Example of polynomial regression with continuous and categorical
predictorsmain topic interpreting results session command see also
Example of polynomial regression with continuous and categorical
predictorsAn engineer at a computer manufacturing company wants to understand the relationship between the strength of plastic and the predictors temperature and manufacturer. He suspects the relationship between temperature and strength is quadratic and also wants to test for an interaction between manufacturer and temperature. He collects 14 samples of plastic from Manufacturer A and 13 samples from Manufacturer B. The engineer subjects the samples to various temperatures, then measures the strength of the plastic.
1 Open the worksheet PLASTIC.MTW.
2 Choose Stat > Regression > Regression > Fit Regression Model.
3 In Responses, enter Strength.
4 In Continuous predictors, enter Temp.
5 In Categorical predictors, enter Manufacturer.
6 Click Model.
7 Under Predictors, choose Temp and Manufacturer.
8 To the right of Interactions through order, choose 2, and click Add.
9 Under Predictors, choose Temp.
10 To the right of Terms through order, choose 2, and click Add.
11 Click OK.
12 Click Graphs.
13 Under Residuals for Plots, choose Standardized.
14 Under Residuals Plots, choose Four in one.
15 Click OK in each dialog box.
Session window output
Regression Analysis: Strength versus Temp, Manufacturer
Method
Categorical predictor coding (1, 0)
Analysis of Variance
Source DF Adj SS Adj MS F-Value P-Value Regression 4 276060 69015.1 58.28 0.000 Temp 1 22256 22255.9 18.79 0.000 Manufacturer 1 21 21.2 0.02 0.895 Temp*Temp 1 22581 22580.8 19.07 0.000 Temp*Manufacturer 1 21 20.6 0.02 0.896 Error 22 26054 1184.3 Lack-of-Fit 17 22644 1332.0 1.95 0.237 Pure Error 5 3411 682.1 Total 26 302115
Model Summary
S R-sq R-sq(adj) R-sq(pred) 34.4134 91.38% 89.81% 86.31%
Coefficients
Term Coef SE Coef T-Value P-Value VIF Constant -39905 10470 -3.81 0.001 Temp 472 109 4.34 0.000 12400.05 Manufacturer B -152 1136 -0.13 0.895 7350.60 Temp*Temp -1.234 0.283 -4.37 0.000 12866.50 Temp*Manufacturer B 0.76 5.75 0.13 0.896 7636.31
Regression Equation
Manufacturer A Strength = -39905 + 472 Temp - 1.234 Temp*Temp
B Strength = -40057 + 473 Temp - 1.234 Temp*Temp
Fits and Diagnostics for Unusual Observations
Obs Strength Fit Resid Std Resid 27 4760.0 4836.6 -76.6 -2.62 R
R Large residual |
Graph window output
Minitab displays two regression equations, one for each level of the categorical predictor Manufacturer. You can use these equations to predict for specific values of temperature.
The analysis of variance table indicates that the quadratic relationship between temperature and plastic strength is significant (P = 0.000).
The VIFs are very high. VIF values greater than 5-10 suggest that the regression coefficients are poorly estimated due to severe multicollinearity.
In this case, the VIFs are high because of the higher-order terms. Higher-order terms are correlated with main effect terms because they include the main effects terms. To reduce the VIFs, you can fit the model with one of the standardize predictors options in the Coding subdialog that subtracts the mean.
The R-Sq value shows that regression model explains 91.38% of the variance in strength, indicating that the model fits the data fairly well. R-Sq(pred) is 86.31%.
Observation 27 is identified as an unusual observation. This could indicate that this observation is an outlier. See Checking your model, Identifying outliers, and Choosing a residual type.
The histogram shows one potential outlier in the data.
The normal probably plot shows an approximately linear pattern consistent with a normal distribution.
The plot of residual versus the fitted values shows a random pattern, which suggests that the residuals have constant variance. See [11] for information on non-constant variance. This plot also shows one potential outlier.
The residuals versus order plot shows the order that the data was collected and can be used to find non-random error, especially of time-related effects. The residuals versus order plot does not display a pattern.