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Chapter 18: Split-Plot, Repeated Measures, and Crossover Designs

1 Introduction

When the experiment involves a factorial treatment structure, the implementation of one or two

factors may be more time-consuming, more expensive, or require more material than the other

factors. In situations such as these, a split-plot design is often implemented. For example, in an

educational research study involving two factors, teaching methodologies and individual tutorial

techniques, the teaching methodologies would be applied to the entire classroon of students. The

tutorial techniques would then be applied to the individual students within the classroom. In

an agricultural experiment involving the factors, levels of irrigation and varieties of cotton, the

irrigation systems must apply the water to large sections of land which would then be subdivided

into smaller plots. The different varieties of cotton would then be plantted on the smaller plots.

In a crossover designed experiment, each subject receives all treatments. The individual subjectsin the study are serving as blocks and hence decreasing the experimental error. This provides an

increased precision of the treatment comparisons when compared to the design in which each subject

receives a single treatment.

In the repeated measures designed experiment, we obtain t different measurements correspond-

ing to t different time points following administration of the assigned treatment. The multiple

observations over time on the same subject often yield a more efficient use of experimental re-

sources than using a different subject for each obsevation time. Thus, fewer subjects are required,

with a subsequent reduction in cost. Also, the estimation of time trends will be measured with

a greater degree of precision. Medical researchers, ecological studies, and numerous other areas

of research involve the evaluation of time trends and hence may find the repeated measure design

useful.

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2 Split-Plot Designed Experiments

The yields of three different varieties of soybeans are to be compared under two different levels

of fertilizer application. If we are interested in getting n = 2 observations at each combination of

fertilizer and variety of soybeans, we would need 12 equal-sized plots. Taking fertilizer as factor A

and varieties as a treatment factor T, one possible design would be an 2 × 3 factorial treatment

structure with n = 2 observations per factor-level combination. However, since the application of

fertilizer to a plot occurs when the soil is being prepared for planting, it would be difficult to first

apply fertilizer A1 to six of the plots dictated by the factorial arrangement of factors A and T and

then fertilizer A2 to the other six plots before planting the required varieties of soybeans in each

plot.

An easier design to execute would have each fertilizer applied to two larger “wholeplots” andthen the varieties of soybeans planted in three “subplots” within each whole plot.

This design is called a split-plot design, and with this design there is a two-stage randomization.

First, levels of factor A (fertilizers) are randomly assigned to the wholeplots; second, the levels of

factor T (soybeans) are randomly assigned to the subplots within a wholeplot. Using this design,

it would be much easier to prepare the soil and to apply the fertilizer to the larger wholeplots.

Consider the model for the split-plot design with a levels of factor A, t levels of factor T, and

n repetitions of the i levels of factor A. If yijk denotes the kth response for the ith level of factor

A, jth level of factor T, then

yijk = µ + τ i + δik + γ j + τ γ ij + ijk ,

where

• τ i: Fixed effect for ith level of A.

• γ j : Fixed effect for jth level of T.

• τ γ ij : Fixed effect for ith level of A, jth level of T.

• δik: Random effect for the kth wholeplot receiving the ith level of A. The δik are independent

normal with mean 0 and variance σ2δ .

• ijk : Random error.

The δik and ijk are mutually independent.

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Table 1: An ANOVA table for a completely randomized split-plot design.

Source SS df EMS

A SSA a − 1 σ2 + tσ2

δ + tnθτ

Wholeplot Error SS(A) a(n − 1) σ2 + tσ

2δ

T SST t − 1 σ2 + anθγ

AT SSAT (a − 1)(t − 1) σ2 + nθτ γ

subplot error SSE a(n − 1)(t − 1) σ2

Total TSS atn − 1

The ANOVA for this model and design is shown in Figure 1. The sum of squares can be

computed using our standard formulas.

Wholeplot analysis H 0 : θτ = 0 (or, equivalently, H 0: all τ i = 0), F = M SAM S (A) .

Subplot Analysis H 0 : θτ γ = 0 (or, equivalently, H 0: All τ γ ij = 0), F = MSAT M SE

.

H 0 : θγ = 0 (or, equivalently, H 0: All γ j = 0), F = M ST M SE

.

A variation on this design introduces a blocking factor (such as farms). Thus for our example,

there may be b = 2 farms with a = 2 wholeplots per farm and t = 3 subplots per wholeplot. The

model for this more general two-factor split-plot design laid off in b blocks is as follows:

yijk = µ + τ i + β j + τ β ij + γ k + τ γ ik + ijk ,

where yijk denotes the measurement receiving the ith level of factor A and the kth level of factor

T in the jth block. The parameters τ i, γ k, and τ γ ik are the usual main effects and interaction

parameters for a two-factor experiment, whereas β j is the effect due to block j and τ β ij is the

interaction between the ith level of factor A and the jth block. The analysis corresponding to this

model is shown in Table 2.

• Wholeplot analysis.

H 0 : θτ = 0 (or, equivalently, H 0: all τ i = 0), F = M SAMSAB

.

• Subplot analysis.

H 0 : θτ γ = 0 (or, equivalently, H 0: all τ γ ik = 0), F = MSAT M SE

.

H 0 : θγ = 0 (or, equivalently, H 0: all γ k = 0), F = M ST M SE

.

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Table 2: An ANOVA table for a randomized split-plot design (A, T fixed; block random.

Source SS df EMS

Blocks SSB b − 1 σ2 + atσ2

β

A SSA a − 1 σ2 + tσ2δ + btθτ

AB(Wholeplot Error) SSAB (a − 1)(b − 1) σ2 + tσ2

τ β

T SST t − 1 σ2 + abθγ

AT SSAT (a − 1)(t − 1) σ2 + bθτ γ

subplot error SSE a(b − 1)(t − 1) σ2

Total TSS abt − 1

Example 2.1 Soybeans are an important crop throughout the world. A study was designed to

determine if additional phosphorus applied to the soil would increase the yield of soybean. There

are three major varieties of soybeans of interest (V 1, V 2, V 3) and four levels of phosphorus (0, 20,

40, 65, pounds per acre). The researchers have nine plots of land available for the study which

are grouped into blocks of three plots each based on the soil characteristics of the plots. Because

of the complexities of planting the soybeans on plots of the given size, it was decided to plant a

single variety of soybeans on each plot and then divide each plot into four subplots. The researchers

randomly assigned a variety to one plot within each block of three plots and then randomly assigned the levels of phosphorus to the four subplots within each plot. The yields (bushels/acre) froom the

36 plots are given in Table 18.5 of the textbook.

For this study, we have a randomized complete block design with a split-plot structure. Variety,

with 3 levels, is the wholeplot treatment and amount of phosphorus is the split-plot treatment. The

ANOVA analysis is as follows.

The results indicate that there is a significant variety by phosphorus interaction from which we

can conclude that the relationship between average yield and amount of phosphorus added to the

soil is not the same for the three varieties.

The distinction between this two-factor split-plot design and the standard two-factor experi-

ments discussed in Chapter 14 lies in the randomization. In a split-plot design, there are two stages

to the randomization process; first levels of factor A are randomized to the wholeplots within each

block, and then levels of factor B are randomized to the subplot units within each wholeplot of

every block. In contrast, for a two-factor experiment laid off in a randomized block design, the

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Table 3: An ANOVA table for a randomized split-plot design (A, T fixed; block random).

Source df SS MS F p-value

Blocks 2 763.25 381.63 * *

V 2 671.81 335.90 232.60 < .0001

BV(Wholeplot Error) 4 6.56 1.64 * *

P 3 408.37 136.12 601.04 < 0.0001

PV 6 117.41 19.57 86.40 < 0.0001

subplot error 18 4.08 0.23

Total 35 1971.48

randomization is a one-step procedure; treatments (factor-level combinations of the two factors)

are randomized to the experimental units in each block.

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3 Single-Factor Experiments with Repeated Measures

In Section 18.1, we discussed some reasons why one might want to get more than one observation

per patient. Consider a design, three compounds are administered in sequence to each of the n

patients. A compound is administered to a patient during a given treatment period. After a

sufficiently long “washout” period, another compound is given to the same patient. This procedure

is repeated until the patient has been treated with all three compounds. The order in which the

compounds are administered would be randomized. The data is shown below.

multicolumn4cPatient

Compound 1 2 · · · n

1 y11 y12 · · · y1n

2 y21 y22 · · · y2n

3 y31 y32 · · · y3n

The model for this experiment can be written as

yij = µ + τ i + δ j + ij ,

where µ is the overall mean response, τ i is the effect of the ith compound, δ j is the effect of jth

patient, and ij is the experimental error for the jth patient receiving the ith compound.

For this model, we make the following assumptions:

1. τ is are constants with τ a = 0.

2. The δ j are independent and normally distributed N (0, σ2δ ).

3. The ij s are independent of the δ j s.

4. The ij s are normally distributed N (0, σ2 ).

5. The ij s have the following correlation relationship: ij and i

j are correlated for i = i

; andij and i j are independent for j = j.

That is, two observations from the same patient are correlated but observations from different

patients are independent. From these assumptions it can be shown that the variance of yij is

σ2δ + σ2

. A further assumption is that the covariance for any two observations from patient j,

yij and yi j , is constant. These assumptions give rise to a variance-covariance matrix for the

observations, which exhibits compound symmetry.

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The ANOVA table for the experiment is shown below.

Source SS df EMS (A fixed, patients random)

Patients SSP n − 1 σ2 + aσ2

δ

A SSA a − 1 σ2 + nθτ

Error SSE (a − 1)(n − 1) σ2

Totals TSS an − 1

When the assumptions hold, and hence compound symmetry holds, the statistical test on factor

A (F = MSA/MSE ) is appropriate. The conditions under which the F test for factor A is valid

are often not met because observations on the same patient taken closely in time are more highly

correlated than are observations taken farther apart in time. So be careful about this.

In general, when the variance-covariance matrix does not follow a pattern of compound sym-

metry, the F test for factor A has a positive bias, which allows rejection of H 0 : all τ i = 0 more

often than is indicated by the critical F -value.

Example 3.1 An exercise physiologist designed a study to evaluate the impact of the steepness of

running courses on the peak heart rate (PHR) of well-conditioned runners. There are four five-mile

courses that have been rated as flat, slightly steep, moderately steep, and very steep with respect to

the general steepness of the terrain. The 20 runners will run each of the four courses in a randomly

assigned order. There will be sufficient time between the runs so that there should not be any

carryover effect and the weather conditions during the runs were essentially the same. Therefore,

the researcher felt confident that the model

yij = µ + τ i + δ j + ij

would be appropriate for the experiment.

The ANOVA table for the experiment is shown below:

Source SS df EMS (A fixed, patients random) F Prob

Runner 4048.44 19 213.08 11.21 < 0.0001

Course 3619.25 3 1206.41 63.47 < 0.0001

Error 1083.51 57 19.01

Totals 8751.19 79

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From the output we have that the p-value associated with the F test of

H 0 : µ1 = µ2 = µ3 = µ4 versus H 1 : not H 0

has p-value< 0.0001. Thus, we conclude that there is significant evidence of a difference in themean heart rates over the four levels of steepness.

The estimated variance components are given by

σ2Error = M SE = 19.01

σ2Runner =

M S Runner − M SE

4=

213.08 − 19.01

4= 48.52

Therefore, 72% of the variation in the heart rates was due to the differences in runners and 28%

was due to all other sources.

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4 Two-factor Experiments with Repeated Measures on One of the

Factors

We can extend our discussion of repeated measures experiments to two-factor settings. For ex-

ample, in comparing the blood-pressure-lowering effects of cardiovascular compounds, we could

randomize the patients so that n different patients receive each of the three compounds. Repeated

measurements occur due to taking multiple measurements across time for each patient. For exam-

ple, we might be interested in obtaining blood pressure readings immediately prior to receiving a

single dose of the assigned and then every 15 minutes for the first hour and hourly thereafter for

the next 6 hours.

This experiment can be described generally as follows. There are m treatments with n exper-

imental units randomly assigned to the treatments. Each experimental unit (EU) is assigned to a

single treatment with t measurements taken on each of the EUs. The form of the data is shown

below. Note that this is a two-factor experiment (treatment and time) with repeated measurements

taken over the time factor.

Time Period

Treatment EU 1 2 · · · t

1 1 y111 y112 · · · y11t

... · · · · · · · · · · · ·

n y1n1 y1n2 · · · y1nt

...

m 1 ym11 ym12 · · · ym1t

... · · · · · · · · · · · ·

n ymn1 ymn2 · · · ymnt

The analysis of a repeated measurement design can, under certain conditions, be approximated

by the methods used in a split-plot experiment.

• Each treatment is randomly assigned to an EU. This is the wholeplot in the split-plot design.

• Each EU is then measured at t time points. This is considered the split-plot unit.

• The major difference is that in a split-plot design, the levels of factor B are randomly assigned

to the split-plot EUs. In the repeated measurement design, the second randomization does

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not occur, and thus there may be strong correlation between the measurements across time

made on the same EU.

The split-plot analysis is an appropriate analysis for a repeated measurement experiment only

when the covariance matrix of the measurements satisfy a particular type of structure: Compound

Symmetry:

Cov(yijk , yi jk) =

σ2 when i = i, j = j

ρσ2 when i = i, j = j

0 when i = i

where yijk is the measurement from the kth EU receiving treatment i at time j. Thus we have

Corr(yijk , yij

k) = ρ.

This implies that there is a constant correlation between observations no matter how far apart

they are taken in time. This may not be realistic in many applications. One would think that

observations in adjacent time periods would be more highly correlated than observations taken two

or three time periods apart.

The model can be written as

yijk = µ + τ i + dik + β j + (τ β )ij + ijk

where i = 1, . . . , m, j = 1, . . . , t, k = 1, . . . , n, τ i is the ith treatment effect, β j is the jth time

effect, (τ β )ij is the treatment-time interaction effect, dik is the subject-treatment interaction effect

(random, independent, N (0, σ2d), ijk independent N (0, σ2

), and dik and ijk are independently

distributed.

Let λ = tρ/2(1 − ρ). The ANOVA table for the split-plot analysis of a repeated measures

experiment is given in Table 4, where the treatment and time effects are fixed.

Based on Table 4, the following tests can be performed:

• H 0 : θτ β = 0

F =M S trt∗time

M SE

• H 0 : θβ = 0

F =M S time

M SE

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Table 4: An ANOVA table for a two-factor experiment, repeated measures on one factor.

Source df Expected Mean Squares

TRT m − 1 σ2 (1 + 2λ) + tσ2

d + ntθτ

EU(TRT) (n − 1)m σ2 (1 + 2λ) + tσ

2d

Time t − 1 σ2 + nmθβ

TRT*Time (m − 1) ∗ (t − 1) σ2 + nθτ β

Error m(t − 1)(n − 1) σ2

Total mnt − 1

• H 0 : θτ = 0

F =M S T rt

M S EU (trt)

Example 4.1 In a study, three levels of a vitamin E supplement, zero (control), low, and high,

were given to guinea pigs. Five pigs were randomly assigned to each of the three levels of the vitamin

E supplement. The weights of the pigs were recorded at 1, 2, 3, 4, 5, and 6 weeks after the beginning

of the study. This is a repeated measurement experiment because each pig, the EU, is given only

one treatment but each pig is measured six times.

The ANOVA table for the example is as follows.

Source df SS MS F p-value

TRT 2 18548.07 9274.03 1.06 0.3782

PIG(TRT) 12 105,434.20

Week 5 142,554.50 28510.90 52.55 < 0.0001

TRT*Week 10 9762.73 976.27 1.80 0.0801

Error 60 32,552.60 542.54

From this table, we find that there is not significant evidence of an interaction between the

treatment and time factors.

Since the interaction was not significant, the main effects of treatment and time can be analyzed

separately. The p-value=0.3782 for treatment differences and p-value< 0.0001 for time differences.

The mean weights of the pigs vary across the 6 weeks but there is not significant evidence of a

difference in the mean weights for the three levels of vitamin E feed supplements. Therefore, the

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two levels of vitamin E supplement do not appear to provide an increase in the mean weights of

the pigs in comparison to the control, which was a zero level of vitamin E supplement.

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5 Crossover Designs

In a crossover design, each experimental unit (EU) is observed under each of the t treatments

during t observation times. It is important to emphasize the difference between a crossover design

and the general repeated measurement design. In a repeated measurement design, the EU receives

receives a treatment and then the EU has multiple observations or measurements made on it over

time or space. The EU does not receive a new treatment between successive measurements.

The crossover designs are often useful when a latin square is to be used in a repeated mea-

surement study to balance the order positions of treatments, yet more subjects are required than

called for by a single latin square. With this type of design, the subjects are randomly assigned to

the different treatment order patterns given by a latin square. Consider an experiment in which

treatments A, B, and C are to be administered to each subject, and the three treatment orderpattern are given by the latin square

Order Position

pattern 1 2 3

1 A B C

2 B C A

3 C A B

Suppose that 3n subjects are available for the study. Then n subjects will be assigned at

random to each of the three order patterns in a latin square crossover design. Note that this design

is a mixture of repeated measures (within subjects) and latin square (order patterns from a latin

square).

For this experiment, the model can be written as

yijkm = µ + ρi + κ j + τ k + ηm(i) + ijkm,

where i = 1, . . . , t, j = 1, . . . , t, k = 1, . . . , t, and m = 1, . . . , m. The term ρi denotes the effect

of the ith treatment order pattern, κ j denotes the effect of the jth order position, τ k denotes the

effect of the kth treatment, and ηm(i) denotes the effect of subject m which is nested within the ith

treatment order pattern. Here we assume ηm(i) are independent N (0, σ2η), ijkm are independent

N (0, σ2 ) and independent of the ηm(i).

The ANOVA table for the experiment is as follows.

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Source of Variation SS df EMS

Patterns(P) SSP t − 1 σ2 + rσ2

η + nrθρ

Order Position(O) SSO t − 1 σ2 + nrθκ

Treatments(TR) SSTR t − 1 σ2 + nrθτ

Subjects SSS t(n − 1) σ2 + rσ2

η

Error SSE (t − 1)(nt − 2) σ2

Total SST nt2 − 1

The formulas for the sums of squares follow the usual pattern:

SS T =

i

j

m

(yijkm − y...)2

SS P = nt

i

(yi... − y....)2

SS O = nt

j

(y.j.. − y....)2

SSTR = nt

k

(y..k. − y....)2

SS S = t

i

m

(yi..m − y....)2

SS E = SS T − SS P − SSO − SSTR − SSS.

Example 5.1 The following table contains data for a study of three different displays on the sale

of apples, using a latin square crossover design. Six stores were used, with two assigned at random

to each of the three treatment order patterns shown. Each display was kept for two weeks, and the

observed variable was sales per 100 customers.

Two-week Period(j)

Pattern(i) Store 1 2 3

1 m=1 9(B) 12(C) 15(A)m=2 4(B) 12(C) 9(A)

2 m=1 12(A) 14(B) 3(C)

m=2 13(A) 14(B) 3(C)

3 m=1 7(C) 18(A) 6(B)

m=2 5(C) 20(A) 4(B)

The ANOVA table for the data is as follows:

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Source of Variation SS df MS

Patterns 0.33 2 0.17

Order positions 233.33 2 116.67

Displays 189.00 2 94.50Stores (within patterns) 21.00 3 7.00

Error 20.33 8 2.54

The test for the treatment effect is

F =M S T R

M SE =

94.50

2.54= 37.2

which is greater than F 0.05,2,8 = 4.46. Therefore, we conclude that there are differential sales effects

for the three displays. Tests for pattern effects, order position effects, and store effects were also

carried out. They indicated that order position effects were present, but no pattern or store effects.

Order position effects here are associated with the three time periods in which the displays were

studied, and may reflect seasonal effects as well as the results of special events, such as unusually

hot weather in one period.

If the order position effects are not approximately constant for all subjects (stores, etc.), a

crossover design is not fully effective. It may then be preferable to place the subjects into homoge-

neous groups with respect to the order position effects and use independent latin squares for each

group.

Carryover Effects If carryover effects from one treatment to another are anticipated, that is, if

not only the order position but also the preceding treatment has an effect, these carryover effects

may be balanced out by choosing a latin square in which every treatment follows every other

treatment an equal number of times. For t = 4, an example of such a latin square is

Period

Subject 1 2 3 4

1 A B D C

2 B C A D

3 C D B A

4 D A C B

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Note that treatment A follows each of the other treatments once, and similarly for the other

treatments. This design is appropriate when the carryover effects do not persist for more than one

period.

When t is odd, the sequence balance can be obtained by using a pair of latin squares with the

property that the treatment sequences in one square are reversed in the other square.

For the earlier apple display illustration in which three displays were studied in six stores, the

two latin squares might be as shown in the next table. The stores should first be placed into two

homogeneous groups and these should then be assigned to the two lattin squares.

Two-week Period(j)

Square Store 1 2 3

1 A B C1 2 B C A

3 C A B

4 C B A

2 5 A C B

6 B A C

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