Simple Linear Regression Lecture for Statistics 509 November-December 2000

Week of 11/27/2000Stat Regression Lecture2 Correlation and Regression Study of association and/or relationship between variables. Useful for determining the effect of changes in one variable (called the independent or control variable) on another variable (called the dependent or response variable). Regression models could be utilized to determine optimal operating conditions [these conditions specified by the control variables] in order to achieve a certain specified value or yield on the response variable. Regression models could also be utilized to predict the value of the response given a value of the independent variable, or could be used for “calibrating” the value of the independent variable to achieve a certain response.

Week of 11/27/2000Stat Regression Lecture3 Some Examples Control variable is X = Average Speed of a Car and response variable is Y=Fuel Efficiency of the Car. Goal is to determine speed to optimize the efficiency of the car. Control variable is X = Temperature, while the response variable is Y = Yield in a chemical reaction. Control variable is X = amount of fertilizer applied on a plant, while the response variable is Y = yield of this plant. Control variable is X = thickness of a stack of bond paper, while the response variable is Y = number of sheets in this stack. Control variable is X = average time of studying, while the response variable is Y = GPA.

Week of 11/27/2000Stat Regression Lecture4 Population Model Each member of the population will have a value for the independent variable X and the response variable Y, usually represented by the vector (X,Y). For a given value X = x, the variable Y has a certain distribution whose conditional mean is  (x) and whose conditional variance is  2 (x). This could be visualized as follows: When you consider the subpopulation consisting of units whose values of X equal x, then their Y-values has a certain distribution whose mean is  (x) and whose variance is  2 (x). When you pick a unit from this subpopulation, then the Y-value that you will observe is governed by this particular distribution. In particular, this observation could be expressed via Y =  (x) + , where e is some “error term.”

Week of 11/27/2000Stat Regression Lecture5 Assumptions for Simple Linear Regression  (x) = E(Y|X=x) =  +  x. This means that the mean of Y, given X = x, is a linear function of x.  is called the regression coefficient or the slope of the regression line;  is the y-intercept.  2 (x) =   does not depend on x. This is the assumption of “equal variances” or homoscedasticity. Furthermore, for the sample data (x 1, Y 1 ), (x 2, Y 2 ), …, (x n, Y n ): Y 1, Y 2, …, Y n are independent observations, and their conditional distributions are all normal. In shorthand notation: Y i =  (x i ) +  i =  +  x i +  i, i=1,2,…,n, where  1,  2, …,  n are independent and identically distributed (IID) N(0,  2 ).

Week of 11/27/2000Stat Regression Lecture6 Regression Problem Given the sample (bivariate) data (x 1, Y 1 ), (x 2, Y 2 ), …, (x n, Y n ), satisfying the linear regression model Y i =  +  x i +  i with  1,  2, …,  n IID N(0,  2 ) we would like to address the following questions: How should the data be summarized graphically? What are the estimators of the parameters , , and  2 ? What will be an estimate of the prediction line? What are the properties of the estimators of the model parameters? How do we test whether the fitted regression model is a significant model? How do we construct CIs or test hypotheses concerning parameters? How do we perform prediction using the prediction model?

Week of 11/27/2000Stat Regression Lecture7 Illustrative Example: On Plasma Etching Plasma etching is essential to the fine-line pattern transfer in current semiconductor processes. The paper “Ion Beam- Assisted Etching of Aluminum with Chlorine” in J. Electrochem. Soc. (1985) gives the data below on chlorine flow (x, in SCCM) through a nozzle used in the etching mechanism, and etch rate (y, in 100A/min)

Week of 11/27/2000Stat Regression Lecture8 The Scatterplot

Week of 11/27/2000Stat Regression Lecture9 Least-Squares Prediction Line

Week of 11/27/2000Stat Regression Lecture10

Week of 11/27/2000Stat Regression Lecture11

Week of 11/27/2000Stat Regression Lecture12 Analysis of Variance Table

Week of 11/27/2000Stat Regression Lecture13

Week of 11/27/2000Stat Regression Lecture14 Excel Worksheet for Regression Computations

Week of 11/27/2000Stat Regression Lecture15 Regression Analysis from Minitab

Week of 11/27/2000Stat Regression Lecture16 Fitted Line in Scatterplot with Bands