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notes509fall11sec61

# notes509fall11sec61 - STAT 509 Sections 6.1-6.2 Linear...

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STAT 509 – Sections 6.1-6.2: Linear Regression • Mostly we have studied the behavior of a single random variable. • Often, however, we gather data on two random variables. Response Variable ( Y ): Measures the major outcome of interest in the study (also called the dependent variable). Independent Variable ( X ): Another variable whose value explains, predicts, or is associated with the value of the response variable (also called the predictor or the regressor ). • We wish to determine: Is there a relationship between the two r.v.’s? • Can we use the values of one r.v. to predict the other r.v.? Observational Studies vs. Designed Experiments • In observational studies, we simply measure or observe both variables on a set of sampled individuals. • In a designed experiment, we manipulate the predictors ( factors ), setting them at specific values of interest. We then observe what values of the response correspond to the fixed predictor values.

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Example 1 (Table 6.1): We observe the Rockwell Hardness ( X ) and Young’s modulus ( Y ) for seven high- density metals. The resulting data were: X: 41 41 44 40 43 15 40 Y: 310 340 380 317 413 62 119 Example 2 (Table 6.3): A chemical engineering class studied the effect of the reflux ratio ( X ) on the ethanol concentration ( Y ) of an ethanol-water distillation. For a variety of settings of the reflux ratio, the ethanol concentration was measured: X: 20 30 40 50 60 Y: 0.446 0.601 0.786 0.928 0.950 We assume there is random error in the observed response values, implying a probabilistic relationship between the 2 variables. • Often we assume a straight-line relationship between two variables. • This is known as simple linear regression . Y i = β 0 + β 1 x i + ε i Y i = i th response value β 0 = Intercept of regression line x i = i th predictor value β 1 = slope of regression line ε i = i th random error component
• We assume the random errors ε i have mean 0 (and variance σ 2 ), so that E( Y ) = β 0 + β 1 x. • Typically, in practice, β 0 and β 1 are unknown parameters. We estimate them using the sample data. Fitting the Model (Least Squares Method) • If we gather data ( X i , Y i ) for several individuals, we can use these data to estimate β 0 and β 1 and thus estimate the linear relationship between Y and X .

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notes509fall11sec61 - STAT 509 Sections 6.1-6.2 Linear...

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