Interaction regression modelsWhat is an additive model?Slide 3What is an interaction model?A two-predictor interaction regression functionSlide 6Slide 7Slide 8Slide 9Data analysis exampleSlide 11Slide 12Slide 13Slide 14Interaction models in MinitabInteraction regression modelsWhat is an additive model?A regression model with p-1 predictor variables contains additive effects if the response function can be written as a sum of functions of the predictor variables: 112211 ppXfXfXfYE 222111110XXXYEFor example:105055504540353025201510X1E(Y)E(Y(X2=1))E(Y(X2=5))E(Y(X2=8)) 213210 XXYE What is an interaction model?Two predictor variables interact when the effect on the response variable of one predictor variable depends on the value of the other.A two-predictor interaction regression function 211222110XXXXYE• β0 = the expected response when X1 = 0 and X2 = 0• But now, β1 and β2 can no longer be interpreted as the change in the mean response with a unit increase in the predictor variable, while the other predictor variable is held constant at a given value. 211222110XXXXYE 12121220XxxYEIf we hold X2 = x2 constant:• The intercept depends on the value of x2.• The slope coefficient of X1 depends on the value of x2. 211222110XXXXYE 21122110XxxYEIf we hold X1 = x1 constant:• The intercept depends on the value of x1.• The slope coefficient of X2 depends on the value of x1. 2121215210 XXXXYE 1050908070605040302010X1E(Y)E(Y(X2=1))E(Y(X2=3))E(Y(X2=6)) 2121215210 XXXXYE 1050403020X1E(Y)E(Y(X2=1))E(Y(X2=3))E(Y(X2=6))Data analysis example•Quality score, y, of a product. Score is number between 0 and 100.•Predictor, x1, is temperature (degrees F) at which product was produced.•Predictor, x2, is pressure (pounds per square inch) at which product was produced.•Designed experiment, sample size of n = 27 items.82.72553.375958557.552.5910073003325277582.72553.37555004500958557.552.5910073003325277555004500qualitytemppressuretempsqpresssqtpThe regression equation is quality = - 5128 + 31.1 temp + 140 pressure - 0.133 tempsq - 1.14 presssq - 0.145 tpPredictor Coef SE Coef T PConstant -5127.9 110.3 -46.49 0.000temp 31.096 1.344 23.13 0.000pressure 139.747 3.140 44.50 0.000tempsq -0.133389 0.006853 -19.46 0.000presssq -1.14422 0.02741 -41.74 0.000tp -0.145500 0.009692 -15.01 0.000S = 1.679 R-Sq = 99.3% R-Sq(adj) = 99.1%Analysis of VarianceSource DF SS MS F PRegression 5 8402.3 1680.5 596.32 0.000Residual Error 21 59.2 2.8Total 26 8461.4Source DF Seq SStemp 1 1510.7pressure 1 279.3tempsq 1 1067.6presssq 1 4909.7tp 1 635.110090801301201101009080TemperatureE(Y)E(Y(P=50))E(Y(P=55))E(Y(P=60))6055501301201101009080PressureE(Y)E(Y(T=80))E(Y(T=100))E(Y(T=90))Interaction models in Minitab•Use Calc >> Calculator to create interaction predictor variables in worksheet.•Use Stat >> Regression >> Regression as
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