Computer Solution

Performing the regression analysis computations without the help of a computer can be quite time consuming. In this section we discuss how the computational burden can be minimized by using a computer software package such as JMP or Excel. Although the layout of the information may differ by computer software, the informa­tion shown in

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Residual Analysis: Outliers and Influential Observations

In Section 14.8 we showed how residual analysis could be used to determine when viol­ations of assumptions about the regression model occur. In this section, we discuss how residual analysis can be used to identify observations that can be classified as outliers or as being especially influential in determining the estimated regression equation. Some

Multiple Regression Model

Multiple regression analysis is the study of how a dependent variable y is related to two or more independent variables. In the general case, we will use p to denote the number of independent variables. 1. Regression Model and Regression Equation The concepts of a regression model and a regression equation introduced in the

Least Squares Method for Multiple Regression

In Chapter 14, we used the least squares method to develop the estimated regression equation that best approximated the straight-line relationship between the dependent and independent variables. This same approach is used to develop the estimated multiple re­gression equation. The least squares criterion is restated as follows: The predicted values of the dependent variable

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Multiple Coefficient of Determination in Multiple Regression

In simple linear regression, we showed that the total sum of squares can be partitioned into two components: the sum of squares due to regression and the sum of squares due to error. The same procedure applies to the sum of squares in multiple regression. Because of the computational difficulty in computing the three

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Multiple Regression Model Assumptions

In Section 15.1 we introduced the following multiple regression model. The assumptions about the error term e in the multiple regression model parallel those for the simple linear regression model. To obtain more insight about the form of the relationship given by equation (15.11), consider the following two-independent-variable multiple regression equation. The graph of

Testing for Significance for Multiple Regression

In this section we show how to conduct significance tests for a multiple regression rela­tionship. The significance tests we used in simple linear regression were a t test and an F test. In simple linear regression, both tests provide the same conclusion; that is, if the null hypothesis is rejected, we conclude that b1

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Using the Estimated Multiple Regression Equation for Estimation and Prediction

The procedures for estimating the mean value of y and predicting an individual value of y in multiple regression are similar to those in regression analysis involving one independent variable. First, recall that in Chapter 14 we showed that the point estimate of the expected value of y for a given value of x

Categorical Independent Variables in Multiple Regression

Thus far, the examples we have considered involved quantitative independent variables such as student population, distance traveled, and number of deliveries. In many situations, however, we must work with categorical independent variables such as gender (male, female), method of payment (cash, credit card, check), and so on. The purpose of this sec­tion is to

Residual Analysis in Multiple Regression

In Chapter 14 we pointed out that standardized residuals are frequently used in residual plots and in the identification of outliers. The general formula for the standardized residual for observation i follows. The general formula for the standard deviation of residual i is defined as follows. As we stated in Chapter 14, the leverage

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Logistic Regression

In many regression applications, the dependent variable may only assume two discrete values. For instance, a bank might want to develop an estimated regression equation for predicting whether a person will be approved for a credit card. The dependent variable can be coded as y = 1 if the bank approves the request for

General Linear Model

Suppose we collected data for one dependent variable y and k independent variables x1, x2, . . . , xk. Our objective is to use these data to develop an estimated regression equation that provides the best relationship between the dependent and independent variables. As a general framework for developing more complex relationships among

Determining When to Add or Delete Variables

In this section we will show how an F test can be used to determine whether it is advant­ageous to add one or more independent variables to a multiple regression model. This test is based on a determination of the amount of reduction in the error sum of squares resulting from adding one or

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Analysis of a Larger Problem

In introducing multiple regression analysis, we used the Butler Trucking example extensively. The small size of this problem was an advantage in exploring introductory concepts but would make it difficult to illustrate some of the variable selection issues involved in model building. To provide an illustration of the variable selection procedures discussed in the

Variable Selection Procedures

In this section we discuss four variable selection procedures: stepwise regression, for­ward selection, backward elimination, and best-subsets regression. Given a data set with several possible independent variables, we can use these procedures to identify which in­dependent variables provide the best model. The first three procedures are iterative; at each step of the procedure a

Multiple Regression Approach to Experimental Design

In Section 15.7 we discussed the use of dummy variables in multiple regression analysis. In this section we show how the use of dummy variables in a multiple regression equation can provide another approach to solving experimental design problems. We will demonstrate the multiple regression approach to experimental design by applying it to the

Autocorrelation and the Durbin-Watson Test

Often, the data used for regression studies in business and economics are collected over time. It is not uncommon for the value of y at time t, denoted by y, to be related to the value of y at previous time periods. In such cases, we say autocorrelation (also called serial correlation) is present

Time Series Patterns

A time series is a sequence of observations on a variable measured at successive points in time or over successive periods of time. The measurements may be taken every hour, day, week, month, or year, or at any other regular interval.[1] The pattern of the data is an important factor in understanding how the

Forecast Accuracy with Time Series Analysis

In this section we begin by developing forecasts for the gasoline time series shown in Table 17.1 using the simplest of all the forecasting methods: an approach that uses the most recent week’s sales volume as the forecast for the next week. For instance, the dis­tributor sold 17 thousand gallons of gasoline in week

Moving Averages and Exponential Smoothing in Time Series Analysis

In this section, we discuss three forecasting methods that are appropriate for a time series with a horizontal pattern: moving averages, weighted moving averages, and exponential smoothing. These methods also adapt well to changes in the level of a horizontal pattern such as we saw with the extended gasoline sales time series (Table 17.2

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