1 Statistical Analysis Professor Lynne Stokes Department of Statistical Science Lecture 5 Complete Factorial Experiments, Completely Randomized Designs, Main Effects & Interactions

2 Complete Factorial Experiments Does not require that each combination occur an equal number of times Does not state what type of statistical design is being used All combinations of the factor levels appear in the design at least once All combinations of the factor levels appear in the design at least once “Factorial Design” is not a correct statistical term

3 Completely Randomized Design Can be any type of complete or fractional factorial experiment Randomize the sequence of test runs, assignment to experimental units Randomize the sequence of test runs, assignment to experimental units

4 Randomization Inexpensive Insurance Bias due to changes in uncontrollable Factors or in experimental conditions Bias due to premature termination of the experiment Bias due to machine drift, fatigue, wear Validates key assumptions (Independence, Randomization Distributions) Validates key assumptions (Independence, Randomization Distributions)

5 Changes in Experimental Conditions : Oil Viscosity Tests Experiment :Two Oils One Test Run Per Day Six Test Runs Per Week DayWeek #1Week # Average t-Test: averages are not significantly different (A) (B)

6 Changes in Experimental Conditions : Oil Viscosity Tests DayWeek #1Week # Average Average viscosity for Oil B is significantly greater than for Oil A Environmental or equipment change between weeks : add 5 unit bias to all measurements in Week #2 (A) (B)

7 Changes in Experimental Conditions : Oil Viscosity Tests DayWeek #1Week # Average Randomize the test sequence, 3 oils each week, add 5 unit bias to all measurements in Week #2 (B) (A) (B) (A) (B) (A) (B) (A) (B) (A) Original Biased Randomized Oil B - Oil A

8 Premature Termination : Oil Viscosity Tests Oil TypeLubricantViscosity A # # # # B #160.6 #2 #3 #4 Equipment Failure

9 Background Noise Time Test Runs ResponseDrift Figure 4.7 Influence of machine drift; test runs indicated by arrows. 50 Gallon Drum of Chemicals

10 Torque Study Goal :Investigate the effects of three factors on torque forces on rotating shafts FactorLevels Shaft AlloySteel, Aluminum Sleeve MetalPorous, Nonporous Lubricant TypeLub 1, Lub 2, Lub 3, Lub 4 Stationary SleeveRotating Shaft MGH Figure 5.1 Lay Out the Design Lay Out the Design

11 Construction of Completely Randomized Designs List the factor-level combinations All combinations, if complete factorial; only those to be tested, if a fractional factorial Include repeat tests, if any Number the combinations (including repeats) from 1 to N Obtain one or more random number sequences of numbers from 1 to N Randomize the test run sequence, if testing is performed sequentially (one after another) Randomize the assignment of factor-level combinations to experimental units, if any

12 Completely Randomized Design for Torque Study Random Number Sequence : 8, 13, 4, 7, 5, 1, 11, 15 9, 3, 12, 10, 6, 14, 16, 2 MGH Table 5.1

13 Completely Randomized Design for Torque Study Repeats : Same Procedure MGH Table 5.2

14 Completely Randomized Design for Torque Study Randomly Selected Repeat Tests cf. Table 5.3 CRD with 2 Repeats

15 Lubricant Deposit Study What are the Main Factor Effects ?

16 Notation (One-Factor Experiment) 123k... Factor Level Average Data: y ij i = Factor Level j = Repeat Overall Average

17 Factor Main Effects Factor Level12...k Mean  1  2...  k Average... Are the means different ? Are the averages significantly different ? (Main Effects) Change in the average response due to changes in levels of one factor Change in the average response due to changes in levels of one factor

18 Factor Main Effects Fixed Effects : Constant (mean) changes (Pre-selected levels, systematic changes) Random Effects : Random changes (Standard Deviation > 0) (Effects sampled from a probability distribution)

19 Factor Main Effects Models One factor model Averages Assumption E(e ij ) = 0 Conventions  i =  +  i   i = 0 Assumption E(e ij ) = 0 Conventions  i =  +  i   i = 0 Cell Means Model Effects Model

20 Factor Main Effects Models Main Effects Note:  i =  j  i =  j Main Effects Theoretical: changes in factor-level means Empirical : changes in factor-level averages Note:  i =   i = 0

21 Pilot Plant Experiment Temperature Concentration C1 C2 Catalyst MGH Figure

22 Pilot Plant Experiment Temperature Concentration C1 C2 Catalyst

23 Pilot Plant Experiment Temperature Concentration C1 C2 Catalyst

24 Pilot Plant Experiment Temperature Concentration C1 C2 Catalyst

25 Pilot Plant Experiment Main Effects Main effects do not measure joint factor effects Main effects are averaged across levels of the other factors Main effects do not measure joint factor effects Main effects are averaged across levels of the other factors Change in the average response due to changes in levels of one factor Change in the average response due to changes in levels of one factor (High – Low)

26 Interactions Effects of the levels of one factor on the response depend on the levels of one or more other factors

27 No Interaction Response Factor #2 Factor #1 Change in average response for factor #1 is constant for all levels of factor #2 Level #1 Level #2 Change

28 Pilot Plant Experiment Temperature Concentration C1 C2 Catalyst Change with concentration roughly the same for each catalyst

29 Pilot Plant Experiment Temperature Concentration C1 C2 Catalyst Change with temperature greater for catalyst C2 than for catalyst C

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31 Pilot Plant Experiment Interaction Plot Average Yield (%) 2040 Concentration (%) MGH Figure 5.4 Catalyst C1 Catalyst C2 No Interaction

32 Pilot Plant Experiment Interaction Plot Average Yield (%) Temperature (deg C) MGH Figure Catalyst C1 Catalyst C2 Interaction

33 Cutting Tool Life Lathe Speed (rpm) Tool Life (hrs) Tool Type A Tool Type B 15 Hours } No Interaction MGH Figure 5.2

34 Plasticity Study Purpose : Study “Plastic-Like” Properties of Friction-Reducing Lubricants and Additives Stationary Platform Moveable Test Sample Lubricant + Additive Responses: Plastic Viscosity Gel

35 Plasticity Experiment Design Factors Lubricant Acme XLT, Monarch1, Standard Additive None, 1%, 5% Design Factorial Experiment 2 Repeat Tests / Combination Completely Randomized Design

36 Factor Effects on Plastic Viscosity Effects of Lubricants Acme XLT - Standard : = 1.41 Monarch 1 - Standard : = Main Effects for Lubricants Effects of Additives 1% - None : = % - None : = 1.00 Main Effects for Additives

37 Factor Effects on Plastic Viscosity None1 %5 %Additive Amount Average Plastic Viscosity Acme XLT Monarch1 Standard Primary Conclusion Acme XLT has a greater mean plastic viscosity, especially with 5% concentration Primary Conclusion Acme XLT has a greater mean plastic viscosity, especially with 5% concentration If Verified by statistical analysis Weak, if any, interaction effects

38 Factor Effects on Gel Effects of Lubricants Acme XLT - Standard : = 1.44 Monarch 1 - Standard : = 0.09 Main Effects for Lubricants Effects of Additives 1% - None : = % - None : = Main Effects for Additives

39 Factor Effects on Gel None1 %5 %Additive Amount Average Gel Acme XLT Monarch1 Standard Primary Conclusion Acme XLT has a lower mean gel with no additive, greater with additives Primary Conclusion Acme XLT has a lower mean gel with no additive, greater with additives If verified by statistical analysis Strong Interaction Effects

40 Completely Randomized Design for Torque Study

41 Interaction Effects : Torque Study Average Torque (in-oz) Lubricant Type 1234 Porous Sleeve Aluminum Shaft

42 Interaction Effects : Torque Study Average Torque (in-oz) Lubricant Type 1234 Porous Sleeve Nonporous Sleeve Aluminum Shaft

43 Interaction Effects : Torque Study Average Torque (in-oz) Lubricant Type 1234 Porous Sleeve Nonporous Sleeve Aluminum Shaft Lubricant Type 1234 Porous Sleeve Nonporous Sleeve Average Torque (in-oz) Steel Shaft

44 Interaction Effects Interaction effects cannot be properly evaluated if the design does not permit their estimation Interaction effects cannot be properly evaluated if the design does not permit their estimation Complete factorials permit the evaluation of all main effects and all interaction effects