Diploma in Statistics Design and Analysis of Experiments Lecture 2.11 Design and Analysis of Experiments Lecture Review of Lecture Randomised Block Design and Analysis –Illustration –Explaining ANOVA –Interaction? –Effect of Blocking –Matched pairs as Randomised blocks 3.Introduction to 2-level factorial designs –A 2 2 experiment –Set up –Analysis –Application

Diploma in Statistics Design and Analysis of Experiments Lecture 2.12 Minute Test - How Much

Diploma in Statistics Design and Analysis of Experiments Lecture 2.13 Minute Test - How Fast

Diploma in Statistics Design and Analysis of Experiments Lecture 2.14 Was the blocking effective?

Diploma in Statistics Design and Analysis of Experiments Lecture 2.15 Comparing several means Membrane A:standard Membrane B:alternative using new material Membrane C:other manufacturer Membrane D:other manufacturer Burst strength (kPa) of 10 samples of each of four filter membrane types

Diploma in Statistics Design and Analysis of Experiments Lecture 2.16 Comparing several means Tukey 95% Simultaneous Confidence Intervals All Pairwise Comparisons among Levels of Membrane Membrane = A subtracted from: Membrane Lower Center Upper B (---*----) C (----*---) D (----*----) Membrane = B subtracted from: Membrane Lower Center Upper C (----*---) D (----*----) Membrane = C subtracted from: Membrane Lower Center Upper D (---*----)

Diploma in Statistics Design and Analysis of Experiments Lecture 2.17 Comparing several means Membrane B mean is significantly bigger than Membranes C and D means and close to significantly bigger than Membrane A mean. Membrane C mean is significantly smaller than the other three means. Membranes A and D means are not significantly different.

Diploma in Statistics Design and Analysis of Experiments Lecture 2.18 Comparing several means; Conclusions Membrane C can be eliminated from our inquiries. Membrane D shows no sign of being an improvement on the existing Membrane A and so need not be considered further. Membrane B shows some improvement on Membrane A but not enough to recommend a change. It may be worth while carrying out further comparisons between Membranes A and B.

Diploma in Statistics Design and Analysis of Experiments Lecture 2.19 Characteristics of an experiment Experimental units: entities on which observations are made Experimental Factor: controllable input variable Factor Levels / Treatments: values of the factor Response: output variable measured on the units

Diploma in Statistics Design and Analysis of Experiments Lecture Randomised blocks Illustration Manufacture of an organic chemical using a filtration process Three step process: –input chemical blended from different stocks –chemical reaction results in end product suspended in an intermediate liquid product –liquid filtered to recover end product.

Diploma in Statistics Design and Analysis of Experiments Lecture Randomised blocks Illustration 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)

Diploma in Statistics Design and Analysis of Experiments Lecture Results

Diploma in Statistics Design and Analysis of Experiments Lecture Exercise What were the experimental units factor factor levels response blocks randomisation procedure

Diploma in Statistics Design and Analysis of Experiments Lecture Minitab Analysis General Linear Model ANOVA General Linear Model: Loss, per cent versus Blend, Block Analysis of Variance for Loss,%, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P Blend Block Error Total S = R-Sq = 65.38% R-Sq(adj) = 39.41% Unusual Observations for Loss, per cent Loss, per Obs cent Fit SE Fit Residual St Resid R

Diploma in Statistics Design and Analysis of Experiments Lecture 2.115

Diploma in Statistics Design and Analysis of Experiments Lecture Conclusions (prelim.) F(Blends) is almost statistically significant, p = 0.07 F(Blocks) is not statistically significant, p = 0.4 Prediction standard deviation:S = 0.93

Diploma in Statistics Design and Analysis of Experiments Lecture Deleted diagnostics

Diploma in Statistics Design and Analysis of Experiments Lecture Iterated analysis: delete Case 12 General Linear Model: Loss versus Blend, Block Analysis of Variance for Loss Source DF Seq SS Adj SS Adj MS F P Blend Block Error Total S =

Diploma in Statistics Design and Analysis of Experiments Lecture Deleted diagnostics

Diploma in Statistics Design and Analysis of Experiments Lecture Conclusions (prelim.) F(Blends) is highly statistically significant, p = 0.01 F(Blocks) is not statistically significant, p = 0.65 Prediction standard deviation:S = 0.67

Diploma in Statistics Design and Analysis of Experiments Lecture Explaining ANOVA ANOVA depends on a decompostion of "Total variation" into components: Total Variation = Blend effect + Block effect + chance variation;

Diploma in Statistics Design and Analysis of Experiments Lecture Decomposition of results

Diploma in Statistics Design and Analysis of Experiments Lecture Decomposition of results

Diploma in Statistics Design and Analysis of Experiments Lecture Decomposition of results

Diploma in Statistics Design and Analysis of Experiments Lecture Decomposition of results

Diploma in Statistics Design and Analysis of Experiments Lecture Decomposition of results

Diploma in Statistics Design and Analysis of Experiments Lecture Decomposition of results

Diploma in Statistics Design and Analysis of Experiments Lecture Decomposition of results

Diploma in Statistics Design and Analysis of Experiments Lecture Interaction? Blend x Block interaction?

Diploma in Statistics Design and Analysis of Experiments Lecture Interaction? Blend x Block interaction?

Diploma in Statistics Design and Analysis of Experiments Lecture Exercise Calculate fitted values: Overall Mean + Blend Deviation + Block deviation

Diploma in Statistics Design and Analysis of Experiments Lecture Exercise (cont'd) Make a Block profile plot

Diploma in Statistics Design and Analysis of Experiments Lecture Fitted values; NO INTERACTION

Diploma in Statistics Design and Analysis of Experiments Lecture Actual plot: Interaction? Blend effects (the contributions of each blend to Loss) are similar for Blocks 1 and 2 but quite different for Block 3.

Diploma in Statistics Design and Analysis of Experiments Lecture Effect of Blocking Analysis of Variance for Loss (one run deleted) Source DF Seq SS Adj SS Adj MS F P Blend Block Error Total Analysis of Variance for Loss (one run deleted) unblocked Source DF Seq SS Adj SS Adj MS F P Blend Error Total

Diploma in Statistics Design and Analysis of Experiments Lecture Matched pairs as Randomised blocks Wear of shoe soles made of two materials, A and B, worn on opposite feet by each of 10 boys

Diploma in Statistics Design and Analysis of Experiments Lecture Pairing equals Blocking Paired T for Material B - Material A T-Test of mean difference = 0 (vs not = 0): T-Value = 3.35 P-Value = Two-way ANOVA: Wear versus Material, Boy Source DF SS MS F P Material Boy Error Total

Diploma in Statistics Design and Analysis of Experiments Lecture t and F

Diploma in Statistics Design and Analysis of Experiments Lecture t and F

Diploma in Statistics Design and Analysis of Experiments Lecture More on t

Diploma in Statistics Design and Analysis of Experiments Lecture More on F

Diploma in Statistics Design and Analysis of Experiments Lecture Paired Comparison: Effect of Pairing / Blocking Paired T for Material B - Material A T-Test of mean difference = 0 (vs not = 0): T-Value = 3.35 P-Value = Two-sample T for Material B vs Material A T-Value = 0.37 P-Value = 0.716

Diploma in Statistics Design and Analysis of Experiments Lecture Paired Comparison: Effect of Pairing / Blocking Two-way ANOVA: Wear versus Material, Boy Source DF SS MS F P Material Boy Error Total One-way ANOVA: Wear versus Material Source DF SS MS F P Material Error Total

Diploma in Statistics Design and Analysis of Experiments Lecture Introduction to 2-level factorial designs A 2 2 experiment Project: optimisation of a chemical process yield Factors (with levels): operating temperature (Low, High) catalyst (C1, C2) Design: Process run at all four possible combinations of factor levels, in duplicate, in random order.

Diploma in Statistics Design and Analysis of Experiments Lecture Exercise What were the experimental units factors factor levels response blocks randomisation procedure

Diploma in Statistics Design and Analysis of Experiments Lecture Set up

Diploma in Statistics Design and Analysis of Experiments Lecture Set up: Randomisation

Diploma in Statistics Design and Analysis of Experiments Lecture Set up: Run order

Diploma in Statistics Design and Analysis of Experiments Lecture Results (run order)

Diploma in Statistics Design and Analysis of Experiments Lecture Results (standard order)

Diploma in Statistics Design and Analysis of Experiments Lecture Analysis (Minitab) Main effects and Interaction plots Pareto plot of effects ANOVA results –with diagnostics Calculation of t-statistic

Diploma in Statistics Design and Analysis of Experiments Lecture Main Effects and Interactions

Diploma in Statistics Design and Analysis of Experiments Lecture Pareto plot of effects Bar height = t value (see slide 31) Reference line is at critical t value (4 df)

Diploma in Statistics Design and Analysis of Experiments Lecture Minitab DOE Analyze Factorial Design Estimated Effects and Coefficients for Yield (coded units) Term Effect Coef SE Coef T P Constant Temperature Catalyst Temperature*Catalyst S = R-Sq = 95.83% R-Sq(adj) = 92.69% Analysis of Variance for Yield (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects Way Interactions Residual Error Pure Error Total

Diploma in Statistics Design and Analysis of Experiments Lecture 2.155

Diploma in Statistics Design and Analysis of Experiments Lecture Minitab DOE Analyze Factorial Design Estimated Effects and Coefficients for Yield (coded units) Term Effect Coef SE Coef T P Constant Temperature Catalyst Temperature*Catalyst S = R-Sq = 95.83% R-Sq(adj) = 92.69% Analysis of Variance for Yield (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects Way Interactions Residual Error Pure Error Total

Diploma in Statistics Design and Analysis of Experiments Lecture ANOVA results ANOVA superfluous for 2 k experiments "There is nothing to justify this complexity other than a misplaced belief in the universal value of an ANOVA table". BHH (2nd ed.), Section 5.10 "a convenient method of arranging the arithmetic" R.A. Fisher

Diploma in Statistics Design and Analysis of Experiments Lecture Diagnostic Plots

Diploma in Statistics Design and Analysis of Experiments Lecture Calculation of t-statistic Results (Temperature order)

Diploma in Statistics Design and Analysis of Experiments Lecture Exercise Calculate a confidence interval for the Temperature effect. All effects may be estimated and tested in this way. Homework Test the statistical significance of and calculate confidence intervals for the Catalyst effect and the Temperature × Catalyst interaction.

Diploma in Statistics Design and Analysis of Experiments Lecture Application Finding the optimum More Minitab results Least Squares Means for Yield Mean SE Mean Temperature Low High Catalyst Temperature*Catalyst Low High Low High

Diploma in Statistics Design and Analysis of Experiments Lecture 

Diploma in Statistics Design and Analysis of Experiments Lecture Optimum operating conditions Highest yield achieved with Catalyst 2 at High temperature. Estimated yield: 81.5% 95% confidence interval: 81.5 ± 2.78 × 2.622, i.e., 81.5 ± 7.3, i.e., ( 74.2, 88.8 )

Diploma in Statistics Design and Analysis of Experiments Lecture Homework As part of a project to develop a GC method for analysing trace compounds in wine without the need for prior extraction of the compounds, a synthetic mixture of aroma compounds in ethanol- water was prepared. The effects of two factors, Injection volume and Solvent flow rate, on GC measured peak areas given by the mixture were assessed using a 2 2 factorial design with 3 replicate measurements at each design point. The results are shown in the table that follows. What conclusions can be drawn from these data? Display results numerically and graphically. Check model assumptions by using appropriate residual plots.

Diploma in Statistics Design and Analysis of Experiments Lecture Peak areas for GC study (EM, Exercise 5.2)

Diploma in Statistics Design and Analysis of Experiments Lecture Reading EM §5.3, §7.4.2 DCM §§4-1, 5-1, 5-2, 6-1, 6-2