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Linear Regression - Topics Basics of Linear Regression Variation in Linear Regression Linear Regression Analysis Goodness of Fit Standard Error terms for.

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Presentation on theme: "Linear Regression - Topics Basics of Linear Regression Variation in Linear Regression Linear Regression Analysis Goodness of Fit Standard Error terms for."— Presentation transcript:

1 Linear Regression - Topics Basics of Linear Regression Variation in Linear Regression Linear Regression Analysis Goodness of Fit Standard Error terms for Linear Regression Hypothesis testing

2 Regression - Types

3 Linear Regression A statistical technique that uses a single, independent variable (X) to estimate a single dependent variable (Y). Based on the equation for a line: Y = b + mX

4 Linear Regression - Model i  X Y YX    YiYi XiXi ? (the actual value of Y i )

5 Linear Regression - Model Regression Coefficients for a... Population Sample ˆ Y = b 0 + b 1 X i + e Y = b 0 + b 1 X i ˆ

6 ANOVA CRD - Variation SST SSE SSTR SST = SSTR + SSE SST is a measure of the total variation of observations. A measure of the differences in observations. Due to treatments. Random/unexplained.

7 Linear Regression - Variation Ice Cream Example Y Y = 2.53

8 Linear Regression - Variation Ice Cream Example Sample Regression Line

9 Linear Regression - Variation SST SSE SSR SST = SSR + SSE Due to regression. Random/unexplained.

10 Linear Regression - Variation XiXi Y X Y SST =  (Y i - Y) 2 SSE =  (Y i - Y i ) 2  SSR =  (Y i - Y) 2  _ _ _

11 Determining the Regression Line/Model Use Excel (or any other popular statistical software)  Select Tools, Data Analysis, Regression  Provide the X range  Provide the Y range  Output the analysis to a new sheet Manual Calculations

12 Determining the Regression Line/Model using Excel

13 Determining the Regression Line/Model Manual Calculations SST =  (Y i - Y) 2 SSE =  (Y i - Y i ) 2 SSR =  (Y i - Y) 2 __ SSx =  (X i - X ) 2 _ SSy =  (Y i - Y) 2 _ SSxy =  (X i - X )(X i - Y ) __  b 1 =SSxy/SSx b 0 = Y – b 1 X __ MSE = SSE / df MSR = SSR / df R 2 = SSR/SST YX SSE S n-2  t-test = b 1 / S b1

14 Measures of Model Goodness 1. R 2 – Coefficient of Determination 2. F-test > F-crit or p-value less than alpha 3. Standard Error 4. t-test

15 Hypothesis testing for Testing to see if the linear relationship between X and Y is significant at the population level. t-test Follow the 5-step process  H 0 : H A :  t-crit, alpha or alpha/2, n-2 df

16 Standard Error Terms in Linear Regression S e (standard error of the estimate) A measure of variation around the regression line If the S e is small… Standard deviation Of the Errors S b1 (standard error of the the sampling distribution of b 1 ) Standard deviation of the slopes A measure of the variation of the slopes from different samples If the S b1 is small…our b 1 estimate is probably very accurate Estimates of … b1b1 b1b1 b1b1

17 Linear Regression Example Petfood, Estimate Sales based on Shelf Space Two sets of samples, 12 observations each Perform a Regression Analysis on both sets of data Sample1Sample2


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