unit 10: split-plot designs, repeated measures, and...

Unit 10: Split-Plot Designs,Repeated Measures, and Expected

Mean Squares

STA 643: Advanced Experimental Design

Derek S. Young

1

Learning Objectives

I Become familiar with whole plots and subplots as they pertain tosplit-plot designs

I Know how to analyze a split-plot design

I Understand the different sources of experimental error in split-plotdesigns

I Know how to calculate standard errors for treatment factor meansin a split-plot design

I Become familiar with split-split-plot and split-block designs

I Understand repeated measures designs and their connection tosplit-plot designs

I Know how to handle ranked data in a repeated measures design

I Become familiar with the steps for determining expected meansquares

2

Outline of Topics

1 Split-Plot Designs

2 Split-Split-Plot Designs

3 Split-Block Designs

4 Repeated Measures Designs

5 Handling Ranked Data

6 Expected Mean Squares

3

Outline of Topics







4

Example: Agricultural Experiment

We motivate our discussion by considering an agricultural investigation tostudy the effects of two irrigation methods (factor A) and two fertilizers (factorB) on yield of a crop. Let A1 and A2 be the two levels of factor A and B1 andB2 be the two levels of factor B. Four fields are available as the EUs. The fourtreatments – A1B1, A1B2, A2B1, and A2B2 – are assigned at random to thefour fields in a CRD. Since there are four treatments and four EUs, there willbe no df for estimation of error as shown in the abbreviated ANOVA tablebelow. However, if the fields could be divided into smaller fields, replicates ofeach factor-level combination could be obtained and the error variance couldthen be estimated. In this case, the smaller fields become the MUs.Unfortunately, in this investigation, it is not possible to apply differentirrigation methods (factor A) in areas smaller than a field (the EU), althoughdifferent fertilizer types (factor B) could be applied in relatively small areas.Such a situation is where a split-plot design can be used.

Source of Variation dfFactor A 1Factor B 1

AB Interaction 1Error 0Total 3

5

Background on Split-Plot DesignsI As noted in the motivating example, one factor sometimes requires more

experimental material for its evaluation than a second factor in factorialexperiments.

I In agricultural studies, a factor – such as irrigation method – may only beapplicable (or best-suited) for large plots, while another factor – such asfertilizers – may be applied easily to smaller plots.

I The larger treatment is applied to what we call the whole plots, which is then

split into smaller subplots. Designs that accommodate this allocation of

treatments are called split-plot designs.I The usage of the term plots stems from split-plot designs being developed

for agricultural studies; while still commonly found in agriculture,split-plot designs are also used in laboratory, industrial, and social scienceexperiments.

I The primary advantage of a split-plot design is that it allows us to design anexperiment when one factor requires considerably more experimental materialthan another factor, or accommodate the situation where there is an opportunityto study responses to a second factor while efficiently utilizing resources.

I The primary disadvantage of a split-plot design is the potential loss in treatmentcomparisons and an increase in complexity of the statistical analysis (which wewill see).

6

Randomization for Split-Plot Designs

I In split-plot designs, the usual randomization of treatmentcombinations to the EUs is altered to accommodate theparticular requirements of the experiment.

I In the usual CRD, all treatment combinations are assignedrandomly to the EUs.

I However, in the split-plot design, each level of the secondfactor is combined with a single level of the first factor (i.e.,the factor requiring more experimental material), which hasbeen assigned to the whole plot.

I In other words, at the first level of randomization, the wholeplot treatments are randomly assigned to whole plots; at thesecond level, the subplot treatments are randomly assigned tothe subplots.

7

Example: Agricultural Experiment

As we noted in the agricultural experiment, the irrigation methods(factor A) can only be applied to one field as a whole. In thefigure below, we see that the four fields have been randomlyassigned to one of the two factor levels of A. Within each field(the whole plot), it is divided into two subplots. Each of thesesubplots is then randomly assigned to one of the two levels offactor B. Through this split-plot design, we have increased ourtotal number of EUs N from 4 to 8.

Field&1 Field&2 Field&3 Field&4

B1

B1

Whole&Plot&Factor&A

Subplot&Factor&B

A2 A1 A1 A2

B1

B2

B2

B2

B2

B1

8

Two Experimental Errors for Split-Plot Designs

I In the analysis of split-plot designs, we must account for thepresence of two different sizes of EUs used to test the effect of thewhole plot treatment and subplot treatment.

I Factor A effects are estimated using the whole plots and factor Band the A×B interaction effects are estimated using the subplots.

I Since the size of whole plot and subplots are different, they havedifferent precisions.

I Moreover, two separate errors account for the fact that observationsfrom different subplots within each whole plot may be positivelycorrelated, since it is natural to assume that EUs adjacent to oneanother are likely to respond similarly.

I For example, consider neighboring field plots, students in aclassroom, or the batch of some raw material in an industrialexperiment.

9

Two Experimental Errors for Split-Plot Designs

I We assume that

I there is a correlation ρ between observations on any two subplots inthe same whole plot, and

I observations from two different whole plots are uncorrelated.

I Given the above assumptions and assuming that there are b subplots for

each whole plot, then

I the error variance for the main effects of A on a per-subplot basis isσ2[1 + (b− 1)ρ], and

I the error variance per subplot for the main effects of B and theinteraction A×B is σ2(1− ρ).

I These different error variances result in a different SS partitioning for theANOVA.

I The partitions for blocking and factor effects remain the same as for afactorial design, but the experimental error is partitioned into twocomponents: one for the whole-plot treatment factor and one for thesubplot treatment factor and interaction.

10

Split-Plot Model

I For the split-plot design, a mixed-model formulation is used with separaterandom error effects for the whole-plot units and subplot units.

I Placing the whole-plot treatment in an RCBD, a linear model for the split-plotdesign is

Yijk = µ+ αi + ρk + dik + βj + (αβ)ij + εijk, (1)

whereI µ is the overall mean (a constant);I i = 1, . . . , a, j = 1, . . . , b, and k = 1, . . . , n;I αi are the (fixed) effects of factor A subject to the constraint

∑ai=1 αi = 0;

I ρk are the (fixed) effects of block k subject to the constraint∑nk=1 ρk = 0;

I dik are the (random) effects for the whole-plot random error and are iid normal with mean 0 and

variance σ2d;

I βj are the (fixed) effects of factor B subject to the constraint∑bj=1 βj = 0;

I (αβ)ij are the (fixed) effects due to the interaction of A and B subject to the constraint∑ai=1(αβ)ij =

∑bj=1(αβ)ij = 0;

I εijk is the subplot (random) error and are iid normal with mean 0 and variance σ2; andI dik and εijk are assumed independent of one another.

I Note that randomization of treatments is what justifies the assumption ofindependence for the two random errors and the equal correlation between theerrors for subplot units on the same whole-plot unit.

11

SS Decomposition

I The total deviation for the split-plot model on the previousslide is

total deviation: (yijk − y···),

which is the sum of the following components:

block deviation: (y··k − y···)

factor A deviation: (yi·· − y···)

whole-plot error: (yi·k − yi·· − y··k + y···)

factor B deviation: (y·j· − y···)

interaction AB deviation: (yij· − yi·· − y·j· + y···)

subplot error: (yijk − yij· − yi·k + yi··)

12

SS Decomposition

I The SS of the decomposition on the previous slide (where the sums ofcross-products drop out) yields

Block effect: =

Factor A effect: =

Whole-plot error: =

Factor B effect: =

Interaction AB effect: =

Subplot error: =

13

ANOVA for Split-Plot Designs

I Finally, we can obtain the following ANOVA table:Source df SS MS E(MS)Whole Plots

Block (n− 1) SSBlk MSBlk σ2 + bσ2d + ab

∑k ρ

2k

n−1

A (a− 1) SSA MSA σ2 + bσ2d + nb

∑i α

2i

a−1

Whole-Plot Error (a− 1)(n− 1) SSWP MSWP σ2 + bσ2d

Subplots

B (b− 1) SSB MSB σ2 + na

∑j β

2j

b−1

AB (a− 1)(b− 1) SSAB MSAB σ2 + n

∑i∑j(αβ)

2ij

(a−1)(b−1)

Subplot Error a(b− 1)(n− 1) SSSP MSSP σ2

Total abn− 1 SSTot

I The E(MS) for the whole-plot error and the subplot errorreflect the differences in the variability for the two differenttypes of EUs – namely, the expected error variances for thewhole plots are larger than those for the subplots.

14

Tests of Hypotheses About Factor EffectsI First we test for interactions with the following hypotheses:

H0 : (αβ)ij = 0 for all i, j

HA : (αβ)ij 6= 0 for some i, j,

which has test statistic

F∗

=

I Next we test for factor B main effects with the following hypotheses:

H0 : β1 = β2 = · · · = βb = 0

HA : at least one βj is different,


F∗

=

I Finally we test for factor A main effects with the following hypotheses:

H0 : α1 = α2 = · · · = αa = 0

HA : at least one αi is different,


F∗

=

15

Computational Notes

I Split-plot ANOVA calculations are standard with most statisticalcomputing packages.

I The instructions for the analysis are similar to those for a two-factorfactorial treatment design, but for the split-plot design you need tocompute two separate error terms.

I The SSWP is numerically equivalent to

I the error SS for the experimental design utilized for the wholeplots;

I the SS for whole plots within factor A treatments in a CRD;I The SS computation for blocks by factor interaction in a

RCBD.

I The SSSP is ordinarily the residual SS computed automatically bythe statistical program.

16

Example: Turfgrass ExperimentConsider the general set-up we introduced for the agricultural experiment at thebeginning of the lecture. A soil scientist wants to investigate the effects of nitrogensupplied in different chemical forms and evaluate those effects combined with those ofthatch accumulation on the quality of turfgrass. Four forms of nitrogen fertilizer areused: (1) urea, (2) ammonium sulphate, (3) isobutylidene diurea (IBDU), and (4)sulphur-coated urea, (urea (SC)). Any differences in response to the fertilizers thencould be attributed to the mode of nitrogen release because an equivalent amount ofnitrogen was supplied by each form. A golf green was constructed and seeded withPenncross creeping bent grass on the experimental plots. The nitrogen treatmentplots were arranged on the golf green in an RCBD with two replications. After twoyears, the second treatment factor – years of thatch accumulation – was added to theexperiment. Each of the eight experimental plots was split into three subplots towhich levels of the second treatment factor were randomly assigned. The lengths oftime the thatch was allowed to accumulate on the subplot were 2, 5, or 8 years. Thus,the split plot design had the whole-plot treatment factor of nitrogen source (A) in anRCBD with years of thatch accumulation (B) as the subplot treatment factor. Thedata for this experiment are presented on the next slide.

17

Example: Turfgrass Experiment

NitrogenBlock

YearsSource 2 5 8

Urea1 3.8 5.3 5.92 3.9 5.4 4.3

Ammonium Sulphate1 5.2 5.6 5.42 6.0 6.1 6.2

IBDU1 6.0 5.6 7.82 7.0 6.4 7.8

Urea (SC)1 6.8 8.6 8.52 7.9 8.6 8.4

*The measurements made on the turfgrass plots (which are reported in thetable above) is the chlorophyll content of clippings (mg/g).

18


Error: Field

Df Sum Sq Mean Sq F value Pr(>F)

Block 1 0.51 0.510 1.217 0.3505

Nitrogen 3 37.32 12.442 29.672 0.0099 **

Residuals 3 1.26 0.419

---

Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

Error: Within


Years 2 3.816 1.9079 8.891 0.00927 **

Nitrogen:Years 6 4.154 0.6924 3.227 0.06460 .

Residuals 8 1.717 0.2146

---

Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

I The test for interaction between nitrogen and thatch is F∗ = 0.692/0.215 = 3.227, which yields a ap-value of 0.065; therefore, the interaction is moderately significant at the 0.05 level.

I The test for differences among thatch means is F∗ = 1.908/0.215 = 8.891, which yields a a p-valueof 0.009; therefore, the differences are significant at the 0.05 level.

I The test for differences among the whole-plot nitrogen treatment means isF∗ = 12.442/0.419 = 26.672, which yields a a p-value of 0.010; therefore, the differences aresignificant at the 0.05 level.

I Notice in the output above how the sources of variation are partitioned based on the error source that theyare tested against; i.e., Field and Within.

19


Nitrogen Years NitrogenSource 2 5 8 Means

Urea 3.85 5.35 5.10 4.77Ammonium Sulphate 5.60 5.85 5.80 5.75

IBDU 6.50 6.00 7.80 6.77Urea (SC) 7.35 8.60 8.45 8.13

Thatch Means 5.83 6.45 6.79

I The above table gives the cell means (i.e., the mean of the two blocks forthe given treatment combination) and the marginal means of nitrogenand years of thatch accumulation.

I Turf that received urea (SC) had the highest chlorophyll content followed

by IBDU, ammonium sulphate, and finally urea.

I This relative ranking of chlorophyll content was the same for eachyear of thatch accumulation.

20


45

67

8

Factor Plot for Grass Experiment

Years of Thatch Accumulation

Mea

n C

hlor

ophy

ll C

onte

nt

2 5 8

Nitrogen Source

urea_SCIBDUammsulphurea

I Above is a factor plot of the mean chlorophyll content versus the years ofthatch accumulation, separated by nitrogen source.

I The marginal means in the table on the previous slide indicate an increasein chlorophyll content as the years of thatch accumulation increase;however, there is not an increase in the cell means for each nitrogensource as the years of thatch accumulation increase.

21

Standard Errors for Treatment Factor Means

I We next present the standard errors for tests of hypotheses forcomparisons among the estimated treatment means.

I The df associated with each of the standard errors are the dffor the MS that is used in the standard error calculation,except for the difference between two factor A means at thesame or different level of factor B.

I The df for this latter setting is obtained using theSatterthwaite approximation as follows:

f3 =[(b− 1)MSSP + MSWP]

2

[(b−1)MSSP]2

f2+ [MSWP]2

f1

,

where f1 = (a− 1)(n− 1) and f2 = a(b− 1)(n− 1) are the dffor MSWP and MSSP, respectively.

22

Standard Errors for Treatment Factor Means

Treatment Comparison SE Estimator df

Difference between two A means

yu·· − yv··

√2MSWP

nb f1

Difference between two B means

y·u· − y·v·

√2MSSPna f2

Difference between two B means at same A level

yju· − yjv·

√2MSSPn f2

Difference between two A means at same or different B levels

yuk· − yvk· or √2[(b−1)MSSP+MSWP]

nbf3yuk· − yvm·

23

Example: Turfgrass ExperimentUsing the standard error formulas on the previous slide, we can calculate them for the turfgrass experiment:

I two nitrogen means: √2MSWP

nb=

√2(0.419)

6= 0.374 df = 3

I two thatch means: √2MSSP

na=

√2(0.215)

8= 0.232 df = 8

I two thatch means for the same nitrogen level:√2MSSP

n=

√2(0.215)

1= 0.464 df = 8

I two nitrogen means at same or different thatch levels:√2 [(b− 1)MSSP + MSWP]

nb=

√2[2(0.215) + 0.419]

6= 0.532,

with df:[2(0.215) + 0.419]2

[2(0.215)]2

8+

[0.42]2

3

= 8.830 (or 9)

24

Efficiency of Split-Plot Designs

I We can also calculate the efficiency of the split-plot designrelative to the RCBD for each of the subplot and whole-plotcomparisons.

I These efficiencies are

REsub = Ja(b− 1)MSSP + (a− 1)MSWP

(ab− 1)MSSP

REwhole = Ka(b− 1)MSSP + (a− 1)MSWP

(ab− 1)MSWP

respectively, where

J =(a(b− 1)(r − 1) + 1)((ab− 1)(r − 1) + 3)

(a(b− 1)(r − 1) + 3)((ab− 1)(r − 1) + 1)

K =((a− 1)(r − 1) + 1)((ab− 1)(r − 1) + 3)

((a− 1)(r − 1) + 3)((ab− 1)(r − 1) + 1).

25

Replicates

I There are alternative forms of split-plot models, just like wewould have with other designs.

I Another common model is one where blocking is not present,but rather the blocking term is replaced by a random effectdue to replicates.

I On the next slide is a model for such a design, but there are acouple of notes regarding the model:

I No df will be left to estimate a variance component for εijk;i.e., no additional index l is available to include in the subscriptof εijk.

I We break our convention of using Roman letters for randomeffects because the “bookkeeping” of the notation for theindices and the effects becomes quite cumbersome!

26

Split-Plot Design with Replicates

I The linear model is

Yijk = µ+ γi + αj + (γα)ij + βk + (γβ)ik + (αβ)jk + (γαβ)ijk + εijk, (2)

whereI µ is the overall mean (a constant)I i = 1, . . . , r, j = 1, . . . , a, and k = 1, . . . , bI γi are the (random) effects due to replicates and are iid normal with mean 0 and variance σ2

γI αj are the (fixed) effects of factor A subject to the constraint

∑aj=1 αj = 0

I (γα)ij are the (random) effects for the whole-plot random error and are iid normal with mean 0

and variance σ2γα

I βk are the (fixed) effects of factor B subject to the constraint∑bk=1 βj = 0

I (γβ)ik are the (random) effects for the replicates by factor B interaction and are iid normal with

mean 0 and variance σ2γβ

I (αβ)jk are the (fixed) effects due to the interaction of A and B subject to the constraint∑aj=1 αjβk =

∑bk=1 αjβk = 0

I (γαβ)ijk are the (random) effects due to the subplot error and are iid normal with mean 0 and

variance σ2γαβ

I εijk is the (random) error term and are iid normal with mean 0 and variance σ2

I γi, (γα)ij , (γβ)ik , (γαβ)ijk , and εijk are all assumed independent of one another

27

Expected Mean Squares

I The E(MS) for the model in (2) are as follows:

Model Term E(MS)Whole Plotsγi σ2 + abσ2

γ

αj σ2 + abσ2γα +

rb∑j α

2j

a−1

(γα)ij σ2 + abσ2γα

Subplots

βk σ2 + aσ2γβ +

ra∑k β

2k

b−1

(γβ)ik σ2 + aσ2γβ

(αβ)jk σ2 + σ2γαβ +

r∑j

∑k(αβ)2

jk

(a−1)(b−1)

(γαβ)ijk σ2 + σ2γαβ

εijkl σ2 (not estimable)

28

Outline of Topics







29

Split-Split-Plot Designs

I The restriction on randomization mentioned in the split-plotdesign can be extended to more than one factor.

I The convenience of introducing a third factor into thetreatment designs requires a subdivision of the subplots suchthat all levels of the third factor are administered to these newsubdivisions, called sub-subplots.

I The design is called a split-split-plot design and has threedifferent sizes (or types) of EUs.

I The ANOVA follows the same logic as that for the split-plotdesign, but now we have a third factor (C) at c levels, whichresults in additional sources of variation (and hence differentlayers) in the ANOVA table.

30

Split-Split-Plot Design Model

I Placing the whole-plot treatment in an RCBD, a linear model for thesplit-split-plot design is

Yijkl = µ+ αi + ρl + dil + βj + (αβ)ij + fijl

+ γk + (αγ)ik + (βγ)jk + (αβγ)ijk + εijkl(3)

whereI µ is the overall mean (a constant);I i = 1, . . . , a, j = 1, . . . , b, k = 1, . . . , c, and l = 1, . . . , n;I αi, βj , and γk are the (fixed) effects of factors A, B, and C, respectively, while ρl are the (fixed)

effects of block l, all of which are subject to the constraint that they some to 0 over theirrespective indices;

I (αβ)ij , (αγ)ik , (βγ)jk , and (αβγ)ijk are the interaction terms subject to the usualconstraints;

I dil are the (random) effects for the whole-plot random error and are iid normal with mean 0 and

variance σ2d;

I fijl are the (random) effects for the subplot random error and are iid normal with mean 0 and

variance σ2f ;

I εijkl is the sub-subplot (random) error and are iid normal with mean 0 and variance σ2; andI dil, fijl, and εijkl are assumed independent of one another.

I Since the SS calculations are quite lengthy, we merely present an abbreviatedANOVA table on the following slide.

31

ANOVA for Split-Split-Plot Designs

Source df SS MSWhole PlotsBlock (n− 1) SSBlk MSBlkA (a− 1) SSA MSAWhole-Plot Error (a− 1)(n− 1) SSWP MSWPSubplotsB (b− 1) SSB MSBAB (a− 1)(b− 1) SSAB MSABSubplot Error a(b− 1)(n− 1) SSSP MSSPSub-subplotsC (b− 1) SSC MSCAC (a− 1)(c− 1) SSAC MSACBC (b− 1)(c− 1) SSBC MSBCABC (a− 1)(b− 1)(c− 1) SSABC MSABCSub-subplot Error ab(c− 1)(n− 1) SSSSP MSSSPTotal abcn− 1 SSTot

32

Example: Antibiotic Absorption TimeWe do not perform a full data analysis of a split-split-plot designed experiment, butsimply discuss an example where one has been used. A researcher is interested instudying the effect of technicians (A), dosage strength (B), and capsule wall thickness(C) on absorption time of a particular type of antibiotic. Three technicians, threedosage strengths, and four capsule wall thicknesses results in N = 36 observations perreplicate needed, but the experimenter wants to perform four replicates on differentdays. To do so, first, technicians are randomly assigned to units of antibiotics, whichare the whole plots. Next, the three dosage strengths are randomly assigned tosplit-plots. Finally, for each dosage strength, the capsules are created with differentwall thicknesses, which is the split-split factor and then tested in random order. Noticethe restrictions that exist on randomization. We cannot simply randomize the 36 runsin a single block (or replicate), because technician is a hard-to-change factor. Next,after selecting a level for this hard-to-change factor (say technician 2 is selected) wecannot randomize the 12 runs under this technician because dosage strength is ahard-to-change factor. After we select a random level for this second factor, saydosage strength of level 3, we can then randomize the four runs under thesecombinations of two factors and randomly run the experiments for different wallthicknesses (third factor).

33

Example: Antibiotic Absorption Time

Step 1: Lay out the EUs (antibiotic units)

Unit%1 Unit%2 Unit%3

34


Step 2: Randomly assign whole plot factor A (technicians)


Technician%3 Technician%1 Technician%2

Whole%Plot%Factor%A

35


Step 3: Randomly assign subplot factor B (dosage)

Dose%2

Dose%3

Dose%3

Dose%1

Dose%1


Technician%3 Technician%1 Technician%2

Subplot%Factor%B

Dose%2

Dose%3

Dose%1 Dose%2

36


Step 4: Randomly assign sub-subplot factor C (thickness)

Dose%1,%Thickness%3

Dose%1,%Thickness%2

Dose%1,%Thickness%1

Sub$subplot*Factor*C

Dose%2,%Thickness%2

Dose%3,%Thickness%2

Dose%3,%Thickness%4

Dose%3,%Thickness%3

Dose%3,%Thickness%1

Dose%3,%Thickness%3

Dose%3,%Thickness%1

Dose%3,%Thickness%2

Dose%3,%Thickness%4

Dose%1,%Thickness%3

Dose%2,%Thickness%4

Dose%2,%Thickness%3

Dose%2,%Thickness%1

Dose%2,%Thickness%2

Dose%3,%Thickness%3

Dose%3,%Thickness%1

Dose%3,%Thickness%2

Dose%3,%Thickness%4

Dose%1,%Thickness%2

Dose%1,%Thickness%4

Dose%1,%Thickness%3

Dose%1,%Thickness%1

Dose%2,%Thickness%4

Dose%2,%Thickness%3

Dose%1,%Thickness%2

Dose%1,%Thickness%4

Dose%1,%Thickness%1

Technician*3 Technician*1 Technician*2

Dose%2,%Thickness%1

Dose%2,%Thickness%2

Dose%2,%Thickness%4

Dose%2,%Thickness%3

Dose%2,%Thickness%1

Dose%1,%Thickness%4

Unit*1 Unit*2 Unit*3

37

Remarks on Split-Split-Plot Designs

I Note that all of the standard errors applied to the A and Beffects in the split-plot design are valid in the split-split-plotdesign, and need only to have the value of c included in thedivisor.

I For example, the standard error of the difference between twofactor A means, yu··· − yv···, is

√2MSWP/nbc.

I Those comparisons involving factor C effects are given on thenext slide.

38

Treatment Comparisons Involving Factor C

Treatment Comparison SE Estimator

Difference between two C means

y··u· − y··v·√

2MSSSPnab

Difference between two C means at same A level

yi·u· − yi·v·√

2MSSSPnb

Difference between two C means at same B level

y·iu· − y·iv·√

2MSSSPna

Difference between two C means at same A and B levels

yiju· − yijv·√

2MSSSPn

Difference between two B means at same or different C levelsy·ui· − y·vi· or √

2[(c−1)MSSSP+MSSP]nbcy·ui· − y·vj·

Difference between two B means at same A and C levels

yiuj· − yivj· or

√2[(c−1)MSSSP+MSSP]

nc

Difference between two A means at same or different C levelsyu·i· − yv·i· or √

2[(c−1)MSSSP+MSWP]nbcyu·i· − yv·j·

Difference between two A means at same or different B and C levelse.g., yuij· − yvij· or √

2[b(c−1)MSSSP+(b−1)MSSP+MSWP ]nbcyuij· − yvlk·

39

Replicates

I There are alternative forms of split-split-plot models, just likewe showed with the split-plot design.

I We can, again, replace the blocking term by a random effectdue to replicates.

I On the next slide is a linear model for such a design, followedby a table of the expected mean squares.

I Note that, again, no df will be left to estimate a variancecomponent for εijkh; i.e., no additional index l is available toinclude in the subscript of εijkh.

40

Split-Split-Plot Design with ReplicatesI The linear model is

Yijkh = µ+ τi + βj + (τβ)ij + γk + (τγ)ik + (βγ)jk + (τβγ)ijk

+ δh + (τδ)ih + (βδ)jh + (τβδ)ijh + (γδ)kh + (τγδ)ikh

+ (βγδ)jkh + (τβγδ)ijkh + εijkh,

(4)

whereI µ is the overall mean (a constant)I i = 1, . . . , r, j = 1, . . . , a, k = 1, . . . , b, and l = 1, . . . , cI βj , γk, and δh are the (fixed) effects of factors A (whole plot), B

(subplot), and C (sub-subplot)I (βγ)jk, (βδ)jh, (γδ)kh, and (βγδ)jkh are the (fixed) interaction terms,

subject to the usual constraintsI τi are the (random) effects due to replicates and are iid normal with

mean 0 and variance σ2

I (τβ)ij , (τγ)ik, (τβγ)ijk, (τδ)ih, (τβδ)ijh, (τγδ)ikh, and (τβγδ)ijkhare all iid normal with mean 0 and variances, respectively, σ2

τβ , σ2τγ ,

σ2τβγ , σ2

τδ, σ2τβδ , σ2

τγδ, and σ2τβγδ

I εijkh is the (random) error term and are iid normal with mean 0 andvariance σ2

I All random error terms all assumed independent of one another

41

Expected Mean Squares

Model Term E(MS)Whole Plots

τi σ2 + abcσ2τ

βj σ2 + bcσ2τβ +

rbc∑j β

2j

(a−1)

(τβ)ij σ2 + bcσ2τβ

Subplots

γk σ2 + acσ2τγ +

rac∑k γ

2k

(b−1)

(τγ)ik σ2 + acσ2τγ

(βγ)jk σ2 + cσ2τβγ +

rc∑j∑h(βγ)2jh

(a−1)(b−1)

(τβγ)ijk σ2 + cσ2τβγ

Sub-subplots

δh σ2 + abσ2τδ +

rab∑k δ

2k

(c−1)

(τδ)ih σ2 + abσ2τδ

(βδ)jh σ2 + bσ2τβδ +

rb∑j∑h(βδ)2jh

(a−1)(c−1)

(τβδ)ijh σ2 + bσ2τβδ

(γδ)kh σ2 + aσ2τγδ +

ra∑k

∑h(γδ)2kh

(b−1)(c−1)

(τγδ)ikh σ2 + aσ2τγδ

(βγδ)jkh σ2 + aσ2τβγδ +

r∑i∑j∑k(βγδ)2ijk

(a−1)(b−1)(c−1)

(τβδγ)ijkh σ2 + aσ2τβγδ

εl(ijkh) σ2 (not estimable)

42

Outline of Topics







43

Split-Blocks

I The split-block design (also called strip-plot design orstrip-split-plot design) is used when the subunit treatments occurin a strip across the whole-plot units.

I The notion of “strips” are really just another set of whole plots.

I The split-block design can be useful in agricultural field studieswhen two treatment factors require the use of large field plots.

I The levels of one treatment factor A are randomly assigned tothe plots in an RCBD.

I The plots for the second factor B are constructed in the samemanner, but are laid out perpendicular to the plots for factor A.

I The levels of the second factor B are then randomly assignedto this second array of plots across the same block.

I A schematic showing this design is given on the following slide.

44

Split-Block Design Schematic

A1B3 A2B3

Subplot(Interaction(AB

A3 A1 A2

A3B1 A1B1 A2B1

A3B2 A1B2 A2B2

A3B3

B1 B1 B1

B2 B2 B2

B3 B3 B3

Whole(Plot(Factor(A

A3 A1 A2

B1

B2

B3

Strip4Plot(Factor(B

A3 A1 A2

A3 A1 A2

45

Split-Block DesignI We can begin by placing this design into an RCBD and construct the linear

model as follows:

Yijkl = µ+ ρk + αi + dik + βj + gjk + (αβ)ij + εijk (5)

whereI µ is the overall mean (a constant)I i = 1, . . . , a, j = 1, . . . , b, and k = 1, . . . , nI αi, and βj are the (fixed) effects of factors A (whole plot) and B (strip

plot), respectively, while ρk are the (fixed) effects of block k, all of whichare subject to the constraint that they some to 0 over their respectiveindices

I (αβ)ij are the interaction terms subject to the usual constraintsI dik are the (random) effects for the whole-plot random error and are iid

normal with mean 0 and variance σ2d

I gjk are the (random) effects for the strip-plot random error and are iidnormal with mean 0 and variance σ2

gI εijk is the subplot (random) error and are iid normal with mean 0 and

variance σ2

I dik, gjk, and εijk are assumed independent of one another

46

Abbreviated ANOVA for Split-Block Designs

Source df SS MSWhole PlotsBlock (n− 1) SSBlk MSBlkA (a− 1) SSA MSAWhole-Plot Error (a− 1)(n− 1) SSWP MSWP

Strip PlotsB (b− 1) SSB MSBStrip-plot Error (b− 1)(n− 1) SSStP MSStP

SubplotsAB (a− 1)(b− 1) SSAB MSABSubplot Error (a− 1)(b− 1)(n− 1) SSSP MSSP

Total abn− 1 SSTot

47

Outline of Topics







48

Overview of Repeated Measures Designs

I Repeated measures designs are used when it is desired to take repeatedmeasurements of the response variable on each EU.

I Repeated measures designs utilize the same subject (e.g., person, store,

plant, animal, etc.) for each of the treatments in the study.I Therefore, the subject serves as a block and the EU within a block

may be viewed as the different occasions when a treatment isapplied to the subject.

I A repeated measures study may involve several treatments or a singletreatment that is evaluated at different points in time.

I Repeated measures on each EU provide information on the time trend of

the response variable under different treatment conditions.I Time trends can reveal how quickly units respond to different

treatment conditions or simply how they are manifested over thecourse of the design.

I Note that the study units (e.g., person, store, plant, animal, etc.) areoften referred to as subjects in repeated measures designs.

49

Examples of Repeated Measures Designs

1 15 test markets are to be used to study each of two differentadvertising campaigns. In each test market, the order of the twocampaigns will be randomized. A sufficient time lapse between thetwo campaigns is allotted so that the effects of the initial campaignwill not carry over into the second campaign. What are the subjectsin this study?

2 200 individuals with persistent migraine headaches are each to begiven two different drugs and a placebo for two weeks each. Theorder of how the drugs are administered is randomized. Who are thesubjects in this study?

3 In a weight loss study, 100 overweight persons are to be given thesame diet (i.e., a single treatment) and their weights measured atthe end of each week for 12 weeks to assess their weight loss overtime. Who are the subjects in this study?

50

Example: Sales of Athletic Shoes

I Suppose a manufacturer ofathletic shoes is interested in theeffect of different advertisingstrategies on shoe sales.

I Four different advertisingcampaigns (T1, T2, T3, T4) areconsidered over the period of fourmonths (one campaign used permonth) in five different testmarkets (i.e., the subjects).

I To the right is a hypotheticallayout for this single-factorrepeated measures design.

1 2 3 4

Subject,1 T4 T3 T2 T1





Treatment,Order

51

Advantages of Repeated Measures Designs

I Repeated measures designs provide good precision forcomparing treatments because all sources of variabilitybetween subjects are excluded from the experimental error.

I Only variation within subjects enters the experimental errorsince any two treatments can be compared directly for eachsubject.

I Moreover, one can view subjects as serving as their owncontrols.

I Repeated measures designs also economize on the selectedsubjects.

I This is particularly important when only a few subjects (e.g.,stores, test markets, or test sites) can be utilized for theexperiment.

I When effects of a treatment over time are of interest, then it isusually desirable to observe the same subject at different timepoints.

52

Disadvantages of Repeated Measures Designs

I Disadvantages of repeated measures designs usually involve various forms ofinterference.

I One type of interference is an order effect, which is connected with the position

in the treatment order.I For example, in evaluating multiple different advertisements, subjects tend

to give higher (or lower) ratings for advertisements shown at the end ofthe sequence than at the beginning.

I Another type of interference is a carryover effect, which is connected with the

effects on the measured response from the previous treatment(s).I For example, in evaluating multiple different soup recipes, a bland recipe

may get a higher (or lower) rating when preceded by a soup that is muchspicier than the current soup.

I Oftentimes, interference effects can be mitigated by randomization of the orderof the treatments and allowing sufficient time between treatments (i.e., awashout period).

I Randomization in a repeated measures design involves taking each subject,performing a random permutation to define the treatment order, and selectingindependent permutations for the different subjects.

53

Repeated Measures vs. Repeated Observations

I Repeated measures designs are not the same as when we havedesigns with repeated observations.

I In repeated measures designs, several or all of the treatmentsare applied to the same subject.

I Designs with repeated observations, on the other hand, aredesigns where several observations on the response variableare made for a given treatment applied to an EU – usually toquantify measurement error.

I However, it is possible to develop a repeated measures designwith repeated observations, as when a given subject is exposedto each treatment under study and a number of observationsare made at the end of each treatment application.

54

Profile Plots

I Profile plots can be a helpful visualization tool in repeatedmeasures designs.

I They can help reveal trends under two primary settings:I If there is a significant upward (downward) trend in the

response across time for different treatments.I If there is a significant upward (downward) trend in the

response if treatment order is fixed; i.e., the treatments areessentially an ordinal variable.

I Noticeable trends within the profile plots can also help identifymeaningful contrasts for study (although these would, ideally,be specified during the design of the experiment).

55

Single-Factor Experiments with Repeated Measures

I We first consider repeated measures designs where treatments are basedon a single factor.

I Almost always, the subjects in repeated measures designs are viewed as arandom sample from a population, hence they are random effects.

I When treatment effects are fixed, the following model is appropriate for asingle-factor repeated measures design:

Yij = µ+ ρi + τj + εij , (6)

where

I µ is the overall mean (a constant);I i = 1, . . . , n and j = 1, . . . , r;I ρi are the random effects due to subject i and are iid normal with mean 0

and variance σ2ρ;

I τj are the (fixed) treatment effects subject to the constraint∑rj=1 τj = 0;

I εij is the (random) error and are iid normal with mean 0 and variance σ2;and

I εij and ρi are assumed independent of one another.

56

Further Assumptions About yij

I The model we just introduced makes the following additional assumptions aboutthe observations:

E(Yij) = µ+ τj

Var(Yij) = σ2Y = σ2

ρ + σ2

Cov(Yij , Yij′ ) = σ2ρ + ωσ2

Y , j 6= j′

Cov(Yij , Yi′j′ ) = 0, i 6= i′

I In the above, ω = σ2ρ/σ

2Y , which is a measure (specifically, a coefficient of

correlation) between any two observations for the same subject relative to that

between subjects.I This is akin to the intraclass correlation coefficient, but for a repeated

measures design.

I Thus, we assume in advance that any two treatment observations Yij and Yij′for a given subject are correlated in the same fashion for all subjects.

I Moreover, we assume that once the subjects have been selected, then any twoobservations for a given subject are independent (i.e., no interference effects).

57

Variance-Covariance MatrixI More generally, the correlation between two variables, Yi and Yj , is defined as

ρij =σij

σiσj,

where σi and σj are the standard deviations of Yi and Yj , respectively, and σijis the covariance between the two variables.

I The above gives a general format to define a variance-covariance matrix.I For example, the theoretical variance and covariances for repeated measures

taken successively as y1, y2, y3, and y4 is

Σ =

σ2

1 σ12 σ13 σ14

σ21 σ22 σ23 σ24

σ31 σ32 σ23 σ34

σ41 σ42 σ43 σ24

I Equal variances for the treatment groups and independent, normally distributed

observations are the usual ANOVA assumptions, which would mean all of thevariance terms on the diagonal in the matrix above are the same (i.e., σ2) andall of the off-diagonals are 0 (independence).

58

Compound Symmetry

I A particular experiment with randomization of treatments to EUs is onlya random sample of all possible randomized experiments that could havebeen used.

I Randomization does not remove the correlation between the observationson EUs; however, the expected correlation between the EUs is assumedconstant under all possible randomizations.

I If the variances and correlations are constant, the covariances will havethe constant value σij = ρσ2, which is called compound symmetry.

I The matrix on the previous slide – when compound symmetry is present –is

Σ =

σ2 ρσ2 ρσ2 ρσ2

ρσ2 σ2 ρσ2 ρσ2

ρσ2 ρσ2 σ2 ρσ2

ρσ2 ρσ2 ρσ2 σ2

59

Split-Plot Treatments are Randomized

I The assumption of compound symmetry was implicitly used for the errorsof observations in the split-plot design because treatments were randomlyassigned to the subplots.

I The subject in the repeated measures design is equivalent to thewhole-plot in the split-plot design, while the between-subjects treatmentfactor is equivalent to the whole-plot treatment factor in the split-plotdesign.

I The repeated measure on a subject is analogous to the subplot in thesplit-plot design.

I The difference between the subplot observations and the repeatedmeasures is that treatments are randomized to the subplots in thesplit-plot design, whereas there is no randomization for the repeatedmeasures.

I If all repeated measures on a subject are equally correlated, then there iscompound symmetry and the repeated measures design can be analyzedas a split-plot design with time of measurement as the subplot treatmentfactor.

60

Huynh-Feldt Condition

I The Huynh-Feldt condition is a less stringent requirement for ANOVAthan compound symmetry.

I The condition is to have the same variance of the difference for allpossible pairs of observations taken at different time periods, say yi andyj , or

σ2(yi−yj) = 2λ, for i 6= j (7)

for some λ > 0.

I This condition can also be stated as

σij =1

2(σ2i + σ2

j )− λ, for i 6= j (8)

I The matrix of variances and covariances satisfying the Huynh-Feldtcondition is called the Type H matrix.

I The mean squares from an ANOVA can be used to test hypotheses aboutthe within-subjects treatments if the Huynh-Feldt condition is satisfied.

61

Tests for the Huynh-Feldt Assumption

I A common test for the Huynh-Feldt assumption is the Mauchly W testfor a Type H matrix.

I The test statistic is approximately χ2-distributed.

I The null hypothesis is that the Huynh-Feldt condition is appropriate, thuswe hope to obtain a large p-value.

I However, the calculation of the test statistic is handled differently indifferent software and textbooks.

I We will not dive much into the details of this test statistic, other than tonote that you might find discrepancies based on the approximations usedor the software used.

I When reporting results from this test, you should note the softwareand/or calculation being used.

62

SS for Repeated Measures

I The ANOVA SS for repeated measures has a SS for subjects (SSS) and aninteraction SS between treatments and subjects (SSTr.S).

I We use the following identity to break out the above effects:

(yij − y··) = (yi· − y··) + (y·j − y··) + (yij − yi· − y·j + y··)

I The components in the above decomposition are:

total deviation: (yij − y··)subject deviation: (yi· − y··)

treatment deviation: (y·j − y··)treatment by subject error: (yij − yi· − y·j + y··)

I Note that the above is analogous to the identity used for RCBDs.

63

SS for Repeated MeasuresI Taking the SS of the decomposition on the previous slide (where the sums of

cross-products drop out), we get

Total deviation: SSTot =n∑i=1

r∑j=1

(yij − y··)2

Subject effect: SSS = rn∑i=1

(yi· − y··)2

Treatment effect: SSTr = nr∑j=1

(y·j − y··)2

Treatment by subject effect: SSTr.S =n∑i=1

r∑j=1

(yij − yi· − y·j + y··)2,

where no SSE is present here because there are no replications.I Note that sometimes SSTr and SSTr.S are sometimes combined into a

within-subjects SS (SSW) such that SSTot = SSS + SSW.

64

ANOVA and Testing

I The ANOVA table is given as follows:Source df SS MS E(MS)Subjects n− 1 SSS MSS σ2 + rσ2

ρ

Treatments r − 1 SSTr MSTr σ2 + n

∑j τ

2j

r−1

Error (n− 1)(r − 1) SSTr.S MSTr.S σ2

Total nr − 1 SSTot

I The appropriate test statistic for the test on treatment effects:

H0 : τ1 = τ2 = · · · = τr = 0

HA : not all τj equal 0

is

F ∗ =MSTr

MSTr.S∼ F(r−1),(r−1)(n−1)

65

Example: Wine-Judging CompetitionIn a wine-judging competition, j = 4 Chardonnay wines of the same vintage werejudged by i = 6 experienced judges. Each judge tasted the wines in a blind fashion;i.e., without knowing the wine they were tasting. The order of the wine presentationwas randomized independently for each judge. To reduce carryover and otherinterference effects, the judges did not drink the wines and thoroughly rinsed theirmouths between tastings. Each wine was scored on a 40-point scale with larger scoresindicating greater excellence of the wine. The judges are considered to be a randomsample from a population of possible judges while the wines in the study are ofinterest in themselves. Hence, we will use a single-factor repeated measures modelwith the effects of subjects (judges) treated as random and the effects of treatments(wines) treated as fixed. The data are below:

Judge Wine (j)(i) 1 2 3 41 20 24 28 282 15 18 23 243 18 19 24 234 26 26 30 305 22 24 28 266 19 21 27 25

66

Example: Wine-Judging Competition

I To the right is a profile plot of thewine data.

I We see that there are somedistinct differences in ratingsbetween judges, but the ratingsfor wines 3 and 4 are consistentlybest, while wine 1 is generally theworst.

I We also see that the rating curvesfor the judges do not appear toexhibit substantial departures frombeing parallel, hence an additivemodel is appropriate.

1520

2530

Wine-Judging Comptetion

Wine

Score

1 2 3 4

Judge

123456

67


Mauchly Tests for Sphericity

Test statistic p-value

wine 0.35156 0.5772

I Above is the Mauchly W test for sphericity.

I The p-value is 0.5772, so the test is not statisticallysignificant against the Huynh-Feldt condition (i.e., sphericityis an appropriate assumption).

I The above results were obtained in R, so calculations indifferent packages might differ from the above.

68


Analysis of Variance Table

Response: score


judge 5 173.33 34.667 32.5 1.549e-07 ***

wine 3 184.00 61.333 57.5 1.854e-08 ***

Residuals 15 16.00 1.067

---

Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

I Above is the ANOVA table for the wine-tasting competition data.

I The test of treatment effects has the following test statistic:

F ∗ =MSTr

MSTr.S=

61.333

1.067= 57.5,

which follow an F3,15 distribution.

I The p-value for the above test is considerably lower than 0.05, so we concludethat the mean wine ratings for the four wines differ.

69

Using Split-Plot ANOVA

I If we can reasonably assure ourselves that the ANOVA assumptions are valid forrepeated measures on each subject in the treatment groups, then the split-plotanalysis of variance mean squares can be used to test hypotheses about thetreatment means and their interactions with time.

I This can be used when we have more than one subject to which each treatmentis applied.

I The split-plot design linear model of interest for our set-up is

Yijk = µ+ αi + dik + βj + (αβ)ij + εijk, (9)

whereI µ is the overall mean (a constant);I i = 1, . . . , a, j = 1, . . . , b, and k = 1, . . . , n;I αi are the (fixed) effects of treatment subject to the constraint

∑ai=1 αi = 0;

I dik are the (random) experimental errors for subjects within treatment and are iid normal with

mean 0 and variance σ2d;

I βj are the (fixed) effects of time subject to the constraint∑bj=1 βj = 0;

I (αβ)ij are the (fixed) effects due to the interaction of subject and treatment subject to the

constraint∑ai=1(αβ)ij =

∑bj=1(αβ)ij = 0;

I εijk are the (random) errors and are iid normal with mean 0 and variance σ2; andI dik and εijk are assumed independent of one another.

70

Outline of Topics







71

Ranking Observations

I In repeated measures studies, the observations are sometimes ranks, aswhen a number of tasters are each asked to rank recipes or when severaluniversity admissions officers are each asked to rank applicants foradmission.

I When the data in a repeated measures study are ranks, we can use thenonparametric rank F -test for testing whether or not the treatmentmeans are equal.

I The observations Yij for each subject need to be ranked first in ascendingorder.

I Let Rij denote the rank of Yij when the observations for the ith subject

are ranked from 1 to r.I In the case of ties, each of the tied observations is given the mean

of the ranks involved.I We are interested in testing the alternatives

H0 : µ·1 = µ·2 = . . . = µ·r

HA : at least one µ·j differs.

72

Test Statistic

I The F -statistic is now based on the ranks as follows:

F ∗R =MSTR

MSRM, (10)

where

MSTR =n∑rj=1(R·j − R··)2

r − 1

MSRM =

∑ni=1

∑rj=1(Rij − R·j)2

(n− 1)(r − 1)

R·j =

∑ni=1 Rij

n

R·· = Ri· =r + 1

2.

I MSTR is the mean square due to the treatment while MSRM can bethought of as a mean square due to the repeated measures.

73

Interpreting the Test

I If there are no treatment differences, all ranking permutationsfor a subject are assumed to be equally likely and the statisticF ∗R will be distributed approximately as Fr−1,(n−1)(r−1),provided the number of subjects is not too small.

I Large values of the test statistic lead to the conclusion thatthe treatments have unequal effects.

I We can also address multiple comparisons in a way similar tohow we handled multiple comparisons for the continuousresponse settings.

74

Large-Sample Bonferroni Pairwise Comparisons

I We can use a large-sample testing analog of the Bonferroni pairwisecomparison procedure to obtain information about the comparativemagnitudes of the treatment means when the nonparametric rank F -testindicates that the treatment means differ.

I Testing limits for all g = r(r − 1)/2 pairwise comparisons using the meanranks R·j are set up as follows for the joint confidence level ofsignificance α:

(R·j − R·j′)± z1−α/(2g)

√r(r + 1)

6n. (11)

I If the testing limits include 0, then we conclude that thecorresponding treatment means µ·j and µ·j′ do not differ.

I If they do not include 0, then we conclude that the twocorresponding treatment means differ.

I We can then setup groups of treatments whose means do not differaccording to this simultaneous testing procedure.

75

Example: Sweetener Rankings

n = 6 subjects were each asked to rank r = 5 coffee sweetenersaccording to their taste preference, with a rank of 5 being theirmost preferred sweetener. The data are given below and suggestthat a sweetener effect may be present; e.g., no judge rankedsweetener B higher than 2.

Subject Sweetener (j)

(i) A B C D E

1 5 1 2 4 32 4 2 1 5 33 3 2 1 4 54 5 2 3 4 15 4 1 2 3 56 4 1 3 5 2

R·j 4.17 1.50 2.00 4.17 3.17

76

Example: Sweetener Rankings

Analysis of Variance Table

Response: rank


sweetener 4 36 9.00 9.375 9.019e-05 ***

Residuals 25 24 0.96

---

Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

Note that we only use the above single-factor ANOVA table to obtain the SSTR (36)and the SSE (24), which are the numerators for MSTR and MSRM, respectively. Therest of the table is not used for our test. We wish to test

H0 : µ·1 = µ·2 = µ·3 = µ·4 = µ·5

HA : at least one µ·j differs.

The test statistic here is

F ∗R =MSTR

MSRM=

36/(5− 1)

24/((6− 1)(5− 1))=

9.00

1.20= 7.50 ∼ F4,20,

which has a p-value of 0.0007, indicating highly significant evidence of a treatmenteffect.

77

Example: Sweetener RankingsSo which sweeteners differ? We will use a joint level of significance of α = 0.20.To construct the Bonferroni pairwise comparisons, we have g = 10, so that

z1−α/(2g)

√r(r + 1)

6n= 2.33

√0.91 = 2.12

The comparisons are given below:

Comparisons Difference Bonferroni CI Different?

A B -2.67 (-4,79, -0.54) YesC -2.17 (-4.29, -0.04) YesD 0.00 (-2.12, 2.12) NoE -1.00 (-3.12, 1.12) No

B C 0.50 (-1.62, 2.62) NoD 2.67 (0.54, 4.79) YesE 1.67 (-0.46, 3.79) No

C D 2.17 (0.04, 4.29) YesE 1.17 (-0.96, 3.29) No

D E -1.00 (-3.12, 1.12) No

78

Example: Sweetener RankingsFrom the table on the previous slide, we can see that A and B differ, Aand C differ, B and D differ, and C and D differ. Therefore, all otherpairings do not differ. Note that we can further group these sweetenersinto treatments that do not differ:

Group 1 Group 2Sweetener B R·2 = 1.50 Sweetener E R·5 = 3.17Sweetener C R·3 = 2.00 Sweetener A R·1 = 4.17Sweetener E R·5 = 3.17 Sweetener D R·4 = 4.17

Thus, we conclude that with joint confidence level 0.20, that sweetenersA and D are preferred to sweeteners B and C, and that it is not clear asto which group sweetener E would belong.

79

Outline of Topics







80

Expected Mean SquaresI In the ANOVA tables we have presented thus far, we have often reported

expected mean square (E(MS)) values, but did not emphasize how to obtainthose expressions.

I There is a basic recipe that can be followed to obtain these expressions, and

how to calculate them is a matter of practice.I The E(MS) for the MSE is σ2.I In the case of a restricted model, for every other model term, the E(MS)

contains σ2 plus either the variance component or the fixed effectcomponent for that term, plus those components for all other modelterms that contain the effect in question and that involve no interactionswith other fixed effects.

I The coefficient of each variance component or fixed effect is the numberof observations at each distinct value of that component.

I A full outline of the steps is given in Appendix D.3 of Applied Linear StatisticalModels, Fifth Edition (2005) by Kutner et al.

I We will illustrate the E(MS) calculations with some concrete examples of

models we have already seen.I For all models, let i = 1, . . . , a, j = 1, . . . , b, and k = 1, . . . , n, so that we

are working with a balanced design.

81

Example: Two-Factor Fixed EffectsFor the two-factor fixed effects model (with interaction), we have

Yijk = µ+ αi + βj + (αβ)ij + εijk,

where all of the usual assumptions are in place. Let us find the E(MSAB), i.e.,the E(MS) of the interaction term. The E(MS) will contain only the fixedeffect for the AB interaction (because no other model terms contain AB) plusσ2, and the fixed effect for AB will be multiplied by n because there are nobservations at each distinct value of the interaction component (the nobservations in each cell). Thus, the E(MS) for AB is

E(MSAB) = σ2 +n∑ai=1

∑bj=1(αβ)2

ij

(a− 1)(b− 1)

As a further illustration, consider the E(MS) for the main effect A:

E(MSA) = σ2 +bn

∑ai=1 α

2i

(a− 1)

The multiplier in the numerator is bn because there are bn observations at eachlevel of A. The AB interaction term is not included in the E(MS) because whileit does include the effect in question (A), factor B is a fixed effect.

82

Example: Two-Factor Random EffectsTo illustrate the calculations for random effects, consider the two-factorrandom effects model

Yijk = µ+ ai + bj + (ab)ij + εijk,

where all of the usual assumptions are in place. The E(MS) for the ABinteraction would then be

E(MSAB) = σ2 + nσ2ab

and the E(MS) for the main effect of A would be

E(MSA) = σ2 + nσ2ab + bnσ2

a

Note that the variance component for the AB interaction term isincluded because A is included in AB and B is a random effect.

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Example: Two-Factor Mixed Effects (Unrestricted)Next consider the unrestricted form of the two-factor mixed effects model

Yijk = µ+ αi + bj + (αb)ij + εijk,

where A is a fixed effect, B is a random effect, and all of the usual assumptionsare in place. In the unrestricted version, the interaction term is treated as fixed.The E(MS) for the AB interaction is once again

E(MSAB) = σ2 + nσ2αb

but now the E(MS) for the main effect of A would be

E(MSA) = σ2 + nσ2αb + bn

∑ai=1 α

2i

(a− 1)

The interaction variance component is included because A is included in ABand B is a random effect. For the main effect of B, the E(MS) is

E(MSB) = σ2 + nσ2αb + anσ2

b

Here, the interaction variance component is not included, because while B isincluded in AB, A is a fixed effect.

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Example: Two-Factor Mixed Effects (Restricted)

Finally, consider the restricted form of the two-factor mixed effectsmodel, where the interaction term is random, but has the furtherconstraint that

∑ai=1(αb)ij = 0. For the E(MS), simply include

the term for the effect in question, plus all the terms that containthis effect as long as there is at least one random factor. Toillustrate, the E(MS) for the interaction AB is

E(MSAB) = σ2 + nσ2αb

For the main effect A – the fixed factor – the E(MS) is

E(MSA) = σ2 + nσ2αb +

bn∑a

i=1 α2i

a− 1

and for the main effect B – the random factor – the E(MS) is

E(MSB) = σ2 + anσ2b

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Remark on Nested Effects

I The same recipe used for the previous examples also applieswhen we have nested designs.

I If any level of the nested hierarchy is random, then that makesthe entire effect random.

I Thus, the E(MS) would be calculated using the E(MS)formulation for random effects given in the examples.

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This is the end of Unit 10.

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unit 10: split-plot designs, repeated measures, and...

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