/ department of mathematics and computer science DS01 Statistics 2 for Chemical Engineering

/ department of mathematics and computer science 1212 Lecturers Marko Boon Dr. A. Di Bucchianico Ir. G.D. Mooiweer Drs. C.M.J. Rusch – Groot (

/ department of mathematics and computer science 1212 Important to remember Web site for this course: No textbook, but handouts + Powerpoint sheets through web site Bring notebook to fourth lecture (12 th of April) and self-study Software: –Statgraphics (version 5.1). If not installed, install through –Java (at least version 1.4). Install through Java is needed to run Statlab ( Important: In order to run Statlab during the exams, security settings have to be adjusted!

/ department of mathematics and computer science 1212 Goals of this course teach students need for statistical basis of experimentation teach students statistical tools for experimentation –design of experiments (factorial designs, optimal designs) –analysis of experiments (ANOVA) –use of statistical software give students short introduction to recent developments

/ department of mathematics and computer science 1212 Week schedule Week 1: Introduction to Analysis of Variance (ANOVA) Week 2: Factorial designs: screening Week 3: Factorial designs: optimisation Week 4: Optimal experimental design and mixture designs (by A. Di Bucchianico – Bring your laptop!)

/ department of mathematics and computer science 1212 Detailed contents of week 1 statistics and experimentation short recapitulation of regression analysis one-way ANOVA one-way ANOVA with blocks multiple comparisons

/ department of mathematics and computer science 1212 Statistics and experimentation Chemical experiments often depend on several factors (pressure, catalyst, temperature, reaction time,...) Two important questions: which factors are really important? what are optimal settings for important factors?

/ department of mathematics and computer science 1212 Use of statistical experimentation in chemical engineering Chemical synthesis (synthetic steps; work up and separation; reagents, solvents, catalysts; structure, reactivity and properties,...) Biotech industry (drug design, analytical biochemistry, process optimization – fermentation, purification,...) Process industry (process optimization and control -yield, purity, through put time, pollution, energy consumption; product quality and performance - material strength, warp, color, taste, odour;...)...

/ department of mathematics and computer science 1212 Short history of statistics and experimentation 1920’s -... introduction of statistical methods in agriculture by Fisher and co-workers 1950’s -... introduction in chemical engineering (Box,...) 1980’s -... introduction in Western industry of Japanese approach (Taguchi, robust design) 1990’s -... combinatorial chemistry, high througput processing

/ department of mathematics and computer science 1212 Link to Statistics 1 for Chemical Engineering introduction to measurements –data analysis –error propagation regression analysis use of statistical software (Statgraphics)

/ department of mathematics and computer science 1212 Types of regression analysis Linear means linear in coefficients, not linear functions! Simple linear regression Multiple linear regression Non-linear regression

/ department of mathematics and computer science 1212 Model:  ssumptions: the model is linear (+ enough terms) the  i 's are normally distributed with  =0 and variance  2 the  i 's are independent. Linear regression

/ department of mathematics and computer science 1212 Specific warmth specific warmth of vapour at constant pressure as function of temperature data set from Perry’s Chemical Engineers’ Handbook thermodynamic theories say that quadratic relation between temperature and specific warmth usually suffices:

/ department of mathematics and computer science 1212 Scatter plot of specific warmth data

/ department of mathematics and computer science 1212 Regression output specific warmth data Polynomial Regression Analysis Dependent variable: Cp Standard T Parameter Estimate Error Statistic P-Value CONSTANT T T^ Analysis of Variance Source Sum of Squares Df Mean Square F-Ratio P-Value Model Residual Total (Corr.) R-squared = percent R-squared (adjusted for d.f.) = percent Standard Error of Est. = Mean absolute error = Durbin-Watson statistic = (P=0.0000) Lag 1 residual autocorrelation =

/ department of mathematics and computer science 1212 Issues in regression output significance of model significance of individual regression parameters residual plots: –normality (density trace, normal probability plot) –constant variance (against predicted values + each independent variable) –model adequacy (against predicted values) –outliers –independence influential points

/ department of mathematics and computer science 1212 Residual plot specific warmth data This behaviour is visible in plot of fitted line only after rescaling!

/ department of mathematics and computer science 1212 Plot of fitted quadratic model for specific warmth data

/ department of mathematics and computer science 1212 Conclusion regression models for specific warmth data we need third order model (polynomial of degree 3) careful with extrapolation original data set contains influential points original data set contains potential outliers

/ department of mathematics and computer science 1212 Analysis of variance name refers to mathematical technique, not to goal comparison of means (!!) using variances (extension of t- test to more than 2 samples) samples usually are groups of measurements with constant factor settings

/ department of mathematics and computer science 1212 Example: ANOVA production of yarns: influence of fibre composition on breaking tension simplification: one factor: % cotton fixed factor levels: 15%, 20%, 25%, 30%, 35% experimental design: produce on the same machine 5 threads of each type of fibre composition in random order

/ department of mathematics and computer science 1212 Statistical setting Basis model: Y ij =  +  i +  ij influence factor levels i=1,2,…k error term: normal  =0,  2 independent replications j=1,2,…,n Basis hypotheses: H 0 :  i = 0 for all i H 1 :  i  0 for at least one i overall mean

/ department of mathematics and computer science 1212 Expectation under H 0 (= no effect of factor level) spread observations with respect to group means spread group means with respect to overall mean chance

/ department of mathematics and computer science 1212 Expectation under H 1 spread observations with respect to group means chance systematic spread group means with respect to overall mean

/ department of mathematics and computer science 1212 Illustration of group means

/ department of mathematics and computer science 1212 Group means versus overall mean

/ department of mathematics and computer science 1212 Conclusion Comparison of both spreads yields indication for H 0 vs H 1. total treatment: between groups rest: within groups =+

/ department of mathematics and computer science 1212 Conclusion Comparison of both spreads yields indication for H 0 vs H 1. total treatment: between groups rest: within groups =+ Spreads are converted into sums of squares:

/ department of mathematics and computer science 1212 Mean Sums of Squares sums of squares differ with respect to number of contributions. for fair comparison: divide by degrees of freedom. we expect under H 0 : MS between  MS within we expect under H 1 : MS between >> MS within summary in ANOVA table

/ department of mathematics and computer science 1212 Completely Randomized One-factor Design Experiment, in which one factor varies on k levels. At each level n measurements are taken. The order of all measurements is random.

/ department of mathematics and computer science 1212 Multiple comparisons ANOVA only indicates whether there are significantly different group means ANOVA does not indicate which groups have different means (although we may construct confidence intervals for differences) various methods exist for correctly performing pairwise comparisons: –LSD (Least Significant Difference) method –HSD (Honestly Significant Difference) method –Duncan –Newman – Keuls –...

/ department of mathematics and computer science 1212 Randomized one-factor block design In each block all treatments occur equally often; randomization within blocks Experiment with one factor and observations in blocks Blocks are levels of noise factor.

/ department of mathematics and computer science 1212 Example testing method for material hardness : force pressure pin/tip strip testing material practical problem: 4 types of pressure pins  do these yield the same results?

/ department of mathematics and computer science 1212 Experimental design pin 1 pin 2 pin 4pin 3 testing strips Problem: if the measurements of strips 5 through 8 differ, is this caused by the strips or by pin 2?

/ department of mathematics and computer science 1212 Experimental design 2 Take 4 strips on which you measure (in random order) each pressure pin once : strip 1strip 2strip 4strip 3 pressure pins

/ department of mathematics and computer science 1212 Blocking Advantage of blocked experimental design 2: differences between strips are filtered out Model: Y ij =  +  i +  j +  ij Primary goal: reduction error term factor pressure pin block effect strip error term

/ department of mathematics and computer science 1212 Summary completely randomized design randomized block design multiple comparisons Reading material: Statgraphics lecture notes section 4.1 through 4.3