Question: The article The Ames Salmonell Microsome Mutagenicity Assay Issues

The article “ 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 = ∑ 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.

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