STA2004F: Design of experiments
Solutions to the final homework, although I haven't had a chance to do the bonus question yet.
Final homework is here.
It is due on December 21.
Data for Question 1.
Data for Question 5.
Sketch of solutions
- 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
- Example K from CS, see also Ch 5 of CR
- 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;
- Efficiency of randomized block design relative to CR design;
- 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
here is a link to the MS of chapter 1.