Applied Regression Modeling. Iain Pardoe

Chapter 1 Foundations

This chapter provides a brief refresher of the main statistical ideas that are a useful foundation for the main focus of this book, regression analysis, covered in subsequent chapters. For more detailed discussion of this material, consult a good introductory statistics textbook such as Freedman et al. (2007) or Moore et al. (2018). To simplify matters at this stage, we consider univariate data, that is, datasets consisting of measurements of a single variable from a sample of observations. By contrast, regression analysis concerns multivariate data where there are two or more variables measured from a sample of observations. Nevertheless, the statistical ideas for univariate data carry over readily to this more complex situation, so it helps to start out as simply as possible and make things more complicated only as needed.

      After reading this chapter you should be able to:

1.1 Identifying and Summarizing Data

One way to think about statistics is as a collection of methods for using data to understand a problem quantitatively—we saw many examples of this in the introduction. This book is concerned primarily with analyzing data to obtain information that can be used to help make decisions in real‐world contexts.

      The process of framing a problem in such a way that it is amenable to quantitative analysis is clearly an important step in the decision‐making process, but this lies outside the scope of this book. Similarly, while data collection is also a necessary task—often the most time‐consuming part of any analysis—we assume from this point on that we have already obtained data relevant to the problem at hand. We will return to the issue of the manner in which these data have been collected—namely, whether we can consider the sample data to be representative of some larger population that we wish to make statistical inferences for—in Section 1.3.

      For now, we consider identifying and summarizing the data at hand. For example, suppose that we have moved to a new city and wish to buy a home. In deciding on a suitable home, we would probably consider a variety of factors, such as size, location, amenities, and price. For the sake of illustration, we focus on price and, in particular, see if we can understand the way in which sale prices vary in a specific housing market. This example will run through the rest of the chapter, and, while no one would probably ever obsess over this problem to this degree in real life, it provides a useful, intuitive application for the statistical ideas that we use in the rest of the book in more complex problems.

      For this example, identifying the data is straightforward: the units of observation are a random sample of size single‐family homes in our particular housing market, and we have a single measurement for each observation, the sale price in thousands of dollars ($), represented using the notation . Here, is the generic letter used for any univariate data variable, while is the specific variable name for this dataset. These data, obtained from Victoria Whitman, a realtor in Eugene, Oregon, are available in the HOMES1 data file on the book website—they represent sale prices of 30 homes in south Eugene during 2005. This represents a subset of a larger file containing more extensive information on 76 homes, which is analyzed as a case study in Chapter 6 (refer www.wiley.com/go/pardoe/AppliedRegressionModeling3e).

      The particular sample in the HOMES1 data file is random because the 30 homes have been selected randomly somehow from the population of all single‐family homes in this housing market. For example, consider a list of homes currently for sale, which are considered to be representative of this population. A random number generator—commonly available in spreadsheet or statistical software—can be used to pick out 30 of these. Alternative selection methods may or may not lead to a random sample. For example, picking the first 30 homes on the list would not lead to a random sample if the list were ordered by the size of the sale price.

      We can simply list small datasets such as this. The values of in this case are as follows: