3 Linear RegressionA 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 Y? (the actual value of Yi)YXb b+=YiieXXi
5 Linear Regression - Model PopulationRegression Coefficients for a . . .ˆY = b0 + b1Xi + eSampleˆY = b0 + b1Xi
6 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.SSTRDue to treatments.SSTSSESST = SSTR + SSERandom/unexplained.
7 Linear Regression - Variation Ice Cream ExampleYY= 2.53
8 Linear Regression - Variation Ice Cream ExampleSample Regression LineRegression Model
9 Linear Regression - Variation SSRDue to regression.SSTSSESST = SSR + SSERandom/unexplained.
10 Linear Regression - Variation YSSE =(Yi - Yi )2_SST = (Yi - Y)2_SSR = (Yi - Y)2_YXXi
11 Determining the Regression Line/Model Use Excel (or any other popular statistical software)Select Tools, Data Analysis, RegressionProvide the X rangeProvide the Y rangeOutput the analysis to a new sheetManual Calculations
12 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.
13 Determining the Regression Line/Model Manual Calculations __SSE =(Yi - Yi )2SSR = (Yi - Y)2SST = (Yi - Y)2_SSx =(Xi - X )2b1=SSxy/SSxSSy =(Yi - Y)2___SSxy =(Xi - X )(Xi - Y )_b0 = Y – b1XMSE = SSE / dfMSR = SSR / dfR2 = SSR/SSTYXSSESn-2=t-test = b1 / Sb1
14 Measures of Model Goodness R2 – Coefficient of DeterminationF-test > F-crit or p-value less than alphaStandard Errort-test
15 Hypothesis testing for Testing to see if the linear relationship between X and Y is significant at the population level.t-testFollow the 5-step processH0:HA:t-crit, alpha or alpha/2, n-2 df
16 Standard Error Terms in Linear Regression Se (standard error of the estimate)A measure of variation around the regression lineIf the Se is small…Standard deviation Of the ErrorsSb1 (standard error of the the sampling distribution of b1)Standard deviation of the slopesA measure of the variation of the slopes from different samples If the Sb1 is small…our b1 estimate is probably very accurateEstimates of …b1b1b1
17 Linear Regression Example Petfood, Estimate Sales based on Shelf SpaceTwo sets of samples, 12 observations eachPerform a Regression Analysis on both sets of dataSample1Sample2