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CH-15 PPT

# CH-15 PPT - Multiple Regression Analysis Introduction We...

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Multiple Regression Analysis November 30, 2010

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Introduction We can use the same ideas from simple linear regression to analyze relationships between a dependent variable and several independent variables Multiple regression is an extension of the simple linear regression for investigating how a response y is affected by several independent variables x 1 , ..., x k Our objectives are: - find relationships between y and x 1 , ..., x k - predict y using x 1 , ..., x k
Examples Monthly sales ( y ) of a retail store may depend on x 1 = advertising expenditure x 2 = time of year x 3 = size of inventory x 4 = state of economy Body fat ( y ) may depend on x 1 = age x 2 = sex x 3 = body type

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Pertinent Questions Which of the independent variables are useful and which are not? How could we create a prediction equation which allows us to predict y using knowledge of x 1 , x 2 , x 3 etc? How strong is the relationship between y and the independent variables? How good is the prediction?
The General Linear Model y = β 0 + β 1 x 1 + β 2 x 2 + ... + β k x k + y is the dependent variable (response) x 1 , ..., x k are independent variables (predictors) The deterministic part of the model, E ( y ) = β 0 + β 1 x 1 + β 2 x 2 + ... + β k x k , describes average value of y for any fixed values of x 1 , ..., x k . The observation y deviates from the deterministic model by an amount is random error. We assume random errors are independent normal random variables with mean zero and a constant variance σ 2

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The Method of Least Squares Data: n observations of the response y and the independent variables x 1 , ..., x k The best fitting prediction equation is ˆ y = b 0 + b 1 x 1 + b 2 x 2 + ... + b k x k We choose our estimates b 0 , ..., b k to minimize SSE = X ( y - β 0 - β 1 x 1 - β 2 x 2 - ... - β k x k ) 2 The computation is usually done by a computer
Example A data set contains the college GPA ( y ), high school GPA ( x 1 ) and study time ( x 2 ) of 59 randomly selected students. We are interested in regressing college GPA on high school GPA and study time.

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