What is the natural model for design and how many df are

Question

Question #1:

In the manufacture of integrated circuits, it is often necessary to coat silicon wafers with photoresist for later lithographic imaging. To learn more about a specific resist, an experiment was run to determine the effects of several factors on the photoresist coating process. The three most important factors were thought to be development time, prebake temperature, and post-exposure bake temperature. Based on current theories, the scientists thought that the single most important factor would be development time-this was thought to increase the Gamma value (explained below), but in a non-linear way. The second most important factor was expected to be pre-bake, and the third most important factor was thought to be post-exposure bake.

Three responses were measured:

• Gamma ("Gamma"). A measure of how fast the resist will clear. A higher gamma refers to a sharper image and this is desired. This is the most important response.
• Dose to clear ("DTC"). Amount of energy needed to expose the resist that will develop away cleanly. Minimum amount is desired. This is the second most important response.
• Thickness loss ("ThkLoss"). This is a loss of photoresist after development. Minimal loss is desired. This is third most important response.

The factors levels were

• Development time ("DevTime"). 37, 45 (standard), 53, and 61 seconds.
• Prebake temperature ("PreBake"). 100 (standard) and 110^oC.
• Post-Exposure bake ("PEB"). 110 (standard) and 120^oC.

All combinations were run twice. The experiment was run in a completely random order, over a one-day period. One wafer was used per run, so 32 wafers were used in all. The wafers were selected at random from a set of wafers that were available for the experiment. The wafer # is the same as the time order-for example, Wafer #1 was run first, and Wafer #32 was run last.

The  Treatment Structure is:4x2x2 factorial

The Error-Control Structure is: Completely Randomized Design, with 2 replicates

The extended structure diagram is:

Where DT = DevTime, Fixed with 4 treat levels, Pr = PreBake, Fixed with 2 Treat levels, and PE = Post Bake, Fixed with 2 treat levels, and R = Wafers, Random Experimental Units

Q. Critique the error-control structure by explaining a simple way to create a better error-control structure (without using any additional resources). Show the structure diagram for this improved design. (Of course, the experiment was already run, so it is too late to change the design! What I want you to think about is how you would have improved the design while the design was still being created.)

Question #2:

If you run a 2³treatment design in a Latin-Square type design with 14 people as the levels for the first blocking factor and 8 time points as the levels for the second blocking factor, (a) what is the natural model for this design and (b) how many d.f. are there for each term in the model?