Experimental Design IENG581 Design and Analysis of Experiments INTRODUCTION Experiments are performed by investigators in virtually all fields of inquiry, usually to discover something about a particular process or system. Experiment is a test. Designed experiment is a test or series of tests in which purposeful changes are made to the input variables of a process or system so that we may observe and identify the reasons for changes in the output response. The process or system under study can be represented by the model shown in the figure below: Figure: General model of a process or system We can visualize the process as a combination of machines, methods, people, and other resources that transforms some input (often a material) into an output that has one or more observable responses.
Some of the process variables x 1,x 2,,x p are controllable, whereas other variables z 1,z 2,,z q are uncontrollable (although they may be controllable for purposes of a test). The objectives of the experiment may include the following: 1. Determining which variables are most influential on the response y 2. Determining where to set the influential x s so that y is almost always near the desired nominal value 3. Determining where to set the influential x s so that variability in y is small. 4. Determining where to set the influential x s so that the effects of the uncontrollable variables z 1,z 2,,z q are minimized. Experimental design methods play an important role in process development and process troubleshooting to improve performance. The objective in many cases may be to develop a robust process, that is, a process affected minimally by external sources of variability (the z s ). *In any experiment, the results and conclusions that can be drawn depend to a large extent on the manner in which the data were collected. Applications of Experimental Design: Experimental design methods, have found broad application in many disciplines. In fact we may view experimentation as part of the scientific process and as one of the ways we learn about how systems or processes work. Generally, we learn through a series of activities in which we make conjectures about a process, perform experiments to generate data from the process, and then use the information from the experiment to establish new conjectures, which lead to new experiments, and so on. Experimental design is a critically important tool in the engineering world for improving the performance of a manufacturing process. It also has extensive application in the development of new processes.
The application of experimental design techniques early in process development can result in: 1. Improved process yields 2. Reduced variability and closer conformance to nominal or target requirements. 3. Reduced development time. 4. Reduced overall costs. Experimental design also plays a major role in engineering design activities, where new products are developed and existing ones are improved. Some applications of Experimental design are: 1. Evaluation and comparison of basic design configurations. 2. Evaluation of material alterations. 3. Selection of design parameters so that the product will work well under a wide variety of field conditions, that is, so that the product is robust. 4. Determination of key product design parameters that impact product performance. The use of experimental design in these areas can result in: - products that are easier to manufacture, - products that have enhanced field performance and reliability, - lower product cost, - and shorter product design and development time. Basic Principles of Experimental Design If any experiment is to be performed most efficiently, then a scientific approach to planning the experiments must be employed. By the statistical design of experiments, we refer to the process of planning the experiment so that appropriate data that can be analyzed by statistical methods will be collected, resulting in valid and objective conclusions.
*The statistical approach to experimental design is necessary if we wish to draw meaningful conclusions from the data. When the problem involves data that are subject to experimental errors, statistical methodology is the only objective approach to analysis. Thus, there are two aspects to any experimental problem: - The design of the experiment and - the statistical analysis of the data. These two subjects are closely related since the method of analysis depends directly on the design employed. Both topics will be addresses in this course (IE-581). The three basic principles of experimental design are: - Replication, - Randomization, and - Blocking. Replication has two important properties: First, it allows the experimenter to obtain an estimate of the experimental error. This estimate of error becomes a basic unit of measurement for determining whether observed differences in the data are really statistically different. Second, if the sample mean (e.g. ÿ) is used to estimate the effect of a factor in the experiment, then replication permits the experimenter to obtain a more precise estimate of this effect. e.g. If 2 is the variance of the data, and there are n replicates, then the variance of the sample mean is: 2 ÿ = 2 /n - If n=1, the experimental error will be high and we will be unable to make satisfactory inferences, since the observed difference in the sample mean could be the result of experimental error.
- If n is reasonably large, the experimental error will be sufficient small and we would be reasonably safe in conclusion. Randomization is the cornerstone underlying the use of statistical methods in experimental design. By randomization we mean that both, the allocation of the experimental material and the order in which the individual runs or trials of the experiment are to be performed, are randomly determined. Statistical methods require that the observations (or errors) be independently distributed random variables. Randomization usually makes this assumption valid. By properly randomizing the experiment, we also assist in averaging out the effects of extraneous factors that may be present. Blocking is a technique used to increase the precision of an experiment. A block is a portion of the experimental material that should be more homogeneous than the entire set of material. Blocking involves making comparisons among the conditions of interest in the experiment within each block. Guidelines for Designing Experiments: To use the statistical approach in designing and analyzing, it is necessary that everyone involved in the experiment have a clear idea in advance of: - Exactly what is to be studied, - How the data are to be collected, and - At least a qualitative understanding of how these data are to be analyzed.
An outline of the recommended procedure is as follows: 1. Recognition and statement of the problem In practice it is often not simple neither to realize that a problem requiring experimentation exists, nor is it simple to develop a clear and generally accepted statement of this problem. It is necessary to develop all ideas about the objectives of the experiment. Usually, it is important to solicit input from all concerned parties: - Engineering, - Quality assurance, - Manufacturing, - Marketing, - Management, - The customer, and operating personnel - Etc. A clear statement of the problem often contributes substantially to a better understanding of the phenomena and the final solution of the problem. 2. Choice of factors and levels The experimenter must choose the factors to be varied in the experiment, the ranges over which these factors will be varied, and the specific levels at which runs will be made. Thought must also be given to how these factors are to be controlled at the desired values and how they are to be measured. 3. Selection of the response variable In selecting the response variable, the experimenter should be certain that this variable really provides useful information about the process under study. Most often, the average or standard deviation (or both) of the measured characteristic will be the response variable. Multiple responses are not unusual.
4. Choice of experimental design If the first three steps are done correctly, this step is relatively easy. Choice of design involves: - The consideration of sample size (number of replicates), - The selection of a suitable run order for the experimental trials, - And the determination of whether or not blocking or other randomization restrictions are involved. In selecting the design, it is important to keep the experimental objectives in mind. 5. Performing the experiment When running the experiment, it is vital to monitor the process carefully to ensure that everything is being done according to plan. Errors in experimental procedure at this stage will usually destroy experimental validity. Up-front planning is crucial to success. 6. Data analysis Statistical methods should be used to analyze the data so that results and conclusions are objective rather than judgmental in nature. If the experiment has been designed correctly and if it has been performed according to the design, then the statistical methods required are not elaborate. There are many excellent software packages designed to assist in data analysis, and simple graphical methods play an important role in data interpretation. Residual analysis and model adequacy checking are also important analysis techniques.
Note: Remember that: - Statistical methods cannot prove that a factor (or factors) has a particular effect. They only provide guidelines as to the reliability and validity of results. - Properly applied, statistical methods do not allow anything to be proved, experimentally, but they do allow us to measure the likely error in a conclusion or to attach a level of confidence to a statement. - The primary advantage of statistical methods is that they add objectivity to the decision-making process. - Statistical techniques coupled with good engineering or process knowledge and common sense will usually lead to sound conclusions. 7. Conclusions and recommendations Once the data have been analyzed, - The experimenter must draw practical conclusions about the results and recommend a course of action. - Graphical methods are often useful in this stage, particularly in presenting the results to others. - Follow-up runs and confirmation testing should also be performed to validate the conclusions from the experiment. Throughout this entire process, it is important to keep in mind that experimentation is an important part of the learning process, where; - We tentatively formulate hypotheses about a system, - Perform experiments to investigate these hypotheses, and - On the basis of the results formulate new hypotheses, and so on. This suggests that experimentation is iterative.
Using statistical techniques in experimentation Much of the research in engineering, science and industry is empirical and makes extensive use of experimentation. Statistical methods can greatly increase the efficiency of these experiments and often strengthens the conclusions so obtained. The intelligent use of statistical techniques in experimentation requires that the experimenter keep the following points in mind: 1. Use your non-statistical knowledge of the problem. 2. Keep the design and analysis as simple as possible. 3. Recognize the difference between practical and statistical significance. 4. Experiments are usually iterative
SIMPLE COMPARATIVE EXPERIMENTS We ll consider experiments to compare two conditions (sometimes called treatments). We ll begin with an example of an experiment performed to determine whether two different formulations of a product give equivalent results. The discussion leads to a review of several basic statistical concepts. Example: The tension bond strength of Portland cement mortar is an important characteristic of the product. An engineer is interested in comparing the strength of a modified formulation in which polymer latex emulsions have been added during mixing to the strength of the unmodified mortar. The experimenter has collected 10 observations of strength for the modified formulation and another 10 observations for the unmodified formulation. The data are shown in table 2-1. We could refer to the two different formulations as two treatments or as two levels of the factor formulations.
The data from this experiment are plotted in figure 2-1, below: This display is called a dot diagram. Visual examination of these data gives the immediate impression that the strength of the unmodified mortar is greater than the strength of the modified mortar. This impression is supported by comparing the average tension bond strengths, Ÿ1=16.76 kgf/cm2 for the modified mortar and Ÿ2=17.92 kgf/cm 2 for the unmodified mortar. The average tension bond strengths in these two samples differ by what seems to be a nontrivial amount. However, it is not obvious that this difference is large enough to imply that the two formulations really are different. Perhaps this observed difference in average strengths is the result of sampling fluctuations and the two formulations are really identical. Possibly another two samples would give opposite results, with the strength of the modified mortar exceeding that of the unmodified formulation. A technique of statistical inference called hypothesis testing (or significance testing) can be used to assist the experimenter in comparing these two formulations. Hypothesis testing allows the comparison of the two formulations to be made on objective terms, with knowledge of the risks associated with reaching the wrong conclusion.
To present procedures for hypothesis testing in simple comparative experiments, however, it is first necessary to develop and review some elementary statistical concepts. 2-2 Basic Statistical Concepts Each of the observations in the Portland cement experiment described previously would be called a run. Notice that the individual runs differ, so there is fluctuation or noise in the results. This noise is usually called experimental error or simply error. It is a statistical error, meaning that it arises from variation that is uncontrolled and generally unavoidable. The presence of error or noise implies that the response variable, tension bond strength, is a random variable. A random variable may be either discrete or continuous. If the set of all possible values of the random variable is either finite or countably infinite, then the random variable is discrete. If the set of all possible values of the random variable is an interval, then the random variable is continuous. Graphical Description of variability The Dot Diagram, illustrated in figure 2-1, is a very useful device for displaying a small body of data. The Dot Diagram enables the experimenter to see quickly the general location or central tendency of the observations and their spread. In our example; the Dot Diagram reveals that the two formulations probably differ in mean strength but that both formulations produce about the same variation in strength. If the data is numerous, then a histogram may be preferable.
Figure 2-2. Presents a histogram for 200 observations on the metal recovery or yield from a smelting process. The Histogram shows the central tendency, spread, and general shape of the distribution of the data. Recall Histogram is constructed by dividing the horizontal axis into intervals (usually of equal length) and drawing a rectangle over the j th interval with area of the rectangle proportional to nj, the number of observations that fall in that interval. A Box Plot (or Box and Whiskers plot), displays the minimum, the maximum, the lower and upper quartiles (the 25 th and the 76 th percentiles, respectively), and the median (the 50 th percentile) on a rectangular box aligned either horizontally or vertically. The box extends from the lower quartile to the upper quartile, and a line is drawn through the box at the median lines (or whiskers) extend from the ends of the box to the minimum and maximum values.
Figure 2-3 presents the Box Plots for the two samples of tension bond strength in the Portland cement mortar experiment. This display clearly reveals the difference in mean strength between the two formulations. It also indicates that both formulations produce reasonably symmetric distributions of strength with similar variability or spread. Summary: Dot diagrams, histograms, and Box plots are useful for summarizing the information in a sample of data. To describe the observations that might occur in a sample more completely; we use the concept of the probability distribution.
Probability Distributions: The probability structure of a random variable, say y, is described by its probability distribution. If y is discrete, we often call the probability distribution of y, say p(y), the probability function of y. If y is continuous, the probability distribution of y, say f(y), is often called the probability density function for y. The figures below, illustrate a hypothetical discrete and continuous probability distributions. Notice: In the discrete probability distribution, it is the height of the function p(y) that represents probability, whereas in the continuous case, it is the area under the curve f(y) associated with a given interval that represents probability.
The properties of probability distributions may be summarized quantitatively as follows: y discrete: 0 p(yj) 1 yj P(y= yj) = p(yj) yj P(yj) = 1 y continuous: 0 f(y) P( a y b) = f(y) dy f(y) dy = 1 - Mean, Variance, and Expected Values The mean of a probability distribution is a measure of its central tendency or location. yf(y) dy y continuous - = (2-1) yjp(yj) y discrete all j We may also express the mean in terms of the expected value or the long-run average value of the random variable y as = E(y) (2-2) The spread or dispersion of a probability distribution can be measured by the variance, defined as (y- ) 2 f(y) dy y continuous - 2 = (2-3) (yj - ) 2 P(yj) all j y discrete Note that the variance can be expressed entirely in terms of expectation since; 2 = E[(y- ) 2 ] (2-4)
The variance is used so extensively that it is convenient to define a variance operator V such that V(y) = E[(y- ) 2 ]= 2 (2-5) Elementary results concerning the mean and variance operators: If y is a random variable with mean and variance 2 and c is a constant, then 1. E(c)=c 2. E(y)= 3. E(cy)=cE(y)=c 4. V(c)=0 5. V(y)= 2 6. V(cy)=c 2 V(y)=c 2 2 If there are two random variables, for example, y1 with E(y1) = 1 and V(y1)= 1 2 and y2 with E(y2) = 2 and V(y2)= 2 2, then we have 7. E(y1+y2) = E(y1) + E(y2) = 1+ 2 It is possible to show that 8. V(y1+y2) = V(y1) + V(y2) + 2 Cov(y1,y2) 9. V(y1-y2) = V(y1) + V(y2) - 2 Cov(y1,y2) Where ; Cov(y1,y2) = E[(y1-1)(y2-2)] (2-6) Is the covariance of the random variables y1 and y2. *The covariance is a measure of the linear association between y1 and y2. If y1 and y2 are independent, the Cov(y1,y2)=0. 10. V (y1±y2) = V (y1) + V (y2)= 1 2 + 2 2 And 11. E (y1.y2) = E (y1). E (y2)= 1. 2
11. E (y1/y2) E(y1)/E(y2) Regardless of whether or not y1 and y2 are independent. Sampling and Sampling distributions: The objective of statistical inference is to draw conclusions about a population using a sample from that population. Most of the methods that we will study assume that random samples are used. That is, if the population contains N elements and a sample of n of them is to be selected, then if each of the N!/(N-n)!n! possible samples has an equal probability of being chosen, the procedure employed is called random sampling. In practice, it is sometimes difficult to obtain random samples, and tables of random numbers may be helpful. Statistical inference makes considerable use of quantities computed from the observations in the sample. A statistic is defined as any function of the observations in a sample that does not contain unknown parameters. e.g. Suppose that y1, y2,, yn represent a sample. Then the sample mean and the sample variance are both statistics. ў = yi /n S 2 = (yi- ў) 2 /(n-1) These quantities are measures of the central tendency and dispersion of the sample, respectively. Sometimes S= S 2, called the sample standard deviation, is used as a measure of dispersion. Engineers often prefer to use the standard deviation to measure dispersion because its units are the same as those for the variable of interest y.
Properties of the Sample Mean and Variance: The sample mean ў is a point estimator of the population mean, and the sample variance S 2 is a point estimator of the population variance σ 2. In general, an estimator of an unknown parameter is a statistic that corresponds to that parameter. Note: A point estimator is a random variable. A particular numerical value of an estimator, computed from sample data, is called an estimate. Properties of good point estimators: 1. The point estimator should be unbiased. That is, the long-run average or expected value of the point estimator should be the parameter that is being estimated. 2. An unbiased estimator should have minimum variance. This property states that the minimum variance point estimator has a variance that is smaller than the variance of any other estimator of that parameter. Degrees of Freedom: If y is a random variable with variance σ 2 and SS= Σ (yi- ў) 2 has ν degrees of freedom, then E (SS/ν) = σ 2 The number of degrees of freedom of a sum of squares is equal to the number of independent elements in that sum of squares. In fact SS has (n-1) degrees of freedom.
Some Sampling Distributions: Often, we are able to determine the probability distribution of a particular statistic if we know the probability distribution of the population from which the sample was drawn. The probability distribution of a statistic is called a sampling distribution. Normal Distribution: If y is a normal random variable, then the probability distribution of y is: Where - < < is the mean of the distribution and σ 2 > 0 is the variance. Often, sample runs that differ as a result of experimental error, are well described by the Normal distribution. The normal plays a central role in the analysis of data from designed experiments.
Many important sampling distributions may also be defined in terms of normal random variables. We often use this notation: y~n(, σ 2 ) A special case of the normal distribution is the standard normal distribution. That is =0 and σ 2 =1 y~n(0, 1). Then, the random variable z = (y- )/ σ ~ N(0, 1). Many statistical techniques assume that the random variable is normally distributed. The Central Limit theorem is often a justification of approximate Normality. Chi-square Distribution: If z1, z2,.., zk are NID(0,1) then, the random variable χ 2 = (z1) 2 + (z2) 2 + (z3) 2 +..+ (zk) 2 follows Chi-square distribution with k degrees of freedom. The density function of chi-square is:
NOTE: = k and σ 2 =2k Example: As an example of a random variable that follows the Chisquare distribution, suppose that y1, y2,,yn is a random sample from a N~(µ, σ 2 ) distribution. Then, SS/σ 2 = ( n (yi-ỹ) 2)/σ 2 ~ χ 2 (n-1) Is distributed as Chi-square with (n-1) degrees of freedom. The sample variance is: S 2 =SS / (n-1) If observations in the sample are NID µ, σ 2 ), then the distribution of S 2 is [σ 2 /(n-1)] χ 2 (n-1). Thus the sampling distribution of the sample variance is a constant times the Chi-square distribution if the population is normally distributed. t-distribution: If z and χ 2 k are independent standard normal and Chi-square random variables, respectively, then the random variable tk= z/ (χ 2 k /k) follows the t-distribution with k degrees of freedom, denoted tk. The density function of t is:
And the mean and variance of t are : µ=0 and σ 2 =k/(k-2) for all k>2. If y1, y2,.., yn is a random sample from the N(µ,σ 2 ) distribution, then the quantity t= ( ỹ - µ )/(S/ n) ~ t(n-1) F-distribution: If χ 2 u and χ 2 v are two independent Chi-square random variables with u and v degrees of freedom, respectively, the ratio Fu,v = [χ 2 u/ u] / [χ 2 v/ v] Follows the F-distribution with u numerator degrees of freedom and v denominator degrees of freedom. The probability distribution of F is:
Example: Suppose we have two independent normal populations with common variance σ 2. If y1,1, y1,2,.., y1,n1 is a random sample of n1 observations from the first population, and if y2,1, y2,2,.., y2,n2 is a random sample of n2 observations from the second, then (S 2 1/S 2 2) ~ F(n1-1), (n2-1) Inferences About Differences in Means, Randomized Designs: Assumptions: 1- A completely randomized design is used. 2- The data are viewed as if they were a sample from a Normal distribution. Hypothesis testing: A statistical hypothesis is a statement about the parameters of a probability distribution. Example: The Portland cement experiment) We may think that the mean tension bond strengths of the two mortar formulations are equal. Ho: µ1= µ2 H1: µ1 µ2 The alternative hypothesis specified here is called a two sided alternative hypothesis since it would be true either if µ1<µ2 or if µ1>µ2.
To test a hypothesis: 1- Take a random sample. 2- Compute an appropriate test statistic. 3- Reject or failing to reject Ho. When testing hypothesis, two kinds of errors may be committed: Type I Error: α = P(Type I Error)= P(Reject Ho Ho: True) Type II Error: β = P(Type II Error)= P(Failed to Reject Ho Ho: False) Power of the Test = (1- β) = P(Reject Ho Ho: False) The general Procedures in Hypothesis Testing: 1- Specify a value for α. 2- Design the test procedures so that β has a suitable small value. Suppose that we could assume the variances of tension bond strengths were identical for both mortar formulations. Then a test statistic to=( ỹ1- ỹ2)/ Sp [(1/n1) + (1/n2)] Where; Sp 2 =[(n1-1)s1 2 + (n2-1)s2 2 ]/(n1+n2-2) Then, Reject Ho if to > t α /2, n1+n2-2 In some problems, one wish to reject Ho only if one mean is larger than the other. Thus, one would specify a one-sided alternative hypothesis H1: µ1>µ2 and would reject Ho if to > t α, n1+n2-2. If one wants to reject Ho only if µ1 is less than µ2, then the alternative hypothesis H1: µ1<µ2, and one would reject Ho if to < -t α, n1+n2-2. Example: Modified Mortar: ỹ1 = 16.76 kgf/cm 2, S1 2 = 0.316, n1=10 Unmodified Mortar: ỹ2 = 17.92 kgf/cm 2, S2 2 = 0.061, n2=10 Sp 2 = [(n1-1) S1 2 + (n2-1)s2 2 ]/(n1+n2-2) = 0.081 Thus: Sp =0.284 and to = (ỹ1- ỹ2) / Sp [(1/n1) + (1/n2)]=-9.13 Therefore, reject Ho since to > t α /2, n1+n2-2= t 0.25,18 = 2.101.
Choice of Sample Size: The choice of sample size and the probability of type II error β are closely connected. Suppose that we are testing the hypothesis Ho: µ1= µ2 H1: µ1 µ2 And the means are not equal so that = µ1- µ2. Since Ho is not true, we are concerned about wrongly failing to reject Ho. The probability of type II error depends on the true difference in means. A graph of β versus for a particular sample size is called the operating characteristic curve (or O. C. curve) for the test. If β error is also a function of sample size, generally, for a given value of, the β error decreases as the sample size increases.