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notes515fall10chap11 - STAT 515 - Chapter 11: Regression...

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STAT 515 -- Chapter 11: Regression • Mostly we have studied the behavior of a single random variable. • Often, however, we gather data on two random variables. • 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.? • Often we assume a straight-line relationship between two variables. • This is known as simple linear regression . Probabilistic vs. Deterministic Models If there is an exact relationship between two (or more) variables that can be predicted with certainty, without any random error, this is known as a deterministic relationship . Examples:
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In statistics, we usually deal with situations having random error, so exact predictions are not possible. This implies a probabilistic relationship between the 2 variables. Example: Y = breathalyzer reading X = amount of alcohol consumed (fl. oz.)
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• We typically assume the random errors balance out – they average zero. • Then this is equivalent to assuming the mean of Y , denoted E( Y ), equals the deterministic component. Straight-Line Regression Model Y = 0 + 1 X + Y = response variable (dependent variable) X = predictor variable (independent variable) = random error component 0 = Y-intercept of regression line 1 = slope of regression line Note that the deterministic component of this model is E( Y ) = 0 + 1 X Typically, in practice, 0 and 1 are unknown parameters. We estimate them using the sample data. Response Variable (Y): Measures the major outcome of interest in the study. Predictor Variable (X): Another variable whose value explains, predicts, or is associated with the value of the response variable.
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Fitting the Model (Least Squares Method) If we gather data ( X , Y ) for several individuals, we can use these data to estimate 0 and 1 and thus estimate the linear relationship between Y and X . First step: Decide if a straight-line relationship between Y and X makes sense. Plot the bivariate data using a scattergram (scatterplot). Once we settle on the “best-fitting” regression line, its equation gives a predicted Y-value for any new X-value. How do we decide, given a data set, which line is the best-fitting line?
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Note that usually, no line will go through all the points in the data set. For each point, the error
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notes515fall10chap11 - STAT 515 - Chapter 11: Regression...

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