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www.downloadslide.com 14.6 Other **Applications** 735 statistical tests, several additional applications of the χ 2 test are worth mention as a matter of interest. For example, suppose that we wish to test a hypothesis stating that a population possesses a normal probability distribution. The cells of a sample frequency histogram would correspond to the k cells of the multinomial experiment, and the observed cell frequencies would be the number of measurements falling into each cell of the histogram. Given the hypothesized normal probability distribution for the population, we could use the areas under the normal curve to calculate the theoretical cell probabilities and hence the expected cell frequencies. MLEs must be employed when µ and σ are unspecified for the normal population, and these parameters must be estimated to obtain the estimated cell probabilities. The construction of a two-way table to investigate dependency between two classifications can be extended to three or more classifications. For example, if we wish to test the mutual independence of three classifications, we would employ a threedimensional “table.” The reasoning and methodology associated **with** the analysis of both the two- and three-way tables are identical although the analysis of the three-way table is a bit more complex. A third and interesting application of our methodology would be its use in the investigation of the rate of change of a multinomial (or binomial) population as a function of time. For example, we might study the problem-solving ability of a human (or any animal) subjected to an educational program and tested over time. If, for instance, the human is tested at prescribed intervals of time and the test is of the yes or no type, yielding a number of correct answers y that would follow a binomial probability distribution, we would be interested in the behavior of the probability of a correct response p as a function of time. If the number of correct responses was recorded for c time periods, the data would fall in a 2 × c table similar to that in Example 14.4 (Section 14.5). We would then be interested in testing the hypothesis that p is equal to a constant—that is, that no learning has occurred—and we would then proceed to more interesting hypotheses to determine whether the data present sufficient evidence to indicate a gradual (say, linear) change over time as opposed to an abrupt change at some point in time. The procedures that we have described could be extended to decisions involving more than two alternatives. You will observe that our change over time example is common to business, to industry, and to many other fields, including the social sciences. For example, we might wish to study the rate of consumer acceptance of a new product for various types of advertising campaigns as a function of the length of time that the campaign has been in effect. Or we might wish to study the trend in the lot-fraction defective in a manufacturing process as a function of time. Both these examples, as well as many others, require a study of the behavior of a binomial (or multinomial) process as a function of time. The examples just described are intended to suggest the relatively broad application of the χ 2 analysis of categorical data, a fact that should be borne in mind by the experimenter concerned **with** this type of data. The statistical test employing X 2 as a test statistic is often called a goodness-of-fit test. Its application for some of these examples requires care in the determination of the appropriate estimates and the number of degrees of freedom for X 2 , which for some of these problems may be rather complex.

736 Chapter 14 Analysis of Categorical Data www.downloadslide.com 14.7 Summary and Concluding Remarks The material in this chapter has been concerned **with** tests of hypotheses regarding the cell probabilities associated **with** multinomial experiments (Sections 14.2 and 14.3) or several independent multinomial experiments (Section 14.5). When the number of observations n is large, the test statistic X 2 can be shown to possess, approximately, a χ 2 probability distribution in repeated sampling, the number of degrees of freedom depending on the particular application. In general, we assume that n is large and that the minimum expected cell frequency is equal to or greater than five. Several words of caution concerning the use of the X 2 statistic as a method of analyzing categorical data are appropriate. The determination of the correct number of degrees of random associated **with** the X 2 statistic is critical in locating the rejection region. If the number is specified incorrectly, erroneous conclusions might result. Notice, too, that nonrejection of the null hypothesis does not imply that it should be accepted. We would have difficulty in stating a meaningful alternative hypothesis for many practical applications, and therefore we would lack knowledge of the probability of making a type II error. For example, we hypothesize that the two classifications of a contingency table are independent. A specific alternative must specify a measure of dependence that may or may not possess practical significance to the experimenter. Finally, if parameters are missing and the expected cell frequencies must be estimated, missing parameters should be estimated by the method of maximum likelihood in order that the test be valid. In other words, the application of the χ 2 test for other than the simple applications outlined in Sections 14.3– 14.5 will require experience beyond the scope of this introductory presentation of the subject. References and Further Readings Agresti, Alan. 2002. Categorical Data Analysis, 2d ed. New York: Wiley- Interscience. Agresti, Alan. 2007. An Introduction to Catagorical Data Analysis, 2d ed. New York: Wiley-Interscience. Cochran, W. G. 1952. “The χ 2 Test of Goodness of Fit,” Annals of **Mathematical** **Statistics** 23: 315–345. Conover, W. J. 1999. Practical Nonparametric **Statistics**, 3d ed. New York: Wiley. Daniel, W. W. 1990. Applied Nonparametric **Statistics**, 2d ed. Boston: PWS-Kent. Fienberg, Stephen E. 1980. The Analysis of Cross-Classified Categorical Data, 2d ed. Cambridge, Mass.: MIT Press. Kendall, M. G., A. Stuart, J. K. Ord, and S. Arnold. 1999. Kendall’s Advanced Theory of **Statistics**: Volume 2A—Classical Inference and the Linear Model, 6th ed. London: Arnold.

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