Lecture 2.11 © 2015 Michael Stuart Design and Analysis of Experiments Lecture Review –Minute tests 1.2 –Homework –Randomized Blocks Design 2.Randomised blocks analysis 3.Two design factors –a 3 x 3 experiment Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture 2.12 © 2015 Michael Stuart Minute Test: How Much Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture 2.13 © 2015 Michael Stuart Minute Test: How Fast Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture 2.14 © 2015 Michael Stuart Exercise Process Development Study Formal test: Numerator measures change effect, Denominator measures chance effect. Carry out the test using the results from the first two runs at each speed. Compare with test using complete data Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture 2.15 © 2015 Michael Stuart Process Development Study Variable N Mean StDev Speed B Speed A Speed BSpeed A Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture 2.16 © 2015 Michael Stuart Randomized block design Where replication entails increased variation, replicate the full experiment in several blocks so that non-experimental variation within blocks is as small as possible, –comparison of experimental effects subject to minimal chance variation, variation between blocks may be substantial, –comparison of experimental effects not affected Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture 2.17 © 2015 Michael Stuart Illustrations of blocking variables Agriculture: fertility levels in a field or farm, moisture levels in a field or farm, genetic similarity in animals, litters as blocks, etc. Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture 2.18 © 2015 Michael Stuart Illustrations of blocking variables Clinical trials (stratification) age, sex, height, weight, social class, medical history etc. Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture 2.19 © 2015 Michael Stuart Illustrations of blocking variables Clinical trials different treatments applied to the same individual at different times, cross-over, carry-over, correlation, body parts as blocks, hands, feet, eyes, ears, etc. Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Illustrations of blocking variables Industrial trials similar machines, time based blocks, time of day, day of week, shift etc. Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Case Study Reducing yield loss in a chemical process Process: chemicals blended, filtered and dried Problem:yield loss at filtration stage Proposal:adjust initial blend to reduce yield loss Plan: –prepare five different blends –use each blend in successive process runs, in random order –repeat at later times (blocks) Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Classwork 1.2.5: What were the response: experimental factor(s): factor levels: treatments: experimental units: observational units: unit structure: treatment allocation: replication: Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Classwork 1.2.5: What were the response: experimental factor(s): factor levels: treatments: experimental units: observational units: unit structure: treatment allocation: replication: yield loss Blend A, B, C, D, E process runs 3 blocks of 5 units random order of blends within blocks 3 Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Unit Structure Block 1Block 2Block 3 Postgraduate Certificate in Statistics Design and Analysis of Experiments Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 1 Unit 2 Unit 3 Unit 4 Unit 5

Lecture © 2015 Michael Stuart Unit Structure Block 1Block 2Block 3 Postgraduate Certificate in Statistics Design and Analysis of Experiments Unit 1_1 Unit 1_2 Unit 1_3 Unit 1_4 Unit 1_5 Unit 2_1 Unit 2_2 Unit 2_3 Unit 2_4 Unit 2_5 Unit 3_1 Unit 3_2 Unit 3_3 Unit 3_4 Unit 3_5 Blocks Units Units nested in Blocks

Lecture © 2015 Michael Stuart Randomization procedure 1.enter numbers 1 to 5 in Column A of a spreadsheet, headed Run, 2.enter letters A-E in Column B, headed Blend, 3.generate 5 random numbers into Column C, headed Random 4.sort Blend by Random, 5.allocate Treatments as sorted to Runs in Block I, 6.repeat Steps for Blocks II and III. Go to Excel Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Part 2 Randomised blocks analysis Exploratory analysis Analysis of Variance Block or not? Diagnostic analysis –deleted residuals Analysis of variance explained Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Results Ref: BlendLoss.xls Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Initial data analysis Little variation between blocks More variation between blends Disturbing interaction pattern; see later Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Formal Analysis Analysis of Variance: Loss vs Block, Blend Source DF SS MS F P Block Blend Error Total Classwork 1.2.6: Confirm the calculation of Total DF, Total SS, MS(Block), MS(Blend), MS(Error) F(Block), F(Blend) Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Formal Analysis Analysis of Variance: Loss vs Block, Blend Source DF SS MS F P Block Blend Error Total Classwork 1.2.6: Confirm the calculation of Total DF, Total SS, MS(Block), MS(Blend), MS(Error) F(Block), F(Blend) Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Assessing variation between blends Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Assessing variation between blends F(Blends)= 3.3 F 4,8;0.1 = 2.8 F 4,8;0.05 = 3.8 p = 0.07 F(Blends) is "almost statistically significant" Multiple comparisons: All intervals cover 0; Blends B and E difference "almost significant" Ref: Lecture Note 1.2, pp Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Assessing variation between blocks F(Blocks) = 0.94 < 1; MS(Blocks) < (MS(Error) differences between blocks consistent with chance variation; Source DF SS MS F P Block Blend Error Total Source DF SS MS F P Blend Error Total Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Was the blocking effective? Source DF SS MS F P Block Blend Error Total S = Source DF SS MS F P Blend Error Total S = Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Was the blocking effective? F(Blocks) < 1 Blocks MS smaller than Error MS When blocks deleted from analysis –Residual standard deviation almost unchanged and –F(Blends) almost unchanged Blocking NOT effective. Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Block or not? Source DF SS MS F P Block Blend Error Total Source DF SS MS F P Blend Error Total Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Block or not? Source DF SS MS F P Block Blend Error Total Source DF SS MS F P Blend Error Total Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Block or not? Omitting blocks increases DF(Error), therefore increases precision of estimate of , and increases power of F(Blends) F 4,8:0.10 = 2.8; F 4,8:0.05 = 3.8 F 4,10:0.10 = 2.6F 4,10:0.05 = 3.5 Smaller critical value easier to exceed, more power. Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Block or not? Statistical theory suggests no blocking. Practical knowledge may suggest otherwise. Quote from Davies et al (1956): "Although the apparent variation among the blocks is not confirmed (i.e. it might well be ascribed to experimental error), future experiments should still be carried out in the same way. There is no clear evidence of a trend in this set of trials, but it might well appear in another set, and no complication in experimental arrangement is involved". Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Diagnostic plots The diagnostic plot, residuals vs fitted values –checking the homogeneity of chance variation The Normal residual plot, –checking the Normality of chance variation Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Diagnostic analysis Postgraduate Certificate in Statistics Design and Analysis of Experiments One exceptional case –likely to be related to interaction pattern. see Slide 19 −resist deletion and refitting!

Lecture © 2015 Michael Stuart Initial data analysis Little variation between blocks More variation between blends Disturbing interaction pattern; see later Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Deleted residuals Residual –observed – fitted Standardised Residual –divide by the usual estimate of  Standardised Deleted Residual –residual calculated from data with suspect case deleted –  estimated from data with suspect case deleted Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Deleted residuals For each potentially exceptional case: –delete the case –calculate the ANOVA from the rest –use the deleted fitted model to calculate a deleted fitted value –calculate deleted residual = obseved value – deleted fitted value –calculate deleted estimate of s –standardise Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Deleted residuals Minitab does this automatically for all cases! They are used to allow each case to be assessed using a criterion not affected by the case. It is not the residuals are deleted, it is the case that is deleted to facilitate calculation of the "deleted" residuals Simple linear regression illustrates: Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Scatterplot Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Scatterplot Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Scatterplot Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Deleted residual Given an exceptional case, deleted residual> residual using all the data deleted s< s using all the data deleted standardised residual >> standardised residual using all the data Using deleted residuals accentuates exceptional cases Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Corresponding residual plots, Standardized vs Deleted Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Randomized Blocks Example Standardized vs Deleted Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Randomized Blocks Example Standardized vs Deleted Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Analysis of Variance Explained Decomposing Total Variation Expected Mean Squares Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Decomposing Total Variation Analysis of Variance: Loss vs Block, Blend Source DF SS MS F P Block Blend Error Total SS(TO) = SS(Block) + SS(Blend) + SS(Error) Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Model for analysis Yield loss includes –a contribution from each blend plus –a contribution from each block plus –a contribution due to chance variation. Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Model for analysis Y =  +  +  +  where  is the overall mean,  is the blend effect, above or below the mean, depending on which blend is used,  is the block effect, above or below the mean, depending on which block is involved  represents chance variation Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Estimating the model Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Estimating the model Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Estimating the model Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Decomposing Total Variation statistical residual format mathematically simplified format SSTO = SS(Blocks) + SS(Blends) + SS(Error) Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Expected Mean Squares F(Blends) = tests equality of blend means F(Blocks) = assesses effectiveness of blocking Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Part 3Factorial Design a 3 x 3 experiment Iron-deficiency anemia the most common form of malnutrition in developing countries contributory factors: –cooking pot type Aluminium (A), Clay (C) and Iron (I) –food type Meat (M), Legumes (L) and Vegetables (V) Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Study design and results 4 samples of each food type were cooked in each pot type, iron content in each sample measured in milligrams of iron per 100 grams of cooked food. Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Classwork What were the –response –experimental factors –factor levels –treatments –experimental units –unit structure –treatment assignment –replication Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Initial Data Analysis Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Model for analysis Iron content includes –a contribution for each food type plus –a contribution for each pot type plus –a contribution for each food type / pot type combination plus –a contribution due to chance variation. Minitab:Pot Food Pot * Food Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Analysis of Variance Analysis of Variance for Iron Source DF SS MS F P Pot Food Pot*Food Error Total S = Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Summary Cooking in iron pots adds substantially to the average iron content of all cooked foods. However, it adds considerably more to the iron content of meat, – around 2.5 to 2.6 mgs per 100gms on average, than to that of legumes or vegetables, – around 1.2 to 1.5 mgs per 100gms on average The iron content is very similar using aluminium and clay for all three food types. Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Interaction Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Interaction Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Diagnostic plots Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Diagnostic plots Slight suggestion of skewness, but conclusions are sufficiently strong to ignore this Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Minute test –How much did you get out of today's class? –How did you find the pace of today's class? –What single point caused you the most difficulty? –What single change by the lecturer would have most improved this class? Postgraduate Certificate in Statistics Design and Analysis of Experiments

Lecture © 2015 Michael Stuart Reading EM §7.3.2, §7.4.2 DCM §§4-1, 5-1 to 5-3 DV §6.2, 6.4 Postgraduate Certificate in Statistics Design and Analysis of Experiments