# Question

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.

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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