Randomized block designs are similar to two-way randomized group designs except that one of the factors is a control variable instead of an experimental variable. This control variable is frequently referred to as a blocking factor.  The important difference is that individuals or groups are not randomized across the control (or organismic) variable. This means that individuals or groups will be related to each other across the levels of the experimental variable. This will become clear in the following description of the design.

The randomized block design is used to reduce the amount of experimental error and/or to explore the effects of the experimental variable on levels of some organismic variable. Blocks are formed on the basis of the control variable--which may take the form of  intellectual aptitude, social class, gender, age, etc. For example, an experiment comparing three different teaching methods with ninety subjects, might first order the subjects according to academic ability as measured by some test. The ninety subjects could then be divided into anywhere from three to thirty groups or blocks on the basis of their ability. Next, each group of subjects would be distributed at random among the three teaching methods (members of the block are distributed randomly accross the levels of the experimental factor). The experimental variable in this case would be teaching method.  In summary, the randomized block design is created by forming blocks of individuals that are alike with respect to the control condition, and the individulas within a block are then distributed at random across the treatment conditions, thus creating a two-way analysis of variance. Where all assumptions are met, and all conditions are equal, the randomized block design can be a more powerful design than a comparable randomized group design.

The simple randomized block design will receive special consideration because it plays an important role in many other designs and because it involves the case where their is just one observation per cell. Because it has only one observation per cell it differs in significant ways from the randomized groups design.

In the simple randomized block design the 90 students mentioned above would be divided into 30 groups of three subjects each. Each group of three subjects would then be randomely assigned to one of the three treatment conditions resulting in one individual per cell The design for a standard randomized block design would look as follows:
 
 

	B1	...	Bj	...	Bb
A1	X11	...	X1j	...	X1b	1.
...		...		...
Ai	Xi1	...	Xij	...	Xib	i.
...		...		...
Aa	Xa1	...	Xaj	...	Xab	a.
	.1		.j		.b	..

Where A is the blocking factor and B is the treatment or experimental factor. The experiment contains "a" blocks each with b individuals or experimental units which have been distributed at random across the levels of factor B.

The sum of squares

The sum of squares total can be divided into three components corresponding to the elements in the model. They are as follows:
SSA = b(_i.- _..)²  with (a-1) degrees of freedom. [This sum of squares is also called sum of squares Between blocks.]

SSB = a(_.j- _..)²  with (b-1) degrees of freedom.

SSAB = (X_ij - _i. -_.j + _..)² with (a-1)(b-1) degrees of freedom.

A "within" sum of squares is also traditionally computed.
The sum of squares within subjects is given by: (X_ij - _i. )² with a(b-1) degrees of freedom. [The sum of squares within subjects is equal to SSB plus SSAB]
The sum of squares total is given by: (X_ij -_..)²  with (ab-1) degrees of freedom.

The Models

There are two possible models for this design. The first model allows for an interaction between subjects and treatments and the second model assumes no interaction is present. The first model is defined as follows:

X_ij = A_i + B_j +[AB_ij + e_ij] + constant.  (This is called the non-additive model)

The  AB_ij and e_ij effects are confounded in this model. There can be no separate measure of randomness or error in this model as all factor are crossed.  If an interaction exists between the individuals and treatments,  then AB_ij and  e_ij will both contribute to MSAB. This leads to one of the problems with this design which will be discussed below.

When it can be assumed that no intrinsic interaction is present, then the second and  following model is appropriate:

X_ij = A_i+ B_j+ e_ij+ constant.   (This is called the additive model)

The AB_ij term drops out of the model, as no interaction is present and MSAB becomes the measure of error or randomness. In either case, MSAB will be the error term or denominator for the F ratios.

The first possible problem with this design has to do with choosing between the two models and the limitations of the second model. It is not easy to choose conclusively between the two models,  although Tukey's test of non-additivity can be used to test for some forms of non-additivity. (See Roger E Kirk, Experimental Design. pages 250-253.). If the first model is selected, and an intrinsic interaction is present, then the power of the test to detect a false hypothesis is reduced, and the effects of the treatments will not be generalizable across individuals. If the second model is appropriate, then the analysis is straightforward.

The second problem with this design results directly from the expected intra-subject correlations across the treatments, as similiar subjects are tested under each condition. (The data in one column will be correlated with the data in the others, rather than independent as in the one-way analysis of variance.). Because the scores are correlated the randomized block design has an additional assumption. The design assumes not only equal population variances (homogeneity of variance), but equal population covariances (homogeneity of covariance) as well, across the treatment conditions. This assumption is usually reasonable in the randomized block design, but it presents problems in the repeated measures design where the block represents a single subject who recieves all levels of factor B (For a more complete discussion of the problem, see Kirk, Experimental Design, second edition pages 259-262).

In the analysis of variance blocks (factor A) will be considered a random factor and factor B a fixed factor. The additive model and that the existence homogeneity of covariance will also be assumed.

The null hypotheses are that the variances and or differential effects assssociated with  A_i, and B_j  are both equal to zero.

The ANOVA

The analysis can be summarized in the following table:
 

The above Fcv values are appropriate if all the assumptions for a valid randomized block analysis of variance have been met. If the assumption of homogeneity of covariance can not be met then a more conservative F test, such as the Box adjustment, should be used to test the significance of MSB. A very conservative test called the Geisser-Greenhouse procedure might also be used (this procedure uses F((1); (b-1)) instead of F((a-1); (a-1)(b-1)) as the critical value for testing the significance of the F ratio). Most statistical software provides tests for homogeneity of covariances and appropriate tests of significance when heterogeneity of covariance exists, so I will not go into the details here (See Hayes or Kirk for a more complete discussion of these procedures).

We would expect MSA to be significant as we would expect individuals to differ from each other. It is usually not of interest to the researcher.

It is interesting to note that MSAxB must be used for the error term or denominator in testing the significance of the main or treatment effects. There is no MSC(AB). There is no problem in doing this as long as the additive model is appropriate.  If an interaction is present (a non-additive model), it reduces the effectiveness of the design--the test is positively biased toward type two errors.

In summary, the assumptions for the randomized block design include all those of the randomized group designs (although randomization occurs within and between  blocks). While the disadvantages of the randomized block design include (1) the additional assumption of homogeneity of covariance, (2) the use of an interaction mean squared for an error term, and (3) some loss of degrees of freedom as compared with the randomized groups design, they have one great redeeming quality--the possibility of greatly reducing the amount of experimental error. By using matched groups of subjects, they have the potential to greatly reduce random variation between groups.

Finally, an applet for determining power with respect to sample size for a randomized block design can also be  found at:
http://www.stat.uiowa.edu/~rlenth/Power/index.html

Then generalized randomized block design differs from the standard randomized block design in that it contains more than one subject per cell. In the above example, if the ninety subjects were divided into into three groups of 30 subjects each who were then randomly assigned to the three treatment conditions it would represent a generalized block design. The analysis of variance for a generalized randomized block design is essentially the same as for a randomized groups design so it will not be presented here.

An example of complex randomized block design might include two treatment factors labeled A and B and one blocking factor labeled C. For example, factor A might be two different teaching methods and factor B might be two different evaluation methods and factor C (the blocking factor) might be ability level. If forty subjects were available, they might be divided into ten blocks of four subjects each based on their ability, with the four most able subjects being randomly assigned to the four treatment combinations of the first block, and then the next four most able subjects assigned to the second block and so on until the tenth block received the four least able subjects randomly assigned across the four treatment combinations. The design would look like the  following:
 
 

	A1		A2
	B1	B2	B1	B2
Block 1    (C1)	X111	X121	. . .	. . .	..1
Block 2   (C2)	. . .	. . .	. . .	. . .	..2
Block 3   (C3)	. . .	. . .	. . .	. . .	..3
Block 4   (C4)	. . .	. . .	. . .	. . .	..4
Block 5   (C5)	. . .	. . .	. . .	. . .	..5
Block 6   (C6)	. . .	. . .		. . .	..6
Block 7   (C7)	. . .	. . .	. . .	. . .	..7
Block 8   (C8)	. . .	. . .	. . .	. . .	..8
Block 9   (C9)	. . .	. . .	. . .	. . .	..9
Block 10  (C10)	. . .	. . .	X21(10)	X22(10)	..(10)
	11.	12.	21.	22.
	1..		2..		...

Summary tables

Three summary tables would need to be constructed in order to perform an analysis of variance. The summary tables are the AB, AC and BC summary tables where "C" stands for blocks.

The AB summary table:
 

	B1	B2
A1	11.	12.	1..
A2	21.	22.	2..
	.1.	.2.	...

The AC summary table:
 

	A1	A2
C1	1.1	2.1	..1
C2	1.2	2.2	..2
C3	1.3	. . .	. . .
C4	. . .	. . .	. . .
C5	. . .	. . .	. . .
C6	. . .	. . .	. . .
C7	. . .	. . .	. . .
C8	. . .	2.8	. . .
C9	1.9	2.9	..9
C10	1.(10)	2.(10)	..(10)
	1..	1..	...

The BC summary table:
 

	B1	B2
C1	.11	.21	..1
C2	.12	.22	..2
C3	.13	. . .	. . .
C4	. . .	. . .	. . .
C5	. . .	. . .	. . .
C6	. . .	. . .	. . .
C7	. . .	. . .	. . .
C8	. . .	.28	. . .
C9	.19	.29	..9
C10	.1(10)	.2(10)	..(10)
	.1.	.2.	...

In performing the ANOVA, the means for each on the blocks (C) are computed, as well as the means corresponding to factors A and B, and the AB, AC, BC, and ABC interactions.

The sum of squares

Letting
X_ijk  stand for any individual score,
_... stands for the grand mean,
_i.. stands for any factor A mean,
_.j. stands for any factor B mean,
_..k stands for any block mean,
_ij. stands for any mean in the AB summary table,
_i.k stands for any mean in the AC summary table, and
_.jk stands for any mean in the BC summary table;
then the appropriate sums of squares would then be determined as follows:

SSA = cb(_i..-_...)²

SSB = ca(_.j.-_...)²
SSAB =c(_ij.-_i..-_.j.+_...)²

SSC =ab(_..k-_...)²
SSAC= b(_i.k-_i..-_..k+_...)²
SSBC=a(_.jk- _.j.-_..k+_...)²
SSABC=(X_ijk- _ij.- _i.k-_.jk+_i..+_.j.+ _..k- _...)²

The model

The basic model for a three-way randomized block design assuming the nonadditive model is:

X_ijkl = A_i + B_j + C_k + AB_ij + AC_ik + BC_jk + ABC_ijk + Constant.

If the additive model is assumed the model is:

X_ijkl= A_i + B_j + C_k + AB_ij + e + Constant

And the corresponding null hypotheses are that the differential effects and/or variances associated with  A_i, B_j, C_k, and AB_ij, are all equal to zero.

The ANOVA

Assuming that factors A and B are Fixed and factor C (blocks) is random, the ANOVA table based on the nonadditive model would be constructed as follows:
 
 

Source	SS	df	MS	E(ms)	F
Ai	SSA	a-1	SSA/a-1	e +bAC + bcA	MSA/MSAC
Bj	SSB	b-1	SSB/b-1	e +aBC + acA	MSB/MSBC
ABij	SSAB	(a-1)(b-1)	SSAB/ (a-1)(b-1)	e +ABC + cAB	MSAB/MSABC
Ck	SSC	c-1	SSC/c-1	e + abC
ACik	SSAC	(a-1)(c-1)	SSAC/ (a-1)(c-1)	e +bAC
BCjk	SSBC	(b-1)(c-1)	SSBC/ (b-1)(c-1)	e +aBC
ABCijk	SSABC	(a-1)(b-1) (c-1)	SSABC/(a-1) (b-1)(c-1)	e +ABC
See below	SSBlocksX Treatments	(c-1)(ab-1)		e

Note that because factor C is random, MSA, MSB and MSAB have different denominators or error terms for their corresponding F ratios. The tests of significance based on these F ratios often lack power. The remedy is to construct a composite mean square where the additive model is tenable. If it can be assumed that (_AC), (_BC) and (_ABC)  all have the same value (often zero), then SSAC, SSBC and SSABC can be pooled into one composite sum of squares called SSBlocksXTreatments (or SSBXT) with (ab-1)(c-1) degrees of freedom.

[SSBlocksXTreatments( or SSBXT) = SSAC + SSBC + SSABC =

(X_ijk- _ij.- _..k+ _...)² with (c-1)(ab-1) degrees of freedom.^]

The corresponding mean square called MSBlocksXTreatments (or MSBXT) can in turn be used as a common denominator or error term for MSA, MSB and MSAB when computing the corresponding F tests of significance. MSBlocksXTreatments would also be the appropriate denominator if a test of significance were performed for MSC. If MSBlocksXTreatments were used as a common error term, the ANOVA table based on the additive model would be constructed as follows:
 
 

Source	SS	df	MS	E(ms)	F
Ai	SSA	a-1	SSA/a-1	e  + bcA	MSA/ MSBXT
Bj	SSB	b-1	SSB/b-1	e  + acA	MSB/ MSBXT
ABij	SSAB	(a-1)(b-1)	SSAB/ (a-1)(b-1)	e  + cAB	MSAB/ MSBXT
Ck	SSC	c-1	SSC/c-1	e + abC	MSC/ MSBXT
Error	SSBlocksX Treatments	(c-1)(ab-1)	SSBXT/ (c-1)(ab-1)	e

There are many advantages to the randomized block designs. By  grouping individuals into blocks, the effects of the experimental conditions can be measured with greater precision. A source of variation that otherwise would become part of mean squares within in the one-way analysis of variance is removed by including it as a control variable. The advantage of using a control variable or blocking factor, is that the groups receiving the experimental conditions are more alike than they would be in a simple one way analysis of variance.

The disadvantages  associated with using the randomized block design include:
(1) The loss of some degrees of freedom  from the error term (the denominator in the F test of significance for the treatment or experimental variable).
(2) The problems associated with choosing between additive and non-additive models and using interactions mean squares as error terms are also present.
(3) The assumption of homogeneity of covariance across treatments.

How the advantage and disadvantages are weighted will depend on the circumstances--especially how effective the control variable is in reducing the amount of randomness (experimental error) in the experiment and how well the assumptions can be met.

The one-way analysis of variance, discussed in lecture two, involves applying treatments to different individuals. The simple repeated measures design, in contrast, involves applying all treatments to the same individuals. Another way to think of the simple repeated measures design is as a two-way analysis of variance where the columns are the treatments and the rows the individuals, with one observation per cell. Also, the repeated measures and randomized group designs have much in common. The only difference between them is that individuals replace the blocks. The advantage of the repeated measures design is that each individual serves as his or her own control--thus a source on experimental error resulting from individual difference is controlled for. The disadvantage of the design is that it introduces dependencies across the treatments (intra-subject dependencies).

The repeated measures design has come under much criticism in recent years, and to some extent has been replaced by the multi-variate analysis of variance design, which has fewer restrictions. The repeated measures designs can still can be used, but under very restrictive conditions as will be discussed later. It is an important design, as it introduces the idea of individuals as a random factor that crosses or intersects other factors in the design. It also illustrate a design with no sum of squares within.

The simple repeated measures design is essentially the same as the randomized block design. It can be represented as follows:
 

	B1	...	Bj	...	Bb
A1	X11	...	X1j	...	X1b	1.
...		...		...
Ai	Xi1	...	Xij	...	Xib	i.
...		...		...
Aa	Xa1	...	Xaj	...	Xab	a.
	.1		.j		.b	..

Here A stands for the individuals or subjects who receive each the treatments corresponding to each level of factor B. In other words, A₁would be the first individual, and A_a the last individual. Factor B is the treatment variable. The presentations of the levels of factor B to the individuals would be done at random. For example, which treatment individual A₁ received first would not necessarily be B₁. The order in which individual A₁ would received the levels of factor would be randomized. Randomization is an attempt to control for the order or carry-over effects. In randomized block designs individuals are assigned at random to treatments, and in repeated measures designs, treatments are assigned at random to individuals.

The sum of squares

The sum of squares for the simple repeated measures design involves dividing the sum of squares Total into three components, with each component corresponding to one of the elements in the model. The three components are as follows:
SSA = b(_i.- _..)²  with (a-1) degrees of freedom. [This sum of squares is also called sum of squares Between individuals.]

SSB = a(_.j- _..)²  with (b-1) degrees of freedom.

SSAB = (X_ij - _i. -_.j + _..)² with (a-1)(b-1) degrees of freedom.

A "within" sum of squares is also traditionally computed, but not used in the analysis as it is a redundant sum of squares.
The sum of squares Within individuals is given by: (X_ij - _i. )² with a(b-1) degrees of freedom. [The sum of squares within subjects is equal to SSB plus SSAB]
The sum of squares total is given by: (X_ij -_..)²  with (ab-1) degrees of freedom.

The models

As in the case of the single-subject-per-cell randomized block design, there are two possible models. They will be discussed in detail here. The first model allows for an interaction between subjects and treatments and the second model assumes no interaction is present. The first model is defined as follows:

X_ij = A_i + B_j +[AB_ij + e_ij] + Constant.  (This is called the non-additive model)

The  AB_ij and e_ij effects are confounded in this model. There can be no separate measure of randomness or error in this model as all factors are crossed.  If an interaction exists between the individuals and treatments,  then  AB_ij and  e_ij will both contribute to MSAB. This leads to one of the problems with this design which will be discussed below.

When it can be assumed that no intrinsic interaction is present, then the second and  following model is appropriate:

X_ij = A_i + B_j + e_ij + constant.   (This is called the additive model)

The AB_ij term drops out of the model, as no interaction is present and MSAB becomes the measure of error or randomness. In either case, MSAB will be the error term or denominator for the F ratios.

The first problem with this design (as with the randomized block design) is in having to choosing between these the two models and the limitations of the second model. It is not easy to choose conclusively between the two models, although Tukey's test of non-additivity can be used to test for some forms of non-additivity. (See Roger E Kirk, Experimental Design. pages 250-253.). If the first model is selected, and an intrinsic interaction is present, then the power of the test to detect a false hypothesis is reduced, and the effects of the treatments will not be generalizable across individuals. If the second model is appropriate, then the analysis is straightforward.

The second problem with this design is that the treatments are able to have carry-over effects on the individuals. When carryover effects are not controlled for, and a significant F ratio is obtained for the treatments, it will not be possible to determine whether or not the significant F ratio was the result of the treatment effects, or the carry-over effects, or both. This makes interpretation of the results difficult.

The third problem with this design results directly from the expected intra-subject correlations across the treatments, as the same subjects are tested under each condition. (The data in one column will be correlated with the data in the others, rather than independent as in the one-way analysis of variance.). Because the scores are correlated, the repeated measures design involves the additional assumption homogeneity of covariance-- that the scores under the treatment conditions are all equally correlated with each other. Because the same subjects are used over and over again this condition is more difficult to meet than in the case of the randomized blocks design where matched subjects are used. This assumption limits the usefulness of the repeated measures design. A less restrictive assumption can be made for this design called the circularity assumption, but it is still quite restrictive. An adjustment for lack of compound symmetry has been developed by G. E. P. Box, called the Box adjustment. It reduces the number of degrees of freedom for both numerator and denominator in determining the critical value, Fcv, for the F ratio for MSB. For a discussion of  the circularity assumption, the Box adjustment, and possible decision making strategies regarding it see William L. Hays (1988), Statistics, Fourth Edition, (New York: Holt, Rinehart & Winston) pages 520-527.

In the remainder of this section we will generally assume the additive model-- that carry-over effects have been controlled for and the existence homogeneity of covariance.

The null hypotheses are that the differential effects and/or variances associated with  A_i and B_j are equal to zero.

The ANOVA

The analysis of variance for the simple repeated measures design is presented below. In the analysis of variance individuals (factor A) will be considered a random factor and factor B a fixed factor. The analysis can be summarized in the following table:
 
 

The above Fcv values are appropriate if all the assumptions for a valid repeated measures analysis of variance have been met. If the assumption of homogeneity of covariance, or the circularity assumption, has not been met , then a more conservative F test, such as the Box adjustment, should be used to test the significance of MSB. A very conservative test called the Geisser-Greenhouse procedure might also be used (this procedure uses F((1); (b-1)) instead of F((a-1); (a-1)(b-1)) as the critical value for testing the significance of the F ratio). Most statistical software provides tests for homogeneity of covariances and appropriate tests of significance when heterogeneity of covariance exists, so I will not go into the details here (See Hayes or Kirk for a more complete discussion of these procedures).

We would expect MSA to be significant as we would expect individuals to differ from each other. It is usually not of interest to the researcher. Also, javascript programs for doing repeated measures ANOVAS can be found at: http://faculty.vassar.edu/~lowry/corr3.html and at http://faculty.vassar.edu/~lowry/corr4.html

Multiple comparisons with repeated measures

The procedures for making multiple comparisons among the treatment means follow directly from those used in the one-way analysis of variance. The formula for a sum of squares (using totals instead of means) corresponding to a comparison among the treatment means is as follows:
 

  with one degree of freedom.
MSAB is the appropriate error term in computing an F ratio if all the conditions for the repeated measures analysis of variance have been met.

Trend analysis is of special interest in the simple repeated measures design. In research on learning,  we are often interested in improvement in performance over a series of trials. In this case, each level of factor B (the experimental variable) is the same treatment, and our interest is in the carry-over effect from treatment (trial) to treatment (trial). Here an overall significant F ratio indicates either an improvement or decrement in performance over a given number of trials.

The formulas for determining the linear, quadratic, cubic, etc. components of sum of squares is the same as for one-way analysis of variance. MSAB is the appropriate error term in computing the F ratios.

Repeated measures designs, like the randomized block designs, can become more complex as more treatment factors are added. A more complex repeated measures design involves two treatment factors. It corresponds directly to the complex randomized block design. In this case, each individual would receive each combination of each of the the treatments in random order. A design with two levels for each of the factors and five individuals would look as follows:
 
 

The treatment combinations are assigned at random to the individuals or blocks.

The resulting data could be organized by the following table:
 
 

In actually performing the analysis it would be helpful to create AB and AC and BC summary tables as in three-way analysis of variance. The major difference between this analysis and the three-way analysis of variance is that here there is no within sum of squares and factor C corresponds to individuals.

Assuming that factors A and B are fixed and factor C is random, the nonadditive model model for this design would be as folloes:  X_ijk = A_i + B_j + C_k + AB_ij + AC_ik + BC_jk + ABC_ijk + e_ijk + Constant, where e_ijk is not independently measured.

[The additive model would be X_ijk = A_i + B_j + C_k + AB_ij  + e_ijk + Constant.]

The expected values of the mean squares would be as follows:
 

Source	E(MS)
Between Individuals
Ck	e + abC
Within Individuals
Ai	e +bAC + bcA
ACik	e +bAC
Bj	e +aBC + acB
BCjk	e +aBC
ABij	e  + ABC + cAB
ABCijk	e + ABC

As in the case of the nonadditive randomized three-way block design, the denominator for MSA is MSAC, and for MSB  is MSBC, and for MSAB it is MSABC. If it can be assumed that there are no interactions between the treatments and subjects (the additive model), then SSAC, SSBC, and SSABC might all be combined to form a SSres with (a-1)(c-1) + (b-1)(c-1) + (a-1)(b-1)(c-1) degrees of freedom. The corresponding MSres could then be used as the denominator for MSA, MSB, and MSAB in testing their significance as it was in the corresponding randomized block design.

The limitations of repeated measures designs have been discussed above.
 

Still more complex forms of repeated measures designs are possible. Because they have the same form as split-plot designs, which will be the subject matter of the next chapter, they will only be presented in outline form here. A fuller development of the design principles along with examples can be found in section III of the next chapter on split-plot designs.

In the following (first) example factors B and C are within factors, and factor A is a between factor. Factor D stands for individuals. There are two levels of factor B, two levels of factor C, four levels of factor A and four individuals within each level of factor C. The design has been kept unrealistically simple to keep the notation easy to follow.
 
 

		B1		B2
		C1	C2	C1	C2
A1	D1	.	.	.	.
	D2	.	.	.	.
	D3	.	.	.	.
	D4	.	.	.	.
A2	D1'	.	.	.	.
	D2'	.	.	.	.
	D3'	.	.	.	.
	D4'	.	.	.	.
A3	D1''	.	.	.	.
	D2''	.	.	.	.
	D3''	.	.	.	.
	D4''	.	.	.	.
A4	D1'''	.	.	.	.
	D2'''	.	.	.	.
	D3'''	.	.	.	.
	D4'''	.	.	.	.

Individuals would be assigned at random to levels of factor A and combinations of factors B and C would be assigned at random to individuals.

Assuming that factors A, B and C are fixed and factor D (individuals) is random, the nonadditive model model for this design would be as follows:

X_ijkl = A_i + B_j + C_k + D(A)_l(a) + AB_ij + AC_ik + BC_jk + ABC_ijk + BD(A)_jl(i) + CD(A)_kl(i) + BCD(A)_jkl(i) + e_ijkl + Constant,  where e_ijkl is not independently measured.

[The "additive" model , where interactions of  factors B, C and BC with D are assumed to be non-existant, would be as follows:  X_ijk =  A_i + B_j + C_k + D(A)_l(a) + AB_ij  + AC_ik + BC_jk + ABC_{ijk +}  e_ijkl + Constant.]

Assuming factors A, B and C are fixed and factor D (individuals) is random the ANOVA table would be as follows:
 

Source	df	E(MS)
Between individuals
Ai	a-1	e +  bcD(A)  + dbcA
A(D)l(i)	a(d-1)	e + bcD(A)
Within individuals
Bj	b-1	e  +  cBD(A)+ acdB
ABij	(a-1)(b-1)	e  +  cBD(A)+ cdAB
BD(A)jl(i)	a(b-1)(d-1)	e  +  cBD(A)
Ck	c-1	e  +  bCD(A)+ abdC
ACik	(a-1)(c-i)	e  +  bCD(A) + bdAC
CD(A)kl(i)	a(c-1)(d-1)	e  +  bCD(A)
BCjk	(b-1)(c-1)	e  + BDC(A) + daBC
ABCijk	(a-1)(b-1)(c-1)	e  + BCD(A) + dABC
BCD(A)jkl(i)	a(b-1)(c-1)(d-1)`	e + BCD(A)

If the interactions between BD, CD and BCD can be considered to be non-existant, then SSBD(A), SSCD(A), and SSBCD(A)  could be pooled as a basis for a common within individuals error term.

In the following (second) example factors C is a within factor, and factors A and B are between factors. Factor D once again stands for individuals. There are two levels of factors A and B, four levels of factor C, and three individuals within each combination of factors A and B. The design has been kept unrealistically simple to keep the notation easy to follow.
 
 

			C1	C2	C3	C4
A1	B1	D1	.	.	.	.
		D2	.	.	.	.
		D3	.	.	.	.
	B2	D1'	.	.	.	.
		D2'	.	.	.	.
		D3'	.	.	.	.
A2	B1	D1''	.	.	.	.
		D2''	.	.	.	.
		D3''	.	.	.	.
	B2	D1'''	.	.	.	.
		D2'''	.	.	.	.
		D3'''	.	.	.	.

Individuals would be assigned at random to the 12 levels of factors A and B, as in a two way ANOVA, and levels of factor C would be assigned at random to individuals, as in a simple repeated measures ANOVA.

Assuming that factors A, B and C are fixed and factor D (individuals) is random, the model for this design would be as follows:

X_ijkl = A_i + B_j + C_k +D(AB)_l(ab) + AB_ij + AC_ik + BC_jk + ABC_ijk +  CD(AB)_kl(ij) + e_ijkl + Constant,  where e_ijkl is not independently measured.

Assuming factors A, B and C are fixed and factor D (individuals) is random the ANOVA table would be as follows:
 

Source	df	E(MS)
Between individuals
Ai	a-1	e  +  cD(AB)+  bcdA
Bj	b-1	e  +  cD(AB)+ acdB
ABij	(a-1)(b-1)	e  +  cD(AB)+ cdAB
D(AB)l(ij)	ab(d-1)	e  +  cD(AB)
Within individuals
Ck	c-1	e + CD(AB)  + abdC
ACik	(a-1)(c-1)	e + CD(AB)  + bdAC
BCjk	(b-1)(c-1)	e + CD(AB)  + adBC
ABCijk	(a-1)(b-1)(c-1)	e + CD(AB)  + dABC
CD(AB)kl(ij)	ab(c-1)(d-1)	e + CD(AB)

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