IE 429 DESIGNS OF EXPERIMENTS

IE 429 DESIGNS OF EXPERIMENTS Dr. Özlem Türker Bayrak A 206-B, ext: 4053 ozlemt@cankaya.edu.tr

Course Outline • Experiments with a single factor: The analysis of variance • Randomized Blocks, Latin squares and related designs • Design of experiments with several factors: Factorial experiments • The 2k factorial design • Blocking and confounding in the 2k factorial design • Two-level fractional factorial designs • Nested and Split Plot Designs • Taguchi’s contributions to experimental design and quality engineering

Text Book & Exams • Text Book: Montgomery, D. C., Design and Analysis of Experiments, 5th eds., John Wiley & Sons, Inc • Exams and Evaluation: • Midterm 1 250 points • Midterm 2 250 points • Final Exam 300 points • Class Exercises 200 points • TAKING MAKE UP EXAM IS NOT RECOMMENDED.

Attandence • Students are expected to attend all lectures. • Attendance must be at least 70%; else the grade NA is given. • The attendance to the lectures are not a regularity for the students taken this course before and got F*. • However, be aware that there is a grade for class exercises which requires attendance.

Office Hours • TBA • Students can consult with the instructor during her office hour, if they have difficulty in understanding any material after they have tried their best. • Other than the office hours, students have to make an appointment via email or phone at least one day advance to consult with the instructor.

INTRODUCTION IE 429 Lecture 1

Introduction • Example of a medical experiment • Two headache remedies are to compared, to see which is more effective at alleviating the symptoms. • Control (Drug 1) and New (Drug 2) - Drug type is the factor whose effect on the outcome is to be studied.

Introduction • Outcome variable: rate of blood flow to the brain, one hour after taking the drug. • Other factors may well affect the outcome – gender of the patient (M or F), dosage level (LO or HI), etc.

Introduction • Single factor experiment: here the drug type will be the only factor in our statistical model. • Patients suffering from headaches volunteer for the study, and are randomly divided into two groups. One group receives Drug 1, the other receives Drug 2. Within each group the dosage levels are randomly assigned. • We hope that the randomization will control for the other factors, in that (on average) there should be approximately equal numbers of men and women in each group, equal numbers at each dosage level, etc.

A statistical model: Suppose that n patients receive Drug 1 (i = 1) and n receive Drug 2 (i = 2). • Define yij to be the response of the jth subject (j = 1, ..., n) in the ith group. A possible model is where the parameters are: • μ =‘overall mean’ (mean effect of the drugs, irrespective of which one is used), estimated by the overall average

i =‘ith treatment effect’, estimated by, ,where =‘random error’ - measurement error, and effects of factors which are not included in the model, but perhaps should be.

We might instead control for the dosage level as well. • We could obtain 4n volunteers, split each group into four (randomly) of size n each and assign these to the four combinations • Drug 1/LO dose, • Drug 1/HI dose, • Drug 2/LO dose, • Drug 2/HI dose.

This is a 22 factorial experiment: 2 factors - drug type, dosage - each at 2 levels, for a total of 2 × 2 = 22 combinations of levels and factors. • Now the statistical model becomes more involved - it contains parameters representing the effects of each of the factors, and possibly interactions between them (e.g., Drug 1 may only be more effective at HI dosages, Drug 2 at LO dosages).

We could instead split the group into men and women (the 2 blocks) of subjects and run the entire experiment, as described above, within each block. • We expect the outcomes to be more alike within a block than not, and so the blocking can reduce the variation of the estimates of the factor effects.

There are other experimental designs which control for the various factors but require fewer patients - Latin squares, Graeco-Roman squares, repeated measures, etc. • Designing an experiment properly takes a good deal of thought and planning.

Stages of DOE • Experiment • Design • Analysis

Stages of DOE: 1. Experiment • Statement of the Problem: It is important to bring out all points of view to establish just what the experiment is intended to do. A careful statement of the problem gives a long way toward its solution. • Choice of Response Variable(s): i.e. criteria to be used in assessing the results of the study. • may be qualitative or quantitative. • Knowledge of it is essential because the shape of its distribution often dictates what statistical tests can be used in the subsequent data analysis. • One must also ask: Is the criterion measurable, and if so, how accurately can it be measured? What instruments are necessary to measure it?

Stages of DOE: 1. Experiment • Selection of Factors to be Varied: i.e. factors (indep. var.) that may affect the dep. or response variable. • Are these factors to be held constant, to be manipulated at certain levels, or to be averaged out by a process of randomization? • Choice of Levels of These Factors • Quantitative or Qualitative • Fixed or Random: Are levels of the factors to be set at certain fixed values, such as temp at 20, 35, 50 C, or are such levels to be chosen at random from among all possible levels? • How factor Levels to be Combined

Stages of DOE: 2. Design • Number of Observations to be Taken To decide on the size of the sample to be taken, it is important to consider: • how large a difference is to be detected, • how much variation is present, and, • what size risks are to be tolerated. • Order of Experimentation Should be random. Once a decision has been made to control certain variables at specified levels, there are always a number of other variables that cannot be controlled. • Randomization of the order of experimentation will tend to average out the effect of these uncontrolled variables. • Randomization will also permit the experimenter to proceed as if the errors of measurement were independent, a common assumption in most statistical analysis .

Stages of DOE: 2. Design • Method of Randomization to be Used What is the randomization procedure? and How are the units aranged for testing? • Math. Model to Describe the Experiment This model will show the response variable as a function of all factors to be studied and any restrictions imposed on the experiment as a result of the method of randomization. • Hypotheses to be Tested Since the objective of the research project is to shed light on a stated problem, one should now express the problem in terms of a testable hypothesis or hypotheses. A research hypothesis states what the experimenter expects to find in the data. It can usually be expressed in terms of the mathematical model set up.

Stages of DOE: 3. Analysis • Data collection and Processing • Computation of Test Statistics Test statistics are used in making decisions about various aspects of an experiment. Therefore analysis involves the computation of test statistics and their corresponding decision rules for testing hypotheses about the mathematical model. • Interpretation of Results For the Experimenter Decisions should be expressed in graphical or tabular form, in order that they be clearly understood by the experimenter and by those persons who are to be “sold” by the experiment. Statistical tests results also be used as “feedback” to design a better experiment, once certain hypotheses seem tenable.

IE 429 DESIGNS OF EXPERIMENTS