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Lecture 6

MGSC 372 Lecture 6: Fall 2016 Semester Notes 6

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McGill University
Management Science
MGSC 372
Brian Smith

If NumBid = 5, Price = βˆ’146.5 + 7.3 Age If NumBid = 10, Price = βˆ’613 + 13.8 Age If NumBid = 15, Price = βˆ’1079.5 + 20.3 Age Interpretation of Interaction For this example, we conclude that there is a significant interaction between Age and NumBid. This means that the response of Price to Age depends on the number of bidders. We see that more bidders always results in a higher auction price. Furthermore, as the number of bidders increases, the rate of increase of Price as a function of Age also increases. In practical terms, this implies that the antique dealer should reserve older clocks for larger auctions, as each extra year of age results in a large increase in price when there is a larger number of bidders. Warning about predictions Assume a clock aged 150 years and an auction with 5 bidders. Model ignoring interaction: 𝑃 = βˆ’133.9 + 12.74 150 + 85.99 5 = $2206.85 Model including interaction: 𝑃 = 320 + 0.88 150 βˆ’ 93.3 5 + 1.3 150 5 = $960.5 If we ignore interaction in this model, the predicted price will be seriously overestimated. Caution on Interaction If the interaction term π‘₯ = π‘₯ π‘₯ in the model 3 1 2 𝐸 𝑦 = 𝛽 +0𝛽 π‘₯ 1 1 π‘₯ + 2 2 π‘₯ 3 1 2 Is significant, you should not conduct t-tests on the coefficients of the first-order terms π‘₯ and π‘₯ . 1 2 These terms must be retained in the model regardless of their associated p-values. Note: Interaction β‰  Multicollinearity The example of the next slide shows a data set with two variables that are chosen to have no multicollinearity but do exhibit interaction. Logarithmic Transformations Example: Number of cell phones in new rural community The linear function π‘π‘’π‘šπΆπ‘’π‘™π‘™π‘  = βˆ’12471 + 4878𝑑 Forecast for 2015 (𝑑 = 10): ( ) π‘π‘’π‘šπΆπ‘’π‘™π‘™π‘  = βˆ’12471 + 4878 10 = 36309 The log-linear function ln π‘π‘’π‘šπΆπ‘’π‘™π‘™π‘  = 5.524 + 0.5835𝑑 5.524+0.5835𝑑 0.5835𝑑 π‘π‘’π‘šπΆπ‘’π‘™π‘™π‘  = 𝑒 = 250.64𝑒 Forecast for 2015 (𝑑 = 10): 0.5835 10 π‘π‘’π‘šπΆπ‘’π‘™π‘™π‘  = 250.64𝑒 = 85735 Note on model fit Note that the linear model seriously underestimates the predicted number of cell phones for 2015 assuming the historical exponential rate of growth continues at the same rate. Log-log function lnπ‘Œ = 𝛽 + 𝛽 ln(𝑑) 0 1 lnπ‘Œ = 𝛽 0 ln 𝑑( )𝛽1 lnπ‘Œ βˆ’ ln𝑑 𝛽1= 𝛽 0 π‘Œ ln(𝑑𝛽 1 = 𝛽 0 π‘Œ = 𝑒 𝛽0 𝑑𝛽1 𝛽 π‘Œ = 𝐢𝑑 , where 𝐢 = 𝑒 0 Lack of Fit Tests A regression model exhibits lack-of-fit when it fails to adequately describe the functional relationship between the experimental factors and the response variable. Lack-of-fit can occur if important terms from the model such as interactions or quadratic terms are not included. It can also occur if several, unusually large residuals result from fitting the model. Lack of Fit Test in Minitab Minitabdisplays the lack-of-fit test when your data contain replicates (multiple observations with identical π‘₯-values). Replicates represent "pure error" because only random variation can cause differences between the observed response values. To determine whether the model accurately fits the data, compare the p-value to your si
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