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Chapter 3Design & Analysis of Experiments1Design of Engineering ExperimentsChapter 3 Experiments with a Single Factor: The Analysis of VarianceContrastsScheffes method for comparing all contrastsTukeys testFishers Least Significance Difference (LSD) Method Determining sample sizeThe general regression significance testChapter 3Design & Analysis of Experiments2Post-ANOVA Comparison of MeansThe analysis of variance tests the hypothesis of equal treatment means

Assume that residual analysis is satisfactory

If that hypothesis is rejected, we dont know which specific means are different

Determining which specific means differ following an ANOVA is called the multiple comparisons problem2Chapter 3Design & Analysis of Experiments3ContrastsEXAMPLE

A manufacturer of paper used for making grocery bags is interested in improving the tensile strength of the product. Product engineering thinks that tensile strength is a function of the hardwood concentration in the pulp. A team of engineers responsible for the study decides to investigate four levels of hardwood concentration: 5%, 10%, 15%, and 20%. They decide to make up six test specimens at each concentration level, using a pilot plant. All 24 specimens are tested on a laboratory tensile tester, in random order.EXAMPLE: ANALYSIS REVIEWContrasts

One-way ANOVA: 5, 10, 15, 20

Source DF SS MS F PFactor 3 382.79 127.60 19.61 0.000Error 20 130.17 6.51Total 23 512.96

Level N Mean StDev 5 6 10.000 2.828 10 6 15.667 2.805 15 6 17.000 1.789 20 6 21.167 2.639 Compute the F-ratio for the example4Chapter 3Design & Analysis of Experiments5ContrastsIn the example, you would like to test if there a significant difference between the mean tensile strength obtained from the 10% and 15% hardwood concentration (From the graphs, it seems they are not very different).

Hence you would like to test the following hypotheses

Chapter 3Design & Analysis of Experiments6ContrastsA contrast is a linear combination of parameters of the form

Where the contrast constants sum to zero; that is

Hypotheses can be expressed in terms of a contrast if

EXAMPLE. Are the hypotheses expressing a contrast?

You have to get the means from the ANOVA to do this 6Chapter 3Design & Analysis of Experiments7ContrastsTo test the set of hypotheses

Compute the sum of squares of the contrasts

And the F statistic is , where

The null hypothesis is rejected if

HYPOTHESIS TESTING INVOLVING CONTRASTS (F-METHOD)

There are two ways of testing hypotheses involving contrasts. With t-tests and with F-tests. We will cover the procedure for the F-test

Note that the mean Square of the contrast has only one degree of freedom7Chapter 3Design & Analysis of Experiments8ContrastsEXAMPLE: For the hardwood concentration problem, test the set of hypotheses

HYPOTHESIS TESTING INVOLVING CONTRASTS (F-METHOD)

There are two ways of testing hypotheses involving contrasts. With t-tests and with F-tests. We will cover the procedure for the F-test

Note that the mean Square of the contrast has only one degree of freedom8Chapter 3Design & Analysis of Experiments9ContrastsThis is a technique for partitioning the treatment sum of squares of the ANOVA

In a balanced design, two contrast with coefficients and are orthogonal if

If there are more than two contrasts, they are mutually orthogonal if each contrast in the set is orthogonal to any other contrast.

ORTHOGONAL CONTRASTS

9Chapter 3Design & Analysis of Experiments10ContrastsTHEOREM: The maximum number of linear contrasts in a set of Mutually Orthogonal Linear Contrasts of the quantities is .

Linear contrasts are making comparisons amongst the values

Orthogonal Linear Contrasts are making independent comparisons amongst the values

The number of independent comparisons amongst the values I is

EXAMPLE: Find a set or orthogonal contrasts for the hardwood concentration experiment

ORTHOGONAL CONTRASTS

There are two ways of testing hypotheses involving contrasts. With t-tests and with F-tests. We will cover the procedure for the F-test

Note that the mean Square of the contrast has only one degree of freedom10Chapter 3Design & Analysis of Experiments11ContrastsThe contrast sum of squares completely partition the treatment sum of Squares

EXERCISE: Test the hypotheses of a set of orthogonal contrast that completely partition the treatment sum of squares. Use the F-test and a=0.05

HYPOTHESIS TESTING WITH ORTHOGONAL CONTRASTSDo the whole ANOVA for the orthogonal contrasts11Chapter 3Design & Analysis of Experiments12Scheffes method for comparing all contrastsGenerally, the method of contrasts and orthogonal contrasts is useful for what are called preplanned comparisons

These methods are not controlling the type I error

Scheffe (1953) has proposed a method for comparing any and all possible contrasts between treatment means

In the Scheffe method, the type I error is at most a for any of the possible comparisons

Preplanned comparisons: Contrast are specified prior to running the experiment and examining the data

Methods are not controlling the type I error: If comparisons are selected after data is collected and analyzed, the researcher can select the largest differences to compare, and since some of these differences can be the results of random error (not of the treatment), the type I error can be increased

Any and all: You can check any contrast that you want (no matter if it is orthogonal or not). You can check them all if you want12Chapter 3Design & Analysis of Experiments13Scheffes method for comparing all contrastsAssume the following set of m contrasts in the treatment means

The corresponding contrast in the treatment averages is

The standard error of this contrast is

The critical value against which should be compared is

If , the hypothesis that the contrast is rejected

13Chapter 3Design & Analysis of Experiments14Scheffes method for comparing all contrastsEXAMPLE: In the hardwood concentration problem, test the following hypotheses using the Scheffes method

14Chapter 3Design & Analysis of Experiments15The Tukey-Kramer testSuppose that you have rejected that means are equal in an ANOVA test.

Now, it is desired to test all pairwise mean comparisons: for all

The Tukey-Kramer test declares two means significantly different if the absolute value of their sample differences exceeds

Where is the studentized range statistic, which depends on the number of treatment means , and the degrees of freedom associated with the

The Tukey-Kramer (for unbalanced) is the same Tukeys for the balanced case

Here f=an-a or N-a15Chapter 3Design & Analysis of Experiments16The Tukey-Kramer testWe can also construct a set of 100(1-a) percent confidence intervals for all means as follows:

The Tukey-Kramer test controls the a error for the whole experiment (family error rate)

The Tukey-Kramer procedure is equivalent to the Tukeys test in the balanced experiment case

EXAMPLE: Perform the Tukey-Kramer comparisons for the hardwood concentration experiment (by hand and MINITAB)

The Tukey-Kramer (for unbalanced) is the same Tukeys for the balanced case16Chapter 3Design & Analysis of Experiments17The Tukey-Kramer testMINITAB OUTPUT

The Tukey-Kramer (for unbalanced) is the same Tukeys for the balanced case17Chapter 3Design & Analysis of Experiments18The Fisher Least Significant Difference (LSD) MethodHypotheses:

Test statistic

Decision criteria: would be rejected if

The Fisher LSD controls a for each individual pairwise comparison but it does not control the experimentwise or family error rate

EXAMPLE: Perform the Fisher LSD comparisons for the hardwood concentration experiment (by hand and MINITAB)

The Fisher controls the error rate alfa for each individual pairwise comparison but it does not control the experimentwise or family error rate18Chapter 3Design & Analysis of Experiments 8E 2012 Montgomery19Sample Size Determination

FAQ in designed experiments

Answer depends on lots of things; including what type of experiment is being contemplated, how it will be conducted, resources, and desired sensitivity

Sensitivity refers to the difference in means that the experimenter wishes to detect

Generally, increasing the number of replications increases the sensitivity or it makes it easier to detect small differences in meansChapter 3Design & Analysis of Experiments 8E 2012 Montgomery20Sample Size DeterminationFixed Effects CaseChoose the sample size to detect a specific difference in means and achieve desired values of type I and type II errorsType I error reject H0 when it is true ( )Type II error fail to reject H0 when it is false ( )Power = 1 - Operating characteristic curves plot against a parameter where

The parat20Chapter 3Design & Analysis of Experiments 8E 2012 Montgomery21Sample Size DeterminationFixed Effects Case---use of OC CurvesThe OC curves for the fixed effects model are in the Appendix, Table V

A very common way to use these charts is to define a difference in two means D of interest, then the minimum value of is

Typically work in term of the ratio of and try values of n until the desired power is achieved

Recall that n is the sample size per treatment level21Chapter 3Design & Analysis of Experiments 8E 2012 Montgomery22Sample Size DeterminationFixed Effects Case---use of OC CurvesEXAMPLE: Assume you are yet to determine the sample size per treatment in the hardwood concentration problem. Find the sample size needed if you would like an a=0.01, a power of 90%, and a difference in means of at most 30 psi. The experimenter does not think that the standard deviation is higher than 20 psi.

Perform the sample size calculation by hand and by MINITAB

Chapter 3Design & Analysis of Experiments 8E 2012 Montgomery23Power and Sample Size from MINITAB

One-way ANOVA

Alpha = 0.01 Assumed standard deviation = 20

Factors: 1 Number of levels: 4

Maximum Sample Target Difference Size Power Actual Power 30 19 0.9 0.909124

The sample size is for each level.Sample Size Determination

Chapter 3Design & Analysis of Experiments 8E 2012 Montgomery24The general regression significance test

It is a procedure to find the values for the sums of squares required to develop the ANOVA

In the single factor ANOVA, the procedure estimates the parameters m and ti for the full model

With the full model parameters, it is possible to find the variability explained by the treatments ti in the full model

It is also necessary to find the parameters and the variability explained by the reduced model . Note that this model occurs when

The procedure is the least square estimation. Yij is the data we collect and eij^2 is what we minimize. For each i and j: eij^2=(yij-miu-taoi)^224Chapter 3Design & Analysis of Experiments 8E 2012 Montgomery25The general regression significance test

The difference between the variability explained by the full model and the variability explained by the restricted model is equal to the treatment sum of squares

The variability explained by the full model is also used to determine the error sum of squares

EXERCISE: Using MINITAB, perform the ANOVA for the hardwood concentration experiment using the general regression significance approach

The procedure is the least square estimation. Yij is the data we collect and eij^2 is what we minimize. For each i and j: eij^2=(yij-miu-taoi)^225Chapter 3Design & Analysis of Experiments 8E 2012 Montgomery26q-table