linear correlation between T values and coefficients for two-level factorial designs

#1
Hi all

Since I'm not sure if it is a problem of Minitab or not I decided to post this question here.

I'm new in multivariate statistics and I would like to understand a possible "linear correlation" between T values and the model coefficients in two-level factorial designs.

The problem is simple: a two-level factorial design with 2 variables and three replicates. Let's see this example from Minitab manual "Meet Minitab 16" (previous versions of the manual use the same example):

HTML:
<pre>
 A    B   Hours
-1   -1   14.72
 1    1    9.62
-1    1   13.81
 1    1    7.97
 1   -1   12.52
-1    1   13.78
-1   -1   14.64
 1    1    9.41
-1   -1   13.89
-1    1   13.89
 1   -1   12.57
 1   -1   14.06
</pre>
In Minitab example the factor A is the "OrderSystem" and the factor B is the "Pack".

When analysing the factorial design including the interaction and using alpha=0.05, we should get:

HTML:
<pre>
Estimated Effects and Coefficients for Hours (coded units)
Term               Effect    Coef   SE Coef      T      P  Coef/T
Constant                   12.573   0.1929   65.20  0.000   0.193
OrderSystem         3.097   1.548   0.1929    8.03  0.000   0.193
Pack               -2.320  -1.160   0.1929   -6.01  0.000   0.193
OrderSystem*Pack    1.730   0.865   0.1929    4.49  0.002   0.193

S = 0.668069     PRESS = 8.0337
R-Sq = 93.79%    R-Sq(pred) = 86.02%  R-Sq(adj) = 91.46%
</pre>
As I understood, T values here should be obtained from the Student's t-distribution and If |T| > 2 I should reject the null hypothesis for a factor or interaction.

The last column from this analysis was inserted by me and computes the value Coef/T = 0.193 for both the factors and the interaction and also for the constant.

Since I could not reproduce all the factorial design calculations by hand yet and I'm also unable to reproduce a factorial design in R I would like to know the general expression for this T value estimator. From these examples and anothers that I've "googled" and also from my own experimental data I believe that these T values in some conditions could behaves like a linear function of the coefficients but to me it appears to be totally wrong.

If I calculate all the contrasts, the mean over all the experimental data and the standard deviation of the sample, could I calculate this T value estimator or do I need to know anything else?

Could anyone suggest me a good reference to understand how to reproduce all these calculations by hand?

Thank you guys for your attention!

Regis
 
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