chapter 15prepared by samantha gaies, m.a.1 let’s start with an example … a high school gym...

Chapter 15 Prepared by Samantha Gaies, M.A. 1

Let’s start with an example …• A high school gym instructor would

like to test the effectiveness of a behavior modification inter-vention in controlling aggressive acts during gym class. The DV is the number of outbursts for each student.

Chapter 15: Repeated-Measures

ANOVA


The One-Way RM ANOVA as a Two-Way Independent ANOVA

• With the matched-pairs t test, we took advantage of the consistency of the difference scores.

• Treating the one-way RM design as a two-way independent-groups design allows us to attain the same advantage.

• We compute the RM ANOVA using the formulas of the two-way ANOVA, with slightly modified notation:

where c is the number of different levels of the RM factor, and r is the number of different subjects


Completing the Calculation of the F Ratio for RM ANOVA

– The dfs for the one-way RM ANOVA are:

– Only two MS components are needed as shown:

– Finally, the F ratio is:

– The critical F for the RM ANOVA is:

Fcrit (dftreat, dfinter) = Fcrit [c – 1, (c – 1)(r – 1)]

– In the example on the first slide, r and c both happen to be 4, so for α = .05, Fcrit = F.05 (3, 9) = 3.86.

– The calculation of the example on the first slide is shown on the next slide.


Calculations for the Aggressive Gym Behavior Example

• First, calculate SStot:

• Note that the correction factor was 121 in the previous calculation and is used again for SStreat and SSsub :

• Then, SSinter is found by subtraction:

• The MSs and F ratio are as follows:


Plot of the Cell Means (i.e., Scores) from the First Slide

(shows the Subject by Treatment Interaction)

A B C D

Subject

BeforeOne Week

One monthSix Months

Time

0.00

2.00

4.00

6.00

8.00

Nu

mb

er

of

ac

ts


Rationale for the RM ANOVA Error Term

• MSinter is a measure of how closely subjects follow the same pattern over the different levels of the treatment; it is the interaction you can see in the graph on the previous slide.

• If all the subjects exhibit nearly the same profile over the treatment levels, the lines would be almost parallel and the amount of subject by treatment interaction would be small.

• A smaller value for MSinter leads to a larger F ratio; a smaller value for MSinter in the population means greater power for the RM ANOVA.

• MSinter is also referred to as MSresidual, or MSerror.


Assumptions of the RM ANOVA

– The dependent variable (DV) has been measured on an interval or ratio scale.

– The DV follows a normal distribution in each population.

– All of the observations are mutually independent within each sample, but the scores are related across samples.

– Homogeneity of Variance (HOV): The DV has the same amount of variance for each level of the RM factor. This assumption is usually replaced by the sphericity assumption described on the next slide.


The Sphericity Assumption

– MSW is found by averaging all of the sample variances; the variances must be assumed to be the same in the population (HOV) for this to be appropriate.

– Similarly, MSinter is an average of the variances of the difference scores for every possible pair of levels of the RM factor; the assumption that these are all equal in the population is called the sphericity assumption.

– Lack of sphericity can lead to a higher Type I error rate than expected, if you do not adjust α or the df of your critical value.


Dealing with a Lack of Sphericity• If your F does not exceed the ordinary

critical value for RM ANOVA, there is nothing that needs to be (or can be) done.

• If your F is significant, you can test the sphericity assumption with Mauchly’s W, but this is not often heeded.

• It is common to reduce df in order to increase the critical value of the RM ANOVA, and thus be more conserva-tive. Here are two possibilities:

– The Greenhouse & Geiser df adjustment. This is best computed by statistical software.

– Worst-case critical F: Use F (1, r – 1) to find the critical value corresponding to the maxi-mum possible violation of sphericity. If your calculated F exceeds this very conservative value, you can reject the null without concern about the sphericity assumption.


Follow-Up Tests– If you are confident that the

sphericity assumption applies to your population, you can use:• LSD or HSD by substituting MSinter for

MSw in the appropriate formulas.

– If you cannot be confident that the sphericity assumption applies to your population, you should:• conduct pairwise comparisons as

separate matched t tests, and• use the Bonferroni correction to adjust

your alpha for each comparison.


Problems with the RM Design• Simple order effects (e.g., practice,

fatigue) can bias the results if conditions are repeated in the same order for all Ss.

• One solution: complete counterbalancing– Convenient only for two or three conditions – Requires 24 different orders to counterbalance

four conditions– Not feasible if there are more than 4 conditions

• Solution for 4 or more conditions: The Latin Square Design– The number of orders required equals the

number of sequentially repeated conditions. – Differential carryover effects can still be a

problem.– Sometimes carryover can be minimized by

separating conditions by sufficient time or a distracter task.

– If carryover effects are still a problem, a matched or independent-groups design should be considered.


Three Experimental Designs That Increase Power by Reducing the Error Term of the F Ratio

(without repeated measures)• 1. Add a Grouping Factor that affects the

DV to a One-Way Independent-Samples ANOVA.

• 2. Create a Treatment-by-Blocks design by classifying subjects into one of several broad levels of a quantitative factor that is relevant to the DV (e.g., age, IQ, income). The number of subjects per block should be a multiple of the number of conditions, and subjects are randomly assigned to the conditions.

• 3. Create a Randomized-Blocks (RB) design by matching subjects into “blocks” on relevant measures; the number of subjects in a block should equal the number of different conditions.


More about the RB Design– It is an extension of the matched-

pairs design for dealing with more than two experimental conditions.

– It eliminates the possibility of carryover effects, as well as simple order effects.

– The better the matching, the higher the power of the test. With good matching, the RB design will have almost as much power as an RM design without the drawbacks.

– Unless the matching is very poor, the RB design will have more power than a corresponding independent-samples design.


The Two-Way Mixed Design– There are matching or repeated

measures for one of the factors, but not for the other. Examples:• Two or more different groups are

measured at several points in time.• A grouping variable (e.g., gender) is

added to a repeated-measures study.• One factor in a two-way ANOVA lends

itself to repeated measures, but the other factor does not.

– Two different error terms are needed:• One for the between-groups factor• One for both the RM/RB factor and its

interaction with the between-groups factor


Consider this example: A researcher has designed an experiment to explore whether artists differ from writers when identifying three simple shapes by touch alone. The dependent variable is the number of each type of shape that is correctly identified.

Sub Circle Square Triang S Means Row1 2 12 7 72 8 10 9 93 4 15 3 7.3334 6 9 7 7.3335 9 13 8 106 7 14 8 9.667

Cell Means 6 12.17 7 8.397 13 3 15 178 21 4 10 18.3339 26 10 15 23.667

10 22 8 10 2011 20 9 8 1912 19 8 7 18

Cell Means 20.17 7 10.83 12.67Col Means 13.083 9.583 8.92

Wri

ters

Art

ists


Calculating the SSs of theMixed-Design ANOVA

• First, calculate the SS for each of the two factors and their interaction as you would with an ordinary two-way ANOVA.– The Between-Groups factor is labeled SSgroups

– The Repeated-Measures factor is labeled SSRM

– The Interaction is simply SSinter

• If you have the raw data, calculate SSTot and subtract SSBet-cells to get SSwithin-cells.

• Calculate the between-subjects sum of squares (SSsub) as in RM ANOVA (it is based on the subject means).

• Subtract SSgroup from SSsub to get SSW, which is used to form the error term for the between-groups factor.

• Subtract SSW from SSwithin-cell to obtain SSerror, which is used to form the error term for the RM factor.


Calculating the MSs and Fs of the Mixed-Design ANOVA

• Degrees of Freedom:

dfT = NT – 1 = nkc – 1 dfRM = c – 1

dfgroup = k – 1 dfinter = (c – 1)(k – 1)

dfW = k (n – 1) dferror = k (n – 1)(c – 1)

• Numerator Mean Squares:

• Denominator (Error) Mean Squares:

• F ratios:

SSMS

df groups

groupsgroups


• Assumptions of the Mixed-Design ANOVA– All of the assumptions of the RM

ANOVA should be met, plus … – Multisample sphericity: The amount of

interaction in the population between any two treatment levels is the same for each subgroup in the design.

• Follow-Up Tests– If the interaction is not significant:

• Focus on significant main effects with more than two levels: perform pairwise comparisons using the appropriate error term (unless HOV or sphericity cannot be assumed).

– Interaction significant:• The conservative procedure is to avoid

using either error term from the mixed-design ANOVA; base your error term in just the data in the comparison.


• The Before-After Multigroup Experiment

– The chief interest in this design is the group by time interaction.

– You could calculate the before-after difference scores and then compute a one-way ANOVA on them.• The F ratio for this one-way ANOVA

would be the same as the F ratio for the interaction effect of the mixed design ANOVA.

• The One-Way RM ANOVA with Counterbalancing– Even if simple order effects are

eliminated (balanced out), they will increase the size of the error term unless you add order as a between-groups factor to create a mixed-design ANOVA.


The Answers to the Shape Identification Example

• Degrees of Freedom:

dfT = 6*2*3 – 1= 35 dfRM = 3 – 1 = 2

dfgroup = 2–1 = 1 dfinter = (3 – 1)(2 – 1) = 2

dfW = 2 (6 – 1) = 10 dferror = 2 (6 – 1)(3 – 1) = 20

• Numerator Mean Squares:

• Denominator (Error) Mean Squares:

• F ratios:

chapter 15prepared by samantha gaies, m.a.1 let’s start with an example … a high school gym...

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