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In Chapter 8 we introduce the algebra and geometry behind the fitting of a linear model relating a response variable to one or more explanatory (predictor) variables using the criterion of least squares. In this chapter we consider in more detail situations where there are two or more predictors.
Abstract. It is commonly believed that slight flexion/extension of the head will reverse the cervical lordosis. The goal of the present study was to determine whether slight head extension could result in a cervical kyphosis changing into a lordosis. Forty consecutive volunteer subjects with a cervical kyphosis and with flexion in their resting head position had a neutral lateral cervical radiograph...
The quality of inferences are commonly conveyed by probabilities. Therefore, before discussing inferential techniques later in this chapter, we briefly digress to discuss probability in this section and random variables in Section 3.2.
In Exercise 3.13 we discover that the probability of simultaneously making three correct inferences, when each of the three individually has P(correct inference) = 1— α= 0.95, is only (1 — α)3 = .953 = 0.857. Alternatively, the probability of making at least one incorrect inference is 1— 0.857 = 0.143 ≈ 3α. In general, the more simultaneous inferences we make at one time, the smaller the probability...
In Chapter 5 we consider ways to compare the means of two populations. Now we extend these procedures to comparisons of means from several populations. For example, we may wish to compare the average hourly production of a company’s six factories. We say that the investigation has a factorfactory that has six levels, namely the six identifiers distinguishing the factories from one another...
Statistics is the science and art of making decisions based on quantitative evidence. This introductory chapter motivates the study of statistics by describing where and how it used in all endeavors. It gives examples of applications, a little history of the subject, and a brief overview of the structure and content of the remaining chapters.
Logistic regression is a technique similar to multiple regression with the new feature that the predicted response is a probability. Logistic regression is appropriate in the often-encountered situation where we wish to model a dependent variable which is either dichotomous: The dependent variable can assume only the two possible values 0 and 1 (often as a coding of a two-valued categorical...
Any analysis of variance model (for example, anything in Chapters 6, 12, 13, or 14) can be expressed as a regression with dummy variables. Many software procedures and functions make explicit use of this form of expression. Here we explore this equivalence. The notation in Chapter 10 is that used in Sections F.4.2, 9.2, and 9.5.1.
Statistics is the field of study whose objective is the transformation of data (usually sets of numbers along with identifying characteristics) into information (usually in the form of tables, graphs, and written and verbal summaries) that can inform sound policy decisions. We give examples of applications of statistics to many fields in Chapter 1. Here we focus on the general concepts describing...
In Chapter 6 we consider situations where a response variable is measured on groups of observations classified by a single factor and look at ways to compare the changes in the mean of the response variable attributable to the various levels of this factor. Here we extend this to situations where there are two factors. In Chapters 13 and 14 we will discuss instances where there are more than two factors.
We usually study more than one variable at a time. When the variables are continuous, and one is clearly a response variable and the others are predictor variables, we usually plot the variables and then attempt to fit a model to the plotted points. With one continuous predictor, the first model we attempt is a straight line; with two or more continuous predictors, we attempt a plane. We plot the...
Most of the statistical procedures we’ve introduced in the previous 15 chapters require an assumption about the form of a probability distribution, often the Normal distribution. When such assumptions are unjustified, the consequences of the procedure are dubious at best. In situations where distributional assumptions cannot be justified, even after a well-chosen data transformation, the analyst should...
In Chapter 9 we show how to set up and produce an initial analysis of a regression model with several predictors. In the present chapter we discuss ways to investigate whether the model assumptions are met and, when the assumptions are not met, ways to revise the model to better conform with the assumptions. We also examine ways to assess the effect on model performance of individual predictors or...
Time series analysis is the technique used to study observations that are measured over time. Examples include natural phenomena (temperature, humidity, wind speed) and business variables (price of commodities, stock market indices) that are measured at regular intervals (hourly, daily).
Designs are often described by the number of factors. Chapter 6, “OneWay Analysis of Variance” , discusses designs with one factor. Chapter 12, “Two-Way Analysis of Variance” , discusses designs with two factors. More generally, we speak of “three-way” or “higher-way” designs and talk about main effects (one factor), two-way interactions (two factors), three-way interactions, four-way interactions,...
Graphs are used to inspect and display patterns in data. Appropriately drawn graphs are, in our opinion, the best way to gain an understanding of what data have to say. In this chapter we present several of the types of graphs and plots we will be using throughout. We discuss the visual impact of the graphs and relate them to the tabular presentation of the same material.
In this chapter we introduce some additional topics in experimental design beyond those discussed in Chapters 6, 12, and 13. The principle of confounding is used to design efficient experiments having many factors but using only a small subset of all possible treatment combinations. Split plot designs involve placing a restriction on the randomization of treatments to experimental units in order to...
In this chapter we discuss bivariate discrete distributions. Bivariate means that there are two factors (categorical variables) defining cells. The response values are frequencies, that is, counts or instances of observations, at each cell.
In this chapter we discuss selected topics and issues dealing with statistical inferences from samples to populations, building upon the brief introduction to these ideas in Chapter 3. The discussion here is at an intermediate technical level and at a speed appropriate for review of material learned in the prerequisite course.
This contemporary presentation of statistical methods features extensive use of graphical displays for exploring data and for displaying the analysis. The authors demonstrate how to analyze data—showing code, graphics, and accompanying computer listings—for all the methods they cover. They emphasize how to construct and interpret graphs, discuss principles of graphical design, and show how accompanying...
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