If NumBid = 5, Price = −146.5 + 7.3 Age
If NumBid = 10, Price = −613 + 13.8 Age
If NumBid = 15, Price = −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 = −133.9 + 12.74 150 + 85.99 5 = $2206.85
Model including interaction:
𝑃 = 320 + 0.88 150 − 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 𝑥 = 𝑥 𝑥 in the model
3 1 2
𝐸 𝑦 = 𝛽 +0𝛽 𝑥 1 1 𝑥 + 2 2 𝑥 3 1 2
Is significant, you should not conduct t-tests on the coefficients of the first-order terms 𝑥 and 𝑥 .
These terms must be retained in the model regardless of their associated p-values.
Note: Interaction ≠ 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
𝑁𝑢𝑚𝐶𝑒𝑙𝑙𝑠 = −12471 + 4878𝑡
Forecast for 2015 (𝑡 = 10):
𝑁𝑢𝑚𝐶𝑒𝑙𝑙𝑠 = −12471 + 4878 10 = 36309
The log-linear function
ln 𝑁𝑢𝑚𝐶𝑒𝑙𝑙𝑠 = 5.524 + 0.5835𝑡
𝑁𝑢𝑚𝐶𝑒𝑙𝑙𝑠 = 𝑒 = 250.64𝑒
Forecast for 2015 (𝑡 = 10):
𝑁𝑢𝑚𝐶𝑒𝑙𝑙𝑠 = 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.
ln𝑌 = 𝛽 + 𝛽 ln(𝑡)
ln𝑌 = 𝛽 0 ln 𝑡( )𝛽1
ln𝑌 − ln𝑡 𝛽1= 𝛽 0
ln(𝑡𝛽 1 = 𝛽 0
𝑌 = 𝑒 𝛽0
𝑌 = 𝐶𝑡 , where 𝐶 = 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 𝑥-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