Chapter 13 ANOVA The Design and Analysis of Single Factor Experiments - Part II Chapter 13B Class will begin in a few minutes. Reaching out to the internet

Today’s Special My gosh, this does look promising.

Fixed versus Random Factors Fixed Effects Model – conclusions only apply to factor levels included in the study Random-Effects Model – conclusions extend to the population of factor levels where the levels in the study are a sample assume population of factor levels is either infinite or large enough to be treated as infinite almost the same model as before but this is different

The Model For a random-effects model, the appropriate hypotheses to test are: The ANOVA decomposition of total variability is still valid:

Expected Values of the Mean Squares under H 0 :

The Estimators of the Variance Components

The Problem (Example 13-4) DOE in the Textile Industry

The Data and ANOVA

The Variances

Figure 13-8 The distribution of fabric strength. (a) Current process, (b) improved process. LSL – Lower Specification Limit

Randomized Block Designs (RBD) Variability among subjects may mask or obscure treatment effect of interest Nuisance variable of individualized differences can be minimized using a RBD Appropriate when one treatment with r > 2 levels assignment of subjects to blocks so that variability among subjects within any block is less than the variability among the blocks random assignment of treatments to subjects within blocks

Our Very First Blocking Example The factor levels: 4 different cockpit designs The experimental units: 12 pilots with varying years of experiences The response variable: Number of pilot errors in flying a simulated program Block on years experience or more Randomly assign treatment levels (i.e. cockpit design) to pilots in each block.

CRD versus RBD RBDyrs 1-2yrs 3-4yrs 5+ Design AJoeTomBob BJillTedBarbara CJohnTerrieBill DJimTeresaBetty CRD Designsubjects AJoe TedTerrie BTeresaBillBob CJimBarbaraJill DJohnTomBetty

Complete Confounding CRDyrs 1-2yrs 3-4yrs 5+ DesignABC JoeTomBob JillTedBarbara JohnTerrieBill The devil made me do it this way.

13-4 Randomized Complete Block Designs The randomized block design is an extension of the paired t-test to situations where the factor of interest has more than two levels. Figure 13-9 A randomized complete block design.

Design and Statistical Analyses For example, consider the situation of Example 10-9, where two different methods were used to predict the shear strength of steel plate girders. Say we use four girders as the experimental units.

Design and Statistical Analyses General procedure for a randomized complete block design:

The Linear Statistical Model We assume treatments and blocks are initially fixed effects blocks do not interact

The Hypothesis The partitioning of the sums of squares SS T = SS Treatements + SS Blocks + SS E d.f.’s ab – 1 = (a – 1) + (b – 1) + (a – 1)(b – 1)

Mean Squares The effect of the blocks may or may not be interesting in and of itself. Notice that the effect of blocks is to reduce the sums of squares left as error. This means that the calculation of F 0 will always produce a larger quantity than otherwise. The test on the  i becomes more sensitive. SS E = SS Totals - SS Treatments - SS Blocks

Expected Values Of the Mean Squares If the  i are zero, this is an unbiased estimate of  2. A Mean Square

Computational Formulae

ANOVA Reject H 0 if

Example 13-5

SS for Example 13-5

More SS for Example 13-5

ANOVA for Example 13-5

Minitab Output for Example 13-5

Fisher’s Least Significant Difference for Example 13-5 Figure Results of Fisher’s LSD method.

CI on chemical Individual 95% CI Chemical Mean (--*---) (--*--) (--*---) (---*--)

CI on samples Individual 95% CI sample Mean (----*----) (----*----) (-----*----) (----*----) (----*----)

An Excel Comparison A computer model simulates depot overhaul of jet engines in order to estimate repair cycle times in days Three probability distributions are used to model task times gamma lognormal Weibull Blocking occurs by using the same random number stream with each treatment (distribution) Go to the next slide to observe the highly interesting results

Highly Interesting Results engine overhaul days

CI on distributions Individual 95% CI distribution Mean gamma 23.0 (-----*-----) lognorma 25.2 (-----*-----) Weibull 8.0 (-----*-----)

CI on Random Number Seeds Individual 95% CI seed Mean (----*----) (----*----) (----*----) (----*----)

Even Higher Interesting Results One-Way ANOVA without blocking F.01,2,9 = 8.02 F.05,2,9 = 4.26 Why it appears that the variability introduced by the random number stream is masking the effect of the task time distributions. This is nothing but garbage.

Randomized Complete Block Design is Now Over Want more? Sign up today for ENM 561: DESIGN AND ANALYSIS OF EXPERIMENTS This course introduces advanced topics in experimental design and analysis, including full and fractional factorial designs, response surface analysis, multiple and partial regression, and correlation. Prerequisite: ENM 500 or equivalent.

The ENM 500 Course is Now Over Want more? Sign up today for ENM 501: APPLIED ENGINEERING STATISTICS Concepts and applications of advanced probability modeling and statistical techniques used in the study and solution of operations research/management science problems. The focus of this course is on the application of probability and statistics in the formulation and solution of models found in operations research studies and in engineering design studies. This course builds upon the foundation established in the ENM 500 course. Prerequisite: ENM 500 or equivalent.

Students rushing to sign up for the ENM 501 class ENM 501: Applied Engineering Statistics Coming soon to a classroom near you.