coefTest and one tailed T-test

Dear all,

I have fitted a linear model to my data as follows: Var6 ~ 1 + groupVar1 +groupVar2 + groupVar3 +Var4 + Var5

the first 3 variables are dummy variables for the three groups that I have, and I aim to compare groups using GeneralizedLinearModel.fit and adjusting for nuisance variables. Using coefTest with this contrast I get the f statistic of "any group different from zero" while controlling for Var4 and Var5: [0 1 0 0 0 0;0 0 1 0 0 0;0 0 0 1 0 0] and I get a p value, and so on. However, when it comes to one tail t tests for the following two contrasts I get the same significant p value for which does not make sense to me:

contrast 2:Group1 >Group2 coefTest(m, [0 1 -1 0 0 0]) contrast 3:Group2 >Group1 coefTest (m, [0 -1 1 0 0 0])

It would be great if someone could let me know where I'm getting this wrong, as there is no way for a group to be greater and lesser than another group at the same time.

-Arman

[Arman]

So probably I'm just doing an F-test, is there a way to do one tailed t-tests in GeneralizedLinearModel in MATLAB?

Answer 1

[Tom Lane]

You are right that the result from coefTest is an F-test. There is no built-in way to carry out a one-sided t test.

Here are some commands to reproduce the calculations for the t statistic and its p-value as they appear int the coefficient table:

load carsmall
d = dataset(MPG,Weight);
d.Year = ordinal(Model_Year);
glm = GeneralizedLinearModel.fit(d,'MPG ~ Year + Weight + Weight^2')
glm.Coefficients.Estimate(3)
glm.Coefficients.Estimate(3)/sqrt(glm.CoefficientCovariance(3,3))
2*(1 - tcdf(glm.Coefficients.Estimate(3)/sqrt(glm.CoefficientCovariance(3,3)),glm.DFE))

You could compute a contrast among the coefficients (instead of just taking the third one as I did), use the covariance matrix to compute the variance of this contrast, and so on. Then you could pick the desired tail of the t distribution.

[Arman]

Thanks a lot for your answer. I managed to understand most of the procedure, so back to my model:

responseVariable ~ 1 + groupVar1 +groupVar2 + groupVar3 +nuisanceVar4 + nuisanceVar5

When my contrast is [0 1 -1 0 0 0] to test the hypothesis of whether the mean of second group is lower than that of first, I have difficulty understanding the appropriate coefficient and covariance for the rest of the analysis. I think the correct coefficient for this contrast is the linear combination of that of the "groupVar1" and "groupVar2", is that right? And with regard to the coefficient covariance for t-statistic calculation, is it the linear combination of covariances for each of the regression coefficients (namely, "groupVar1" and "groupVar2" from glm.CoefficientCovariance matrix provided in MATLAB)?

[Tom Lane]

If you have three groups and you include dummy variables for them all, I'd expect that to be collinear with the constant term. But in general for a contrast such as c=[0 1 -1 0 0 0], coefficient vector b, and covariance matrix V, c*b is your estimated contrast and c*V*c' its variance.