## Statistical Inference Methods

We will now discuss the differences between parametric and nonparametric methods, then describe some of the most common and effective statistical methods being used in the field. For a comprehensive listing of tests based on these methods see Table 14.1, and for a listing of the underlying assumptions for each test see Table 14.2.

1. Parametric vs. nonparametric methods Before applying a particular statistical test of a hypothesis, the analyst must verify that the data being tested meets the requirements (assumptions) for that test (see Table 14.1). In particular, when the hypothesis compares the measures of central tendency of two groups, the researcher has the option of using either parametric or nonparametric methods (see Table 14.2). A parametric method is used for testing hypotheses about parameters in a population described by a specified distributional form such as the normal distribution. Alternatively, a nonparametric method is useful for testing hypotheses using information based on a function of the sample observations (often the ranks) whose probability distribution does not depend on a complete specification of the probability distribution of the population from which the sample was drawn, as is generally the case with parametric methods. Hence, a nonparametric method will be valid under relatively general assumptions about the underlying population.

Parametric methods, on the other hand, are valid only under certain distributional conditions. Most parametric methods, such as the t-test described in more detail below, require that observations be approximately normally distributed, and both groups should have equal sample variances. Implicit in these requirements is that the data should be continuous, although ordinal data may be suitable as well, especially when the sample size is large.

Nonparametric methods ignore many of the distributional aspects of the data in favor of having the flexibility to handle nominal data as well as continuous and ordinal data for which we either (a) have insufficient information about the distribution or (b) know that the distributional requirements for parametric methods have not been adequately satisfied. The nonparametric methods offer a suitable alternative under these conditions, but they are not as powerful and efficient as the analogous parametric methods when the data do not violate the distributional assumptions.

With either method, the data must be from a random sample of the population of interest. In addition to generating samples that adequately represent the population as we have already mentioned, random sampling also gives a sample the desirable property of having independent observations. Independence between observations is a critical assumption for virtually every common statistical test. For more information and in-depth discussion of nonparametric methods we suggest that the reader refer to Conover (1999).

2. Analysis of continuous data The t-test is a parametric statistical method used for testing the difference in a mean from a hypothesized value or the difference in means between two normally distributed populations with equal variances.

Analysis of Variance (ANOVA) is another parametric statistical method. One-way ANOVA is a popular tool for comparing the means from several samples. For example, Weindruch et al. (1986) collected data from mice on the relationship between caloric intake and life expectancy. They randomly assigned female mice to feeding regimen groups where scientists regulated the caloric intake as well as the composition and quality of the nutrition in the diets of the mice. In this situation, comparisons of mean life expectancy between feeding groups can be made by using ANOVA. Keep in mind, however, that ANOVA is a parametric method and the validity of the assumptions underlying this method must be verified prior to performing this analysis.

Ordinary least squares regression (OLS) is a parametric method that is used to test for a linear relationship between a continuous dependent response variable and a set of independent predictor variables that can be either continuous or categorical. The parameters

TABLE 14.1. Statistical methods and their assumptions

 Assumptionb Testa Independent observationsc Equal standard deviationsd Normal distriibutione Adequate model fitf Typical Hypothesis Tested T-test