1

Chapter 6

Split-Plot and Strip-Plot Designs

Analysis of Variance and Analysis of Covariance

An example to motivate the use of split plot design is as follows. Suppose we want to study two factors, say

methods of cultivation and varieties of wheat. Suppose, the first factor has t levels and second factor has s

varieties. The first factor requires the use of large complex equipment and consequently, relatively large

plots of land are needed. This will require higher cost and puts a restriction on the number of plots to be

used. Because of the nature of the equipment used for planting the wheat, the second factor can be

accommodated in much smaller plots. To achieve this, the large plots are split into smaller plots at the

planting stage.

This means that since the plots are close together, so less variability is expected among the plots and in turn,

more plots and less variability among plots is expected which implies that the contrasts will have more

information in terms of smaller standard errors. This suggests that the experiment can be conducted with two

strata. The whole-plot stratum consists of large plots in which the plots can be assigned as per any standard

design, e.g. CRD, RBD, or Latin square design.

Next stratum is the split-plot stratum which consists of the split-plots. There are the smaller plots that are

obtained by splitting each of the large plots into s parts. The treatments assigned to the large whole-plots

are replicated r times, and treatments assigned to the split-plots are replicated rt times. Now much more

information on the split-plot factor is available because of the extra replication, and in turn, a smaller split-

plot-to-split-plot variance is expected. The interaction contrasts between whole- and split-plot treatment also

fall into the split-plot stratum and benefit due to smaller variance.

There are two distinct randomizations in the split plot designs:

(i) The first randomization takes place in stratum 1, when the levels of the whole-plot treatment are

randomly assigned to the whole–plots.

(ii) The second randomization takes place in stratum 2 where the levels in the split-plot treatment are

randomly assigned in the split-plot.

Many split-plot plans can easily be modified to become strip-plot experiments. These have their own

advantages and disadvantages.

2

Examples

Following examples have been opted from Giesbrecht and Gumpertz (2004).

“Consider a hypothetical cake baking study in an industry. Assume that there are r recipes and c baking

conditions are to be studied. A simple split- plot experiment with the recipes as a whole-plot factor and the

baking condition as a split-plot factor can be set up if cake batters are made up using recipes in random

order. Each batch of batter is then split into c portions. The portion are then baked under the c conditions. A

new random baking order is selected for each batter. Replication is provided by repeating recipes.

Another option is to make up enough batter to make one cake from each of the r recipes. All cakes based

on r recipes are then baked at one time in an oven at one of the c conditions. Now we have an experiment

with baking conditions as a a whole-plot factor and recipe as a split-plot factor.

In case of a strip-plot design, the experimenter would make up batches of each of the batters large enough.

Then partition each batch into c cakes and then bake the cakes in sets, with one cake of each recipe in each

set. In terms of row-column structure of design, the rows represent recipes and the columns represent

baking conditions. The advantage here is that in the absence of replication, only r batches of batter need be

mixed and the oven need only be set up c times.

In another example of a split-plot experiment in industrial quality research is as follows: The object of the

project is to develop a packaging material that would give a better seal under the wide range of possible

sealing process conditions used by potential customers. The package manufacturer identifies a number of

factors which can affect the quality of the seal. In the whole-plot part of the experiment, the sample lots of

eight different packaging materials are produced. These lots of material are then sent to a customer’s plant,

where they are each subdivided into six subplots. The subplots are used in six different sealing processes.

This constitutes the split-plot part of the study.

Statistical analysis of split-plot experiments

Split-plot experiment with whole-plots in a CRD

Statistical Model

The statistical model for a split-plot consists of the two randomization steps in the split-plot experiment, one

in each stratum. So it is a model with two terms. We consider an experiment with whole-plots arranged in a

CRD. Suppose W represents the whole-plot treatment and S represents the split-plot treatment, then the

linear statistical model is written as

3

(1) ( ) (2) ,ijk i ij k ik ijky w s w s

where (1) 'ij s and '(2)ijk are identically and independently distributed random errors, each with mean 0 but

different variances 2 21 2and , respectively, 1, 2,..., ; 1, 2,...,i t j s and 1,2,...,k s . Moreover, (1) 'ij s

and (2) 'ij s are mutually independent.

The whole-plot stratum of the model contains the whole-plot treatment effects iw and the whole-plot error

terms (1) .ij If we include the mean , this part of the model is similar to the case in one way model for

CRD. The split-plot stratum contains the split-plot treatment effects ks , the interaction effect of w and s

as ( )ikw s and the experimental error associated with individual split-plots (2)ijk . All the terms on the

right-hand side of the model (except ) are assumed to have observations measured as deviation from

respective mean.

Analysis of variance

The analysis of variance for the split-plot experiment in the CRD is like an extended analysis for the CRD.

This can be considered as two separate analysis of variance for each of two strata with two separate error

terms. This is illustrated in the following table.

ANOVA table for a split-plot experiment with whole-plots arranged in a CRD

Source df SS MS ( )E MS ratioF

W 1t 2( )

t

ioo oooi

rs y y MSW 2 22 1 ws rs

(1)

MSW

MSE

Error(1) ( 1)t r 2( )

t r

ijo iooi j

s y y (1)MSE 2 22 1s

S 1s 2( )

s

ook oook

rt y y MSS 22 srt

(2)

MSS

MSE

W S ( 1)( 1)t s 2( )

t s

iok ioo ook oooi k

r y y y y ( )MS W S 22 w srs

Error(2) ( 1)( 1)t r s 2( )

t r s

ijk ijo iok iooi j k

y y y y (2)MSE 22

( )

(2)

MS W S

MSE

Total (corrected) 1rts 2( )

t r s

ijk oooi j k

y y

4

Note that the sum of squares due to W S is 2 2 ,t s

iok oooi k

ry rsty SSW SSS where SSW is the whole-

plot treatment sum of squares and SSS is the split-plot treatment sum of squares. The error(2) sum of

squares is obtained by subtraction. The mean squares are obtained by dividing the sum of squares entries

by respective degrees of freedom.

The quantities , ,w s and w s represent quadratic forms as follows:

2

2

2

1

1

( )

.( 1)( 1)

jj

w

kk

s

jkj k

w s

w

t

s

s

W S

t s

These quadratic forms will be zero under the appropriate null hypotheses. It is clear from the expected mean

squares that

- error (1) is used to test the hypothesis of no whole-plot treatment effect and error and

- error (2) is used to test hypotheses of no interaction or split–plot treatment effects.

The test for interaction is performed first, otherwise other tests of hypothesis are doubtful.

Note that since all the levels of each factor are tested in combination with every level of the other factor, so

the analysis in the whole-plot stratum and the split-plot stratum are orthogonal. Estimates of interactions

between whole-and split-unit factors are the contrasts that are orthogonal to both whole-plot and split-plot

treatment contrasts.

Standard errors of main-effect contrasts

The standard errors in the split-plot are more complex than in other designs. We first consider the contrasts

among levels of the split-plot treatment. We write the general form of a split-plot contrast as .k ookk

c y

Then

k ook k ookk k

k kk

E c y c E y

c s

5

and

2 2

2

(2) /

/ .

k ook k ijkk k i j

kk

Var c y Var c rt

c rt

Since 2(2)E MSE , it follows that the estimated standard error (s.e) of a split-plot treatment contrast is

of the form

2 (2). .

kk

k ookk

c MSEs e c y

rt

For the case 1kc and ' 1kc , the contrast is 'ook ooky y where 'k k . It follows that

'

2 (2). ( ) .ook ook

MSEs e y y

rt

The confidence intervals computed for the contrasts are based on ( 1)( 1)t r s degrees of freedom.

The general form of a whole-plot treatment contrast takes the form .i iooi

c y We have

2 2 22 1

.

(1) (2)

( ).

i ioo i iooi i

i ii

ij ijkj j k

i ioo i ii i i

i

i

E c y c E y

c w

Var c y Var c cr rs

c s

rs

Since 2 22 1(1)E MSE s , it follows that the estimated standard error of a whole-plot treatment contrast

of the form is

2 (1)

. .ii ioo

k i

c MSEs e c y

rs

For the whole-plot treatment difference, the estimate of standard error is

'

(1). .ioo i oo

MSEs e y y

rs

6

Standard errors of interaction contrasts

When the interaction between whole-plot treatment and split-plot treatment is significant, then first consider

the standard errors for contrasts among the split-plot treatment levels at a given whole-plot treatment level.

Such contrasts is k iokk

c y for any specific choice, e.g., contrast of quadratic effect of split-plot factor B

with whole-plot factor A at a specific level. Then

( ) .

k iok k iokk k

k k ikk

E c y c E y

c s w s

and

2 22

(2)

.

ijkj

k iok kk k

kk

Var c y Var cr

c

r

It follows that, in general,

2 (2).

kk

k iokk

c MSEs e c y

rt

and the standard error of the difference between two split-plot treatment means at a given whole-plot

treatment level is given by

'

2 (2). ( ) .iok iok

MSEs e y y

rt

Also, the contrasts among whole-plot treatment levels for same split-plot treatment level or at different split-

plot treatment levels can also be investigated. These contrasts are of the form ,ik ioki k

c y e.g., contrast of

the quadratic effect of whole-plot factor at a given level of split-plot factor, contrasts of the quadratic effects

of whole-plot factor at two different split-plot factor levels. In this case

( ) .ik iok ik k ik iki k i k i k

E c y c w c w s

and

7

2 2 2 21 2

2 2 21 2

(1) (2)

1 1

1

ij ijkj j

ik iok ik iki k i k i k

ik iki k i k

iki k

Var c y Var c cr r

c cr r

cr

Note that we have separate unbiased estimators of 2 21 2and but not for 2 2

1 2( ) in the analysis of

variance.

However, if we consider the weighted mean of (2) and (1)MSE MSE , then we have an unbiased estimator of

2 21 2 as follows:

2 21 2

( 1) (2) (1).

s MSE MSEE

s

and then the standard error can be obtained as the positive square root of

2 ( 1) (2) (1)

. (1)ikik iok

i k i k

c s MSE MSEVar c y

r s

The estimate of standard errors of the differences between the two whole-plot treatment means at either the

same split-plot treatment level or at the two different split-plot treatment levels , e.g., 1 1 2 2 ,o oy y can be

obtained as

2 ( 1) (2) (1)s MSE MSE

r s

It is difficult to find the exact number of associated degrees of freedom in such an estimate which is

obtained as the weighted mean of the two MSEs . The approximate degrees of freedom in such cases can be

obtained through Satterthwaite’s approach given as follows:

Satterthwaite’s approach

Note that the expression (1) can be expressed in the general form

( ).m

ii

a MSE i

This expression can be viewed as approximately distributed 2 random variable with

8

2

2

( )

( )

m

ii

mi

i i

a MSE i

a MSE i

df

degrees of freedom where MSE ( )i has idf degrees of freedom. If we consider

1 2 12, ( 1), 1, ( 1)( 1) 1,m a s a df r t and 2 ( 1)( 1) 1df t r s in this expression, we have the

expression of variance in (1).

Variance components, variance of y , and variance of treatment means

The variance of y depends on the variance among whole-plots and can be expressed as

2 21 2ijkVar y where 2

1 is the variance among whole-plots and 22 is the variance among split-plots

within a whole-plot. An unbiased estimator of 21 is then given by

21

(1) - (2)ˆ .

MSE MSE

s

The variability of whole-plots also contributes to the variance of a treatment mean as

2 21 2 .iokVar y

r

Error (2) Bigger than Error (1)

It is observed from the model (1) ( ) (2)ijk i ij k ik ijky w s w s and also from the ( )E MS that

error(1) error (2). The split-plot experiment design is based on this basis. In some practical situations, it

may happen that the estimate MSE(2) is larger than the estimate MSE(1). A question arises how to handle

such situation in the statistical analysis. Various solutions are available in the literature and there is no

unique answer. In some practical situations, one option may be to replace (1)MSE by (2)MSE . Another

option is to just ignore the whole plots and utilize factorial experiment. This amounts to simply pooling the

two error terms. Both of these strategies entail a shift in the model. One model provides the basis for the

construction of the plan and the randomizations. Then different model is used for the final statistical

analysis. A general acceptable view is to use the original model which is based on randomization and

complete the analysis. Just accept the fact that estimated standard error of split-plot treatment differences

are larger than the standard errors of whole-plot treatment differences. If this problem occurs frequently,

then one needs to be concerned about this. There can be various reasons for this it may be possible that any

assumption is violated, the randomization is incorrect, there may be negative correlations within whole-

plots, or some unknown interaction may be present etc.

9

Split-Plot experiment with whole-plots in an RBD

Statistical Model

Now we consider the split-plot experiment with whole plots in the set up of a randomized block design.

The whole-plots are organized into r blocks. There are two strata and two randomizations. The model for

the whole-plot stratum will now have block effects and the model is given as

(1) ( ) (2) ,hik h i hi k ik hiky b w s w s

where 'hb s denote the block effects as the differences among blocks. The block effects can be fixed as well

as random. If the block effects are assumed to be random, then they are assumed to be identically and

independently distributed with mean 0 and variance 2.h It is also assumed that , (1)h hib , and (2)hik are all

mutually uncorrelated. We consider the case when the block effects are random.

Analysis of variance

The analysis of variance table in this case is shown in following table:

Analysis of variance for a split-plot experiment with whole-plots in an RBD

Source Degrees of freedom Sum of squares Mean squares ( )E MS F

Blocks 1r 2( )

r

hoo oooh

st y y 2 2 22 1 bs st

W 1t 2( )

t

oio oooi

rs y y MSW 2 22 1 ws rs

(1)

MSW

MSE

Error(1) ( 1)( 1)r t 2( )

r t

hio hoo oio oooh i

s y y y y (1)MSS 2 22 1s

S 1s 2( )

s

ook oook

rt y y MSS 22 srt

(2)

MSS

MSE

W S ( 1)( 1)t s 2( )

t s

oik oio ook oooi k

r y y y y ( )MS W S 22 w srs

( )

(2)

MS W S

MSE

Error(2) ( 1) ( 1)r t s 2( )

r t s

hik hio hok hooh i k

y y y y (2)MSE 22

Total (corrected) 1rts 2( )

r t s

hik oooh i k

y y

10

where

2

2

2

1

1

( )

.( 1)( 1)

jj

w

kk

s

jkj k

w s

w

t

s

s

W S

t s

Notice also that the error(1) sum of squares is usually calculated as

2 2(1) -r t

hio oooh i

SSE s y rts y SS blocks SSW

and (2)SSE is calculated by subtraction.

In this case, the whole-plot factor is tested using / (1).MSW MSE The difference among the split-plot

treatment means is tested using the (2).MSE Formally, the SSE(1) or whole-plot error sum of squares can be

thought as the interaction of whole-plot treatments and blocks. The error is considered as the inability of the

treatments to perform identically across blocks. The split-plot error is thought of comprising two parts the

interaction of split-plot treatments and blocks with ( 1)( 1)s r degrees of freedom and the three-way

interaction of whole-plots, split-plots, and blocks with ( 1)( 1)( 1)t s r degrees of freedom.

Standard errors of contrasts

The standard errors for treatment and interaction contrasts are the same as for the CRD as described earlier.

Split-plot model in the mixed model framework

The assignment of whole-plot factors to plots and split-plot factors to split plots within the whole-plots

introduces a correlated error structure. This error structure is similar to that for randomized block

experiments with random block effects. In a split-plot experiment where the whole-plots arranged as per

CRD, under the model (1) ( ) (2) ,ijk i ij k ik ijky w s w s we have

' '

21

, (1) (2) , (1) (2)

,

ijk ijk ij ijk ij ijkCov y y Cov

11

which is the covariance for the two split-plot observations within the same whole-plot and two different

split-plots are indicated by 'k k . On the other hand, the correlation between the two split-plot

observations within the same whole-plot is given by

2

' 2 21 2

,ijk ijkCorr y y

which is based on the fact that the covariance between the two split-plot observations on different whole-

plots is zero and the variance of an observation is 2 21 2 .

This correlation structure underlies the form for the analysis of variance.

The random blocks of whole-plots an RBD also introduce the correlation structure in which the covariances

are given by

2' ' '

2 21

0 if '

, if ' and '

if ' and '.

hik h i k b

b

h h

Cov y y h h i i

h h i i

These covariance are based on the assumption that the observations in different blocks are uncorrelated and

observations within blocks are correlated as well as the observations in the same whole-plot are more highly

correlated. The expressions derived for ( )E MS and F remains valid when no observation is missing. If

some observations are missing, then these forms are complex.

Split-plot experiment with whole-plots in a Latin square

Now we consider the split-plot experiment with whole-plot in a Latin-square.

Statistical Model

The whole-plots are organized into a Latin square with t rows and t columns. The experiment is conducted

with two strata and with two randomizations. The model for the whole-plot stratum now has terms for row

and column effects. The model in case of Latin square design is following by

( , ) , 1,..., ; 1,... ; ( , ) 1,..., ; ,ijk i j d i j ijy r c i t j t d i j t

where 'ij s are identically and independently distributed following normal distribution with mean 0 and

variance 2 and ( , )d i j indicates the effect of treatment assigned to (i, j) th cell and incorporate the split-plot

feature as in the given model:

( , ) ( , )(1) ( ) (2) .ijk i j d i j ij k d i j k ijky r c s s

12

where 1,..., ; 1,... ; ( , ) 1,..., ,i t j t d i j t and 1,..., .k s Also, the 'ir s denote the row effects, 'jc s the

column effects, ( , ) 'd i j s denote the whole-plot treatment effects, 'ks s denote the split-plot treatment

effects, ( , )( ) 'd i j ks s are the interaction effect of whole-plot treatment and split-plot treatment, (1) 'ij s

are the whole-plot errors and (2) 'ijk s are the split-plot errors. The analysis of variance and standard errors

for treatment comparisons follow the usual pattern. The basic analysis of variance is detailed in the

following table.

Expected mean squares and F -ratios for a split-plot experiment with whole-plots arranged in a Latin square

Source Degrees of freedom [ ]E MS F ratio

Rows ( 1)t

Columns ( 1)t

W ( 1)t 2 22 1 ws rs

(1)

MSW

MSE

(1)Error ( 1)( 2)t t 2 22 1s

S 1s 2 22 st

(2)

MSS

MSE

W S ( 1)( 1)t s 22 w st

( )

(2)

MS W S

MSE

(2)Error ( 1)( 1)t t s 22

Total

13

Strip-plot experiments

The strip-block experiment is a variation of split-plot experiment. The following two examples are from

opted from Giesbrecht and Gumpertz (2004):

Example 1:

Consider a field experiment in agriculture. A large rectangular plot is available for experimentation. The

two treatments planned are s modes of seedbed preparation with a large piece of equipment replicated cr

times and seeding v varieties of some crop with a large mechanical planer replicated rr times. To conduct

the experiment, the field is divided into cr s strips in one direction. The modes of seedbed preparation are

randomly assigned to the strips as in the following figure:

Variety Seedbed Preparation

3 1 4 3 1 2 4 2

3

2

2

1

3

1

Each seedbed preparation mode is assigned to cr strips, termed as columns. This is one randomization. It

defines one stratum in the experiment.

Next, the field is divided into rvr strips at right angles to the original. The varieties of the crop are randomly

assigned to these strips called as rows. The rr rows are assigned at random to each variety. This establishes

a stratum for variety. In addition, it establishes a third stratum for the interaction of seedbed preparation and

varieties. Both randomizations affect the assignment in this stratum. If fertility gradients are suspected, then

the strips (either one or both) can be grouped into sets, i.e., the blocking factors can be introduced in one or

both the directions.

This is an example of a strip-plot or strip-block experiment where the stripping is dictated by the nature of

the experimental treatments. It is a convenient way to organize things if one needs to use large pieces of

equipment. Note that eventually the experimenter harvests the subplots defined by the intersection of the

row and column strips. A feature of this design is that it provides most information on the interaction.

14

Example 2:

Consider a a cake baking study. It involves a slightly different experimental procedure, but the design

principles are the same. Suppose a food product developer wants to develop a new cake recipe. There are

a similar recipes and b different baking regimes which are to be tested. This can be achieved by first

mixing the cake batters and use one batch for each recipe. Make each batch large enough to provide b

cakes. Assign one cake from each batch of the ovens and each oven to a baking regime. Assume that the

individual ovens are large enough to hold a cakes. The recipes form rows and the baking regimes form

columns that are the two strata. In addition, there is the third - stratum, the interaction. Observations are

collected on individual cakes. This entire procedure is repeated, i.e. replicated, r times.

An important advantage in this experimental designs is that the amount of work is reduced. The alternative

approach is to mix and bake ab cakes individually. This will constitute one complete replicate. This will

involve much work. The best information is at the interaction level. In many cases this is a good thing since

interactions are often very important. This experiment allows the experimenter to check the robustness of

the recipes to variations in baking routine.

Statistical model for the strip-plot

Consider the following linear statistical model

( ) ( ) ( )hij h i hi j hj ij hijy rep a r b c a b

Where ( ) , ( )hj hjr c and hij are identically and independently distributed, each with mean 0 and variance

2 2 21 2, and , respectively. Moreover, they are mutually independent of each other for all

1,..., , 1,..., , and 1,..., .h r i a j c The replicate effects hrep , effects 'ia s , and effects 'jb s are measured

as deviations from a mean and that the interaction effects are defined to sum to zero in both directions,. The

replicate effects can be assumed to be fixed or random .

15

Analysis of variance and standard errors

The analysis of variance based on this model is given in the following table.

Analysis of variance table for a strip-plot experiment

Source Degrees of freedom Sum of squares Mean squares [ ]E MS

Replication 1r 2( )

r

hoo oooh

ab y y .repMS ---

A 1a 2( )

a

oio oooi

rs y y MSA 2 2

rb rb

( )MSE r ( 1)( 1)r a 2( )

r a

hio hoo oio oooh i

b y y y y ( )MSE r 2 2

rb

B 1b 2( )

b

ooj oooj

ra y y MSB 2 2

c ba ra

( )MSE c ( 1)( 1)r b 2( )

r b

hoj hoo ooj oooh j

a y y y y ( )MSE c 2 2

ca

A B ( 1)( 1)a b 2( )

a b

oio oio ooj oooi j

r y y y y ( )MS AB 2

abr

( )MSE ( 1)( 1)( 1)r a b by subtraction ( )MSE 2

Total (corrected) 1rab 2( )

r a b

hik oooh i j

y y

The standard errors for the row (treatment )A and the column (treatment )B comparisons are

2[ ( )]. ( )

ii

c MSE rs e A

rb

and

2[ ( )]. ( )

ji

c MSE cs e B

ra

,

respectively, where andi jc c are the sets of arbitrary constants that sum to zero. The standard errors for

interaction contrasts are more complex. The general form of these contrasts is , where 'ij oij iji j

c y c s are

arbitrary constant coefficients that sum to zero. For example, one can select:

16

1. To compare the two B treatments at a given level of ,A take '1, 1ij ijc c , for some i and some

'j j and all the other coefficients equal to zero.

2. To compare the two A treatments at a given B level, take 1, 1,ij ijc c for some 'i i and a

specific j and all the other coefficients equal to zero.

3. To compare the two means for different A and B levels, take ' '1, 1ij i jc c , for specific 'i i

and 'j j and all the other coefficients equal to zero.

The variance of the general contrast form is

( ) / ( ) / /ij oij ij hi ij hj ij hiji j i j h i j h i j h

Var c y Var c r r Var c c r Var c r

where all other cross-product terms are zero due to the assumption of independence of errors. Now we must

look at special cases. When '1, 1ij ijc c for some i and ', 0ijj

j j c and then we have

2 2 2

2 2

( )

=

( ) ( 1) ( ).

ij hi ij hiji j h i j h

ij oiji j

ij c

i j

c

c c c

Var c y Var Varr r

c

r

E MSE c a MSE

a

The exact degrees of freedom for conducting the tests and constructing the confidence intervals are

difficult to obtain in this case, we use the approximate degrees of freedom which can be obtained following

the Saitterthwaite approach. In this case, choose

2 1,a a 1 12, 1, (1) ( ), ( 1)( 1),m a MSE MSE c df r b (2) ( )MSE MSE and

and 2 ( 1)( 1)( 1).df r a b

In case 2 with '1, 1ij i jc c for some 'i i and specific j with all the other coefficients zero, we have

2 2 2

2 2

1

( ) ( 1) ( ).

ij r

ij oiji j i j

r

cVar c y

r r

MSE r b MSEE

b

17

The approximate degrees of freedom are obtained using Saitterthwaite approach with

1 12, 1, (1) ( ), ( 1)( 1),m a MSE MSE r df r a 2 1, (2) ( ),a b MSE MSE and

2 ( 1)( 1)( 1).df r a b

In case 3 with '1, 1ij ijc c for specific ', 'i i j j with all the other coefficients zero, we have

2 2 2 2

2 2 2

1

( ) ( ) ( ) ( )

ij oij ij r ci j i j

r c

Var c y cr

aMSE r bMSE c ab a b MSEW

ab

.

The approximate degrees of freedom are obtained using Satterwhaite’s approximation with

13, , (1) ( ),m a a MSE MSE r 1 ( 1)( 1),df r a 2 , (2) ( ),a b MSE MSE c 2 ( 1)( 1),df r b

3 ( ),a ab a b (3) ( ),MSE MSE and 3 ( 1)( 1)( 1).df r a b

Analysis of covariance with one split-plot covariate

There are various possibilities for the analysis of covariance in a split-plot experiment, e.g., there can be a

covariate for the whole-plots and not for the split-plots, a covariate for the split-plots and not for the whole-

plots, or to have different covariate for the whole and split-plots. The adjustments to the treatment means

can be messy and so choose the model carefully. There is no simple unique way to use and adjust for

covariates in split-plot experiments.

Assume that the covariates are part of the experimental units rather than responses to the treatments applied.

This means that treatments do not affect the covariates and so the covariates are available to the experimenter

at planning or execution stage. The covariates are observable constants in the model.

Development of model for one covariate at the split-plot level

Consider a very basic model for a split-plot experiment with whole-plots arranged in an RBD and one

covariate that is associated with the split-plot experimental units,

(1) ( ) (2) ,hij h i hi j ij hij hijy r w s w s x

where 1,..., , 1,..., and 1,...,h r i t j s and the covariate is hijx . Assume that the whole-and split-plot

treatments are fixed effects, implying that o ow s =_________ _________

( ) ( ) 0.oj iow s w s The 'hijx s are observed

18

constants. We further assume that (1)hi and (2)hij are identically and independently normally distributed,

each with mean 0 and variances 21 and 2

2 , respectively. Moreover, they are mutually independent also.

Rewrite the model to isolate the covariate’s contribution to bias and variance. Since there are two types of

experimental units, two sources of error, first to split hijx into hiox , (corresponding to the whole-plots) and

( )hij hiox x (for the split-plots).

The model is generalized to allow different regression coefficients by introducing w for the whole-plot

part of the analysis and s for the split-plot part. The model then becomes

(1) ( ) ( ) (2) .w shij h i hio hi j ij hij hio hijy r w x s w s x x

Now write the model in a form that explicitly shows how the covariate contributes to the bias of the

estimated whole-and split-plot factor effects and the variance components. Using the identities

( ) ( ) ( )hio ooo hoo ooo oio ooo hio hoo oio ooox x x x x x x x x x

and

( ) ( ) ( ) ( )hij hio ooj ooo oij oio ooj ooo hij hio oij oiox x x x x x x x x x x x

the model is rewritten as

* * *

( ) ( ) ( ) (1)

( ) ( ) ( ) ( ) (2) (2)

w w w whij ooo h hoo ooo i oio ooo hio hoo oio ooo hi j

s s sooj ooo ij oij oio ooj ooo hij hio oij oio hij

h i

y x r x x w x x x x x x s

x x w s x x x x x x x x

r w

* *( ) (1) ( ) ( ) (2) ,w shio hoo oio ooo hi j ij hij hio oij oio hijx x x x s w s x x x x

where

*

*

*

*

*

( )

( )

( )

( ) ( ) ( ).

wooo

wh h hoo ooo

wi i oio ooo

sj j ooj ooo

sjk ij oij oio ooj ooo

x

r r x x

w w x x

s s x x

w s w s x x x x

The extra terms in * * **, , ,h i jr w s and *( )ijw s represent the contributions to the bias from the experimental

units via the covariate and ( )whio hoo oio ooox x x x and ( )s

hij hio oij oiox x x x the contributions to

variance. The analysis of covariance provides adjustments to remove all of these.

19

Analysis of covariance table

Once model has been constructed, the whole-plot part of the design matrix is orthogonal to the split-plot part

and it is possible to do the analysis of covariance and estimate all the parameters. The first step is to

construct a compact analysis of covariance table as follows:

Analysis of covariance table

Source y variable Whole-Plot Covariate Split-plot Covariate

Mean yyM wx yM

w wx xM

Blocks yyB wx yB

w wx xB

W yyW wx yW

w wx xW

(1)Error (1) yyE (1)wx yE (1)

w wx xE

S yyS sx yS

s sx xS

W S yyW S sx yW S

s sx xW S

(2)Error (2) yyE (2)sx yE (2)

s sx xE

Total yyT

The quantities in the column labeled “y-variable” are the usual analysis of variance sums of squares. The

columns under the heading “Whole-plot covariate” contains the sums of squares computed using the

whole-plot covariate. The other columns contains the sums of cross-products involving the y -variable and

then whole-plot covariate. Similarly, the two columns under the “Split-Plot Covariate” heading contain the

sums of squares and the cross-products involving the split-plot covariate. The wx and sx subscript identify

terms computed using the whole-plot and split-plot covariates, respectively. For example

2(1) ( )w wx x hio oi hoo ooo

h i

E s x x x x

and

2(2) ( ) .s sx x hij hio oij oio

h i j

E x x x x

The expected values are as follows:

20

2 2 2 22 1

2

2

2 2 2 2 2 2 22 1

(1) ( 1)( 1)( ) ( ) ( )

(1) ( )

( (1) ) ( ) ( ) ( ) ( )

wyy hio oio hoo ooo

h i

wwy hio oio hoo ooo

h i

wwy hio oio hoo ooo hio oio hoo ooo

h i h i

E E r t s s x x x x

E E x s x x x x

E E x s x x x x s s x x x x

22 22

2

22 2 22 2

2

(2) ( 1) ( 1) ( )

(2)

( (2) ) .

s

s

syy hij hio oij oio

h i j

sx y hij hio oij oio

h i j

sx y hij hio oij oio hij hio oij oio

h i j h i j

E E r t s x x x x

E E x x x x

E E x x x x x x x x

Then (1)ˆ(1)

w

w w

x yw

x x

E

E

(2)ˆ(2)

s

s s

x ys

x x

E

E

ˆ( )w wE

ˆ( )s sE

2 22 1

2

2 22 1

( )ˆ[ ]( )

( )

(1)w w

whio oio hoo ooo

h i

x x

sVar

s x x x x

s

E

22

2

22

ˆ[ ]( )

.(2)

s s

shij hio hoj hoo

h i j

x x

Varx x x x

E

Note that there is no wasted degree of freedom if there really is one covariate in the whole-plot stratum and

another in the split-plot stratum and the two effects are additive in the whole-plot stratum.

Adjusting the whole-plot error for the covariate gives

2(1)1(1) (1)

( 1)( 1) 1 (1)w

w w

x yayy

x x

EMSE E

r t E

21

with ( 1)( 1) 1r t degrees of freedom. Notice the superscript “ a ” indicates a mean square adjusted for

the covariate. 2 22 1(1)aE MSE s . Adjusting the split-plot error gives

2(2)1(2) (2)

( 1) ( 1) 1 (1)s

s s

x yayy

x x

EMSE E

r t s E

with 1( 1) ( 1)r t s degrees of freedom. 22(2)aE MSE

The treatment and interaction sums of squares must also be adjusted for the covariates to provide proper tests

of hypotheses. In the whole-plot stratum, the adjusted whole-plot treatment sum of squares with ( 1)t

degrees of freedom is

22

2

(1)1.

( 1) (1)w w

w w w w

x y x yayy

x y x y

W EMSE W

t W E

Similarly, in the split-plot stratum,

2(2)1

( 1) (2)w s

s sw s s

x y x yayy

x x x x

S EMSS S

s S E

22( ) (2) (2)1

( ) ( )( 1)( 1) ( ) (2) (2)

s s s

s s s s s s

x y x y x yayy

x x x x x x

W S E EMS W S W S

t s W S E E

Tests of hypotheses are performed using the adjusted mean squares. For whiole-plot treatments, use

(1)

a

a

MSWF

MSE

with 1t and ( 1)( 1) 1r t degrees of freedom . In the split-plot stratum, test the split-plot treatment

using

(2)

a

a

MSSF

MSE

with 1s and ( 1) ( 1) 1r t s degrees of freedom and the interaction of whole-plot and split-plot

treatments is tested using

( )

(2)

a

a

MS W SF

MSE

with ( 1)( 1)t s and ( 1) ( 1) 1r t s degrees of freedom.

22

Treatment contrasts: Main effect

The usefulness of having covariate in the model provides more accurate and more precise estimates of

treatment contrasts. Now adjust for the covariate to remove biases attributable to differences among

experimental units. Now reduce the bias by adjusting treatment means and contrasts.

Consider the whole-plot part of the design. Consider the model in equation (2) given by

* * *

( ) ( ) ( ) (1)

( ) ( ) ( ) ( ) (2) (2)

w w w whij ooo h hoo ooo i oio ooo hio hoo oio ooo hi j

s s sooj ooo ij oij oio ooj ooo hij hio oij oio hij

h i

y x r x x w x x x x x x s

x x w s x x x x x x x x

r w

* *( ) (1) ( ) ( ) (2) ,w shio hoo oio ooo hi j ij hij hio oij oio hijx x x x s w s x x x x

where

*

*

*

*

( )

( )

( )

wooo

wh h hoo ooo

wi i oio ooo

sj j ooj ooo

x

r r x x

w w x x

s s x x

*( ) ( ) ( ).sjk ij oij oio ooj ooow s w s x x x x

The unbiased estimates of * and *iw are obtained from this model as follows:

* *ˆ ˆoio iy w

and

*ˆi oio i ii i

c y c w

with 0.ii

c But the unbiased estimates of contrasts of the form i ii

c w are needed instead of these

contrasts. Using the definitions implied in equation (2), the unbiased estimates are given as

ˆ ˆˆ ˆ ( )w woio ooo i oio oooy x w x x

and

ˆˆ wi oio i i i oioo

i i i

c y c w c x .

Since 'hijx s are the observed constants, so

ˆˆ ˆ wi oio oiow y x

and

ˆˆ wi i i oio i oio

i i i

c w c y c x .

23

These are the required contrasts as the adjusted main effect contrasts. Further,

22 2 2

2 1

( )( )

ˆ(1)

w w

i oioi i

i ii i x x

c xc s

Var c wr E s

.

2 22 1(1)aE MSE s and (1)aMSE has ( 1)( 1) 1r t degrees of freedom.

Comparisons among the split-plot treatment levels are similar. Then

*ˆ ˆ*ooj jy s

and

*ˆ .j ooj j jj j

c y c s

The unbiased estimates of the contrasts of the form j jj

c s are needed. Using the definitions in equation

(2), the unbiased estimates are given by

ˆ ˆˆ ˆ ( )w sooj ooo j ooj oooy x s x x

and

ˆˆ .sj ooj j j j ooj

j j j

c y c s c x

These expressions can be re-expressed as

ˆ ˆˆ ˆ ˆ( ) wj ooj s ooj ooo ooos y x x x

and

2

222

ˆˆ .

Also,

ˆ .(2)

s s

sj j j ooj j ooj

j j j

j oojjj

j jj j x x

c s c y c x

c xc

Var c srt E

22(2)aE MSE and (2)aMSE has ( 1) ( 1) 1r t s degrees of freedom.

24

Interaction contrasts

Interaction contrasts are of the form ij oijij

c y with 0.ijij

c The standard errors of such contrasts depend

on the nature of the contrasts.

Consider the contrasts among split-plot levels for a fixed whole-plot level. These are of the form ij oijij

c y

with 0ijij

c with a specific i value. Then

* *

*

( ( ) )

( ( ) )( )).

ij oij ij j ijj j

sij j ij oij oio

j

E c y c s w s

c s w s x x

The adjusted contrast is

ˆˆ ( ( ) ) ( ).sij j ij ij oij oij oio

j j

c s w s c y x x

and its variance is

2

2 22

( )1.

(2)s s

oij oioij

j x x

x xc

r E

Next consider the contrasts among whole-plot treatment levels, either at the same split-plot treatment level

or at the different split-plot treatment levels. These contrasts take the general form ,ij oiji j

c y where

0.iji j

c Then

* * *( ( ) )

( ( ) ( ) ( )).

ij oij ij i j iji j i j

w sij i oio ooo j ij oij oio

i j

E c y c w s w s

c w x x s w s x x

The adjusted contrast is

ˆ ˆˆ ˆ( ( ) ) ( ( ) ( ))w sij i j ij ij oij oio ooo oij oio

i j i j

c w s w s c y x x x x

and has variance

222 2

2 2 2 21 21 2 2

( )( )( )( ) .

(1) (2)w w s s

oij oiooio oooij

i j x x x x

x xx xc s

r E E

25

There is no “nice” estimate of this variance. A moderate simplification is obtained by splitting this in two

cases. Consider first whole-plot contrasts at one split-plot level. This implies that the 'ijc s are zero for all

'j j and the general adjusted contrast then becomes

' ' ' ' ' 'ˆˆ ˆ( ( ) ) ,s

ij i j ij ij oij oij oioi i

c w s w s c y x x

and its variance becomes

2 222 2 2 2 ' '

'2 2 21 2 1 2' 2 2

( )( )( ) ( )

.(2) (2)

s s s s

ij oij oiooij oio i

ij iji j i jx x x x

c x xx x

c cr E r E

An unbiased estimate of this variance is given by

2

' 2' '

( )1 1(2) (1)

(2)s s

oij oio a aij ij

i j ix x

x xsc MSE c MSE

sr E sr

.

It is difficult to find its exact degrees of freedom. The Scatterwaite’s formula is used to approximate the

degrees of freedom which is used for test of hypothesis and confidence interval estimation.

For the more general case, the unbiased estimate of the variance is given by

22 222 2( )( ) ( )1 1 1

(2) (1)(1) (2) (1)

w w s s w w

oij oio a aoio ooo oio oooij ij

i j i jx x x x x x

x xx x x xs sc MSE c MSE

sr s E E sr sE

The exact degrees of freedom are difficult to obtain. The degrees of freedom can be obtained using

Scatterthwaite’s approximation.

26

Analysis of covariance with one whole-plot covariate in RBD

We consider illustrate the analysis of covariance for the split-plot experiment with whole-plots in an RBD

and one covariate associated with the whole-plots.

Developing the model

We begin with the model

(1) ( ) (2) ,hij h i hi hi j ij hijy r w x s w s

where 1,..., , 1,..., , and 1,..., ,h r i t j s and the covariate is hix . Assume that both the whole-and split-

plot treatments are fixed effects, implying that _______ _______

( ) ( ) 0.o o oj iow s w s w s The 'hix s are observed

constants. The (1)hi and (2)hij are identically and independently normally distributed each with mean 0

and variances 2 21 2and respectively. Moreover they are mutually independent.

Rewrite the model to isolate the covariate’s contribution to bias and variance. Using

( ) ( ) ( ),hio oo ho oo oi oo hi ho oi oox x x x x x x x x x

the model is rewritten as

* * *

( ) ( ) ( ) (1) ( ) (2)

( ) (1) ( ) (2)

hij oo h ho oo i oi oo hi ho oi oo hi j ij hij

h i hi ho oi oo hi j ij hij

y x r x x w x x x x x x s w s

r w x x x x s w s

where

*

*

* ,

( ),

( ).

oo

h h ho oo

i i oi oo

x

r r x x

w w x x

The terms in * * *, , andh ir w represent the contributions to bias from the experimental units via the covariate,

and ( )hi ho oix x x x represents the contribution to the variance. The analysis of covariance provides

adjustments to remove all of these.

27

Computations in the analysis of covariance

Once the model is constructed, the next step is to construct the compact analysis of covariance table as

follows:

Analysis of covariance table for the split-plot with only a whole-plot covariate

Source y variable Covariate

Mean yyM xyM xxM

Blocks yyB xyB xxB

W yyW xyW xxW

(1)Error (1) yyE (1)xyE (1)xxE

S yyS

W S yyW S

(2)Error (2) yyE

Total yyT xyT xxT

There are no covariance adjustments in the split-plot stratum. In the whole-plot stratum we have

2 22 1

2

2 2

(1)ˆ(1)

( )ˆ( )(1)

(1)1(1) (1)

[( 1)( 1) 1] (1)

[ (1) ] (1)1

( 1) (1) (1)

xy

xx

xx

xyayy

xx

xy xy xyayy

xx xx xx

E

E

sVar

E

EMSE E

r t E

W E EMSW W

t W E E

28

Contrasts

Since there is only a whole-plot covariate, so only the whole-plot treatment contrasts are adjusted. Consider

the contrast

ˆˆ ( )i oio i i i oi ooi i i

c y c w c x x

with 0.ii

c Rewrite this as

ˆˆ .i i i oio i oioi i i

c w c y c x

This contrast has variance

2

2 2 22 1

1.

(1)

i oii

ii xx

c x

c srs E

The estimate of 2 22 1( ) with ( 1)( 1) 1s r t degrees of freedom is given by (1)aMSE .