# STA2004F: Design of experiments

### Announcements

January 17: Solutions to the final homework, although I haven't had a chance to do the bonus question yet.
December 6: Final homework is here. It is due on December 21.
Data for Question 1.
Data for Question 5.

Homework 3
Sketch of solutions

### Lectures

• December 5, 2006
• Solutions to HW 2
• Details on analysis of variance for split plots
• Handout on mixed and random effects models for factorial experiments
• Two handouts from the literature that both used split plot designs: please drop by to get copies if you missed class
• Topics not covered: response surface designs, cost of experimentation (choice of sample size), optimal designs, Taguchi methods, designs for micro-array analysis
• November 28, 2006
• split plot experiments: one whole plot factor and one subplot factor
• analysis of variance and comparison of means
• fractional factorial experiments in split plot designs
• November 21, 2006
• Algebra of 2 level factorials; contrast group and treatment group
• Fractional factorials and alias sets: half and quarter fractions
• Example: Section 5.7 of book
• Confounding effects with blocks, partial confounding
• Confounding main effects --> split plot experiments
• November 14, 2006
• Further thoughts on HW 2
• Factorial treatment structure with 2 levels for each factor
• Estimation of main effects and interactions
• Table of signs
• Example of a 2^4 factorial design and its analysis
• Use of higher order interactions to estimate error
• A replicated 2^3 experiment embedded in the 2^4 experiment
• November 7, 2006
• suggestions re HW 2
• factorial treatment structure
• interpretation of interaction
• October 24, 2006
• Latin squares and Graeco-Latin squares: reminder of definitions
• analysis of Latin squares using Set 9 from Cox and Snell
• the error sum of squares composed of various interactions
• an alternative analysis correcting for days and time of day within each of the two replicates
• (sets of) Latin squares in which each treatment follows each other treatment the same number of times can be used for cross-over designs
• cross-over designs use the same experimental units over time
• gives strong control of block (subject) effects, but needs care
• washout periods can sometimes ensure that order of treatments does not affect response
• if not successful then carry over effects need to be estimated from the data; some details outlined in 4.3.2. and 4.3.3
• balanced incomplete block designs: more treatments than units per block
• treatment means need to be adjusted for block means
• October 17, 2006
• Randomization in design
• Randomization in inference: justifies usual linear model; randomization distribution of summary statistics (handout from Hinkelmann and Kempthorne, Vol 1).
• Analysis of covariance: adjustment for baseline variables or other covariates; least squares estimates and their mean and variance; handout
• Efficiency of randomized block design relative to CR design; handout
• Latin squares: definition and existence; pairs of orthogonal Latin squares
• Article on Latin squares by Ivars Peterson
• Article on Sudoku and Latin squares by Ivars Peterson
• Demonstration of completion of Latin squares (reference from Peterson's first article)
• Picture of a 10 by 10 Graeco-Latin square
• October 10, 2006
• Modified matched pairs analysis: inclusion of pairs in which both units receive T or both units receive C (\S 3.3.2)
• Completely randomized design: linear model and analysis (p.23)
• Randomization: a means of reducing bias, the unit-treatment additivity assumption, causal effects
• Randomization analysis (\S 2.2): justifies the usual linear model
• Handout: Current list of errata for textbook.
• October 3, 2006
• Analysis of randomized block designs: analysis of variance table, comparison of treatment means, orthogonal contrasts for ordered factors, partitioning treatment sum of squares
• Multiple comparisons of a set of treatment means using: Bonferroni, Tukey's studentized range test (references coming)
• The non-central $\chi^2$ distribution
• The error sum of squares in RB design is an interaction SS between treatments and blocks
• Handout on the construction of orthogonal polynomials (from Montgomery)
• September 26, 2006
• Homework 1 is due on October 10.
• A randomized block design with just two treatments is a matched pair design; we developed the usual t-test based on difference by transforming the responses in each pair to sums and differences
• The randomization analysis of the matched pair design will follow after we cover Chapter 2
• Model based analysis of the randomized block design, using the conventional linear model: estimation of parameters under constraints, estimation of residual variance via the analysis of variance table, linear contrasts and their estimates and estimated variances
• September 19, 2006
• Some definitions: experiments, observational studies, units, treatments, response
• Series of experiments (Section 1.8): phase 1, 2 and 3 clinical trials (a good reference is the Cancer Research UK website; evolutionary operation/response surface methodology/Taguchi methods/robust parameter design; variety trials
• A 3 stage design for a case-control study of genetic causes of disease described in Prentice and Qi, Biostatistics 7, 339--354.
• Understanding the mechanisms: causation, intention to treat, baseline and intermediate variables
• Statistical Analysis: randomization as a basis for analysis (called design-based in sample surveys) and model based analysis (called population models in sample surveys)
• Features of a good experiment: Ch. 1 of Planning of Experiments by D.R. Cox
• Introduction to randomized block designs (Section 3.4 of CR)
• Next week: please read Chapter 3 of CR; we will discuss Ch 3 and then go to Ch 2.
• September 12, 2006