1 Dr. Jerrell T. Stracener EMIS 7370 STAT 5340 Probability and Statistics for Scientists and Engineers Department of Engineering Management, Information and Systems Design of Experiments & One Factor Experiments

2 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. Method of Experimental Design

3 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. Some requisites of a good experiment:

4 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. Experimental Units

5 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. Treatment and Treatment Combinations

6 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. Blocking

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

8 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. Balancing

9 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. Experimental Error

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

11 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. Replication

12 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. Randomization

13 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. Local Control

14 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

15 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. Steps in Designing an Experiment continue

16 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 Check List for Planning Test Programs

17 One Factor Designs

18 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 Completely Randomized Designs

19 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 Y ij =  +  i +  ij, i = 1, 2, …, k j = 1, 2, …, n where Y ij is the observation associated with the ith treatment and jth experimental unit,  is the true mean effect (constant)  i is the true effect of the ith treatment and  ij is the experimental error Model

20 The results of a completely random experiment with one observation per experiment unit may be exhibited as follows: Treatment Total 12... Y 11 Y 21...Y  Y 12 Y Y 1n Y 2n...Y n Totals Y 1· Y 2·...Y · Y ·· Number of Obs. nn...n n Means Y 1 Y 2...Y  Y Statistical Layout

21 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

22 where and Analysis of Variance

23 SSE = SST - SSA Analysis of Variance Computational Formulas

24 Sources ofDegrees ofSum of MeanF-Critical VariationFreedomSquares SquareRatioValue of F TreatmentsSSA Error(n-1)SSE Totaln-1SST Analysis of Variance Table (AOV Table or ANOVA Table)

25 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

26 The statistical layout is: Treatment(Brand) Total ABCD … … … … Totals Number of Obs Means Example - Solution

27 overall average= Example - Solution

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

29 Analysis of Variance Table Sources ofDegrees ofSum of Mean of F- Critical VariationFreedomSquares Squares Ratio Value of F Treatments Error Total Example - Solution

30 Since the probability of obtaining an F statistic of 6.93 or larger when the null hypothesis is true is approximately 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. Example - Conclusion