simplelinearRegressionHO

simplelinearRegressionHO - Chapter 1 Simple Linear...

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STAT6220 Chapter 1. Simple Linear Regression 1. Introduction We are interested in establishing the relationship between two variables, especially predicting one variable ( y ) based on the other ( x ). Deterministic Model vs. Probabilistic Model Deterministic model assumes that by knowing x we are able to predict y exactly. This model hypothesizes an exact relationship between the variables and there is no allowance for error in the prediction. A probabilistic model: If y is the variable interest, y = Deterministic component + Random error We assume the mean value of the random error is 0, i.e., E ( y ) = Deterministic component In this chapter we consider the simplest probabilistic model – the deterministic portion of the model graphs as a straight line. Fitting this model to a set of data is called regression. 2 STAT6220 2. Model Suppose we are given observations in pairs: ( X 1 ; Y 1 ) ; : : : ; ( X n ; Y n ) , where X i ; Y i 2 R . Suppose we want to predict variable Y as a function of X because we believe there is some underlying relationship between Y and X , for example, Y can be approximated by a function X , i.e., Y ¼ f ( X ) . We will consider the case when f ( x ) is a linear function of x : f ( x ) = ¯ 0 + ¯ 1 x The probabilistic model is y = ¯ 0 + ¯ 1 x + ² Y : dependent or response variable, X : Independent or predictor variable ² (epsilon) : random error component ¯ 0 : y intercept of the line, the point at which the line intersects of cuts through the y axis ¯ 1 : Slope of the line, the amount of increase (or decrease) in the deterministic component of y for every 1 unit increase in x . Exercise: Draw y = 2 ¡ 1 2 x .
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3 STAT6220 Note that linearity is not always reasonable assumption but a simple and good starting point for more complicated models. Model assumptions: 1) Linearity : E ( ² i ) = 0 for all i . This implies that the mean of y given x is ______________ 2) Homoscedasticity: the errors have the same variance, i.e., Var ( ² i ) = ¾ 2 for all i . This implies that the variance of y is _______________________ 3) Independence: The errors are independent of each other. 4) Normality: ² i is normally distributed for all i . The above model has the following parameters to estimate from the sample: ________________________ 3. Method of Least Squares Example 1: Suppose an experiment involving five subjects is conducted to determine the relationship between the percentage of a certain drug in the bloodstream and the length of time it takes to react to a stimulus. The results are 4 STAT6220 Subject 1 2 3 4 5 Amount of drug x(%) 1 2 3 4 5 Reaction time y(sec) 1 1 2 2 4 First we need to determine if a linear relationship between y and x is plausible. It is helpful to plot the sample data. Such a plot, known as a scatter diagram, locates each of the five data point in the plane of x and y .
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