Representation of Experimental Designs

 

I.  Conceptual Overview.

 

To determine whether some characteristic (whether innate or manipulated) of a subject has an effect on a criterion variable, one may sample a number of subjects that differ on this characteristic, assign them to groups on the basis of the characteristic, measure the criterion values of each subject, and then compare the means of the groups.  These means will reflect differences in the criterion that are related to group membership plus any other differences between the groups of subjects.  Were this procedure to be repeated with a new sample of subjects, one would not expect to obtain exactly the same results because the subjects in the groups would differ.  In a sample, the variance in a criterion attributed to a factor will be a combination of the variance due to that factor plus any variance in the criterion due to interactions between that factor and other sources of variance (e.g., subjects) that would not be duplicated in other samples.  These latter sources are called random variables.

To determine whether a factor has an effect on a criterion variable one can construct a ratio in which the variance attributed to the factor is compared to an estimate of the variance due solely to the random variables (the “random error” variance).   This is the F ratio:  F=MSA/MSerror.  In a sample, the variance attributed to the factor (the numerator in this ratio) is composed of the variance due to the factor and the variance due to interactions between that factor and random variables.  When the variance due to the factor of interest is small relative to the variance due to the random variables, this ratio will be close to 1. 

To obtain an appropriate estimate of the error variance, a model of the relations between the predictor(s) and the criterion variable must be developed.  In this handout, a general method of representing experimental designs is discussed.  By following this method, you will be able to build models to represent most of the commonly encountered experimental designs and obtain appropriate estimates of the error variances.

 

II. Random vs. Fixed factors

 

Before constructing a statistical model, one must determine whether each factor in the model is to be considered "fixed" or "random."

 

A.  A fixed factor is any factor which (conceptually) would be duplicated identically in any future replication of the study.  If a factor is considered fixed, then one can generalize only to the particular items, levels, etc. used in the study. 

 

B.  A random factor is any factor that is represented in the study by items, levels, subjects, etc. which (conceptually) are randomly selected from a larger set of possible items, levels, etc. any of which could have been used in the study.  If a factor is considered random, then one can generalize to the class from which the items, levels, etc. used in the study were drawn.

 

C.  The “levels” of random factors are rarely drawn at random from the class to which they belong. In these cases, one can either treat that factor as random realizing that simply saying doesn't make it so, or treat the factor as fixed and not generalize.

 

D.  Subjects are usually considered to be random because researchers usually want to generalize beyond the subjects examined in a study.  However, subjects may not be the only random factor in a study.  Frequently, the particular items (e.g., words, pictures, sounds) used in a study are selected from a larger conceptual set to which the researcher would like to generalize.  In this case, these items should be considered to be random factors.

 

III. Structural Models

 

A. Crossed Factors: when every level of one factor is paired with every level of another factor, the factors are said to be crossed.

 

1. Example       A is crossed with B; A X B; S(A X B)

                       

 

Brightness (B)

Contrast (A)

Low

High

Low

S1, S2, S3

S7, S8, S9

High

S4, S5, S6

S10, S11, S12

 

S: subject

 

B.  Nested Factors: when each level of one factor is paired with only one level of another factor, then the first factor is said to be nested within the second factor.

 

            1. Example       B is nested within A;     B(A)   

                                   

School (A)

1

2

3

East School (A1)

North School (A2)

West School (A3)

Teachers: B1, B2, B3

Teachers: B4, B5, B6

Teachers: B7, B8, B9

 

 

C.  Some common designs (where F indicates fixed; R indicates random):

            1. Factorial ANOVA:   AF X BF X CF  or  SR(AF X BF X CF)

            2. Repeated Measures:  SR X AF X BF

            3. Nested Design:  AR(BF)

            4. Mixed Design:  SR(AF X BF) X CF

 

IV. Variance Components

 

  1. Sample estimates of the variance attributable to a factor are composed of the variance due to that factor and the variance due to interactions of that factor with other random variables.

 

  1. Note that interactions with other fixed factors will not affect this estimate because they will not vary across samples. One could calculate actual estimates of the true variance attributable to a factor (variance components analysis), but commonly one is only interested in specifying the correct error term for the F ratio. 

 

C.  To determine the appropriate error term, one can use the following heuristic (Cornfield & Tukey, 1956):

 

1.  Construct a structural model of the design; specify whether each factor is fixed or random.

 

            2.  Construct a two-way table with a row for each term in the model and an appropriately labeled column for each factor in the model.

 

3.  The entries in each column are evaluated as follows:

                        a.  For each row determine if the factor in that column helps create a nest for the term represented on that row.  If so, enter a 1; if not:

                        b.  Determine whether the factor is represented in that term.  If so, enter a Df where the f is the letter corresponding to the factor represented by that column; if not:

                        c.  Enter the letter corresponding to the factor represented by that column.

           

            4.  Evaluate the entries in each column.  If the factor is random, then all of the associated Df equal 1.  If the factor is fixed, then the associated Df equal 0.

 

5.      Construct an E(MS) table in which each term in the model is represented on a separate row.

a.       The E(MS) for each term is generated by adding together the variance components generated by every row in the two-way table that includes the term being evaluated. 

b.      The variance component for a factor generated by a row is composed of the variance for the term represented by that row and a coefficient generated by multiplying together all of the entries on the row that are not in a column representing a factor included in the term. 

c.       The error term for each term in the E(MS) table is that term (if one exists) that includes all of the variance components that comprise the E(MS) for the term except the component representing the variance due to the factor of interest.

 

C.  Example: Two-way ANOVA; A, B are fixed; S is random

1.      Model:  SR(AFXBF)

a.       There are 3 factors in this model:  A, B, and S

b.      There are 4 terms:  A, B, AB, S(AB)

 

2.      Two-way Table

 

 

Factors

Terms

a

b

n

A

Da

b

n

B

a

Db

n

A B

Da

Db

n

S(AB)

1

1

Dn

           

Note:  in these tables, n is frequently used to represent the subject factor.

3.      Evaluate entries

S is random, Dn=1; A is fixed, Da=0; B is fixed, Db = 0.

 

Factors

Terms

a

b

n

A

Da=0

b

n

B

a

Db=0

n

A B

Da=0

Db=0

n

S(AB)

1

1

Dn =1

 

4.      E(MS) Table

For A:  nbs2A + 0*ns2AB+ 1*1s2S(AB)+ = nbs2A + s2S(AB)

 

Factors

 

 

 

Terms

a

b

n

 

A

Da=0

b

n

 

B

a

Db=0

n

Skip this row because A not represented in this term

A B

Da=0

Db=0

n

 

S(AB)

1

1

Dn =1

 

 

Skip this column because it represents the factor of interest

 

 

 

 

The first variance component is the variance due to the factor of interest (A).   The coefficient for this component (nb) is composed of letters representing the other factors in the model (S and B) multiplied together.  Added to this component, is the variance due to other factors that are random that interact with A.  In this model, there is only one random factor -- subjects (S).  Because S is nested within the A X B interaction, AB enters the model with S (S(AB)).

 

            Complete E(MS) Table

                                    Source             E(MS)                          Error Term Line

                                    1. A                 nbs2A +s2S(AB)                          4

                                    2. B                  nas2B  + s2S(AB)                           4

                                    3. A X B          ns2AB +s2S(AB)                          4

                                    4. S(AB)          s2 S(AB) 

D.  Alternate Heuristic

 

1. Construct a structural model of the design; specify whether each factor is fixed or random.

 

2.  Construct an E(MS) table with each term listed on a separate row.

 

3. Construct an E(MS) for each term that is composed of:

                        a.  the variance due to that term plus

b.  the variance due to interactions or nestings of that term with other factors that are random (e.g., subjects).

           

4.  Factors which define nests enter an E(MS) when any factor nested within these factors enters.  Remember the mnemonic: random factors cannot leave their nests.

 

            5.  The coefficient for each variance component represents the levels of each of the other sources of variance (including subjects).

 

            6.  The error term for each factor is the term for which the E(MS) is identical to that of the factor of interest except that it lacks the variance component for that factor.

 

            7.  Notation:  variance components are denoted by s2 with subscripts indicating the factor(s). 

a.  Subjects are commonly denoted by an "S" subscript, but an "n" coefficient. 

b.  The within cell error in a factorial design is often denoted by s2e instead of the longer form.