ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II Case Study 3 Trucking Industry What was the impact of deregulation on trucking prices in Florida? What is a good model for predicting prices? Get the data and plot them: truck <- read.table("Text/Cases/TRUCKING.txt", header = TRUE) pairs(truck[, c("DISTANCE", "WEIGHT", "PCTLOAD", "ORIGIN", "MARKET", "DEREG", "LNPRICE")]) 1 / 14 Case Study 3 Trucking Industry
ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II The dependent variable will be LNPRICE = log(price per ton-mile) The study chooses to omit one variable, PRODUCT . 2 / 14 Case Study 3 Trucking Industry
ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II Use stepwise regression to screen the other variables: truck$y <- truck$LNPRICE truck$x1 <- truck$DISTANCE truck$x2 <- truck$WEIGHT truck$x3 <- truck$DEREG truck$x4 <- truck$ORIGIN truck$x5 <- truck$PCTLOAD truck$x6 <- truck$MARKET start <- lm(y ~ 1, truck) firstOrder <- y ~ x1 + x2 + x3 + x4 + x5 + x6 summary(step(start, scope = firstOrder)) 3 / 14 Case Study 3 Trucking Industry
ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II This identifies: x 1 , DISTANCE ; x 2 , WEIGHT ; x 3 , the DEREG indicator; x 4 , the ORIGIN indicator. Stepping down from the full first-order model, instead of stepping up from the empty model, finds the same variables: summary(step(lm(firstOrder, truck), firstOrder)) 4 / 14 Case Study 3 Trucking Industry
ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II The study continues with the full second order model in these variables: lm1 <- lm(y ~ (x1 + x2 + x1:x2 + I(x1^2) + I(x2^2)) * x3 * x4, truck) summary(lm1) Note that none of the 8 squared terms are significant; try dropping them: lm2 <- lm(y ~ (x1 + x2 + x1:x2) * x3 * x4, truck) summary(lm2) anova(lm2, lm1) R 2 drops substantially, and F is highly significant, so the simpler model is rejected. 5 / 14 Case Study 3 Trucking Industry
ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II Next try dropping, from the full second order model, the interactions between quantitative and qualitative variables: lm3 <- lm(y ~ x1 + x2 + x1:x2 + I(x1^2) + I(x2^2) + x3 * x4, truck) summary(lm3) anova(lm3, lm1) Again F is significant, and the simpler model is rejected. 6 / 14 Case Study 3 Trucking Industry
ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II Next try: drop the interactions of the qualitative variables with only the squared terms: lm4 <- lm(y ~ (x1 + x2 + x1:x2) * x3 * x4 + I(x1^2) + I(x2^2), truck) summary(lm4) anova(lm4, lm1) Success! R 2 drops only a little, and R 2 a actually increases; also F is not significant. This simpler model is not rejected. 7 / 14 Case Study 3 Trucking Industry
ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II Next, explore whether x 4 , ORIGIN , can be dropped from this simpler model: lm5 <- lm(y ~ (x1 + x2 + x1:x2) * x3 + I(x1^2) + I(x2^2), truck) summary(lm5) anova(lm5, lm4) F is highly significant, so we reject the simpler model. 8 / 14 Case Study 3 Trucking Industry
ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II Next, explore whether x 3 , DEREG , can be dropped: lm6 <- lm(y ~ (x1 + x2 + x1:x2) * x4 + I(x1^2) + I(x2^2), truck) summary(lm6) anova(lm6, lm4) Again, F is highly significant, so we reject the simpler model. 9 / 14 Case Study 3 Trucking Industry
ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II Finally, explore whether x 3 , DEREG , interacts with x 4 , ORIGIN , by dropping their interaction terms: lm7 <- lm(y ~ (x1 + x2 + x1:x2) * (x3 + x4) + I(x1^2) + I(x2^2), truck) summary(lm7) anova(lm7, lm4) This time, F is not significant, so the simpler model, without the interactions, is not rejected. 10 / 14 Case Study 3 Trucking Industry
ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II Model-building with step() Suppose we begin with the full second order model and simplify it using step() and BIC (same result using AIC): stepLm1 <- step(lm1, direction = "both", k = log(nrow(truck))) summary(stepLm1) The model looks complicated, but the formula is equivalent to y ~ (x1 + x2 + x1:x2) * x3 * x4 + I(x1^2) : summary(lm(y ~ (x1 + x2 + x1:x2) * x3 * x4 + I(x1^2), truck)) This is Model 4 with I(x2^2) dropped. 11 / 14 Case Study 3 Trucking Industry
ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II Without screening: Suppose we skip the screening stage, and just use step() with all six variables; using BIC: secondOrder <- y ~ ((x1 + x2 + x5)^2 + I(x1^2) + I(x2^2) + I(x5^2)) * x3 * x4 * x6 stepBIC <- step(start, secondOrder, k = log(nrow(truck))) summary(stepBIC) This model is the same as y ~ x1 + I(x1^2) + x2 * x3 : quadratic function of x 1 = DISTANCE + interaction of x 2 = WEIGHT with x 3 = DEREG 12 / 14 Case Study 3 Trucking Industry
ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II Using AIC: stepAIC <- step(start, secondOrder) summary(stepAIC) This more complicated model can be written y ~ (x1 + I(x1^2)) * x6 + x2 * x3 . It is similar to the model found with BIC, but now the quadratic function of DISTANCE has different coefficients for each level of x 6 = MARKET . 13 / 14 Case Study 3 Trucking Industry
ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II These models have slightly lower R 2 than Models 4 or 7, but slightly better PRESS statistics. They show the effect of deregulation ( x 3 ) more clearly: intercept reduction of − 0 . 69 ( e − 0 . 69 = 0 . 5); coefficient of WEIGHT ( x 2 ): -0.028 to -0.057. 14 / 14 Case Study 3 Trucking Industry
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