Agresti/FranklinStatistics,1 of 141Chapter 12Multiple RegressionLearn….To use Multiple RegressionAnalysis to predict a responsevariable using more than oneexplanatory variable.

Agresti/FranklinStatistics,2 of 141Section 12.1How Can We Use SeveralVariables to Predict aResponse?

Agresti/FranklinStatistics,3 of 141Regression ModelsThe model that contains onlytwo variables,xandy, is called abivariatemodel

Agresti/FranklinStatistics,4 of 141Regression ModelsThe regression equation for thebivariate model is:xy

Agresti/FranklinStatistics,5 of 141Regression ModelsSuppose there are twopredictors, denoted by x1and x2This is called amultipleregressionmodel

Agresti/FranklinStatistics,6 of 141Regression ModelsThe regression equation for thismultiple regression model with twopredictors is:2211xxy

Agresti/FranklinStatistics,7 of 141Multiple Regression ModelThe multiple regression modelrelates the meanµyof aquantitative response variableyto a set of explanatory variablesx1, x2,….

Agresti/FranklinStatistics,8 of 141Multiple Regression ModelExample:For three explanatoryvariables, the multiple regressionequation is:332211xxxy

Agresti/FranklinStatistics,9 of 141Multiple Regression ModelExample:The sample predictionequation with three explanatoryvariables is:332211ˆxbxbxbay

Agresti/FranklinStatistics,10 of 141Example:Predicting Selling PriceUsing House and Lot SizeThe data set “house selling prices”contains observations on 100 homesales in Florida in November 2003A multiple regression analysis wasdone withselling priceas theresponse variable and withhousesizeandlot sizeas the explanatoryvariables

Agresti/FranklinStatistics,11 of 141Example:Predicting Selling PriceUsing House and Lot SizeOutput from the analysis:

Agresti/FranklinStatistics,12 of 141Example:Predicting Selling PriceUsing House and Lot SizePrediction Equation:wherey = selling price, x1=housesize and x2= lot size2184.28.53536,10ˆxxy

Agresti/FranklinStatistics,13 of 141Example:Predicting Selling PriceUsing House and Lot SizeOne house listed in the data set hadhouse size = 1240 square feet, lot size =18,000 square feet and selling price =$145,000Find its predicted selling price:276,107)000,18(84.2)1240(8.53536,10ˆy

Agresti/FranklinStatistics,14 of 141Example:Predicting Selling PriceUsing House and Lot SizeFind its residual:The residual tells us that the actual sellingprice was $37,724 higher than predicted724,37276,107000,145ˆyy

Agresti/FranklinStatistics,15 of 141The Number of ExplanatoryVariablesYou should not use manyexplanatory variables in a multipleregression model unless you havelots of dataA rough guideline is thatthe samplesize n should be at least 10 times thenumber of explanatory variables

Agresti/FranklinStatistics,16 of 141Plotting RelationshipsAlways look at the data before doinga multiple regressionMost software has the option ofconstructing scatterplots on a singlegraph for each pair of variables•This is called a scatterplot matrix

Agresti/FranklinStatistics,17 of 141Plotting Relationships

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