Inferential statistical methods stem from the distinction between a sample and a population. A sample refers to the data at hand. For example, 100 adults may be asked which of two olive oils they prefer. Imagine that 60 say brand A. But of interest is the proportion of all adults who would prefer brand A if they could be asked. To what extent does 60% reflect the true proportion of adults who prefer brand A?
There are several components to inferential methods. They include assumptions about how to model the probabilities of all possible outcomes. Another is how to model outcomes of interest. Imagine, for example, that there is interest in understanding the overall satisfaction with a particular automobile given an individual’s age. One strategy is to assume that the typical response Y, given an individuals age, X, is given by Y=β0+β1X, where the slope, β1, and intercept, β0, are unknown constants, in which case a sample would be used to make inferences about their values. Assumptions are also made about how the data were obtained. Was this done in a manner for which random sampling can be assumed? There is even an issue related to the very notion of what is meant by probability. Let μ denote the population mean of Y. The frequentist approach views probabilities in terms of relative frequencies and μ is viewed as a fixed, unknown constant. In contrast, the Bayesian approach views μ as having some distribution that is specified by the investigator. For example, it may be assumed that μ has a normal distribution. The point is that the probabilities associated with μ are not based on the notion of relative frequencies and they are not based on the data at hand. Rather, the probabilities associated with μ stem from judgments made by the investigator.
Inferential methods can be classified into three types: distribution free, parametric, and non-parametric. The meaning of the term “non-parametric” depends on the situation as will be explained. The choice between parametric and non-parametric methods can be crucial for reasons that will be outlined. To complicate matters, the number of inferential methods has grown tremendously during the last 50 years. Even for goals that may seem relatively simple, such as comparing two independent groups of individuals, there are numerous methods that may be used. Expert guidance can be crucial in terms of understanding what inferences are reasonable in a given situation.
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