Example of Fit General Linear Model

An electronics design engineer studies the effect of operating temperature and three types of face-plate glass on the light output of an oscilloscope tube.

To study the effect of temperature, glass type, and the interaction between these two factors, the engineer uses a general linear model.

Open the sample data, LightOutput.MTW.
Choose Stat > ANOVA > General Linear Model > Fit General Linear Model.
In Responses, enter LightOutput.
In Factors, enter GlassType.
In Covariates, enter Temperature.
Click Model.
In Factors and covariates, select GlassType and Temperature.
To the right of Interactions through order, select 2, and click Add.
In Factors and covariates, select Temperature.
To the right of Terms through order, select 2, and click Add.
In Factors and covariates, select GlassType and, in Terms in the model, select Temperature*Temperature.
To the right of Cross factors, covariates, and terms in the model, click Add.
Click OK in each dialog.

Interpret the results

In the Analysis of Variance table, the p-values for all of the terms are 0.000. Because the p-values are less than the significance level of 0.05, the engineer can conclude that the effects are statistically significant.

The R² value shows that the model explains 99.73% of the variance in light output, which indicates that the model fits the data extremely well.

The VIFs are very high. VIF values that are 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 the high-order tems also include the main effects terms. To reduce the VIFs, you can standardize the covariates in the Coding sub-dialog box.

Observations with large standardized residuals or large leverage values are flagged. In this example, two values have standardized residuals whose absolute values are greater than 2. You should investigate unusual observations because they can produce misleading results.

General Linear Model: LightOutput versus Temperature, GlassType

Method

Factor coding	(-1, 0, +1)

Factor Information

Factor	Type	Levels	Values
GlassType	Fixed	3	1, 2, 3

Analysis of Variance

Source	DF	Adj SS	Adj MS	F-Value	P-Value
Temperature	1	262884	262884	719.21	0.000
GlassType	2	41416	20708	56.65	0.000
Temperature*Temperature	1	190579	190579	521.39	0.000
Temperature*GlassType	2	51126	25563	69.94	0.000
TemperatureTemperatureGlassType	2	64374	32187	88.06	0.000
Error	18	6579	366
Total	26	2418330

Model Summary

S	R-sq	R-sq(adj)	R-sq(pred)
19.1185	99.73%	99.61%	99.39%

Coefficients

Term	Coef	SE Coef	T-Value	P-Value	VIF
Constant	-4969	191	-25.97	0.000
Temperature	83.87	3.13	26.82	0.000	301.00
GlassType
1	1323	271	4.89	0.000	3604.00
2	1554	271	5.74	0.000	3604.00
Temperature*Temperature	-0.2852	0.0125	-22.83	0.000	301.00
Temperature*GlassType
1	-24.40	4.42	-5.52	0.000	15451.33
2	-27.87	4.42	-6.30	0.000	15451.33
TemperatureTemperatureGlassType
1	0.1124	0.0177	6.36	0.000	4354.00
2	0.1220	0.0177	6.91	0.000	4354.00

Regression Equation

GlassType
1	LightOutput	=	-3646 + 59.47 Temperature - 0.1728 Temperature*Temperature

2	LightOutput	=	-3415 + 56.00 Temperature - 0.1632 Temperature*Temperature

3	LightOutput	=	-7845 + 136.13 Temperature - 0.5195 Temperature*Temperature

Fits and Diagnostics for Unusual Observations

Obs	LightOutput	Fit	Resid	Std Resid
11	1070.0	1035.0	35.0	2.24	R
17	1000.0	1035.0	-35.0	-2.24	R