 61 62 63 67 70 51.7 81.0 1.0 88 90 92 92 102 104 04 106 109 113 117 117 119 119 120 120 121 122 130 133 1109 175.4 3.0 222 233 251 251 253 255 259 275 276 283 284 294 299 301 306 312 315 323 337 340 283.5 1,131.5 10.0 562 587 595 604 623 666 689 692 701 702 703 706 710 714 733 739 763 782 786 789 692.3 4,584.4