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Home > 3.6 - The General Linear Test

3.6 - The General Linear TestThis is just a general representation of an F-test based on a full and a reduced model. We willuse this frequently when we look at more complex models.

Let's illustrate the general linear test here for the single factor experiment:

First we write the full model, Yij = μ + τi + εij and then the reduced model, Yij = μ + εij whereyou don't have a τi term, you just have an overall mean, μ. This is a pretty degenerate modelthat just says all the observations are just coming from one group. But the reduced model isequivalent to what we are hypothesizing when we say the μi would all be equal, i.e.:

H0 : μ1 = μ2 = ... = μa

This is equivalent to our null hypothesis where the τi's are all equal to 0.

The reduced model is just another way of stating our hypothesis. But in more complexsituations this is not the only reduced model that we can write, there are others we could lookat.

The general linear test is stated as an F ratio:

This is a very general test. You can apply any full and reduced model and test whether or notthe difference between the full and the reduced model is significant just by looking at thedifference in the SSE appropriately. This has an F distribution with (df R - df F), df F degreesof freedom, which correspond to the numerator and the denominator degrees of freedom ofthis F ratio.

Let's take a look at this general linear test using Minitab...

Example - Cotton Weight

F = (SSE(R) − SSE(F))/(df R − df F)SSE(F)/df F

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Remember this experiment had treatment levels 15, 20,25, 30, 35 % cotton weight and the observations werethe tensile strength of the material.

The full model allows a different mean for each level ofcotton weight %.

We can demonstrate the General Linear Test byviewing the ANOVA table from Minitab:

STAT > ANOVA > Balanced ANOVA

The SSE(R) = 636.96 with a dfR = 24, and SSE(F) = 161.20 with dfF = 20. Therefore:

This demonstrates the equivalence of this test to the F-test. We now use the General LinearTest (GLT) to test for Lack of Fit when fitting a series of polynomial regression models todetermine the appropriate degree of polynomial.

We can demonstrate the General Linear Test by comparing the quadratic polynomial model(Reduced model), with the full ANOVA model (Full model). Let Yij = μ + β1xij + β2xij

2 + εij bethe reduced model, where xij is the cotton weight percent. Let Yij = μ + τi + εij be the fullmodel.

[1]

The viewlet above shows the SSE(R) = 260.126 with dfR = 22 for the quadratic regressionmodel. The ANOVA shows the full model with SSE(F) = 161.20 with dfF = 20.

Therefore the GLT is:

We reject H0 : Quadratic Model and claim there is Lack of Fit if F* > F1-α (2, 20) = 3.49.

Therefore, since 6.14 is > 3.49 we reject the null hypothesis of no Lack of Fit from thequadratic equation and fit a cubic polynomial. From the viewlet above we noticed that the

=F ∗ (636.96 − 161.20)/(24 − 20)161.20/20

F ∗ =

=

=

==

(SSE(R) − SSE(F))/(df R − df F)SSE(F)/df F

(260.126 − 161.200)/(22 − 20)161.20/20

98.926/28.06

49.468.06

6.14

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cubic term in the equation was indeed significant with p-value = 0.015.

We can apply the General Linear Test again, now testing whether the cubic equation isadequate. The reduced model is:

Yij = μ + β1xij + β2xij2 + β3xij

3 + εij

and the full model is the same as before, the full ANOVA model:

Yij = μ + τi + εij

The General Linear Test is now a test for Lack of Fit from the cubic model:

We reject if F* > F0.95 (1, 20) = 4.35.

Therefore we do not reject Ha: Lack of Fit and conclude the data are consistent with the cubicregression model, and higher order terms are not necessary.

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Source URL: https://onlinecourses.science.psu.edu/stat503/node/17

Links:[1] https://onlinecourses.science.psu.edu/stat503/javascript:popup_window('/stat503/sites/onlinecourses.science.psu.edu.stat503/files/lesson03/L03_cotton_weight_viewlet_swf.html','l03_cotton_weight', 704, 652 );

F ∗ =

=

==

(SSE(R) − SSE(F))/(df R − df F)SSE(F)/df F

(195.146 − 161.200)/(21 − 20)161.20/20

33.95/18.06

4.21