1-way repeated measures ANOVA

Consider the following research description:

Suppose that you are testing the long term effects of a new memory drug. You (the researcher) believe that the drug should improve memory performance, and that continued usage of the drug should lead to continued memory performance. So you give a group of 5 individuals a pre-test for memory. Then you have each start taking the new drug each week. After 4 weeks you test their memory, and again after 16 weeks. The data are presented below. The scores are the number of items recalled in a memory test (out of a possible 10 items).

Person	pre-test	4 week test	16 week test
A	2	4	6
B	0	2	4
C	1	3	5
D	3	6	6
E	4	5	9

One might be tempted to analyze this data set using the ANOVA process that we discussed last time. But that's not the appropriate analysis to do because the data in the different conditions are not independent. This design is referred to as a within-subjects or repeated measures ANOVA design. What this means is that the everybody in the experiment participates in all levels of the factor (independent variable).

So when is an ANOVA the appropriate analysis? Check the decision tree.

Find the string of decisions that lead to a 1-way between groups Analysis of Variance.

Find the string of decisions that lead to a 1-way within groups (repeated measures) Analysis of Variance.

The difference is whether or not the data in the different experimental conditions are independent or not.

Why would we decide to design our experiment as a repeated measures design instead of a between subjects? By using participants as their own comparison group we can remove the variance due to subjects from the overall variance due to treatments.

Now let's consider the sources of variance in our design.

What kinds of things might cause the distributions for each of the different experimental conditions to be different from one another (between treatments variability)?
• Treatment/group effects - the treatment caused the differences
• Random (experimental) error - just what it sounds like, random error

What about the variability within each sample (within treatment variability)?
• Random (experimental) error

Notice the major difference between this breakdown and the breakdown we did for 1-way between groups ANOVA. Because we are using the same participants in all of the experimental conditions, we are able to remove the variance due to individual differences.

The result is that we may dramatically reduce the overall variability in the test statistic, which in turn will dramatically increase the statistical power of our test. This is the main reason why repeated measures designs are used.

1-way between groups ANOVA between treatments variability Treatment/group effects - the treatment caused the differences Individual difference effects - differences due to differences in individuals Random (experimental) error - just what it sounds like, random error within treatment variability Individual difference effects Random (experimental) error

1-way within groups ANOVA between treatments variability Treatment/group effects - the treatment caused the differences Random (experimental) error - just what it sounds like, random error within treatment variability Random (experimental) error

The computations for the two 1 factor ANOVAs are similar. The 1-way between groups ANOVA partitions the total variance into two parts: a between subjects part and a within subjects part.

The 1-way repeated measures ANOVA partitions the total variance into three parts: a between subjects part and a within subjects part and a between treatments part. This is accomplished by doing the same basic computations that we did for the 1-way between groups ANOVA, but then we further partition the within groups variance.

Computing Within groups ANOVA

Same notation as one-way independent ANOVA with one addition

P = person totals (add up all the values for each individual)

Person	pre-test	4 week test	16 week test	person totals
A	2	4	6	12
B	0	2	4	6
C	1	3	5	9
D	3	6	6	15
E	4	5	9	18
Totals	10	20	30
means	2.0	4.0	6.0
SS	10	10	14
n	5	5	5

N = 15
K = 3
G = 60
Grand mean = 60/15 = 4.0
SS_total = S(X - grandmean)² = 74

Step 1: State the hypotheses & set the alpha-level

H₀: m₁ =m₂ = m₃
H_a: not all groups are equal

For this problem let's assume an alpha level = 0.05

Step 2: Figure out the degrees of freedom

We've now got more degrees of freedom to worry about:
• df_total = N - 1
• df_{between treatments} = K - 1 (Notice the name change here)
• df_{between subjects} = n - 1 (Notice the formula change here)
• df_within = N - K
• df_error = df_within - df_{between subjects}

So for our example:

df_total = N - 1 = 15 - 1 = 14

df_{between treatments} = K - 1 = 3 - 1 = 2

df_{between subjects} = n - 1 = 5 - 1 = 4

df_within = N - K = 15 - 3 = 12

df_error = df_within - df_{between subjects} = 12 - 4 = 8

As was the case for degrees of freedom, we've got a lot more Sums of Squares (SS) to compute
• SS_total = S(X - grandmean)²
• SS_{between treatments} = Sn(condition mean - grandmean)² (Notice the name change here)
• SS_{between subjects} = (S[person total]² / K) - (Grand sum)² / N (Notice the formula change here)
• SS_within = SSS_i
• SS_error = SS_within - SS_{between subjects}
So for our example:

SS_total = S(X - grandmean)² = 74

SS_{between treatments} = Sn(condition mean - grandmean)² = 40

SS_{between subjects} = (S[person total]² / K) - (Grand sum)² / N = 12²/3 + 6²/3 + ... + 18²/3 - 60²/15 = 30

SS_within = SSS_i = 34

SS_error = SS_within - SS_{between subjects} = 34 - 30 = 4

Now that we've got our SS's and df's we can compute our Means Squares (variances).

MS_{between treatments} = SS_{between treatments} / df_{between treatments} = 40 / 2 = 20

MS_{between subjects} = SS_{between subjects} / df_{between subjects} = 30 / 4 = 7.5

MS_within = SS_within / df_within = 34 / 12 = 2.83

MS_error = SS_error / df_error = 4 / 8 = 0.5

The two Mean squares in red are the two that are used for the F-ratio
F-ratio = MS_{between treatments}/MS_error = SS_error = 20 / 0.5 = 40.0

You want to use the two degrees of freedom used in the F-ratio.
So that's the df_{between treatments} & df_error SS_error For this example:
df_{between treatments} = 2
df_error = 8

So with an alpha level = 0.05 the critical F(2,8) = 4.46

Since our observed F = 40.0 is greater than the critical F = 4.46, we would reject the null hypothesis.

Source                  SS      df      MS                      
Between treatments      40      2       20.0    F = 40.00
Within treatments       34      12      2.83
   Between subjects	30	4	7.5
   Error		4	8	0.5				
Total                   74      14

Using SPSS to do a 1-way repeated measures ANOVA

As was the case with match-samples t-test, repeated-measures designs require that the data be considered together. That is, all of a single participant's data need be grouped. As a result, the SPSS file needs to have a separate column of data for each level of the within-groups variable.

The repeated measures ANOVA is run through the General Linear Model submenu of the analyze menu.

You will use this option for both 1-way repeated measures designs and factorial (more than 1 factor) repeated measures designs. So you must specify what the factors and levels of the factors are.

You get a lot of output (even more than what is pictured below). For now, this is the only one that we're interested in.

Additional Information about Within subjects designs.

• Advantages:
  • Fewer participants are required
  • Experimental time is shorter
  • Variability between groups is smaller (statistical advantage)
• Disadvantages
  • Carry-over effects - Transfer between conditions is possible
    • Effects persist from one condition into another
    • Eg. Alcohol vs no alcohol experiment on the effects on hand-eye coordination. Hard to know how long the effects of alcohol may persist.
• Counterbalancing is probably necessary
  • This is used to control for "order effects"
    • Practice effects - improvement due to repeated practice
    • Fatigue effects - performance deteriorates as participants get bored, tired, distracted
  • every possible order (n!, e.g., AB = 2! = 2 orders; ABC = 3! = 6 orders, ABCD = 4! = 24 orders, etc).
  • All counterbalancing assumes Symmetrical Transfer
    • the assumption that AB and BA have reverse effects and thus cancel out in a counterbalanced design
  • Partial counterbalancing
    • Latin square designs - a form of partial counterbalancing, so that each group of trials occur in each position an equal number of times
      • 1) each condition appears in each position (unbalanced Latin square)

        A	B	C	D
        B	C	D	A
        C	D	A	B
        D	A	B	C

        2) each condition appears before and after all others (with #1 - balanced Latin square)

        A	B	D	C
        B	C	A	D
        C	D	B	A
        D	A	C	B

    • Randomized block designs - if you've got too many possible orders, then you may just randomize the order of your blocks from participant to participant.
  • Range effects - (context effects) can cause a problem
    • The range of values for your levels may impact performance (typically best performance in middle of range). Since all the participants get the full range of possible values, they may "adapt" their performance (the DV) to this range.

An example

A psychologist is asked by a dog food manufacturer to determine if animals will show a preference among three new food mixes recently developed. The psychologist takes a sample of n = 6 dogs. They are deprived of food overnight and presented simultaneously with three bowls of the mixes on the next morning. After 10 mins, the bowls are rmoved and the amount of food eaten is measured. The data are presented below. Perform the appropriate test with an alpha level = 0.05. Use SPSS or by hand to perform the One-way ANOVA (repeated measures).

Food type
Dog	Mix A	Mix B	Mix C
A	3	2	1
B	0	5	1
C	2	4	3
D	0	7	5
E	0	3	3
F	1	3	5

• What is your computed F-ratio?
• What is your H₀?
• If we assume a = 0.05, what conclusion will we make about our H₀?

  Source SS df MS F Between treatments 28 2 14 4.67 Within treatments 40 15 Between subjects 10 5 Error 30 10 3 Total 68 17 H0: m1 = m2 = m3 Looking at the p-value, we should reject H0.