Principles
We deal with split plot and repeated measures designs in the same More Information page because they can both be described as partially nested designs. Partially nested designs have both crossed and nested factors and include splitplot designs and repeated measures designs. Both types of designs are commonly analyzed with the same family of linear models. Hence you may find data from a repeated measures design being analyzed with a 'split plot' analysis of variance (see one of our examples ). Despite the use of the same family of models, there are some important differences between splitplot and repeated measures designs especially in relation to randomization and assumptions.
The principle of a splitplot design is that different treatments are assigned to sampling units at different scales. So levels of factor A are assigned to mainplots (usually termed blocks), whilst levels of factor B are assigned to plots within each block. Levels of factor C may be assigned to subplots within each plot  and so on... There is commonly only one observation of each treatment combination within a particular sampling unit which invariably means that interactions between the treatment factors and sampling units are confounded (cannot be tested).
There are many different splitplot designs
Repeated measures. subjects by trials design  time is an inherent component of the factor and treatment order cannot be randomized
subjects × treatment designs
For those of you trying to relate the designs below to those given in the rather voluminous literature, we have included the appropriate references for each design.
Splitplot designs
Simple splitplot design
In this design 'n' replicate whole blocks (S) are randomly assigned to each of 'a' levels of treatment A, and each of b levels of treatment B is randomly assigned to one of 'b' plots in each block. This is the splitplot design 11.1 given in Quinn & Keough (2002), the splitplot design 12.4 given in Underwood (1997), the twofactor splitplot model (ii) 5.6 given in Doncaster & Davey (2007) and the splitplot design in section 5.13 in Winer et al (1991).
Splitplot design

Blocks:
 I

A_{1}B_{2
}  A_{1}B_{1
}  A_{1}B_{3
} 

II

A_{2}B_{2
}  A_{2}B_{1
}  A_{2}B_{3
} 

III

A_{1}B_{1
}  A_{1}B_{2
}  A_{1}B_{3
} 

IV

A_{2}B_{3
}  A_{2}B_{2
}  A_{2}B_{1
} 
  
In the figure we have (n=) 2 replicate blocks per treatment level, (a=) 2 levels of treatment A (A_{1}, A_{2}) and (b=) 3 levels of treatment B (B_{1}, B_{2}, B_{3}). We have assumed there is only one replicate of each level of B in each block, although multiple replicates may be used.
Model & expected mean squares
Factors A & B fixed, Factor S random
Y_{ijk} = μ
 +
 α_{i
}  +
 S_{k(i)}
 +
 β_{j
}  +
 αβ_{ij}
 +
 [βS_{jk(i)}]
 +
 ε_{ijk
}   
where:
 Y_{ijk} is the observation of the ith level of factor A (main plot treatment) and the jth level of factor B (subplot treatment) for block k,
 μ is the population (grand) mean, and α_{i} is the fixed effect of level i of factor A,
 S_{k(i)} is the effect of block nested within A, and β_{j} is the fixed effect of level j of factor B,
 αβ_{ij} is the interaction effect between factors A and B,
 ε_{ijk} is the plot random error effect. Note that if there is no replication for each combination of block and factor B, the [β × S] interaction effect cannot be separated from the error term and must be assumed to be zero.

Source of variation
 df
 Expected MS
 Variance ratio


1.  Factor A
 a1
 σ^{2
}  + bσ^{2}_{S{α}}
 + nbΣα^{2}/(a1)
 MS_{1}/MS_{2
} 
2.  S(A)
 a(n1)
 σ^{2
}  + bσ^{2}_{S{α}}



3.  Factor B
 b1
 σ^{2
}  + σ^{2}_{βS(α)
}  + naΣβ^{2}/(b1)
 MS_{3}/MS_{5
} 
4.  A x B
 (a1)(b1)
 σ^{2
}  + σ^{2}_{βS(α)
}  + nΣ
(αβ)^{2}/((a1)(b1))
 MS_{4}/MS_{5
} 
5.  B×S(A)
 (n1)(b1)a
 σ^{2
}  + σ^{2}_{βS(α)
} 


Total variation
 N1




  
where:
 a is the number of levels of factor A, b is the number of levels of factor B,
 n is the number of replicate blocks per treatment, N is the total number of observations,
 S(A) and B×S(A) are also known as the mainplot error and subplot error respectively;
 σ^{2} is the error variance and bσ^{2}_{S{α}} is the blocks within treatments variance component,
 nbΣα^{2}/(a1) is the added treatment component (A),
 σ^{2}_{βS(α)} is the plot variance component (S(A)),
 naΣβ^{2}/(b1) is the added treatment component (B),
 nΣ
(αβ)^{2}/((a1)(b1)) is the interaction component (A×B).

The Fratio for factor A is obtained by dividing MS_{A} by MS_{S(A)} (= mainplot error). The Fratios for factor B and A × B are obtained by dividing their respective mean squares by MS_{S × B(A)}. The S×B(A) interaction cannot be tested because lack of replication means the error term cannot be assessed.
We do not include computational formulae for this design as they are identical to those for the two factor repeated measures design below.
Splitplot design in randomized blocks
In this design each of 'a' levels of treatment A is randomly allocated to one of 'a' plots in each of 's' blocks (S) and each of 'b' levels of treatment B is randomly allocated to one of 'b' subplots in in each plot. This is the splitplot design 12.6 given in Underwood (1997) and the twofactor splitplot model (i) 5.1 given in Doncaster & Davey (2007) It is similar to the splitplot design 11.8 given in Quinn & Keough (2002).
Splitplot design in randomized blocks
  plot 1  plot 2

Blocks:
 I

A_{1}B_{2
}  A_{1}B_{1
}  A_{1}B_{3
} 

A_{2}B_{3
}  A_{2}B_{1
}  A_{2}B_{2
} 


II

A_{2}B_{2
}  A_{2}B_{1
}  A_{2}B_{3
} 

A_{1}B_{3
}  A_{1}B_{1
}  A_{1}B_{2
} 


III

A_{2}B_{3
}  A_{2}B_{2
}  A_{2}B_{1
} 

A_{1}B_{3
}  A_{1}B_{2
}  A_{1}B_{1
} 


IV

A_{1}B_{1
}  A_{1}B_{2
}  A_{1}B_{3
} 

A_{2}B_{2
}  A_{2}B_{3
}  A_{2}B_{1
} 
  
In the figure we have (s=) 4 blocks each with 2 plots allocated to one or other of the (a=) 2 levels of treatment A (A_{1}, A_{2}); each plot has 3 subplots which are each allocated one of the (b=) 3 levels of treatment B (B_{1}, B_{2}, B_{3}).
Model & expected mean squares
Factors A & B fixed, Factor S random
Y_{ijk} = μ
 +
 S_{k
}  +
 α_{i
}  +
 δ_{ik
}  +
 β_{j
}  +
 αβ_{ij}
 +
 ε_{ijk
}   
where:
 Y_{ijk} is the observation of the ith level of factor A (main plot treatment) and the jth level of factor B (subplot treatment) for the kth level of S (block),
 μ is the population (grand) mean,
 S_{k} is the random effect of the kth block, and α_{i} is the fixed effect of level i of factor A,
 δ_{ik} is the main plot random error effect. It includes the confounded [S × α] interaction which is assumed to be zero.
 β_{j} is the fixed effect of level j of factor B,
 αβ_{ij} is the interaction effect between factors A and B,
 ε_{ijk} is the subplot random error effect. It includes [S × β] and [S × α × β] interaction effects, as well as the confounded β × plot(S×A) term which must be assumed to be zero.

Source of variation
 df
 Expected MS
 Variance ratio


1.  Blocks
 s1
 σ^{2
}  + bσ^{2}_{d
}  + abσ^{2}_{S
} 

2.  Factor A
 a1
 σ^{2
}  + bσ^{2}_{d
}  + sbΣα^{2}/(a1)
 MS_{2}/MS_{3
} 
3.  Main plot error
 (s1)(a1)
 σ^{2
}  + bσ^{2}_{d
} 


4.  Factor B
 b1
 σ^{2
} 
 + saΣβ^{2}/(b1)
 MS_{4}/MS_{6
} 
5.  A x B
 (a1)(b1)
 σ^{2
} 
 + sΣ
(αβ)^{2}/((a1)(b1))
 MS_{5}/MS_{6
} 
6.  Subplot error
 (s1)(b1)a
 σ^{2
} 



Total variation
 N1




  
where:
 a is the number of levels of factor A, b is the number of levels of factor B,
 s is the number of blocks, N is the total number of observations,
 σ^{2} is the error variance,
 bσ^{2}_{d} is the main plot variance component,
 asσ^{2}_{S} is the block variance component,
 sbΣα^{2}/(a1) is the added treatment component (A),
 saΣβ^{2}/(b1) is the added treatment component (B),
 sΣ
(αβ)^{2}/((a1)(b1)) is the interaction component (AxB).

The Fratio for factor A is obtained by dividing MS_{A} by MS_{mainplot error}. The Fratios for factor B and A × B are obtained by dividing their respective mean squares by MS_{Subplot error}. The S×B(A) interaction cannot be tested because lack of replication means it cannot be separated from the true subplot error.
Computational formulae
We take a balanced experiment with of 'a' levels of treatment A and 'b' levels of treatment B arranged in a split plot design in 's' blocks . Factor A totals are denoted as TA_{1} to TA_{a}, Factor B totals are denoted as TB_{1} to TB_{b} and block totals as TS_{1} to TS_{s} and the grand total as G.
The sums of squares are calculated as follows:
Algebraically speaking 
SS_{Total
}  =
 Σ(
 Y_{ijk}^{2
}  )
 −
 G^{2
} 

N

 
where:
 SS_{Total} is the total sums of squares,
 Y_{ijk} is the value of the ijkth observation in factor A group i, factor B group j and block k,
 G is the overall total.
SS_{S (Blocks)
}  =  Σ(
 TS_{k}^{2
}  )
 −
 G^{2
} 
 
ab  N

 
where:
 SS_{S} is the blocks sums of squares
 TS_{k} is the sum of the observations in block k,
 a and b are the number of levels of treatment A and B respectively
SS_{A
}  =  Σ(
 TA_{i}^{2
}  )
 −
 G^{2
} 

 
bs  N
  
where:
 SS_{A} is the sums of squares for factor A,
 TA_{i} is the sum of the observations in factor A group i,
 b is the number of levels of treatment b and s is the number of blocks
SS_{Subgrp A×S
}  =  Σ(
 T(AS)_{ik}^{2
}  )
 −
 G^{2
} 

 
b  N
  
where:
 SS_{Subgrp A×S} is the sums of squares for the A×S subgroups,
 T(AS)_{ik} are the totals for each A×S combination
SS_{Main plot error} = SS_{Subgrp A×S}  SS_{A}  SS_{S}

SS_{B
}  =  Σ(
 TB_{j}^{2
}  )
 −
 G^{2
} 

 
as  N
  
where:
 SS_{B} is the sums of squares for factor b,
 TB_{j} is the sum of the observations in factor B group j.
SS_{Subgrp A×B
}  =  Σ(
 T(AB)_{ij}^{2
}  )
 −
 G^{2
} 

 
b  N
  
where:
 SS_{Subgrp A×B} is the sums of squares for the A×B subgroups,
 T(AB)_{ij} are the totals for each A×B combination
SS_{A×B} = SS_{Subgrp A×B}  SS_{A}  SS_{B}

where:
 SS_{A×B} is the sums of squares for the A×B interaction.
SS_{subplot error
}  =  SS_{Total}  SS_{S}  SS_{A }  SS_{Main plot error}  SS_{B}  SS_{A×B}


Subjects x trials designs
Two factor repeated measures design
In this design 'n' replicate subjects (S) are randomly assigned to each of 'a' levels of treatment A, and repeated observations are made on each subject at each of 'b' levels of factor B (time). It is the same as model 6.3 with repeated measures on one cross factor given in Doncaster & Davey (2007) It is identical to the first splitplot design given above, and as such is the same as design 11.1 given in Quinn & Keough (2002).
Two factor repeated measures design

Subjects:
 I

A_{1}B_{1
}  A_{1}B_{2
}  A_{1}B_{3
} 

II

A_{2}B_{1
}  A_{2}B_{2
}  A_{2}B_{3
} 

III

A_{1}B_{1
}  A_{1}B_{2
}  A_{1}B_{3
} 

IV

A_{2}B_{1
}  A_{2}B_{2
}  A_{2}B_{3
} 
  
In the figure we have (n=) 2 replicate subjects per treatment level, (a=) 2 levels of treatment (A: A_{1} & A_{2}) and (b=) 3 levels of time (B: B_{1}, B_{2} & B_{3}).
Model & expected mean squares
Factors A & B fixed, Factor S random
Y_{ijk} = μ
 +
 α_{i
}  +
 S_{k(i)}
 +
 β_{j
}  +
 αβ_{ij}
 +
 [βS_{jk(i)}]
 +
 ε_{ijk
}   
where:
 Y_{ijk} is the observation of the ith level of factor A and the jth level of factor B (time) for subject k,
 μ is the population (grand) mean,
 α_{i} is the fixed effect of level i of factor A,
 S_{k(i)} is the effect of subject nested within A,
 β_{j} is the fixed effect of level j of factor B,
 αβ_{ij} is the interaction effect between factors A and B,
 ε_{ijk} is the random error effect. Since there is no replication for each combination of subject and factor B, the [β × S] interaction effect cannot be separated from the error term and must be assumed to be zero.

Source of variation
 df
 Expected MS
 Variance ratio


1.  Treatment (A)
 a1
 σ^{2
}  +bσ^{2}_{S(α)
}  + nbΣ
α^{2}/(a1)
 MS_{1}/MS_{2
} 
2.  S(A)
 a(n1)
 σ^{2
}  +bσ^{2}_{S(α)
} 


3.  Time (B)
 b1
 σ^{2
}  + σ^{2}_{βS(α)
}  + naΣβ^{2}/(b1)
 MS_{3}/MS_{5
} 
4.  A × B
 (a1)(b1)
 σ^{2
}  + σ^{2}_{βS(α)
}  + nΣ
(αβ)^{2}/((a1)(b1))
 MS_{4}/MS_{5
} 
5.  B × S(A)
 a(b1)(n1)
 σ^{2
}  + σ^{2}_{βS(α)}



Total variation
 N1




  
where:
 a is the number of levels of factor A (treatment), b is the number of levels of factor B (time),
 n is the number of replicate subjects per treatment, N is the total number of observations,
 σ^{2} is the error variance,
 bσ^{2}_{S
(α)} is the subjects within treatments variance component,
 nbΣα^{2}/(a1) is the added treatment component (A),
 naΣβ^{2}/(b1) is the added time component (B),
 nΣ
(αβ)^{2}/((a1)(b1)) is the interaction component (AxB).

Computational formulae
We take a balanced experiment with 'k' replicate subjects (S) randomly assigned to each of 'a' levels of treatment A, and repeated observations are made on each subject at each level of factor B (time). Factor A totals are denoted as TA_{1} to TA_{a}, Factor B totals are denoted as TB_{1} to TB_{b} and subject totals as TS_{1} to TS_{s} and the grand total as G.
The sums of squares are calculated as follows:
Algebraically speaking 
SS_{Total
}  =
 Σ(
 Y_{ijk}^{2
}  )
 −
 G^{2
} 

N

 
where:
 SS_{Total} is the total sums of squares,
 Y_{ijk} is the value of the ijkth observation in treatment (A) level i, time (B) level j and subject k,
 G is the overall total.
SS_{S (Subjects)
}  =  Σ(
 TS_{k}^{2
}  )
 −
 G^{2
} 
 
ab  N

 
where:
 SS_{S} is the subjects sums of squares
 TS_{k} is the sum of the observations in subject k,
 a and b are the number of levels of treatment and time respectively
SS_{A
}  =  Σ(
 TA_{i}^{2
}  )
 −
 G^{2
} 

 
bs  N
  
where:
 SS_{A} is the treatment sums of squares (factor A),
 TA_{i} is the sum of the observations in factor A group i,
 b is the number of levels of time and s is the number of subjects
SS_{Subjects within A} = SS_{SubjectsS}  SS_{A} 
SS_{B
}  =  Σ(
 TB_{j}^{2
}  )
 −
 G^{2
} 

 
as  N
  
where:
 SS_{B} is the sums of squares for factor b,
 TB_{j} is the sum of the observations in factor B group j.
SS_{Subgrp A×B
}  =  Σ(
 T(AB)_{ij}^{2
}  )
 −
 G^{2
} 

 
b  N
  
where:
 SS_{Subgrp A×B} is the sums of squares for the A×B subgroups,
 T(AB)_{ij} are the totals for each A×B combination
SS_{A×B} = SS_{Subgrp A×B}  SS_{A}  SS_{B}

where:
 SS_{A×B} is the sums of squares for the A×B interaction.
SS_{residual
}  =  SS_{Total}  SS_{S}  SS_{A }  SS_{Main plot error}  SS_{B}  SS_{A×B}


Subjects x treatments designs
Changeover design
With the changeover design it is assumed that there is no period effect. The data are analyzed with the randomized complete block ANOVA with subjects as blocks. Hence although time period appears as in the data table, it does not appear in the analysis  it would make no sense to include it because treatments are not balanced between time periods (unlike in the Latin square design below).
Changeover design

Subjects  Time period

1  2  3

1
 47 (A_{1})
 23 (A_{2})
 28 (A_{3})

2
 31 (A_{2})
 24 (A_{3})
 51 (A_{1})

3
 21 (A_{3})
 37 (A_{1})
 18 (A_{2})

4
 17 (A_{2})
 29 (A_{1})
 29 (A_{3})
  
Model & expected mean squares
Factors A fixed, Factor S random
Y_{ij} = μ
 +
 S_{j
}  +
 α_{i
}  +
 [(αS)_{ij}]
 +
 ε_{ij
}   
where:
 Y_{ij} is the observation for treatment i in subject j,
 μ is the population (grand) mean,
 S_{j} is the random effect for the jth subject,
 α_{i} is the fixed effect for the ith level of factor A,
 [(αS)_{ij}] is the interaction effect between treatments and subjects which is assumed to be zero,
 ε_{ij} is the random error effect

Source of variation
 df
 Expected MS
 VC estimate or Fratio


1.  Subjects
 s1
 σ^{2
} 
 + aσ^{2}_{S
}  s_{S}^{2} = (MS_{1}  MS_{3})/a

2.  Treatment
 a1
 σ^{2
}  + [σ^{2}_{αS}]
 + rΣα^{2}/(a1)
 MS_{2}/MS_{3
} 
3.  Remainder
 ar
 σ^{2
}  + [σ^{2}_{αS}]



Total variation
 N1




  
where
 a is the number of levels of the treatment factor (A),
 s is the number of subjects and N is the total number of observations (= as),
 σ^{2} is the error variance
 [σ^{2}_{αS}] is the confounded treatment x subjects interaction component,
 rΣα^{2}/(a1) is the added treatment component,
 aσ^{2}_{S} is the subject variance component.

Two period crossover design
The two period crossover design is analysed with the standard two factor repeated measures ANOVA  not as one might expect with a three factor ANOVA. This is because the two period crossover is a heavily confounded design  we do not have sufficient information to assess all three factors (sequence group, time period and treatment) together with their interactions. Each estimate of a main effect also estimates a twofactor interaction. Hence any difference between groups is confounded (or aliased) with the treatment × period interaction, any difference between treatments is confounded with the group × period interaction and any difference between periods is confounded with the group × treatment interaction. Since subjects are assigned to groups at random, we assume there is no difference between groups nor any significant interactions with groups. Hence a significant group effect is assumed to arise from a treatment × period interaction, a significant group × treatment interaction is assumed to arise from a significant period interaction and a significant treatment effect does indeed result fron significant differences between treatments.
Model
Factors A & B fixed, factor S random
Y_{ijk} = μ
 +
 A_{i
}  +
 S_{k{i}
}  +
 B_{j
}  +
 AB_{ij
}  +
 ε_{ijk
}   
where:
 Y_{ijk} is the observation for the kth subject at the ith level of factor A and the jth level of factor B.
 μ is the population (grand) mean,
 A_{i} is the fixed effect of the sequence group,
 S_{k{i}} is the random effect of subject nested within group,
 B_{j} is the fixed effect of factor A,
 AB_{ij} is the interaction term between factors A and B,
 ε_{ijk} is the error term.

Latin square repeated measures design
Model & expected mean squares
Factors A, B & C fixed, factor S random
Y_{ijkm} = μ
 +
 C_{m
}  +
 S_{k{m}
}  +
 A_{i(s)
}  +
 B_{j(s)
}  +
 AC_{im
}  +
 BC_{im
}  +
 [res_{(s)}]
 +
 ε_{ijkm
}   
where:
 Y_{ijkm} is the observation for the kth position at the ith level of factor A, the jth level of factor B, in the mth level of factor C.
 μ is the population (grand) mean,
 C_{m} is the fixed effect of factor C,
 S_{k{m}} is the random effect of position (subject) k nested within C,
 A_{i(s)} is the fixed effect of factor A estimated from a latin square,
 B_{j(s)} is the fixed effect of factor B estimated from a latin square,
 AC_{im} is the interaction term between factors A and C,
 BC_{jm} is the interaction term between factor B and C,
 res_{(s)} represents the residual component estimated from a latin square,
 ε_{ijk} is the error term.

Source of variation
 df
 Expected MS
 Variance ratio


1.  Factor C
 c1
 σ^{2
}  + aσ^{2}_{S(C)
}  + a^{2}Σ
C^{2}/(c1)
 MS_{1}/MS_{2
} 
2.  Subjects w'in C
 c(a1)
 σ^{2
}  + aσ^{2}_{S(C)
} 


3.  Time (A)
 b1
 σ^{2
} 
 + cbΣ
A^{2}/(a1)
 MS_{3}/MS_{7
} 
4.  Treatment (B)
 a1
 σ^{2
} 
 + caΣ
B^{2}/(a1)
 MS_{4}/MS_{7
} 
5.  C × A
 (c1)(a1)
 σ^{2
} 
 + aΣ
CA/(c1)(a1)
 MS_{5}/MS_{7
} 
6.  C × B
 (c1)(a1)
 σ^{2
} 
 + aΣ
CB/(c1)(a1)
 MS_{6}/MS_{7
} 
7.  residual
 c(a1)(a2)
 σ^{2
}  + [σ^{2}_{res}]



Total variation
 N1




  
where:
 c is the number of levels of factor C (area),
 a is the number of levels of factor A (time), the number of levels of factor B (treatment), and the number of subjects per square
 σ^{2} is the error variance,
 aσ^{2}_{S(C)} is the subjects within area variance component,
 caΣA^{2}/(a1) is the added time component (A),
 caΣB^{2}/(a1) is the added treatment component (B),
 nΣ
(CA)^{2}/((c1)(a1)) is the interaction component (C×A).
 nΣ
(CB)^{2}/((c1)(a1)) is the interaction component (C×B).
