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Unformatted text preview: ISYE2028 Spring 2009 The Linear Model: OLS Regression Dr. Kobi Abayomi April 10, 2009 Introduction Regression Analysis is one of the simplest ways we have in statistics to investigate the relationship between two or more variables related in a non-deterministic way. Examples of a deterministic relationship: F = ma,C = 2 r . If we know exactly everything on the right hand side of the equations, we know exactly everything on the left hand side Examples of a probabilistic relationship: X Bern ( p ) ,Y N ( n,p ). If we know exactly the parameters on the right hand sides of the relationship, there are still probabilistic (or non-definite) values on the left hand side of the relationship. The Normal Model , for example, is implicitly a probabilistic or relationship. In Regression Analysis- we will explicitly state a relationship between two variables, develop proce- dures for making inferences about the parameters of the model, and obtain a quantitative measure (the correlation coefficient) of the extent to which the two variables are related. Ill just say it right out: the attractiveness of the linear model is the interpretability of the parameters - not the plausibility of a linear relationship. 1 The simple linear regression model 1.1 The linear deterministic model A simple deterministic relationship between two variables (here x and y ) is a linear relationship y = + 1 x You remember from algebra and geometry that this is the equation of line with a slope 1 and a 1 y-intercept . What is the value of y when x = 0? How much does the value of y change for each unit change x . Remember all of my comments on populations and parameters? If, say, we all were God (a belief not in contrast with some religions! 1 ), we would know exactly the values of y (because we knew exactly the values of , 1 ) for each given value of x. Conventionally, we call x the independent variable and y the dependent variable . For example, let y the selling price for a house and x the size of the house - in square feet (are you getting a feel for how we assign x and y ?). And you know that the value of y when x = 0 is 25 , 000. And you know that the change in y for each square foot increase is 75. Then you know = 25000 and 1 = 75 and you can write y = 25000 + 75 x as the deterministic relationship between size of house and its selling price Here is a remark: A scatterplot of house and selling price if this relationship is true should look like? In R housesize<-c(1000,1000,1500,1500,1700,1250,2000,2150,2150) #the house sizes in square feet sellingprice<-c(25000000,27000000,38000000,37299999, 42500000,31350003,50000000,53800000,53700000) #the selling prices in dollars plot(housesize,sellingprice,cex=1.5) 1.2 The linear model - with error...
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- Spring '07