Bayesian and Classical Analysis of Multi-Stratum

Response Surface Designs

Steven GilmourQueen Mary, University of London

Peter GoosUniversiteit Antwerpen

Outline

Split-plot and other multi-stratum designs “State-of-the-art” analysis of data

• REML/generalized least squares

• Problems• Estimation of variance components

• Degrees of freedom

Three possible solutions• Fix value(s) of variance-component(s)

• Use randomization-based estimation

• Bayesian analysis

Multi-stratum designs

Randomization of treatments to experimental units is restricted in such a way that particular sets of units must receive the same level of one or more treatment factors• Includes classical orthogonal split-plot, split-split-plot,

criss-cross, etc. designs (regular factorial treatment sets)

• Also includes nonorthogonal designs with similar structures (irregular factorial or response surface treatment sets)

• Are (nested) block designs with at least one main effect totally confounded with block effects

• Often necessary when some factors are hard to change

Multi-stratum designs

I refer to the runs as units and the groups of units defined by the randomization restrictions as blocks, superblocks, …

Randomization is performed by randomly relabelling …, superblocks, blocks and units

Implies random effects for …, superblocks, blocks, units (error) in derived linear model

Fixed treatment effects can be modelled using usual polynomial response surface model

● Response: amount of retained volatile compounds in freeze-dried coffee

● Treatment factors:● Pressure in drying chamber (dial-controlled)

● Heating temperature, Initial solids content, Slab thickness, Freezing rate (all easy to change)

● 5 runs during each of 6 days● Randomization restricted so that all runs in a day

have the same pressure

Freeze-dried coffee experiment

Freeze-dried coffee experiment

Block Press Temp Solids Thickn Rate Block Press Temp Solids Thickn Rate1 1 0 0 0 1 4 1 0 0 -1 01 1 0 0 1 0 4 1 1 0 0 01 1 -1 0 0 0 4 1 0 0 0 -11 1 0 0 0 0 4 1 0 -1 0 01 1 0 1 0 0 4 1 0 0 0 02 0 0 0 0 0 5 -1 0 0 0 02 0 -1 1 -1 1 5 -1 1 1 -1 12 0 1 1 1 -1 5 -1 1 -1 1 -12 0 1 -1 -1 -1 5 -1 -1 1 1 -12 0 -1 -1 1 1 5 -1 -1 -1 -1 13 -1 0 0 0 0 6 0 1 -1 1 13 -1 1 1 1 1 6 0 0 0 0 03 -1 -1 1 -1 -1 6 0 1 1 -1 -13 -1 -1 -1 1 -1 6 0 -1 1 1 13 -1 1 -1 -1 1 6 0 -1 -1 -1 -1

• Model

• Generalized least squares (GLS) estimation

• Variance-covariance matrix

ijiijijy xf '

ZXy

Model and analysis

yVXXVXβ 111 ''ˆ

11')ˆvar( XVXβ

• Model

• Generalized least squares (GLS) estimation

• Variance-covariance matrix

ijiijijy xf '

ZXy

Model and analysis

yVXXVXβ 111 ''ˆ

11')ˆvar( XVXβ 11ˆ')ˆvar(

XVXβ?

? yVXXVXβ 111 ˆ'ˆ'ˆ

• REML: REsidual Maximum Likelihood

• Yields the same answers as ANOVA in orthogonal designs (e.g. standard split-plots)

• Applicable when designs are not orthogonal (e.g. nonorthogonal split-plots)

• State of the art in many disciplines

• Available in many statistical software packages

Variance component estimation

• Different implementations:

• Variance components allowed to be negative or not

• Various methods for obtaining effective degrees of freedom

• Estimates generally consistent with each other, given different implementations

• Effective degrees of freedom can be inconsistent with each other

• All methods can give surprising results

Analysis

• Estimates of whole-plot error variance

• SAS: 0

• GenStat: 0

• R: 0.0051

• Degrees of freedom for testing linear effect of pressure (full model)• SAS proc mixed with Kenward & Roger: 9 df

• SAS proc mixed with containment method: 6 df

• R lme default: 3 df

Freeze-dried coffee experiment

Freeze-dried coffee experiment

Simplified model:

• Data are treated as if they come from a completely randomized experiment

• OLS estimates are obtained

• Degrees of freedom for testing linear effect of pressure are too optimistic• Upper bound for full second-order model: 3 df

• Because of nonorthogonality: less than 3 df• SAS proc mixed with Kenward & Roger: 9 df

• SAS proc mixed with containment method: 6 df

• R lme default: 3 df

Freeze-dried coffee experiment

Artificial exampleBlock X1 X2 Y Block X1 X2 Y Block X1 X2 Y Block X1 X2 Y

1 -1 -1 11 2 -1 -1 10 3 1 -1 31 4 1 -1 401 -1 0 13 2 -1 0 20 3 1 0 38 4 1 0 401 -1 1 18 2 -1 1 23 3 1 1 33 4 1 1 41

Artificial exampleBlock X1 X2 Y Block X1 X2 Y Block X1 X2 Y Block X1 X2 Y

1 -1 -1 11 2 -1 -1 10 3 1 -1 31 4 1 -1 401 -1 0 13 2 -1 0 20 3 1 0 38 4 1 0 401 -1 1 18 2 -1 1 23 3 1 1 33 4 1 1 41

Solution II: Randomization-based analysis

Even nonorthogonal multi-stratum designs have simple orthogonal block structures (if each block/superblock/... is the same size) [Nelder, 1965]

Ignoring treatment structure, randomization-based analysis gives minimum variance unbiased estimators of variance components (pure error)

Only assumption is that treatment and unit effects are additive

Randomization-based analysis

Proposed analysis:• Use discrete treatments defined by combinations

of factor levels (ignoring treatment model)• Anova gives correct estimates of variance components

with correct degrees of freedom

• Use these estimates to fit treatment model using GLS• Base inferences on these estimates

• “Extra sums of squares” represent lack of fit Not clear that GLS is best, but is same as

with REML

Freeze-dried coffee experiment

WP Treat Press Temp Solids Thickn Rate WP Treat Press Temp Solids Thickn Rate1 1 1 0 0 0 1 4 16 1 0 0 -1 01 2 1 0 0 1 0 4 17 1 1 0 0 01 3 1 -1 0 0 0 4 18 1 0 0 0 -11 4 1 0 0 0 0 4 19 1 0 -1 0 01 5 1 0 1 0 0 4 4 1 0 0 0 02 6 0 0 0 0 0 5 11 -1 0 0 0 02 7 0 -1 1 -1 1 5 20 -1 1 1 -1 12 8 0 1 1 1 -1 5 21 -1 1 -1 1 -12 9 0 1 -1 -1 -1 5 22 -1 -1 1 1 -12 10 0 -1 -1 1 1 5 23 -1 -1 -1 -1 13 11 -1 0 0 0 0 6 24 0 1 -1 1 13 12 -1 1 1 1 1 6 6 0 0 0 0 03 13 -1 -1 1 -1 -1 6 25 0 1 1 -1 -13 14 -1 -1 -1 1 -1 6 26 0 -1 1 1 13 15 -1 1 -1 -1 1 6 27 0 -1 -1 -1 -1

Freeze-dried coffee experiment

There are 27 treatments, 3 replicated twice• 0 residual degrees of freedom for blocks

• 3 residual degrees of freedom for runs

Blocks variance component cannot be estimated, unit variance badly estimated• Full polynomial model can be fitted, but no global inference

is possible

• Weak inference is possible for all individual parameters except main effects of pressure

This design is too small for a frequentist analysis

Solution III: Bayesian approach

Advantages:• Takes into account uncertainty in prior beliefs

• Prior beliefs can be contradicted by the data

• No problems determining the appropriate degrees of freedom for hypothesis tests

• WinBUGS software is free

The Bayesian approach

Requires a user-specified (joint) distribution for all model parameters (,

2, 2)

Posterior marginal distributions can be used for inference about parameters

Results:• Similar to REML/GLS if data contain enough

information

• Similar to prior distribution if data don’t contain enough information

The Bayesian approach

Noninformative priors for r: N(0,) Weakly informative priors for r:

• Linear and interaction effects: N(0,25)

• Quadratic effects: N(0,100)

The Bayesian approach

Variance components

The Bayesian approach

Variance components

weakly informative

highly informative

not informative

Results: linear effect of pressure

Linear effect of temperature

Interaction of slab thickness and freezing rate

Summary of results

Prior information on has little impact Prior information on

2 not important at all Some results strongly depend on prior

information about 2

• Hard-to-change factor coefficients

• Sub-plot factor interaction coefficients that are not nearly orthogonal to whole plots

Results for other coefficients insensitive to the choice of the prior for

2

Discussion

REML/GLS analysis can be misleading as it often leads to an analysis that ignores the multi-stratum nature of the design

Likelihood methods have good asymptotic properties, i.e. large numbers of units in each stratum, so should not be expected to work in small experiments

Problem is due to a lack of information in the blocks stratum

We should honestly admit that there is no information and/or provide prior information

Discussion

Randomization based analysis should always be done (in every experiment!) as a first step• Makes very few assumptions, so is much more robust

than any other analysis

• Provides a “reality check”

• Might make extra assumptions unnecessary Bayesian analysis can help

• Prior information is taken into account

• Prior information can be overruled

• Depends heavily on prior assumptions, but these are clearly and honestly expressed

References

Multi-stratum response surface designs Luzia A. Trinca and Steven G. Gilmour Technometrics, 2001

A split-unit response surface design for improving aroma retention in freeze dried coffeeSteven G. Gilmour, J. Mauricio Pardo, Luzia A. Trinca, K. Niranjan and Don Mottram Proceedings of the 6th European Conference on Food-Industry and Statistics, 2000

Analysis of data from unbalanced multi-stratum designsSteven G. Gilmour and Peter Goos Submitted