Assignment 7 - 2014

Due date(s):	07 April 2014, in class.
(PDF)	Assignment questions

<rst> <rst-options: 'toc' = False/> <rst-options: 'reset-figures' = False/> .. note::

Source code will **not be graded** in this assignment. The purpose of this assignment is to show your understanding of the software output.

Assignment objectives

=========

.. question:: :grading: 11

Your company is developing a microgel-hydrogel composite, used for controlled drug delivery with a magnetic field. A previous employee did the experimental work but she has since left the company. You have been asked to analyze the existing experimental data.

The response variable, :math:`y` = sodium fluorescein (SF) released [mg], per gram of gel and the data collected, in the original units are:

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

+-----------+-------+----------------------------+----------------------------+------------+ | Experiment| Order | **M** = microgel weight [%]| **H** = hydrogel weight [%]| :math:`y` | +===========+=======+============================+============================+============+ | 1 | 4 | 4 | 10 | 119 | +-----------+-------+----------------------------+----------------------------+------------+ | 2 | 1 | 8 | 10 | 93 | +-----------+-------+----------------------------+----------------------------+------------+ | 3 | 6 | 4 | 16 | 154 | +-----------+-------+----------------------------+----------------------------+------------+ | 4 | 3 | 8 | 16 | 89 | +-----------+-------+----------------------------+----------------------------+------------+ | 5 | 2 | 6 | 13 | 85 | +-----------+-------+----------------------------+----------------------------+------------+ | 6 | 5 | 6 | 13 | 88 | +-----------+-------+----------------------------+----------------------------+------------+ | 7 | 9 | 3.2 | 13 | 125 | +-----------+-------+----------------------------+----------------------------+------------+ | 8 | 7 | 8.8 | 13 | 111 | +-----------+-------+----------------------------+----------------------------+------------+ | 9 | 10 | 6 | 17.2 | 136 | +-----------+-------+----------------------------+----------------------------+------------+ | 10 | 8 | 6 | 8.8 | 98 | +-----------+-------+----------------------------+----------------------------+------------+

#. What was likely the reason the experimenter added experiments 5 and 6?

#. Why might the experimenter have added experiments 7, 8, 9 and 10 after the first six? Provide a rough sketch of the design, and all necessary calculations to justify your answer.

#. What is the name of the type of experimental design chosen by the employee for *all 10 experiments in the table*?

#. Using these data, you wish to estimate a nonlinear approximation of the response surface using a model with quadratic terms. Write out the equation of such a model that can be calculated from these 10 experiments (*also read the next question*).

#. Write out

* the :math:`\mathbf{X}` matrix, * the corresponding symbolic entries in :math:`\mathbf{b}` * and the :math:`\mathbf{y}` vector

that you would use to solve the set of linear equations :math:`\mathbf{b} = \left(\mathbf{X}^T \mathbf{X} \right)^{-1} \mathbf{X}^T \mathbf{y}` to obtain the parameter estimates of the model you proposed in the previous part. You must use data from all 10 experiments.

#. How many degrees of freedom will be available to estimate the standard error and confidence intervals?

#. Now calculate the coefficients in the linear model using computer software. Which terms in the model are statistically significant?

**Note**: "*linear*" implies the model is linear in the coefficients, not in the terms; that is because the coefficients in front of the nonlinear terms can still be found from solving a set of linear equations).

.. question:: :grading: 7

The following diagram shows data from a central composite design. The factors were run at their standard levels, and there were 4 runs at the center point.

#. Calculate the parameters for a suitable quadratic model in these factors. Show your matrices for :math:`\mathbf{X}` and :math:`\mathbf{y}`.

#. Draw a response surface plot of **A** *vs* **B** over a suitably wide range beyond the experimental region.

#. Where would you move **A** and **B** if your objective is to increase the response value?

#. Report your answer in coded units. #. Report your answer in real-world units, if the full factorial portion of the experiments were ran at:

* **A** = *stirrer speed*, 200rpm and 340 rpm * **B** = *stirring time*, 30 minutes and 40 minutes

.. image:: ../figures/doe/central-composite-question.png :align: center :width: 500px :scale: 60 :alt: ../figures/doe/central-composite-question.svg

You might feel more comfortable setting up the problem in MATLAB. You can use the `contour plot <http://www.mathworks.com/access/helpdesk/help/techdoc/creating_plots/f10-2524.html>`_ functions in MATLAB to visualize the results.

If you are using R, you can use the ``rbind(...)`` or ``cbind(...)`` functions to build up your :math:`\mathbf{X}` matrix row-by-row or column-by-column. The equivalent of meshgrid in R is the ``expand.grid(...)`` function. Pee the `R code on the course website <http://learnche.mcmaster.ca/4C3/Design_and_analysis_of_experiments_(2014)>`_ that shows how to generate surface plots in R.

.. question:: :grading: 30

The next question is `posted here <http://learnche.mcmaster.ca/4C3/Response_surface_project_-_2014>`_. It is due as a separate hand-in. </rst>