Diploma in Statistics Design and Analysis of Experiments Lecture 2.21 © 2010 Michael Stuart Design and Analysis of Experiments Lecture Review of Lecture 2.1 and Laboratory 1 2.Homework Introducing the Design Matrix 4.A 2 3 experiment –3 factors each at 2 levels

Diploma in Statistics Design and Analysis of Experiments Lecture 2.22 © 2010 Michael Stuart Minute Test: How Much

Diploma in Statistics Design and Analysis of Experiments Lecture 2.23 © 2010 Michael Stuart Minute Test: How Fast

Diploma in Statistics Design and Analysis of Experiments Lecture 2.24 © 2010 Michael Stuart Yield Loss Experiment: Blends in Randomised Blocks General Linear Model: Loss, per cent versus Blend, Block Analysis of Variance for Loss,%, Source DF SS MS F P Blend Block Error Total 14

Diploma in Statistics Design and Analysis of Experiments Lecture 2.25 © 2010 Michael Stuart Decomposition of results Effect – – Effect –

Diploma in Statistics Design and Analysis of Experiments Lecture 2.26 © 2010 Michael Stuart Interaction between Factors Case study: Emotional Arousal

Diploma in Statistics Design and Analysis of Experiments Lecture 2.27 © 2010 Michael Stuart Interaction between Factors: main effects of pictures vs gender differentiated effects

Diploma in Statistics Design and Analysis of Experiments Lecture 2.28 © 2010 Michael Stuart Yield loss experiment

Diploma in Statistics Design and Analysis of Experiments Lecture 2.29 © 2010 Michael Stuart Yield loss experiment

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Yield loss experiment

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Laboratory 1 Soybean seed failure rates

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart 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 © 2010 Michael Stuart Set up

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Go to Excel Randomisation

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Set up: Run order NB: Reset factor levels each time

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Results (run order)

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Results (standard order)

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart 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 © 2010 Michael Stuart Main Effects and Interactions

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Bar height = t value (see next slide) Reference line is at critical t value (4 df) df = 7 – 3 = 4

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart 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 = Coef = Effect / 2 SE(Effect) = SE(Coef) x 2

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart 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 © 2010 Michael Stuart 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 © 2010 Michael Stuart 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 © 2010 Michael Stuart 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 © 2010 Michael Stuart 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 © 2010 Michael Stuart Direct Calculation

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart 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 © 2010 Michael Stuart Measurements for GC study (EM, Exercise 5.1, pp )

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Steps in analysis Produce main effects plots, interaction plot, Calculate main effects and interaction effect Calculate standard error of effects Calculate t-tests Produce diagnostic plots Iterate?

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Organising the data for direct analysis

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Organising the data for direct analysis s 2 = average(SD 2 ) = ( ) / 4 = s= 3.85 df(s) = sum[df(SD)] = = 8

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Minitab results Estimated Effects for Measurements Term Effect SE T P Flow rate Volume Flow rate*Volume S =

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Minitab results

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Minitab results

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Minitab results

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart More Minitab results Means for Peak area Mean SE Mean Flow rate Volume Flow rate*Volume

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart More calculations Calculate confidence intervals for Flow Rate effects at Low and High Volumes. Calculate confidence intervals for Volume effects at Low and High Flow Rates.

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Minitab results; diagnostics

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Minitab results; diagnostics

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Part 4Introducing the design matrix Organising the data for calculation Generic notation

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart The design matrix The design matrix displays the range of experimental conditions under which the process is to be run. Each row (design point) designates a set of experimental conditions. With 2 factors each at 2 possible levels, there are 2 2 = 4 sets of experimental conditions, as listed.

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Organising the calculations Main effect of A: average at high A – average at low A = = Main effect of B: average at high B – average at low B = = Columns of design matrix applied to column of means.

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Dual role of the design matrix Prior to the experiment, the rows designate the design points, the sets of conditions under which the process is to be run. After the experiment, the columns designate the contrasts, the combinations of design point means which measure the main effects of the factors. The extended design matrix facilitates the calculation of interaction effects

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Calculating interaction effects AB Interaction=½(A effect at high B – A effect at low B) = = The extended design matrix Check:AB = A × B

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Part 4A 2 3 experiment: 3 factors each at 2 levels An experiment to investigate the effects on yield of a chemical process of changes to operating Temperature, raw material Concentration and type of Catalyst was conducted in a pilot plant set up for experimentation. Details were as follows. Factor settings and codes

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart A three factor example Design matrix (standard order) Run order for design points (in duplicate)

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart A three factor example Results, in standard order Ref: PilotPlant.xls

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Minitab analysis Estimated Effects for Yield Term Effect SE T P T C K T*C T*K C*K T*C*K S =

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Exercise Calculate the T, C and K main effects

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Calculating effects, the extended design matrix Exercise 2.2.3: Complete the missing columns (contrasts). Calculate the TK and TCK interactions

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Calculating s Total 64 s 2 8 s2.828

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Exercise Calculate the t-ratio for the T effect and the 3-factor interaction. What conclusions do you draw?

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Minitab analysis Estimated Effects for Yield Term Effect SE T P T C K T*C T*K C*K T*C*K S =

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Minitab analysis

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Minitab analysis

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Minitab analysis

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Minitab analysis

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Minitab analysis

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Minitab analysis

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Minitab diagnostic analysis

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Minitab diagnostic analysis

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Homework An experiment was run to assess the effects of three factors on the life of a cutting tool A:Cutting speed B:Tool geometry C:Cutting angle. The full 2 3 design was replicated three times. The results are shown in the next slide and are available in Excel file Tool Life.xls. Carry out a full analysis and report. Ref: Tool Life.xls

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Homework Homework 2.2.2: Web Exercises See also Homework Ref: Hardness.xls

Diploma in Statistics Design and Analysis of Experiments Lecture © 2010 Michael Stuart Reading EM §5.3, §5.4, §5.6 DCM §6-2, §6-3 to p.218, §6.5 to p. 235