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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 split-plot 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 split-plot and repeated measures designs especially in relation to randomization and assumptions.

The principle of a split-plot 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 split-plot 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.

 

Split-plot designs

Simple split-plot design

Split-plot design
   Blocks:
I
A1B2 A1B1 A1B3
II
A2B2 A2B1 A2B3
III
A1B1 A1B2 A1B3
IV
A2B3 A2B2 A2B1
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 split-plot design 11.1 given in Quinn & Keough (2002), the split-plot design 12.4 given in Underwood (1997), the two-factor split-plot model (ii) 5.6 given in Doncaster & Davey (2007) and the split-plot design in section 5.13 in Winer et al (1991).

In the figure we have (n=) 2 replicate blocks per treatment level, (a=) 2 levels of treatment A (A1, A2) and (b=) 3 levels of treatment B (B1, B2, B3). 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
Yijk  =  μ  +  αi  +  Sk(i)  +  βj  +  αβij  +  [βSjk(i)]  +  εijk
where:
  • Yijk is the observation of the ith level of factor A (main plot treatment) and the jth level of factor B (sub-plot treatment) for block k,
  • μ is the population (grand) mean, and αi is the fixed effect of level i of factor A,
  • Sk(i) is the effect of block nested within A, and βj is the fixed effect of level j of factor B,
  • αβij is the interaction effect between factors A and B,
  • εijk 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 a-1 σ2 + bσ2S{α} + nbΣα2/(a-1) MS1/MS2
2.S(A) a(n-1) σ2 + bσ2S{α}  
3.Factor B b-1 σ2 + σ2βS(α) + naΣβ2/(b-1) MS3/MS5
4.A x B (a-1)(b-1) σ2 + σ2βS(α) + nΣ (αβ)2/((a-1)(b-1)) MS4/MS5
5.B×S(A) (n-1)(b-1)a σ2 + σ2βS(α)    
Total variation N-1        
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;
  • σ2 is the error variance and 2S{α} is the blocks within treatments variance component,
  • nbΣα2/(a-1) is the added treatment component (A),
  • σ2βS(α) is the plot variance component (S(A)),
  • naΣβ2/(b-1) is the added treatment component (B),
  • nΣ (αβ)2/((a-1)(b-1)) is the interaction component (A×B).

The F-ratio for factor A is obtained by dividing MSA by MSS(A) (= mainplot error). The F-ratios for factor B and A × B are obtained by dividing their respective mean squares by MSS 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.

 

Split-plot design in randomized blocks

Split-plot design
in randomized blocks
 plot 1plot 2
Blocks: I
A1B2 A1B1 A1B3
A2B3 A2B1 A2B2
II
A2B2 A2B1 A2B3
A1B3 A1B1 A1B2
III
A2B3 A2B2 A2B1
A1B3 A1B2 A1B1
IV
A1B1 A1B2 A1B3
A2B2 A2B3 A2B1

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 split-plot design 12.6 given in Underwood (1997) and the two-factor split-plot model (i) 5.1 given in Doncaster & Davey (2007) It is similar to the split-plot design 11.8 given in Quinn & Keough (2002).

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 (A1, A2); each plot has 3 subplots which are each allocated one of the (b=) 3 levels of treatment B (B1, B2, B3).

Model & expected mean squares

Factors A & B fixed, Factor S random
Yijk  =  μ  +  Sk  +  αi  +  δik  +  βj  +  αβij  +  εijk
where:
  • Yijk is the observation of the ith level of factor A (main plot treatment) and the jth level of factor B (sub-plot treatment) for the kth level of S (block),
  • μ is the population (grand) mean,
  • Sk is the random effect of the kth block, and αi is the fixed effect of level i of factor A,
  • δik is the main plot random error effect. It includes the confounded [S α] interaction which is assumed to be zero.
  • βj is the fixed effect of level j of factor B,
  • αβij is the interaction effect between factors A and B,
  • εijk is the subplot random error effect. It includes [S β] and [S α β] interaction effects, as well as the confounded β plot(SA) term which must be assumed to be zero.

Source of variation df Expected MS Variance ratio
1.Blocks s-1 σ2 + bσ2d + abσ2S
2.Factor A a-1 σ2 + bσ2d + sbΣα2/(a-1) MS2/MS3
3.Main plot error (s-1)(a-1) σ2 + bσ2d    
4.Factor B b-1 σ2   + saΣβ2/(b-1) MS4/MS6
5.A x B (a-1)(b-1) σ2   + sΣ (αβ)2/((a-1)(b-1)) MS5/MS6
6.Subplot error (s-1)(b-1)a σ2      
Total variation N-1        
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,
  • σ2 is the error variance,
  • 2d is the main plot variance component,
  • asσ2S is the block variance component,
  • sbΣα2/(a-1) is the added treatment component (A),
  • saΣβ2/(b-1) is the added treatment component (B),
  • sΣ (αβ)2/((a-1)(b-1)) is the interaction component (AxB).

The F-ratio for factor A is obtained by dividing MSA by MSmainplot error. The F-ratios for factor B and A × B are obtained by dividing their respective mean squares by MSSubplot 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 TA1 to TAa, Factor B totals are denoted as TB1 to TBb and block totals as TS1 to TSs and the grand total as G.

The sums of squares are calculated as follows:

Algebraically speaking -
SSTotal   =   Σ( Yijk2 )   −   G2
N
where:
  • SSTotal is the total sums of squares,
  • Yijk is the value of the ijkth observation in factor A group i, factor B group j and block k,
  • G is the overall total.

SSS (Blocks) =Σ( TSk2 ) G2
abN
where:
  • SSS is the blocks sums of squares
  • TSk is the sum of the observations in block k,
  • a and b are the number of levels of treatment A and B respectively

SSA =Σ( TAi2 ) G2  
bsN
where:
  • SSA is the sums of squares for factor A,
  • TAi 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

SSSubgrp A×S =Σ( T(AS)ik2 ) G2  
bN
where:
  • SSSubgrp A×S is the sums of squares for the A×S subgroups,
  • T(AS)ik are the totals for each A×S combination

SSMain plot error = SSSubgrp A×S - SSA - SSS

SSB =Σ( TBj2 ) G2  
asN
where:
  • SSB is the sums of squares for factor b,
  • TBj is the sum of the observations in factor B group j.

SSSubgrp A×B =Σ( T(AB)ij2 ) G2  
bN
where:
  • SSSubgrp A×B is the sums of squares for the A×B subgroups,
  • T(AB)ij are the totals for each A×B combination

SSA×B = SSSubgrp A×B - SSA - SSB
where:
  • SSA×B is the sums of squares for the A×B interaction.

SSsubplot error =SSTotal - SSS - SSA - SSMain plot error - SSB - SSA×B

 

 

Subjects x trials designs

Two factor repeated measures design

Two factor repeated
measures design
   Subjects:
I
A1B1 A1B2 A1B3
II
A2B1 A2B2 A2B3
III
A1B1 A1B2 A1B3
IV
A2B1 A2B2 A2B3

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 split-plot design given above, and as such is the same as design 11.1 given in Quinn & Keough (2002).

In the figure we have (n=) 2 replicate subjects per treatment level, (a=) 2 levels of treatment (A: A1 & A2) and (b=) 3 levels of time (B: B1, B2 & B3).

Model & expected mean squares

Factors A & B fixed, Factor S random
Yijk  =  μ  +  αi  +  Sk(i)  +  βj  +  αβij  +  [βSjk(i)]  +  εijk
where:
  • Yijk 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,
  • αi is the fixed effect of level i of factor A,
  • Sk(i) is the effect of subject nested within A,
  • βj is the fixed effect of level j of factor B,
  • αβij is the interaction effect between factors A and B,
  • εijk 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) a-1 σ2 +bσ2S(α) + nbΣ α2/(a-1) MS1/MS2
2.S(A) a(n-1) σ2 +bσ2S(α)    
3.Time (B) b-1 σ2 + σ2βS(α) + naΣβ2/(b-1) MS3/MS5
4.A × B (a-1)(b-1) σ2 + σ2βS(α) + nΣ (αβ)2/((a-1)(b-1)) MS4/MS5
5.B × S(A) a(b-1)(n-1) σ2 + σ2βS(α)    
Total variation N-1        
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,
  • σ2 is the error variance,
  • 2S (α) is the subjects within treatments variance component,
  • nbΣα2/(a-1) is the added treatment component (A),
  • naΣβ2/(b-1) is the added time component (B),
  • nΣ (αβ)2/((a-1)(b-1)) 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 TA1 to TAa, Factor B totals are denoted as TB1 to TBb and subject totals as TS1 to TSs and the grand total as G.

    The sums of squares are calculated as follows:

    Algebraically speaking -
    SSTotal   =   Σ( Yijk2 )   −   G2
    N
    where:
    • SSTotal is the total sums of squares,
    • Yijk is the value of the ijkth observation in treatment (A) level i, time (B) level j and subject k,
    • G is the overall total.

    SSS (Subjects) =Σ( TSk2 ) G2
    abN
    where:
    • SSS is the subjects sums of squares
    • TSk is the sum of the observations in subject k,
    • a and b are the number of levels of treatment and time respectively

    SSA =Σ( TAi2 ) G2  
    bsN
    where:
    • SSA is the treatment sums of squares (factor A),
    • TAi 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

    SSSubjects within A = SSSubjectsS - SSA

    SSB =Σ( TBj2 ) G2  
    asN
    where:
    • SSB is the sums of squares for factor b,
    • TBj is the sum of the observations in factor B group j.

    SSSubgrp A×B =Σ( T(AB)ij2 ) G2  
    bN
    where:
    • SSSubgrp A×B is the sums of squares for the A×B subgroups,
    • T(AB)ij are the totals for each A×B combination

    SSA×B = SSSubgrp A×B - SSA - SSB
    where:
    • SSA×B is the sums of squares for the A×B interaction.

    SSresidual =SSTotal - SSS - SSA - SSMain plot error - SSB - SSA×B

     

     

    Subjects x treatments designs

    Changeover design

    Change-over design
    SubjectsTime period
    123
    1 47 (A1) 23 (A2) 28 (A3)
    2 31 (A2) 24 (A3) 51 (A1)
    3 21 (A3) 37 (A1) 18 (A2)
    4 17 (A2) 29 (A1) 29 (A3)
    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).

    Model & expected mean squares

    Factors A fixed, Factor S random
    Yij  =  μ  +  Sj  +  αi  +  [(αS)ij]  +  εij
    where:
    • Yij is the observation for treatment i in subject j,
    • μ is the population (grand) mean,
    • Sj is the random effect for the jth subject,
    • αi 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,
    • εij is the random error effect


    Source of variation df Expected MS VC estimate
    or F-ratio
    1.Subjects s-1 σ2   + aσ2S sS2 = (MS1 - MS3)/a
    2.Treatment a-1 σ2 + [σ2αS] + rΣα2/(a-1) MS2/MS3
    3.Remainder ar σ2 + [σ2αS]    
    Total variation N-1        
    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),
    • σ2 is the error variance
    • 2αS] is the confounded treatment x subjects interaction component,
    • rΣα2/(a-1) is the added treatment component,
    • 2S 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 two-factor 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
    Yijk  =  μ  +  Ai  +  Sk{i}  +  Bj  +  ABij  +  εijk
    where:
    • Yijk 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,
    • Ai is the fixed effect of the sequence group,
    • Sk{i} is the random effect of subject nested within group,
    • Bj is the fixed effect of factor A,
    • ABij is the interaction term between factors A and B,
    • εijk is the error term.

     

    Latin square repeated measures design

    Model & expected mean squares

    Factors A, B & C fixed, factor S random
    Yijkm  =  μ  +  Cm  +  Sk{m}  +  Ai(s)  +  Bj(s)  +  ACim  +  BCim  +  [res(s)]  +  εijkm
    where:
    • Yijkm 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,
    • Cm is the fixed effect of factor C,
    • Sk{m} is the random effect of position (subject) k nested within C,
    • Ai(s) is the fixed effect of factor A estimated from a latin square,
    • Bj(s) is the fixed effect of factor B estimated from a latin square,
    • ACim is the interaction term between factors A and C,
    • BCjm is the interaction term between factor B and C,
    • res(s) represents the residual component estimated from a latin square,
    • εijk is the error term.

    Source of variation df Expected MS Variance ratio
    1.Factor C c-1 σ2 + aσ2S(C) + a2Σ C2/(c-1) MS1/MS2
    2.Subjects w'in C c(a-1) σ2 + aσ2S(C)    
    3.Time (A) b-1 σ2   + cbΣ A2/(a-1) MS3/MS7
    4.Treatment (B) a-1 σ2   + caΣ B2/(a-1) MS4/MS7
    5.C A (c-1)(a-1) σ2   + aΣ CA/(c-1)(a-1) MS5/MS7
    6.C B (c-1)(a-1) σ2   + aΣ CB/(c-1)(a-1) MS6/MS7
    7.residual c(a-1)(a-2) σ2 + [σ2res]    
    Total variation N-1        
    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
  • σ2 is the error variance,
  • 2S(C) is the subjects within area variance component,
  • caΣA2/(a-1) is the added time component (A),
  • caΣB2/(a-1) is the added treatment component (B),
  • nΣ (CA)2/((c-1)(a-1)) is the interaction component (CA).
  • nΣ (CB)2/((c-1)(a-1)) is the interaction component (CB).
  •