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COPYRIGHT Abraham, B. and Ledolter, J. Introduction to Regression Modeling Belmont, CA: Duxbury Press, 2006
Abraham Abraham˙C01 November 8, 2004 0:33 1 Introduction to Regression Models 1.1 INTRODUCTION Regression modeling is an activity that leads to a mathematical description of a process in terms of a set of associated variables. The values of one variable frequently depend on the levels of several others. For example, the yield of a certain production process may depend on temperature, pressure, catalyst, and the rate of throughput. The number or the rate of defectives of a process may depend on the speed of the production line. The number of defective seals on toothpaste tubes may depend on the temperature and the pressure of the sealing process. The volume of a tree is related to the diameter of the tree at breast height, the height of the tree, and the taper of the tree. The fuel efficiency of an automobile depends, among others, on the weight of the car and characteristics of its body and engine. Employee efficiency may be related to the performance on employment tests, years of training, and educational background. The salaries of managers, athletes, and college teachers may depend on their seniority, the size of the market, and their performance. Many additional examples can be given, and in Exercise 1.1 we ask you to comment on several other relationships in detail. The supply of a product depends on the price customers are willing to pay; one can expect that more products are brought to market when the price is high. Economists refer to this relationship as the production function . Similarly, the demand for a product depends on the price of the item, the price of the competition, and the amount spent on its advertisement. Economists refer to this relationship as the demand function . One can expect lower sales if the price is high, in- creased sales if the price of the competition is higher, and increased sales if more money is spent on promotion. However, price and advertising may also interact. Advertising may be more effective if the price is low; furthermore, the effect of the competition’s price on sales may depend on one’s own price. Also, seasonal components may have an impact on sales during a certain period because sales of a summer item during winter months will be low in northern states, irrespective of the product’s price. 1
Abraham Abraham ˙ C01 November 8, 2004 0:33 2 Introduction to Regression Models In all these situations we are interested in obtaining a model or a law (i.e., a mathematical description) for the relationship among the variables. Regression analysis deals with modeling the functional relationship between a response variable and one or more explanatory variables . In some instances one has a fairly good idea about the form of these models. Often the laws from physics or chemistry tell us how a response is related to the explanatory variables. These laws may involve complicated mathematical equations that contain functions such as