Regression model for a shearing process 30 Widgets are randomly assigned to a shearing process.
There are 3 such processes, each getting 10 widgets.
The lengths of each widget are recorded before undergoing the shearing.
The amount that was sheared off is recorded after.
$$
\begin{array}{c|lcr}
& \text{Process A Length} & \text{Process B Length} & \text{Process C Length} \\
\hline
& \text{Before} \ / \ \text{Sheared} & \text{Before} \ / \ \text{Sheared} & \text{Before} \ / \ \text{Sheared}\\
1& 10 / 3 & 11 / 2 & 12/4 \\
2& 9.5/2 & 15/7 & 17.5/2 \\
3& etc &etc &etc \\
4& \\
5& \\
6& \\
7& \\
8& \\
9& \\
10& \\
\end{array}
$$
I need a model for this to do an analysis, to the effect of, recommending which process is better for shearing.
Something of the form:
$$  \hat{y} = \beta_0 + \beta_1X_1+\beta_2X_2+\beta_3X_3+\epsilon $$
But I'm not entirely sure what it should be.
Thanks.
 A: This sounds to me like some sort of experiment, unfortunately with few observations per group. The first thing you could do is simply compare the means of your outcome variable (sheared after?), for example with a t-test. Because you have few observations, significant effects will be hard to find (even if there is an effect). Also due to few observations, you should look at the t-test normality assumption again - the central limit theorem does not apply for 10 or 20 observations. If it's not fulfilled then you can go with the Wilcoxon rank sum test, but it is less efficient, so finding an effect there is even harder.
But maybe you don't want to neglect the length before shearing. For $\ge50$ per group, one could ignore this due to random assignment of units and safely go with the above approach, but for 10 per group there may very well be pre-treatment differences. What you can do, then, is some OLS specification that controls for those pre-treatment levels (LENGTH), e.g.,
$$ShearedAfter_i=\beta_0+\beta_1 GROUP1_i\times LENGTH_i+\beta_2 GROUP2_i+\beta_3 GROUP2_i\times LENGTH_i+\varepsilon_i.$$
This allows you to test for differences in outcome, controlling for length before shearing. If $\beta_2\neq 0$, then there are differences between group 1 and 2. The test for differences would also be a t-test, but this time controlling for length. (The specification above can also be amended to include all 3 groups by adding the GROUP3 dummy and the length-group3 interaction.)
Next time: try to get more observations per group, and keep the random assignment. Lab experiments rarely go with less than 30 per group nowadays.
