Blocking In Design Of Experiments
9/4/2012Blocking in design of experiments Blocking is a technique for dealing with nuisancefactors A nuisance factor is a factor that probably has someeffect on the response, but it’s of no interest to theexperimenter however, the variability it transmits to theresponse needs to be minimized Typical nuisance factors include batches of raw material,operators, pieces of test equipment, time (shifts, days,etc.), different experimental units Many industrial experiments involve blocking (or should) Failure to block is a common flaw in designing anexperiment (consequences?)Chapter 41Dealing with nuisance variables If the nuisance variable is known and controllable, weuse blocking If the nuisance factor is known and uncontrollable,sometimes we can use the analysis of covariance (seeChapter 15) to remove the effect of the nuisance factorfrom the analysis If the nuisance factor is unknown and uncontrollable (a“lurking” variable), we hope that randomizationbalances out its impact across the experiment Sometimes several sources of variability are combinedin a block, so the block becomes an aggregate variableChapter 421
9/4/2012Example: Hardness Testing We wish to determine whether 4 different tips producedifferent (mean) hardness reading on a Rockwellhardness tester Assignment of the tips to a test coupon (aka, theexperimental unit) A completely randomized experiment The test coupons are a source of nuisance variability Alternatively, the experimenter may want to test the tipsacross coupons of various hardness levelsChapter 43Example (cont.) To conduct this experiment as a RCBD, assign all 4 tipsto each coupon Each coupon is called a “block”– A more homogenous experimental unit on which to test the tips– Variability between blocks can be large, variability within ablock should be relatively small– In general, a block is a specific level of the nuisance factor– A complete replicate of the basic experiment is conducted ineach block A block represents a restriction on randomization– All runs within a block are randomizedChapter 442
9/4/2012Example (cont.) Suppose that we use b 4 blocks: Notice the two-way structure of the experiment– Once again, we are interested in testing the equality oftreatment means, but now we have to remove the variabilityassociated with the nuisance factor (the blocks)Chapter 45Extension of the ANOVA to RCBD Suppose that there are a treatments (factorlevels) and b blocks A statistical model (effects model) for theRCBD is i 1, 2,., ayij i j ij j 1, 2,., b The relevant (fixed effects) hypotheses areH 0 : 1 2 Chapter 4 a where i (1/ b) j 1 ( i j ) ib63
9/4/2012Extension of the ANOVA to RCBDANOVA partitioning of total variability:abab ( yij y. )2 [( yi. y. ) ( y. j y. )i 1 j 1i 1 j 1 ( yij yi. y. j y. )]2abi 1j 1 b ( yi. y. ) 2 a ( y. j y. ) 2ab ( yij yi. y. j y. ) 2i 1 j 1SST SSTreatments SS Blocks SS EChapter 47Extension of the ANOVA to RCBDThe degrees of freedom for the sums of squares inSST SSTreatments SS Blocks SS Eare as follows:ab 1 a 1 b 1 (a 1)(b 1)Therefore, ratios of sums of squares to their degreesof freedom result in mean squares and the ratio ofthe mean square for treatments to the error meansquare is an F statistic that can be used to test thehypothesis of equal treatment meansChapter 484
9/4/2012ANOVA for the RCBDChapter 4Chapter 49105
9/4/2012Example: Vascular Graft To conduct this experiment as a RCBD, all 4 pressures areassigned to each of the 6 batches of resin Each batch of resin is called a “block”; that is, it’s a morehomogenous experimental unit on which to test theextrusion pressuresChapter 411Minitab OutputTwo-way ANOVA: Flicks versus Pressure, 8.171 59.3904 8.11 0.002192.252 38.4504 5.25 0.006109.886 7.3257480.310S 2.707 R-Sq 77.12% R-Sq(adj) 64.92%Chapter 4Design & Analysis of Experiments7E 2009 Montgomery126
9/4/2012Residual analysisChapter 413Residual Analysis for theVascular Graft Example Basic residual plots indicate that normality,constant variance assumptions are satisfied– No obvious problems with randomization– No patterns in the residuals vs. block Can also plot residuals versus the pressure(residuals by factor)– These plots provide more information about theconstant variance assumption, possible outliersChapter 4Design & Analysis of Experiments7E 2009 Montgomery147
9/4/2012Your turn: Which pressures aredifferent?Chapter 415Other aspects of the RCBD The RCBD utilizes an additive model– Can’t explore interaction between treatments andblocks Treatments and/or blocks as random effects– Analysis is the same but interpretation is different If blocks are random (e.g., selection of raw materialbatches) then assume treatment effect is the samethroughout the population of blocks. Any interaction between treatments and blocks are will notaffect the test on treatment means (interaction will affectboth treatment and error mean squares)Chapter 4168
9/4/2012What about missing values? What happens if one of the measurements in yourexperiment is missing?– An error in measurement gives a result you know isn’t right– Damage to a machine prevents you from completing a test– Etc. Inexact method – estimate the missing value and go onwith the analysis– Reduce error degrees of freedom by 1 for each missing value– Danger – increase in “false” significance Exact method – general regression significance test– Test on unbalanced data (treatment and block not orthogonal)– Use Minitab or other computer application in Minitab, use GLM model in ANOVA menu (2-Way ANOVA requiresbalanced designs)Chapter 417Sample size Sample sizing in the RCBD refers to thenumber of blocks to run Can use Minitab sample size calculator with:– number of levels treatment level– sample size number of blocks Example 4.2, pg. 134– note the difference between the results using the OCcurve approach and MinitabChapter 4189
9/4/2012The Latin square design Latin square designs are used tosimultaneously control (or eliminate) twosources of nuisance variability Latin squares are not used as much as theRCBD in industrial experimentation A significant assumption is that the threefactors (treatments, nuisance factors) do notinteract– If this assumption is violated, the Latin squaredesign will not produce valid resultsChapter 419The rocket propellant problem –A Latin square design This is a 5x5 Latin square design Latin letters (A, B, C, D, E) are the treatment levels The experiment is designed such that every treatmentlevel is tested once at each combination of nuisancefactorsChapter 42010
9/4/2012Other Latin squares designs 4 different fonts (treatments) tested for reading speed ondifferent computer screens and in different ambient lightlevels (4 levels each) 3 different material types (treatments) tested for strengthat different material lengths and measuring device (3levels each) Note that once the design is complete, the order of thetrials in the experiment is randomizedChapter 421Statistical analysis of the Latinsquare design The statistical (effects) model is i 1, 2,., p yijk i j k ijk j 1, 2,., p k 1, 2,., p The statistical analysis (ANOVA) is much like theanalysis for the RCBD. See the ANOVA table, page 140 Using Minitab (GLM) for the analysis Chapter 42211
9/4/2012Chapter 423Checking for model adequacy The residuals in the Latin square are given by: As with any design, check the adequacy byplotting the appropriate residualsChapter 42412
9/4/2012Replication of Latin squares Small Latin squares provide a relatively small numberof error degrees of freedom– makes seeing differences in treatment effects difficult Replication of Latin squares can be done in three ways:– Repeat each combination of row and column variable in eachreplication– Change 1 nuisance variable but keep the other the same foreach replication– Use different levels of both nuisance variables Note, now the number of trials is N np2Chapter 425Other designs Crossover designs– Repeated Latin squares in an experiment in whichorder (or time period) matters– See figure 4.7, pg. 145 Graeco-Latin squares– Extension of the Latin squares design with 3 nuisancevariables Let each Greek letter indicate the 3rd nuisance factor level Each combination of row variable, column variable,Greekletter, and Latin letter appears once and only once. See example 4.4, pg. 147Chapter 42613
9/4/2012Homework: Due Wednesday, 9/12 3.17 3.19 3.24 4.74.194.204.29Chapter 42714
–Use Minitab or other computer application in Minitab, use GLM model in ANOVA menu (2-Way ANOVA requires balanced designs) Chapter 4 17 Sample size Sample sizing in the RCBD refers to the number of blocks to run Can use Minitab sample size calculator with: –number of levels treatment level –sample size number of blocks
26. 2-Point Stance ( WR & DE ) 27. 3-Point Stance 28. 4-Point Stance Section 3: OFFENSIVE SKILLS 29. Under Center Snaps 30. Direct Snaps 31. Bird Dog 32. Sled Blocking 33. Board Drill 34. Double Teams 35. Pulling Progression 36. Crab Blocking 37. Smart Blocking 38. Wedge Progression 39. Splatter Blocking 40. Get a Grip 41. End Run 42. Gauntlet 43.
BD Blocking distance E Empty calibration F Full calibration 2.4 Measuring range 2.4.1 Blocking distance, Nozzle mounting Install the Prosonic M at a height so that the blocking distance BD is not undershot, even at maximum fill level. Use a pipe no zzle if you cannot maintain the blocking distance in any other way.
6.1 Blocking Credit Card Numbers 34 6.2 Blocking Names 35. 3 6.3 Blocking Domain Names 36 6.4 Blocking IP and Class C Addresses 37 6.5 Setting Maximum Purchase Limit 37 6.6 Setting Auto Lockout and Duplicate 38 6.7 Setting a Country Profile 38 . . Virtual Terminal. . , the , Virtual Terminal. processing. .
COMPARA TIVE COST OF THE DIFFERENT BLOCKING AGENTS Table 3 compares the relative cost of the different blocking agents per 100 ml of a 10% solution. Bovine serum albumin and normal goat serum are the two most expensive blocking agents costing US 7·97/100 ml and US 6·22/100 ml respectively. Skimmed milk and both fresh and crude egg
time blocked practice (blocking) is more common. Across four experiments, college students used interleaving or blocking to learn how to conjugate verbs in the Spanish preterite and imperfect past tenses. Interleaving yielded better verb conjugation skills than blocking when it was used to practice those skills across multiple training sessions.
treatment e ects, and propose a novel experimental design in this setting. Our paper adds to both the literature on single-wave and multiple-wave experiments. In the context of single-wave (or two-wave) experiments, existing network experiments include clustered experiments (Eckles et al.,2017;Ugander et al.,2013;Karrer et al.,2021) and satu-
Design of Experiments is a valuable tool to maximize the amount of information with a minimal number of experiments. While numerous DOE frameworks exist, all operate using similar principles. Use sequential series of experiments Screening Design Model Building Design Confirmation Ru
College London (UCL) to investigate the potential uses of experiments in understanding consumer behaviour, and to develop a better understanding of the potential benefits of experiments if used in Ofcom's work. 1.2 The potential uses of experiments . Experiments test the actual behaviour of individuals under different conditions. In an
