Introduction to Linear Regression Analysis. Douglas C. Montgomery. Читать онлайн. Newlib. NEWLIB.NET

Автор: Douglas C. Montgomery
Издательство: John Wiley & Sons Limited
Серия:
Жанр произведения: Математика
Год издания: 0
isbn: 9781119578758
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rel="nofollow" href="#u4064aa44-1729-52c4-85bc-49c35cc5e76f">Appendix E provides a brief introduction to the R statistical software package. We present R code for doing analyses throughout the text. Without these skills, it is virtually impossible to successfully build a regression model.

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      CHAPTER 2

      SIMPLE LINEAR REGRESSION

      2.1 SIMPLE LINEAR REGRESSION MODEL

      This chapter considers the simple linear regression model, that is, a model with a single regressor x that has a relationship with a response y that is a straight line. This simple linear regression model is

      where the intercept β0 and the slope β1 are unknown constants and ε is a random error component. The errors are assumed to have mean zero and unknown variance σ2. Additionally we usually assume that the errors are uncorrelated. This means that the value of one error does not depend on the value of any other error.

      It is convenient to view the regressor x as controlled by the data analyst and measured with negligible error, while the response y is a random variable. That is, there is a probability distribution for y at each possible value for x. The mean of this distribution is

      (2.2a) image

      and the variance is

      (2.2b) image

      The parameters β0 and β1 are usually called regression coefficients. These coefficients have a simple and often useful interpretation. The slope β1 is the change in the mean of the distribution of y produced by a unit change in x. If the range of data on x includes x = 0, then the intercept β0 is the mean of the distribution of the response y when x = 0. If the range of x does not include zero, then β0 has no practical interpretation.

      The parameters β0 and β1 are unknown and must be estimated using sample data. Suppose that we have n pairs of data, say (y1, x1), (y2, x2), …, (yn, xn). As noted in Chapter 1, these data may result either from a controlled experiment designed specifically to collect the data, from an observational study, or from existing historical records (a retrospective study).

      2.2.1 Estimation of β0 and β1

      (2.4) image

      The least-squares estimators of β0 and β1, say in13-1 and in13-2, must satisfy

ueqn13-1

      and

ueqn13-2

      and

      where

ueqn14-1

      Since the denominator of Eq. (2.7) is the corrected sum of squares of the xi and the numerator is the corrected sum of cross products of xi and yi, we may write these quantities in a more compact notation as

      (2.9) Скачать книгу