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Simple+Linear+and+Multiple+Regression+Analysis

Simple+Linear+and+Multiple+Regression+Analysis - Simple...

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Simple Linear Regression Analysis and Multiple Linear Regression Analysis
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What is Regression Analysis? A statistical technique that describes the relationship between a dependent variable and one or more independent variables.
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Examples Consider the relationship between construction permits ( x ) and carpet sales ( y ) for a company. OR Relationship between advertising expenditures and sales There probably is a relationship... ...as number of permits increases, sales should increase. ...set advertising expenditure and we can predict sales But how would we measure and quantify this relationship?
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Simple Linear Regression Model (SLR) Assume relationship to be linear y = 0 + 1x +  Where y = dependent variable x = independent variable 0 = y-intercept 1 = slope  = random error
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Random Error Component () Makes this a probabilistic model ... Represents uncertainty   random variation not explained by x Deterministic Model = Exact relationship Example: Temperature: oF = 9/5 oC + 32 Assets = Liabilities + Equity Probabilistic Model = Det. Model + Error
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Graphically, SLR line is displayed as... 0 5 10 15 0 10 20 30 40 50 X Y line of means
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Model Parameters 0 and 1 Estimated from the data Data collected as a pair (x,y)
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Process of Developing SLR Model Hypothesize the model: E(y) = 0 + 1x Estimate Coefficients Specify distribution of error term How adequate is the model? When model is appropriate, us it for estimation and prediction x ˆ ˆ y ˆ 1 0 β + β =
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Fitting the Straight-Line Model Ordinary Least Squares (OLS) Once it is assumed that the model is y = 0 + 1x +  Next we must collect the data Before estimating parameters, we must ensure that the data follows a linear trend Use scatterplot, scattergram, scatter diagram
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A Scatter Plot of the Data Carpet City Problem 0 5 10 15 0 10 20 30 40 50 Monthly Construction Permits Monthly Carpet Sales
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Assessing Fit Carpet City Problem 0 5 10 15 0 10 20 30 40 50 Monthly Construction Permits Monthly Carpet Sales
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Assessing Fit (Deviations) aka errors or residuals (ri, ei) Difference between the observed value of y and the predicted value of y Want ri to be small i i i i y y r e ˆ - = =
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Assessing Fit (Cont.) NOTE: Sum of the residuals is 0 Can fit many different lines; which one is best? Line that best fits the data is the one that minimizes the sum of squares of the errors (SSE). This is the least squares line.
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Least Squares Line Find the line that minimizes with respect to the parameters Recall that Minimize ( 29 2 i i y ˆ y - i 1 0 i x ˆ ˆ y ˆ β + β = ( 29 [ ] β + β - 2 1 0 i x ˆ ˆ y
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Computational Formulas ˆ β 1 = SS xy SS xx = x i - x ( 29 y i - y ( 29 x i - x ( 29 2 ˆ β 0 = y - ˆ β 1 x
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Example 1 The Central Company manufactures a certain specialty item once a month in a batch production run. The number of items produced in each run varies from month to month as demand fluctuates. The company is interested in the relationship between the size of the production run (x) and the number of man-hours of labor (y) required for the run. The company has collected the following data for the 10 most recent runs:
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Example 1 (Cont.) Run Number of items Labor (man-hours) 1 40 83 2 30 70
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Example 1 (Cont.) Summary Statistics 510 , 78 400 , 39 194 , 1 600 2 = = = = i i i i i y x x y x
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Example 1 (Cont.)
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