Statistical Analysis Professor Lynne Stokes Department of Statistical Science Lecture 17 Block Designs

Lawnmower Stopping Times Complete Factorial Experiment, Repeat Tests MGH Table 9.1

Lawnmower Stopping Times Lawnmowers are experimental units with potentially large unit-to-unit variability

Lawnmower Stopping Times Overall Variation AAAAAA AAAA A A AA BB BBBB BBB BBBB BBBB B Within manufacturer variation Between manufacturer variation Fixed Effect Fixed and Random Effects

Lawnmower Stopping Times Lawnmower Variation AA BBB Between manufacturer Variation: SS M Lawnmower variation (Within Manufacturer): SS L(M) A Fixed Effect Random Effect

H Manufacturer Comparison Repeat Test Variation Cutoff Times (.01 sec) LLLL LL LL L LL HH HH HH HH Speed and Uncertainty Variation Speed variation Variation: SS S Fixed Effect Random Effect

Statistical Model ManufacturerLawnmowerSpeed Random Components of Variation Lawnmowers (within manufacturers) Repeat Test Variation Repeat Test

Combined Error Distribution Error Likelihood Combined Includes All Sources of Variation

Repeat Test Error Distribution Repeat Test Error Likelihood Only Source of Uncertainty for Speed Comparisons

Replicate and Repeat Test Error Distributionss Error Likelihood Repeat Tests Lawnmowers Both Contribute to the Uncertainty in Manufacturer Comparisons

Combined Error Distribution Error Likelihood Repeat Tests Lawnmowers Combined e C = e L + e R  C 2 =  L 2 +  R 2

Controlling Experimental Variability Blocking Matching Experimental Units to obtain Homogeneous Units in each block Grouping Sequencing of test runs to achieve more uniformity within each group Replication Repeating of the entire experiment or portions of it under possibly dissimilar conditions Repeat Tests Two or more factor-level combinations repeated under “identical” conditions

Block Designs Blocks: Groups of homogeneous experimental units or groups of test runs conducted under similar conditions Conditions within blocks are more similar than conditions between blocks Design Issue : Factor changes within a block only subject to repeat test variation Factor changes between blocks also subject to block variation

Blocking Designs Key Assumptions Blocks are not random FACTOR effects Blocks do not interact with the design factors Blocks only contribute to variability Key Assumptions Blocks are not random FACTOR effects Blocks do not interact with the design factors Blocks only contribute to variability If Blocks interact with the design factors, then the block factor should be treated as an additional design factor.

Purposes Reduce experimental variation in the comparisons of factor effects ith Factor EffectError Variation From All Sources

Purposes Reduce experimental variation in the comparisons of factor effects ith Factor Effect Error Variation From All Remaining Sources (Except Blocks) jth Block Effect b j : Random Block Effect

Purposes Reduce experimental variation in the comparisons of factor effects by changing factor levels within each block Obtain estimates of each source of variation Often necessitated by experimental conditions, restrictions Blocks are Not Additional Random Experimental Factors

Randomized Complete Block Designs Group experimental units into homogeneous blocks, if applicable Groups test run sequence into homogeneous blocks, if applicable Randomly assign factor-level combinations to the experimental units and/or test run sequence Use a separate randomization for each block Complete or Fractional Factorial Experiment in Each Block

Allergic-Reaction Study Develop a laboratory protocol to study allergic reactions to environmental pollutants 10 mice from a single strain Two locations (ear, back) on each animal Duplicate skin thickness measurements at each location (with caliper) Measurements immediately after injection with a skin irritant

Allergic-Reaction Study Develop a laboratory protocol to study allergic reactions to environmental pollutants 10 mice from a single strain Two locations (ear, back) on each animal Duplicate skin thickness measurements at each location (with caliper) Measurements immediately after injection with a skin irritant Factor: Location Blocks: Animals Repeats: Duplicate measurements

Design Layout for Allergic- Reaction Study Replicate (Block) Repeat Tests Randomized Complete Block Design

Allergic-Reaction Study

Analysis Blocks: Main Effect, No Interactions Blocks: Main Effect, No Interactions Factors: Main Effects, & Interactions Factors: Main Effects, & Interactions General Model Specification

Allergic-Reaction Study

Designs with Complete or Fractional Factorials Fractional factorial experiments in completely randomized designs Tables of designs : Appendix 7.A.1 Aliasing among all factor effects determined by the defining equations Complete factorial experiments in randomized incomplete block designs Tables of designs : Appendix 9.A.1 Only aliasing is between blocks and the defining equations Fractional factorial experiments in randomized incomplete block designs Tables of designs : Appendix 9.A.2 Aliasing between blocks and the defining equations, and the usual aliasing among factor effects determined by the defining equations

Drilling Tool Experiment FactorsLevels Rotational Drill Speed60, 75 rpm Longitudinal Velocity50, 100 feet/minute Drill Pipe Length200, 400 feet Drilling Angle30, 60 degrees Tool Joint GeometryStraight, Ellipsoidal Edges Drill Pipe Tool Joint Drill Angle

Drilling Tool Experiment Half fraction each day Defining Contrast : I = ABCDE ABCDE aliased with Blocks Randomly select 4 repeat tests each day Randomize Half fractions to days Test run sequence each day Restriction No more than 20 test runs per day Design ?

Drilling Tool Experiment First Day: 20 Test Runs Half Fraction Defining contrast : I = ABCDE Resolution V 4 repeat tests Analysis Main Effects (5 df) Two-factor interactions (10 df) Estimate of error standard deviation (4 df)

Drilling Tool Experiment Complete Factorial: 40 Test Runs Randomized incomplete block design Day alias : ABCDE Resolution not a meaningful property – Complete factorial in incomplete blocks, not a fractional factorial 8 repeat tests Analysis Main Effects (5 df) Day Effect = ABCDE (1 df) Two-Factor interactions (10 df) Three-Factor interactions (10 df) Four-Factor interactions (5 df) Estimate of error standard deviation (8 df) } Combine ?

Drilling Tool Experiment : Second Scenario Restriction No More than 10 Test Runs per Day Quarter fraction each day Defining contrasts : I = ABD=ACE (= BCDE) Assign factors so that ABD, ACE are of little interest (WHY ?) Randomly select 2 repeat tests each day Randomize Quarter fractions to days Test run sequence each day

Randomized Incomplete Block Design Day 1Day 2Day 3Day 4 ABD = -1 ACE = -1 (BCDE = +1) ABD = -1 ACE = +1 (BCDE = -1) ABD = +1 ACE = -1 (BCDE = -1) ABD = +1 ACE = +1 (BCDE = +1) Sequence :

Drilling Tool Experiment First Block: 10 Test Runs Quarter fraction Defining contrast : I = ABD=ACE=BCDE Resolution III 2 repeat tests Analysis Main effects (5 df) Estimate of error standard deviation 2 df -- no assumptions 4 df -- assuming NO interaction effects

Drilling Tool Experiment Second Day: 10 more test runs Half Fraction Choose 2nd day so that BCDE has one sign for both days Defining contrast : I = BCDE Resolution IV (Half Fraction) & ABD, ACE Aliased with Days 4 Repeat tests Analysis (n = 20) Main effects (5 df), Day effect = BCDE (1 df) Two-factor interactions (9 df) : BC = DE Aliased Estimate of error standard deviation (4 df)

Drilling Tool Experiment Third Day: 10 More Test Runs 3/4 Fraction ABD, ACE, BCDE aliased with days Some interactions partially aliased with one another 6 repeat tests Analysis (n = 30) Main effects (5 df), Day effects (2 df) Two-factor interactions (10 df) Other interactions (6) Estimate of error standard deviation (6 df)

Drilling Tool Experiment Complete Factorial: 10 More Test Runs Randomized incomplete block design ABD, ACE, BCDE aliased with days (Complete Factorial) 8 repeat tests Analysis (n = 40) Main effects (5 df) Day effects = ABD = ACE = BCDE (3 df) Two-factor Interactions (10 df) Three-factor interactions (8 df): ABD, ACE aliased with days Four-factor interactions (4 df): BCDE aliased with days Five-Factor interaction (1 df) Estimate of error standard deviation (8 df)

Blocking and Sequential Experimentation Run select fractions in blocks, analyze each block as it is completed can continue or terminate, as warranted by the analysis Run select fractions in blocks, analyze each block as it is completed can continue or terminate, as warranted by the analysis

Balanced Incomplete Block Designs Used when blocks contain fewer experimental units than the number of unique factor-level combinations b blocks f factor-level combinations k < f experimental units per block No interactions with the design factor(s)

Balanced Incomplete Block Designs Blocks = b Factor combinations = f Units per block = k Each combination occurs r times Each pair occurs together in p blocks N = fr = bk p = r(k - 1)/(f - 1) b > f - 1 Note: Cannot select arbitrary values for b, f, k, r, and p

Balanced Incomplete Block Designs Blocks = b Factor combinations = f Units per block = k Each combination occurs r times Each pair occurs together in p blocks Select a basic design based on the values of b, f, and k MGH Table 9.A.3 Randomly order the blocks and/or the testing of blocks, as applicable Randomly assign the combinations to the units and/or test sequence in each block

Oil-Consumption Experiment Measure the fuel consumption associated with four oils Single test engine, single dynamometer test stand -- reduce extraneous variation Test stand must be recalibrated after two test runs Three replicates f = 4, k = 2, r = 3 so that b = 6, p = 1

Oil-Consumption Experiment Recalibration (Block) Oils Tested A,B C,D A,C B,D A,D B,C MGH Table 9.A.3, Design 1 Replicate 1 Replicate 2 Replicate 3 } } } Randomize Replicates, Blocks within replicates, Letters within blocks, Assignment of oils to letters

Latin Square Designs Control two sources of variability Restrictions Factor of interest and two blocking factors each at k levels No Interactions among the experimental and blocking factors Experiment Size Latin Square : n = k 2 Complete Factorial : n = k 3 + r

Latin Square Design Lay out a table with k rows and k columns Assign the letters A, B,..., K to the cells in the first row of the table For the next row, move the first letter to the last position, shift all letters one position to the left Repeat the previous step until all rows are completed Randomize: The levels of one blocking factor to the rows The levels of one blocking factor to the columns The levels of the experimental factor to the letters

Latin Square Design Typical Layout 12341ABCD ABCD234 First Block Levels Second Block Levels Factor Levels MGH Table 8A.2

Latin Square Design Typical Layout 12341ABCD2BCDA3CDAB4DABC12341ABCD2BCDA3CDAB4DABC First Block Levels Second Block Levels Factor Levels MGH Table 9A.4

Tire-Test Study Road test of four tire brands One test run : several hundred miles Several trucks needed Several test days Latin Square Design : 4 Tire Brands 4 Trucks 4 Test Days Expected similar performance of each brand on all trucks, days (Apart from Random Variation)

Tire-Test Study Tire 4Tire 2Tire 1Tire 3 4Tire 2Tire 1Tire 3Tire 4 1Tire 1Tire 3Tire 4Tire 2 2Tire 3Tire 4Tire 2Tire 1 Day Truck Tire Brands

Tire Effects Day Effect Truck Effect Tire Effect Main Effects Only

Tire Effects Day Effect Truck Effect Tire Effect Comparison of Tire 1 with Tire 2 : IF NO INTERACTIONS Main Effects Only