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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 Linear Regression Hypothesis testing

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Regression - Types

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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

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**Linear Regression - Model**

Y ? (the actual value of Yi) Y X b b + = Yi i e X Xi

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**Linear Regression - Model**

Population Regression Coefficients for a . . . ˆ Y = b0 + b1Xi + e Sample ˆ Y = b0 + b1Xi

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**ANOVA CRD - Variation SSTR SST SSE SST = SSTR + SSE**

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

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**Linear Regression - Variation**

Ice Cream Example Y Y = 2.53

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**Linear Regression - Variation**

Ice Cream Example Sample Regression Line Regression Model

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**Linear Regression - Variation**

SSR Due to regression. SST SSE SST = SSR + SSE Random/unexplained.

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**Linear Regression - Variation**

Y SSE =(Yi - Yi )2 _ SST = (Yi - Y)2 _ SSR = (Yi - Y)2 _ Y X Xi

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**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

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**Determining the Regression Line/Model using Excel**

Like in ANOVA, the df for Regression is the number of columns – 1. Total df is always n-1. That leaves Error/Residual df at n-2.

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**Determining the Regression Line/Model Manual Calculations**

_ _ SSE =(Yi - Yi )2 SSR = (Yi - Y)2 SST = (Yi - Y)2 _ SSx =(Xi - X )2 b1=SSxy/SSx SSy =(Yi - Y)2 _ _ _ SSxy =(Xi - X )(Xi - Y ) _ b0 = Y – b1X MSE = SSE / df MSR = SSR / df R2 = SSR/SST YX SSE S n-2 = t-test = b1 / Sb1

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**Measures of Model Goodness**

R2 – Coefficient of Determination F-test > F-crit or p-value less than alpha Standard Error t-test

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**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 H0: HA: t-crit, alpha or alpha/2, n-2 df

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**Standard Error Terms in Linear Regression**

Se (standard error of the estimate) A measure of variation around the regression line If the Se is small… Standard deviation Of the Errors Sb1 (standard error of the the sampling distribution of b1) Standard deviation of the slopes A measure of the variation of the slopes from different samples If the Sb1 is small…our b1 estimate is probably very accurate Estimates of … b1 b1 b1

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**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 Sample1 Sample2

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