A tale of randomization: randomization versus mixed model analysis for single and chain randomizations

Chris BrienPhenomics & Bioinformatics Research Centre, University of South Australia.The Australian Centre for Plant Functional Genomics, University of Adelaide.This work was supported by the Australian Research Council.

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A tale of randomization: outline

1. Once upon a time.

2. Randomization model for a single randomization.

3. Randomization analysis for a single randomization.

4. Randomization model for a chain of randomizations.

5. Randomization analysis for a chain of randomizations.

6. Some issues.

7. Conclusions.

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1. Once upon a time In the 70s I was a true believer:

We are talking randomization inference.

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Purism

These books demonstrate that p-value from randomization analysis is approximated by p-value from analyses assuming normality for CRDs & RCBDs;

Welch (1937) & Atiqullah (1963) show that true, provided the observed data actually conforms to the variance for the assumed normal model (e.g. homogeneity between blocks).

Kempthorne (1975):

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Randomization analysis: what is it? A randomization model is formulated.

It specifies the distribution of the response over all randomized layouts possible for the design.

A test statistic is identified. I will use test statistics from parametric analyses (e.g. F-statistics).

The value of the test statistic is computed from the data for: all possible randomized layouts, or a random sample (with

replacement) of them randomization distribution of the test statistic, or an estimate;

the randomized layout used in the experiment: the observed test statistic.

The p-value is the proportion of all possible values that are as, or more, extreme than the observed test statistic.

Different to a permutation test in that it is based on the randomization employed in the experiment.

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Sex created difficulties … and time Preece (1982, section 6.2): Is Sex a block or a

treatment factor? Semantic problem: what is a block factor? Often Sex is unrandomized, but is of interest – I believe

this to be the root of the dilemma. If it is unrandomized, it cannot be tested using a

randomization test (at all?). In longitudinal studies, Time is similar. Sites also. What about incomplete block designs with

recombination of information? Missing values? Seems that not all inference possible with

randomization analysis.

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Fisher (1935, Section 21) first proposed randomization tests:

It seems clear that Fisher intended randomization tests to be only a check on normal theory tests.

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Fisher (1960, 7th edition) added Section 21.1 that includes:

Less intelligible test nonparametric test.

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Conversion I became a modeller,

BUT, I did not completely reject randomization inference. I have advocated randomization-based mixed

models: a mixed model that starts with the terms that would be in

a randomization model (Brien & Bailey, 2006; Brien & Demétrio, 2009).

This allowed me to: test for block effects and block-treatment interactions; model longitudinal data.

I comforted myself that when testing a model that has an equivalent randomization test, the former is an approximation to the latter and so robust.

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More recently …. Cox, Hinkelmann and Gilmour pointed out, in the

discussion of Brien and Bailey (2006), no one had so far indicated how a model for a multitiered

experiment might be justified by the randomizations employed.

Rosemary Bailey and I have been working for some time on the analysis of experiments with multiple randomizations, using randomization-based (mixed) models; Brien and Bailey (201?) details estimation & testing.

I decided to investigate randomization inference for such experiments, but first single randomizations.

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2. Randomization model for a single randomization

Additive model of constants:y = w + Xht

where y is the vector of observed responses; w is the vector of constants representing the contributions of each

unit to the response; and t is a vector of treatment constants; Xh is design matrix showing the assignment of treatments to units.

Under randomization, i.e. over all allowable unit permutations applied to w, each element of w becomes a random variable, as does each element of y. Let W and Y be the vectors of random variables and so we have

Y = W + Xht. The set of Yn forms the multivariate randomization distribution, our

randomization model. Now, we assume ER[W] = 0 and so ER[Y] = Xht .

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Randomization model (cont’d) Further,

R Rvar .H H H H H HH H H

Y V B S QH H H

H is the set of generalized factors (terms) derived from a poset of factors on the units;

zH is the covariance between variables with the same levels of generalized factor H;

yH is the canonical component of excess covariance for H;

hH is the eigenvalue of VR for H and is its contribution to E[MSq];

BH, SH, and QH are known matrices.

This model has the same terms as a randomization-based mixed model (Brien & Bailey, 2006; Brien & Demétrio, 2009)

However, the distributions differ.

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Randomization by permutation of units & unit factors

       Unit Blocks Units Treatments

1 1 1 12 1 2 23 2 1 14 2 2 2

Permutations for an RCBD with b = 2, k = v = 2. The allowable permutations are:

those that permute the blocks as a whole, and those that permute the units within a block; there are b!(k!)b = 2!(2!)2 = 8.

         Unit Blocks Units Treatments Permutation

1 1 1 1 42 1 2 2 33 2 1 1 14 2 2 2 2

          Permutedunit Blocks Units Treatments Permutation Blocks Units1 1 1 1 4 2 22 1 2 2 3 2 13 2 1 1 1 1 14 2 2 2 2 1 2

Equivalent to Treatments randomization 1, 2, 2, 1.

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Null randomization distribution: RCBD Under the assumption of no treatment effects, Y* = W +

m*1. In which case, the randomization distribution of Y* is termed the null

randomization distribution Actual distribution obtained by applying each unit permutation to y:Permutation Y*11 Y*12 Y*21 Y*22

1 y11 y12 y21 y22

2 y12 y11 y21 y22

3 y11 y12 y22 y21

4 y12 y11 y22 y21

5 y21 y22 y11 y12

6 y21 y22 y12 y11

7 y22 y21 y11 y12

8 y22 y21 y12 y11

Can show that 1st & 2nd order parameters of the distribution, m*, z*G,

z*B and z*

BU, are equal to sample statistics. For example, for all Y*

ij: * * 2

.. BU, .yy s

Y*ij for Unit

j in Block i.

The distribution of gives the distribution of W. * yY 1

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VR for the RCBD example The matrices in the expressions for are known.

* * * *R G 2 2 2 B 2 2 2 BU 2 2

* * * *BU B G G* * * *B BU G G* * * *G G BU B* * * *G G B BU

V J I J I J I I I

* * * *R G 2 2 B 2 2 BU 2 2

* * * * * * *G B BU G B G G

* * * * * * *G B G B BU G G

* * * * * * *G G G B BU G B* * * * * * *G G G B G B BU

V J J I J I I

* * * *1 1 1 1 1R G 2 2 B 2 2 2 BU 2 2 2 22 2 2 2 2V J J I J J I J I J

*RV

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3. Randomization analysis for a single randomization

Estimation and hypothesis testing based on the randomization distribution. Will focus on hypothesis testing.

Propose to use I-MINQUE to estimate the ys and use these estimates to estimate t via EGLS.

I-MINQUE yields the same estimates as REML, but without the need to assume a distributional form for the response.

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Test statistics Have a set R of idempotents specifying a treatment

decomposition. For an R R, to test H0: RXht = 0, use a Wald F, a Wald

test statistic divided by its numerator df:

1 1 1( ) { ( ) ( ) } ( )Wald h h h h h hF traceRX RX X V X RX RX R Numerator is a quadratic form: (est)’ (var(est))-1 (est). For an orthogonal design, FWald is the same as the F from an ANOVA.

Otherwise, it is a combined F test statistic. For nonorthogonal designs, an alternative test statistic is an

intrablock F-statistic. For a single randomization, let QH be the matrix for H that projects

on the eigenspace of V from which RXht is to be estimated. Then and var .ˆ

H H HH

h H h Q R Q Q RQ Q

RX RQ Y RX

The intrablock ˆ' .H HH H traF ce Q R QRQ RY RQ Y

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Randomization distribution of the test statistic To obtain it:

Apply, to the unit factors and y, but not the treatment factors, all allowable unit permutations for the design employed: effects a rerandomization of the treatments;

Compute the test statistic for each allowable permutation; This set of values is the required distribution.

Number of allowable permutations. For our RCBD, there are 8 permutations and so computing the 8

test statistics is easy. For b = 10 and k = 3, there are 1.4 x 1035 — not so easy. An alternative is random data permutation (Edgington, 1995): take a

Monte Carlo sample of the permutations.

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Null distribution of the test statistic under normality

Under normality of the response, the null distribution of FWald is: for orthogonal designs, an exact F-distribution; for nonorthogonal designs, an F-distribution

asymptotically. Under normality of the response, the null

distribution of an intrablock F-statistic is an exact F-distribution.

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Wheat experiment in a BIBD (Joshi, 1987)

Six varieties of wheat are assigned to plots arranged in 10 blocks of 3 plots.

The efficiency factor for this design is 0.80. The ANOVA with the intrablock F and p:

plots tier treatments tier

source d.f. source d.f. MS F p-value

Blocks 9 Varieties 5 39.32 0.58 0.718

Residual 4 67.59 1.17

Plots[B] 20 Varieties 5 231.29 4.02 0.016

Residual 15 57.53

FWald = 3.05 with p = 0.035 (n1 = 5, n2 = 19.1).

Estimates: yB = 14.60 (p = 0.403); yBP = 58.28.

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Test statistic distributions 50,000 randomly selected permutations of blocks

and plots within blocks selected.

Intrablock F-statistic Combined F-statistic

Peak on RHS is all values 10.

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Combined F-statistic

Part of the discrepancy between F- and the randomization distributions is that combined F-statistic is only asymptotically distributed as an F. Differs from Kenward & Rogers (1997) & Schaalje et al (2002) for

nonorthogonal designs.

Randomization distribution Parametric bootstrap

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Two other examples Rabbit experiment using the same BIBD

(Hinkelmann & Kempthorne, 2008). 6 Diets assigned to 10 Litters, each with 3 Rabbits. Estimates: yL = 21.70 (p = 0.002), yLR = 10.08.

Casuarina experiment in a latinized row-column design (Williams et al., 2002). 4 Blocks of 60 provenances arranged in 6 rows by 10

columns. Provenances grouped according to 18 Countries of

origin. 2 Inoculation dates each applied to 2 of the blocks. Estimates: yC = 0.2710; yB, yBR , yBC < 0.06;

yBRC = 0.2711.

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ANOVA for Casuarina experiment

Provenance represents provenance differences within countries.

plots tier treatments tier

source d.f. source d.f. Eff. MS F p-value

Blocks 3 Innoculation 1 11.5411.46

0.077

Residual 2 1.011.17

Columns 9 Country 9 7.25

Rows[B] 20 Country 17 0.90

Provenance 3 0.43

B#C 27 Country 17 0.69

Provenance 10 0.48

R#C[B] 176 Country 170.761

2.4610.25

<0.001

Provenance 410.685

0.291.22

0.235

I#C 170.681

0.130.54

0.917

I#P 410.522

0.150.63

0.938

Residual 60 0.24

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Comparison of p-values

For intrablock F, p-values from F and randomization distributions generally agree.

For FWald, p-values from F-distribution generally underestimates that from randomization distribution: (Rabbit Diets an exception – little interblock contribution).

Example Source Intrablock F FWald (Combined)

n2 F-distri-bution

Randomiz-ation

n2 F-distri-bution

Randomiz-ation

Wheat Varieties 15 0.016 0.012 19.1 0.035 0.096

Rabbit Diets 15 0.038 0.038 16.0 0.032 0.034

Tree Country 60 <0.001 <0.001 79.3 <0.001 0.008

Provenance 60 0.235 0.238 79.0 0.338 0.454

Innoc#C 60 0.917 0.918 84.8 0.963 0.976

Innoc#P 60 0.938 0.938 81.1 0.943 0.966

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4. Randomization model for a chain of randomizations

A chain of two randomizations consists of: the randomization of treatments to the first set of units; The randomization of the first set of units to a second set of units.

For example, a two-phase sensory experiment (Brien & Payne, 1999; Brien & Bailey , 2006, Example 15) involves two randomizations: Field phase: 8 treatments to 48 halfplots using split-plot with 2

Youden squares for main plots. Sensory phase: 48 halfplots randomized 576 evaluations, using

Latin squares and an extended Youden square.

2 Occasions3 Intervals in O6 Judges4 Sittings in O, I4 Positions in O, I, S, J

576 evaluations48 halfplots

2 Squares3 Rows4 Columns in Q2 Halfplots in Q, R, C

8 treatments

4 Trellis2 Methods

(Q = Squares)

Three sets of objects: treatments (G), halfplots () & evaluations (W).

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Randomization model

Additive model of constants:y = z + Xfw + XfXht

where y is the vector of observed responses; z is the vector of constants representing the contributions of each

unit in the 2nd randomization (w W) to the response; w is the vector of constants representing the contributions of each

unit in the 1st randomization (u ) to the response; and t is a vector of treatment constants; Xf & Xh are design matrices showing the randomization

assignments. Under the two randomizations, each element of z and of w

become random variables, as does each element of y.

Y = Z + XfW + XfXht where Y, Z and W are the vectors of random variables. Now, we assume ER[Z] = ER[W] = 0 and so ER[Y] = XfXht .

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Randomization model (cont’d) Further, R

.

H H H HH H

H H H HH H

H H H HH H

V C C

A B

T S

P Q

H H

H H

H H

CW & C are the contributions to the variance arising from W and , respectively.

HW & H are the sets of generalized factors (terms) derived from the posets of factors on W and ;

are the covariances; are the canonical component of excess covariance; are the eigenvalues of CW and C, respectively; are known matrices.

,H H

,H H

,H H

, , , , ,H H H H H H A B T S P Q

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Forming the null randomization distribution of the response

Under the assumption of no treatment effects,

Y* = Z + XfW + m*1. There are two randomizations, G to and to W;

to effect G to , and H are permuted, and

to effect to W, W and HW are permuted.

However, in this model Xf is fixed and reflects the actual randomization employed in the experiment.

Hence, we do not apply the second randomization and consider the null randomization distribution, conditional on the observed randomization of to W.1) Apply the permutations of to H, HW and y, to effect a rerandomization of

G to .o must also be applied to HW so that it does not effect a rerandomization of to W.

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5. Randomization analysis for a chain of randomizations

Again, based on the randomization distribution of the response.

Use the same test statistics as for a single randomization: FWald and intrablock F-statistics.

Obtain or estimate the randomization distributions of these test statistics Based on randomization of G to and is conditional on

the observed randomization of to W.

A Two-Phase Sensory Experiment (Brien & Bailey, 2006, Example 15)

Involves two randomizations:

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(Brien & Payne, 1999)

2 Occasions3 Intervals in O6 Judges4 Sittings in O, I4 Positions in O, I, S, J

576 evaluations48 halfplots

2 Squares3 Rows4 Columns in Q2 Halfplots in Q, R, C

8 treatments

4 Trellis2 Methods

(Q = Squares)

The randomization distribution will be based on the randomization of treatments to halfplots and is conditional on the actual randomization of halfplots to evaluations. Permuting evaluations and y will almost certainly result in unobserved

combinations of halfplots and evaluations, so that the randomization model is no longer valid.

ANOVA table for sensory exp't

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evaluations tier

source df

Occasions 1

Judges 5

O#J 5

Intervals[O] 4

I#J[O] 20

Sittings[OI] 18

S#J[OI] 90

Positions[OISJ] 432

treatments tier

eff source df

1/27 Trellis 3

Residual 3

2/27 Trellis 3

Residual 3

8/9 Trellis 3

Residual 9

Method 1

T#M 3

Residual 20

Intrablock Trellis

Orthogonalsources

halfplots tier

eff source df

Squares 1

Rows 2

Q#R 2

Residual 16

1/3 Columns[Q] 6

Residual 12

2/3 Columns[Q] 6

R#C[Q] 12

Residual 72

Halfplots[RCQ] 24

Residual 408

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Comparison of p-values

Note the difference in denominator df for Trellis.

Source Intrablock F FWald (Combined)

n2 F-distribution

Randomiz-ation

n2 F-distribution

Randomiz-ation

Trellis 9 0.001 0.004 14.9 <0.001 0.004

Method 20 0.627 0.626

Trellis#Method 20 0.009 0.005

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F = 5.10pF = 0.009pR = 0.005

F = 0.24pF = 0.627pR = 0.626

Fcomb = 25.59pF = <0.001pR = 0.004

Fintra = 13.47pF = 0.001pR = 0.004

Comparison of distributions

Trellis

Method

Trellis

Trellis#Method

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6. Some issues Size of permutations sample A controversy: sometimes pooling Unit-treatment additivity

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Size of permutations sample A study of subsamples of the 50,000 randomly

selected permutations revealed that: the estimates of p-values from samples of 25,000 or

more randomized layouts have a range < 0.005. samples of 5,000 randomized layouts will often be

sufficiently accurate – the estimates of p-valueso around 0.01 or less, exhibit a range < 0.005; o in excess of 0.20, show a range about 0.03;o around 0.05, display a range of 0.01.

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Unit-treatment additivity Cox and Reid (2000) allow random unit-treatment

interaction; Test hypothesis that treatment effects are greater than unit-

treatment interaction. Nelder (1977) suggests the random form is questionable.

The Iowa school allows arbitrary (fixed) unit-treatment interactions. Test difference between the average treatment effects over all units,

which is biased in the presence of unit-treatment interaction. Such a test ignores marginality/hierarchy.

Questions: Which form applies? How to detect unit-treatment interaction? Often impossible, but,

when it is possible, cannot be part of a randomization analysis. Randomization analysis requires unit-treatment additivity.

If not appropriate, use a randomization-based mixed model.

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A controversy Should nonsignificant (??) unit sources of variation

be removed and hence pooled with other unit sources?

The point is that effects hypothesized to occur at the planning stage have not eventuated. A modeller would remove them; Indeed, in mixed-model fitting using REML will have no

option if the fitting process does not converge. Some argue, because in randomization model,

must stay. Seems reasonable if doing randomization inference.

Sometimes-pooling may disrupt power and coverage properties of the analysis (Janky, 2000).

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7. Conclusions Fisher was right:

One should employ meaningful models; Randomization analyses provides a check on parametric analyses.

I am still a modeller, with the randomization-based mixed model as my starting point.

I am happy that, for single-stratum tests, the normal theory test approximates an equivalent randomization test, when one exists.

However, the p-values for combined test-statistics from the F-distribution are questionable: novel that depends on ‘interblock’ components; need to do bootstrap or randomization analysis for FWald when

denominator df for intrablock-F and FWald differ markedly; this has the advantage of avoiding the need to pool nonsignificant

(??) unit sources of variation, although fitting can be challenging.

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References Atiqullah, M. (1963) On the randomization distribution and power of the variance

ratio test. J. Roy. Statist. Soc., Ser. B (Methodological), 25: 334-347. Brien, C.J. & Bailey, R.A. (2006) Multiple randomizations (with discussion). J.

Roy. Statist. Soc., Ser. B (Statistical Methodology), 68: 571-609. Brien, C.J. & Demétrio, C.G.B. (2009) Formulating Mixed Models for

Experiments, Including Longitudinal Experiments." J. Agric. Biol. Environ. Statist., 14: 253-280.

Cox, D.R. & Reid, N. (2000). The theory of the design of experiments. Boca Raton, Chapman & Hall/CRC.

Edgington, E.S. (1995) Randomization tests. New York, Marcel Dekker. Fisher, R.A. (1935, 1960) The Design of Experiments. Edinburgh, Oliver and

Boyd. Hinkelmann, K. & Kempthorne, O. (2008) Design and analysis of experiments.

Vol I. Hoboken, N.J., Wiley-Interscience. Janky, D.G. (2000) Sometimes pooling for analysis of variance hypothesis tests:

A review and study of a split-plot model. The Amer. Statist. 54: 269-279. Joshi, D.D. (1987) Linear estimation and design of experiments. Delhi, New Age

Publishers.

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References (cont’d) Kempthorne, O. (1975) Inference from experiments and randomization. A

Survey of Statistical Design and Linear Models. J. N. Srivastava. Amsterdam., North Holland.

Mead, R., S. G. Gilmour & Mead, A.. (2012). Statistical principles for the design of experiments. Cambridge, Cambridge University Press.

Nelder, J.A. (1965) The analysis of randomized experiments with orthogonal block structure. I. Block structure and the null analysis of variance. Proc. Roy. Soc. Lon., Series A, 283: 147-162.

Nelder, J. A. (1977). A reformulation of linear models (with discussion). J. Roy. Statist. Soc., Ser. A (General), 140: 48-77.

Preece, D.A. (1982) The design and analysis of experiments: what has gone wrong?" Util. Math., 21A: 201-244.

Schaalje, B. G., J. B. McBride, et al. (2002). Adequacy of approximations to distributions of test statistics in complex mixed linear models. J. Agric. Biol, Environ. Stat., 7: 512-524.

Welch, B.L. (1937) On the z-test in randomized blocks and Latin squares. Biometrika, 29: 21-52.

Williams, E.R., Matheson, A.C. & Harwood, C.E. (2002). Experimental design and analysis for tree improvement. Collingwood, Vic., CSIRO Publishing.