Design of Experiments & One Factor Experiments

Systems Engineering Program Department of Engineering Management, Information and Systems EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS Design of Experiments & One Factor Experiments Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering

Method of Experimental Design Provides a systematic approach to planning what data is required and the analysis to be performed Designing an experiment - Planning an experiment so that information will be collected which relevant to the problem under investigation. The design of an experiment is the complete sequence of steps taken ahead of time to insure that the appropriate data will be obtained in a way which permits an objective analysis leading to valid inferences with respect to the stated problem.

Some requisites of a good experiment: 1. There must be a clearly defined objective. 2. The effects of the factors should not be obscured by other variables. 3. The results should not be influenced by conscious or unconscious bias in the experiment or on the part of the experimenter. 4. The experiment should provide some measure of precision. 5. The experiment must have sufficient precision to accomplish its purpose.

Experimental Units That unit to which a single treatment, which may be a combination of many factors, is applied in one replication of the basic experiment. A factor refers to an independent variable.

Treatment and Treatment Combinations Implies the particular set of experimental conditions which will be imposed on an experimental condition which will be imposed on an experimental unit within the confines of the chosen design.

Blocking The allocation of the experimental units to blocks in such a manner that the units within a block are relatively homogenous while the greater part of the predictable variation among units has been confounded with the effect of blocks.

Grouping The placing of a set of homogenous experimental units into groups in order that the different groups may be subjected to different treatments

Balancing Obtaining the experimental units, the grouping, the blocking and the assignment of the treatments to the experimental units in such a way that a balanced configuration exists.

Experimental Error The results of experiments are affected not only by the action of the treatments, but also by extraneous variations which tend to mask the effects of the treatments. The term ‘experimental errors’ is often applied to these variations, where the word errors is not synonymous with ‘mistakes’, but includes all types of extraneous variation. Two main sources of experimental errors are: 1. Inherent variability in the experimental material 2. Lack of uniformity in the physical conduct of the experiment, or failure to standardize the experimental technique.

Basic Principles of Experimental Design 1. Replication 2. Randomization 3. Local Control

Replication Repetition of the basic experiment. In order to evaluate the effects of factors, a measure of precision must be available. In situations where the measurement of precision must be obtained from the experiment itself, replication provides the measure. It also provides an opportunity for the effects of uncontrolled factors to balance out, and thus aids randomization as a bias-decreasing tool. Replication will also help to spot gross errors in measurement. Replication makes a test of significance possible.

Randomization By insisting on a random assignment of treatments to the experimental units, we can proceed as though the assumption: ‘Observations are independently distributed’, is true. Randomization makes the test valid by making it appropriate to analyze the data as though the assumption of independent error is true.

Local Control The amount of balancing, blocking, and grouping of the experimental units that is employed in the adopted statistical design. The function of local control is to make the experimental design more efficient. That is, local control makes any test of significance more sensitive. This increase in efficiency (or sensitivity) results because a proper use of local control will reduce the magnitude of the estimate of experimental error.

Steps in Designing an Experiment A statistically designed experiment consists of the following steps: 1. Statement of the problem. 2. Formulation of hypothesis. 3. Devising of experimental technique and design. 4. Examination of possible outcomes and reference back to the reasons for the inquiry to be sure the experiment provides the required information to an adequate extent. 5. Consideration of the possible results from the point of view of the statistical procedures which will be applied to them, to ensure that the conditions necessary for these procedures to be valid are satisfied.

Steps in Designing an Experiment continue 6. Performance of experiment. 7. Application of statistical techniques to the experimental results. 8. Drawing conclusions with measures of the reliability of estimates of any quantities that are evaluated, careful consideration being given to the validity of the conclusions for the population of objects or events to which they are to apply. 9. Evaluation of the whole investigation, particularly with other investigations on the same or similar problem. Note: Frequently, there is a formidable barrier to communications which must be overcome.

Check List for Planning Test Programs A. Obtain a clear statement of the problem. B. Collect available background information. C. Design a test program 1. Hold a conference of all parties concerned 2. Design the program in preliminary form 3. Review the design with all concerned D. Plan and carry out the experimental work. E. Analyze the data F. Interpret the results G. Prepare the report

One Factor Designs

Completely Randomized Designs • A design in which the treatments are assigned • completely at random to the experimental units, • or vice versa. It imposes no restrictions, such as • blocking, or the allocation of the treatments to • the experimental units. • Used because of its simplicity • Restricted to cases in which homogenous • experimental units are available

Model The basic assumption for a completely randomized design with one observation per experimental unit is that the observations may be represented mathematically by the linear statistical model Yij =  + ai + ij , i = 1, 2, …, k j = 1, 2, …, n where Yij is the observation associated with the ith treatment and jth experimental unit,  is the true mean effect (constant) aiis the true effect of the ith treatment and ijis the experimental error

Statistical Layout The results of a completely random experiment with one observation per experiment unit may be exhibited as follows: Treatment Total 1 2 . . . K Y11 Y21 . . . YK1 Y12 Y22 . . . . . . . . . . Y1n Y2n . . . YKn Totals Y1· Y2· . . . YK · Y·· Number of Obs. n n . . . nnK Means Y1 Y2 . . . YkY

Analysis of Variance In partitioning the total variation of the observations into the variation attributable to mean, treatments, and random error, the Sum-of-Squares is used: Total Sum of Squares = Treatment Sum of Squares + Error Sum of Squares SST = SSA + SSE

Analysis of Variance where and

SSE = SST - SSA Analysis of Variance Computational Formulas

Analysis of Variance Table (AOV Table or ANOVA Table) Sources of Degrees of Sum of Mean F- Critical Variation Freedom Squares Square Ratio Value of F Treatments K-1 SSA Error K(n-1) SSE Total nK-1 SST

Example Suppose that an appliance manufacturer is interested in determining whether the brand of laundry detergent used affects the amount of dirt removed from standard household laundry loads. In particular, the manufacturer wants to compare four different brands of detergent (labeled A, B, C, and D). Suppose that, after a random assignment of ten loads to each brand, the amount of dirt removed (measured in milligrams) was determined, with the results summarized below.

Example - Solution The statistical layout is: Treatment(Brand) Total A B C D 11 12 18 11 13 14 16 12 … … … … 14 18 20 18 Totals 139 172 183 149643 Number of Obs. 10 10 10 1040 Means 13.9 17.2 18.3 14.916.075

Example - Solution overall average=16.075

Example - Solution The sum of squares are calculated as follows: The mean squares are: The calculated F-ratio is:

Example - Solution Analysis of Variance Table Sources of Degrees of Sum of Mean of F- Critical Variation Freedom Squares Squares Ratio Value of F Treatments 3 123.275 41.09 6.92 2.866 Error 36 213.5 5.93 Total 39 336.775

Example - Conclusion • Since the probability of obtaining an F statistic of • 6.93 or larger when the null hypothesis is true is • approximately 0.001 and less than the specified • of 0.05, the null hypotheses is rejected. or since the calculated F-ratio of 6.93 is greater than the critical value of 2.87, the null hypotheses is rejected. Therefore conclude that the different brands of detergent are not equally effective.

Design of Experiments & One Factor Experiments