Assignment 7 - 2011

  .. |-| replace:: :math:`-` .. |+| replace:: :math:`+`

.. rubric:: Assignment objectives


 * Learn how to fit a DOE model by hand, and by computer.
 * Become comfortable with fractional factorials, confounding and blocking.

Question 1 [1]

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A concrete slump test is used to test for the fluidity, or workability, of concrete. It's a crude, but quick test often used to measure the effect of polymer additives that are mixed with the concrete to improve workability.

The concrete mixture is prepared with a polymer additive. The mixture is placed in a mold and filled to the top. The mold is inverted and removed. The height of the mold minus the height of the remaining concrete pile is called the "slump".

.. figure:: ../che4c3/Assignments/Assignment-7/images/types_of_concrete_slump.jpg :alt:	http://en.wikipedia.org/wiki/File:Types_of_concrete_slump.jpg :width: 500px :align: center

Your company provides the polymer additive, and you are developing an improved polymer formulation, call it B, that hopefully provides the same slump values as your existing polymer, call it A. Formulation B costs less money than A, but you don't want to upset, or loose, customers by varying the slump value too much.
 * Illustration from* `Wikipedia `_

The following slump values were recorded over the course of the day:

========== ================	Additive	Slump value [cm] ========== ================	A           5.2 A          3.3 B          5.8 A          4.6 B          6.3 A          5.8 A          4.1 B          6.0 B          5.5 B          4.5 ========== ================ In the midterm you derived the 95% confidence interval for the true, but unknown, difference between the effect of the two additives:

.. math:: \begin{array}{rcccl} -c_t &\leq& z	&\leq & +c_t \\ (\overline{x}_B - \overline{x}_A) - c_t \sqrt{s_P^2 \left(\frac{1}{n_B} + \frac{1}{n_A}\right)}	&\leq& \mu_B - \mu_A	&\leq & (\overline{x}_B - \overline{x}_A) + c_t \sqrt{s_P^2 \left(\frac{1}{n_B} + \frac{1}{n_A}\right)}\\ 1.02 - 2.3 \sqrt{0.709 \left(\frac{1}{5} + \frac{1}{5}\right)} 	&\leq& \mu_B - \mu_A	&\leq& 1.02 + 2.3 \sqrt{0.709 \left(\frac{1}{5} + \frac{1}{5}\right)} \\ -0.21	&\leq& \mu_B - \mu_A	&\leq&  2.2 \end{array} Fit a least squares model to the data this time, using an integer variable, :math:`x_A = 0` for additive A, and :math:`x_A = 1` for additive B. The model should include an intercept term also: :math:`y = b_0 + b_A x_A`. *Hint*: use R to build the model, and search the R tutorial with the term *categorical variable* or *integer variable* for assistance.

Show that the 95% confidence interval for :math:`b_A` gives exactly the same lower and upper bounds, as derived in the midterm with the traditional approach for tests of differences.

Question 2 [1]

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A liquid polymer formulation is being made that is applied as a polish to wood surfaces. The group responsible for the product have identified 3 elements to the formulation that have an effect of the liquid polish's final quality attributes (FQAs: this acronym is becoming a standard in most companies these days).


 * **A**: amount of reactive monomer in the recipe (10% at the low level and 30% at the high level)
 * **B**: the amount of chain length regulator (1% at the low level and 4% at the high level)
 * **C**: the type of chain length regulator (regulator P at the :math:`-` level or regulator Q at the :math:`+` level)

In class we have focused on the case where our :math:`y`-variable is continuous, but it could also be *descriptive*. In this question we also see what happens when we have more than one :math:`y`-variable.


 * :math:`y_1` = Milky appearance: either *Yes* or *No*
 * :math:`y_2` = Viscous: either *Yes* or *No*
 * :math:`y_3` = Yellow colour: either *No* or *Slightly*

The following table captures the 8 experiments in standard order, although the experiments were run in a randomized order.

.. tabularcolumns:: |l||c|c|c||c|c|c|

+---++---+++++ +===========+============+===========+============+============+============+============+ +---++---+++++ +---++---+++++ +---++---+++++ +---++---+++++ +---++---+++++ +---++---+++++ +---++---+++++ +---++---+++++
 * Experiment| A         | B         |  C         | :math:`y_1`| :math:`y_2`| :math:`y_3`|
 * 1        | |-|        | |-|       |  P         |  Yes       |  Yes       |  No        |
 * 2        | |+|        | |-|       |  P         |  No        |  Yes       |  No        |
 * 3        | |-|        | |+|       |  P         |  Yes       |  No        |  No        |
 * 4        | |+|        | |+|       |  P         |  No        |  No        |  No        |
 * 5        | |-|        | |-|       |  Q         |  Yes       |  Yes       |  No        |
 * 6        | |+|        | |-|       |  Q         |  No        |  Yes       |  Slightly  |
 * 7        | |-|        | |+|       |  Q         |  Yes       |  No        |  No        |
 * 8        | |+|        | |+|       |  Q         |  No        |  No        |  Slightly  |


 * 1) .	What is the cause of a milky appearance?
 * 2) .	What causes a more viscous product?
 * 3) .	What is the cause of a slight yellow appearance?
 * 4) .	Which conditions would you use to create a product was *not* milky, was of low viscosity, and had no yellowness?
 * 5) .	Which conditions would you use to create a product was *not* milky, was of low viscosity, and had some yellowness?

Question 3 [2]

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Using a :math:`2^3` factorial design in 3 variables (**A** = temperature, **B** = pH and **C** = agitation rate), the conversion, :math:`y`, from a chemical reaction was recorded.

+---++---+++ +===========+============+===========+============+============+ +---++---+++ +---++---+++ +---++---+++ +---++---+++ +---++---+++ +---++---+++ +---++---+++ +---++---+++
 * Experiment| A         | B         |  C         | :math:`y`  |
 * 1        | |-|        | |-|       |  |-|       |  72        |
 * 2        | |+|        | |-|       |  |-|       |  73        |
 * 3        | |-|        | |+|       |  |-|       |  66        |
 * 4        | |+|        | |+|       |  |-|       |  87        |
 * 5        | |-|        | |-|       |  |+|       |  70        |
 * 6        | |+|        | |-|       |  |+|       |  73        |
 * 7        | |-|        | |+|       |  |+|       |  67        |
 * 8        | |+|        | |+|       |  |+|       |  87        |


 * **A** = :math:`\displaystyle \frac{\text{temperature} - 150\text{°C}}{10\text{°C}}`
 * **B** = :math:`\displaystyle \frac{\text{pH} - 7.5}{0.5}`
 * **C** = :math:`\displaystyle \frac{\text{agitation rate} - 50 \text{rpm}}{5 \text{rpm}}`


 * 1) .	Show a cube plot for the recorded data.
 * 2) .	Estimate the main effects and interactions by hand.
 * 3) .	Interpret any results from part 2.
 * 4) .	Show that a least squares model for the full factorial agrees with the effects and interactions calculated by hand.
 * 5) .	Approximately, at what conditions (given in real-world units), would you run the next experiment to improve conversion. Give your settings in coded units, then unscale and uncenter them to get real-world units.

Question 4 [2]

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 * 1) .	Why do we block groups of experiments?
 * 2) .	Write a :math:`2^3` factorial design in two blocks of 4 runs, so that no main effect or 2 factor interaction is confounded with block differences.

Question 5 [3]

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.. note:: This question throws you in the deep end with fractional factorials. You may have to read ahead in the notes to answer all the parts of this question.

Factors related to the shrinkage of plastic film, produced in an injection molding device, are being investigated. The following factors have been identified by the engineer responsible:


 * **A** = mold temperature
 * **B** = moisture content
 * **C** = holding pressure
 * **D** = cavity thickness
 * **E** = booster pressure
 * **F** = cycle time
 * **G** = gate size

+---+-+-+-+-+-+-+-++ +===========+=====+=====+=====+=====+=====+=====+=====+============+ +---+-+-+-+-+-+-+-++ +---+-+-+-+-+-+-+-++ +---+-+-+-+-+-+-+-++ +---+-+-+-+-+-+-+-++ +---+-+-+-+-+-+-+-++ +---+-+-+-+-+-+-+-++ +---+-+-+-+-+-+-+-++ +---+-+-+-+-+-+-+-++
 * Experiment| A  | B   | C   | D   | E   | F   | G   | :math:`y`  |
 * 1        | |-| | |-| | |-| | |+| | |+| | |+| | |-| |  14.0      |
 * 2        | |+| | |-| | |-| | |-| | |-| | |+| | |+| |  16.8      |
 * 3        | |-| | |+| | |-| | |-| | |+| | |-| | |+| |  15.0      |
 * 4        | |+| | |+| | |-| | |+| | |-| | |-| | |-| |  15.4      |
 * 5        | |-| | |-| | |+| | |+| | |-| | |-| | |+| |  27.6      |
 * 6        | |+| | |-| | |+| | |-| | |+| | |-| | |-| |  24.0      |
 * 7        | |-| | |+| | |+| | |-| | |-| | |+| | |-| |  27.4      |
 * 8        | |+| | |+| | |+| | |+| | |+| | |+| | |+| |  22.6      |

You can obtain a copy of this data set if you install the ``BsMD`` package in R. Then use the following commands:

.. code-block:: s

library(BsMD) data(BM93.e3.data) # Use only a subset of the original experiments X <- BM93.e3.data[1:8, 2:10]


 * 1) .	How many experiments would have been required for a full factorial experiment?
 * 2) .	What type of fractional factorial is this (i.e. is it a half fraction, quarter fraction ...)?
 * 3) .	Identify all the generators used to create this design. A table, such as on page 272 in Box, Hunter and Hunter, 2nd edition will help.
 * 4) .	Write out the complete defining relationship.
 * 5) .	What is the resolution of this design?
 * 6) .	Use a least squares approach to calculate a model that fits these 8 experiments.
 * 7) .	What effects would you judge to be significant in this system? The engineer will accept your advice and disregard the other factors, and spend the rest of the experimental budget only on the factors deemed significant.
 * 8) .	What are these effects aliased with (use your defining relationship to find this).
 * 9) .	Why is in necessary to know the confounding pattern for a fractional factorial design.

.. raw:: latex

\vspace{0.5cm} \hrule \begin{center}END\end{center}