

2 :

Mixed designs We’ve discussed between groups designs looking at differences across independent samples
We’ve also talked about within groups designs looking for differences across treatments in which subjects participate in each treatment. 

3 :

Between groups design Typing speed: random assignment to Music or No Music conditions 

4 :

Repeated Measures example Here each person is measured in the Music and No Music conditions 

5 :

B/t groups and RM The research question can often determine the design, however there are some factors that we could not examine in repeated measures design (e.g. ethnicity)
In cases where we might have a choice (as with in the previous example) RM design would most likely be preferred
When subjects are observed only once, their differences contribute to the error term. On repeated occasions we can obtain an estimate of the degree of subject differences and partial that out of the error term
More power
Fewer subjects needed 

6 :

Mixed design A x (B x S)
At least one between, one within subjects factor
Each level of factor A contains a different group of randomly assigned subjects.
On the other hand, each level of factor B at any given level of factor A contains the same subjects 

7 :

Partitioning the variability Partitioning the variance is done as for a standard ANOVA.
Within subjects error term used for repeated measures.
What error term do we use for the interaction of the two factors?


8 :

Partitioning the variability Again we adopt the basic principle we have followed previously in looking for effects. We want to separate between treatment effects and error
A part due to the manipulation of a variable, the treatment part (treatment effects)
A second part due to all other unsystematic or uncontrolled sources of variability (error)
The deviation associated with the error can be divided into two different components:
Between Subjects Error
Estimates the extent to which chance factors are responsible for any differences among the different levels of the between subjects factor.
Within Subjects Error
Estimates the extent to which chance factors are responsible for any differences observed within the same subject 

9 :

How it breaks down
SStotal
SSb/t subjects SSw/in subjects
SSA SSsubj w/in groups SSB SSAxB SSerror (Bxsubject)
a1 a(s1) b1 (a1)(b1) a(b1)(s1) df = 

10 :

Comparing the different designs B/t groups Design W/in groups Design Mixed Design
SSA SSA
SSA/S SSS SSA/S
SSB SSB
SSBxS SSAxB
SSBxS
Note that the between groups outcome (F and pvalue) is the same in the mixed and b/t groups design
In the mixed, the repeated measures are ‘collapsed’, making each subjects score for the between groups factor the mean of those repeated measures
The same is true for the within groups design, except in the mixed the ‘subjects’ are nested within the factor of A, and the interaction of A X B is taken out of the error term
The SSb/t subj in the Within Design is the error term for the between groups factor in the mixed
The Error terms are in blue 

11 :

Comparing the different designs The SSb/t subjects in general reflects the deviation of subjects from the grand mean while the SSw/in in general reflects their deviation from their own mean
The mixed design is the conjunction of a randomized single factor experiment and a single factor experiment with repeated measures 

12 :

Example 2 x 3 mixed factorial design
Gender and tv viewing habits (hours watched per week) drama comedy news
male 4 7 2
male 3 5 1
male 7 9 6
male 6 6 2
male 5 5 1
female 8 2 5
female 4 1 1
female 6 3 4
female 9 5 2
female 7 1 1 

13 :

In SPSS In SPSS, though we have a between groups factor we’ll still use the RM menu 

14 :

Compared to separate designs Between subjects output
If one collapses the RM variables and performs the 1way ANOVA on the resulting dependent variable of subject means the results are the same as in our mixed output 

15 :

Compared to separate designs Similarly, if we ignore gender and run a oneway RM, we can see that this result is contained within the mixed design 

16 :

General Result No main effect for gender
Main effect for tv show, but also gender x tv show interaction 

17 :

Simple effects Comparisons reveal a gender difference in viewing comedy programs but not for others *As mentioned for previously for RM comparisons, SPSS does not use a pooled error term for each comparison.
That would be the approach if sphericity is not met. 

18 :

Assumptions Usual suspects normality, homogeneity of variance, sphericity
For Between subjects effects, variances across groups must be similar
Also for the within subjects effects we have an HoV requirement
That the error (tvshow by subject interaction) is the same for all groups 

19 :

Assumptions In addition, the sphericity assumption extends beyond the within subjects factor
Our var/covar matrices must be similar across groups (gender)
Furthermore, the pooled (average/overall) var/covar matrix of the group var/covar matrices should be spherical
If the first is ok the second will be
Gist: variances of all possible difference scores among the treatments should be similar 

20 :

Post hocs and contrasts If no sig interaction, one may conduct post hoc analysis on the significant main effects factors as described previously
Planned contrasts can be conducted to test specific hypotheses 

21 :

Planned contrasts Focused contrasts can get complicated regarding interactions
Example Age x Therapy
Row and column weights must sum to zero
Does the effect of hospitalization vary as a function of a linear trend with age
Younger benefit more from nonhospitalization Non Hospitalization
Psychoth Companion Traditional Milieu
1 1 1 1
Old 1 1 1 1 1
Middle 0 0 0 0 0
Young 1 1 1 1 1 

22 :

Planned contrasts Example weights for testing a linear trend for age in groups psycoth and traditional (opposite to each other), quadratic for companion and milieu (also opposite)
We could break down the interaction into an orthogonal set of contrasts
Sum up to the interaction (sums of squares)
Non Hospitalization
Psychoth Companion Traditional Milieu
Old 1 1 1 1
Middle 0 2 0 2
Young 1 1 1 1 

23 :

Planned contrasts With mixed designs it can be difficult to determine the appropriate error term
Consult Keppel, or Rosenthal and Rosnow for ideas on how to proceed
Essentially we will have a interaction contrast x subjects error
Furthermore, it has been shown by some that such analyses can be very sensitive to violations of our assumptions (sphericity) 

24 :

More complex mixed designs May have multiple between or within factors
Gist of the approach is pretty much the same for multiple factors of either between or within subjects factors
Interested in interactions involving the two types of factors 

25 :

Two between one within In this case we will have our typical factorial output and with interaction etc. to interpret
Now we will also look to see if the between subjects interaction changes over the levels of the repeated measure 

26 :

Example Anxiety in final weeks of the semester guys A&S 3 1 4 6 7
guys A&S 1 2 5 5 5
guys A&S 4 6 7 7 8
guys Business 0 4 4 7 8
guys Business 2 3 5 7 8
guys Business 0 4 4 4 8
guys Music 1 3 3 4 4
guys Music 1 3 3 5 6
guys Music 1 4 7 7 8
guys Education 3 5 8 7 6
guys Education 0 2 3 6 4
guys Education 2 1 2 5 5
gals A&S 3 3 5 7 7
gals A&S 0 1 3 2 4
gals A&S 2 5 6 6 7
gals Business 1 3 6 5 6
gals Business 0 4 6 7 6
gals Business 2 2 3 5 7
gals Music 2 3 5 7 8
gals Music 0 4 5 8 8
gals Music 1 4 5 7 7
gals Education 1 4 4 5 8
gals Education 1 2 4 6 8
gals Education 2 5 6 7 7 



29 :

Results Regardless of gender or college affiliated with, anxiety increases at approximately the same rate as one approaches finals
Shocking! 

30 :

One between Two within Again we will have our typical output as we would with a two within design
We will also look to see if the within subjects interaction changes over the levels of the between subjects factor


31 :

Example Are there differing effects for age regarding verbal and visuospatial ability?
DV percentage of errors on task
Age x (Verbal/visuospatial ability x Block)
2 x (2 x 6)




34 :

Start simple and build from there
Use visual displays to keep things straight
All three main effects significant 

35 :

2 way interactions
Only type of task by block was close p = .057, PES = .022
Though started out similarly, less improvement over blocks for visuospatial task 

36 :

Significant 3 way interaction
No real interaction for young b/t type of task and rate of improvement
With older folk we see the interaction alluded to in the previous 2way 

37 :

Simple effects In order to test for simple effects we must have the appropriate error term for analysis
Breakdown of general error terms for the previous designs (2 within on left, 2 between subjects factors on right; from Keppel) 

38 :

Simple effects Error terms for simple effects (from Winer)
Comparison to the appropriate critical value with appropriate degrees of freedom for pooled sources of variability from mixed sources can get a little weird
Consult an appropriate text
1 between 2 within: A x (B x C) 2 between 1 within: A x B x C *q and r refer to the number of levels of the repeated measures factors B and/or C
MSA x subj = MSerror(a)
MSB x subj = MSerror(b)
MSC x subj = MSerror(c)
MSBC x subj = MSerror(bc) 

39 :

Summary Mixed design encompasses at least one between subjects factor (independent groups) and one repeated measures factor
The approach is the same as it was for either separately Look for main effects and interactions
In the simplest setting an interaction suggests that the between groups differences are changing over the levels of the repeated measure (or the repeated measure effect is varies depending on which group you are talking about)
With more complex interactions, interactions are changing over the levels of another variable.
The best approach is to start simple (examine main effects) and work your way up, and in the presence of a significant interaction, make sure that your simple effects are tested appropriately 
