MSci_609_simple_regression

MSci_609_simple_regression - 3/20/2011 WEEK 8: SIMPLE...

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3/20/2011 1 WEEK 8: SIMPLE LINEAR REGRESSION 1/2 INSTRUCTOR: AMER OBEIDI 2
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3/20/2011 2 3 Probabilistic Models Regression Models Correlation Models Representation of some phenomenon “an idealized representations for abstracting the essence of 4 an idealized representations…for abstracting the essence of the subject of inquiry.” Mathematical model is a mathematical expression of some phenomenon. Describe relationship between variables . Types Deterministic models Probabilistic models
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3/20/2011 3 Association: An association exists between two variables if a particular value for 5 one variable is more likely to occur with certain values of the other variable. Response variable (dependent ) The variable of interest that is measured or observed (or modeled ). Explanatory variable (independent ) A variable used to explain or predict (or model ) the values of the response variable. Other variables (lurking) A hidden variable that stands behind a relationship and determines it by simultaneously affecting the other two variables. Deterministic model : a model that relate response variable (regression variable) to one or more 6 explanatory variables and does not allow for error in prediction. Probabilistic model : a model that includes both a deterministic component and a random error which represents variations in data. Use statistics to express and understand probabilistic models.
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3/20/2011 4 Answers ‘What is the relationship between variables?’ 7 variables? Equation used ( regression equation ) One numerical response variable What is to be predicted One or more numerical or categorical explanatory variables Used mainly for prediction and estimation 8
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3/20/2011 5 1. Hypothesize deterministic components 2 Estimat unknown model parameter 9 2. Estimate unknown model parameters 3. Specify probability distribution of random error term (use sample data). Estimate standard deviation of error 4. Evaluate model usefulness 5 Use model for prediction and estimation 5. Here is the true or population regression model of the linear line to be used to best fit the data points. 10 Relationship between variables is a linear function. Population Slope Population y -intercept Random Error yx  0 1 Dependent (Response) Variable Independent (Explanatory) Variable
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3/20/2011 6 E(y/x) =E [ β 0 + β 1 x+ E( / ) E β + β +E 11 y Change E(y/x) =E [ 0 + 1 x ] [ E(y) = β 0 + β 1 x (line of means) β 0 =y -intercept x in y Change in x β 1 = Slope Population Random Sampl 12 $ Unknown Relationship 01 yx   Random Sample $ $ ˆˆ ˆ  $ $ $ $
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3/20/2011 7 13 Unknown Relationship 01 yx   ˆˆ ˆ  Estimated Relationship Least squares estimate Line of means 14 y ii i Observed value i = Random error x   E Observed value
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3/20/2011 8 Predicted value Observed value 15 y Unsampled i = Random error of prediction ^ Residual x 01 ˆˆ ˆ ii yx  observation Observed value
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MSci_609_simple_regression - 3/20/2011 WEEK 8: SIMPLE...

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