Multiple-choice example

Solution • A. No, the Mauchly test is

inapplicable when there are only two conditions, as with the Target factor. A is wrong.

• B. No, the p-value is not .090: that’s the value of the Mauchly statistic W itself. The p-value is .300, which is high and gives us no grounds for suspecting heterogeneity of covariance. B is wrong.

• C. The Path factor is within subjects and has more than two levels, so the Mauchly test IS applicable and C is wrong.

• D. Yes! See A.

Second example

Solution

• A. No, a different error term is used for each test. A is wrong.

• B. No, three F tests are reported: one for each main effect; a third for the interaction. B is wrong.

• C. Yes. See B. • D. We have our candidate. Go

for C.

Last week …

• I described a two-factor within subjects factorial experiment. • Each participant is tested under all six combinations of levels of the

Target and Path factors. • In the table above, each row will contain the six scores that one

participant achieves under the six combinations of conditions.

How to define your factors

• When defining your within factors, enter first the factor whose levels change less quickly as you move from left to right.

• The first three conditions (from left to right) involve the Plain target, while we move through all levels of the Path factor.

• In this sense, the Path factor is ‘faster’. Had the order in Variable View been StraightPlain, Straightpatterned, CurvedPlain, CurvedPatterned and so on, the Target factor would have been ‘faster’.

• So your first factor is Target (2 levels: Plain; Patterned) and your second is Path (3 levels: Straight; Curved; Twisting).

Check the order

• The right uppermost panel shows the ANOVA dialog order.

• The two columns of numbers show the ‘slower’ factor on the left and the ‘faster’ on the right.

• The ANOVA order agrees with the order of the variables in Data View (left panel).

Factor defined first

Factor defined second

Filling in the slots

• Use the control button to select all the variables and click on the arrow in the middle pillar to move them en masse to the panel on the right.

• You can only do this if the ordering in the left and right panels is consistent.

Warning!

• This block transfer of the variables in Data View won’t work if the ANOVA order doesn’t agree with the order in Data View.

• Suppose we had defined the factors in reverse order: first Path(3), then Target(2).

Data View and the ANOVA dialog disagree

• The ANOVA system thinks the first two variables in Data View involved the same level of the first factor that was defined. In other words, it will behave as if it thought that the first two variables involved a straight path.

• Actually, the first two variables involved different paths.

• The table below is from the Output Viewer.

• Clearly, the ANOVA procedure doesn’t agree with us about the labelling of the variables in Data View.

Factor defined first

Disaster!

• The upper table shows SPSS’s calculations.

• The lower table shows the correct values from our previous analysis.

• In four cases, the mean values have been misassigned.

Lecture 6

MIXED (OR SPLIT-PLOT) FACTORIAL EXPERIMENTS

So far, we have considered two types of experiment:

1. BETWEEN SUBJECTS experiments in which ALL factors are between subjects: that is, each person is tested under ONLY ONE condition (or combination of conditions);

2. WITHIN SUBJECTS experiments, in which ALL factors are within subjects or have repeated measures. In within subjects experiments, each person is tested under EVERY condition (or combination of conditions).

Mixed (split-plot) factorial experiments

• In MIXED (or SPLIT-PLOT) experiments, SOME (but NOT ALL) factors are within subjects.

• The term ‘split-plot’ derives from the agronomic setting in which the statistician R. A. Fisher devised these experimental designs. The ‘plots’ were areas of ground in which seedlings were grown under systematically controlled conditions.

• Hereafter, I shall drop the term ‘split-plot’ and speak of ‘mixed factorial’ experiments.

Sex differences

• There is evidence to suggest that, while women tend to outperform men on certain verbal tasks, men outperform women on spatial tasks.

• An investigator tests some men and some women on three tasks:

1. A VERBAL task;2. A NEUTRAL task;3. A SPATIAL task.

The experimental design

• Each person attempts all three tasks.In this experiment, therefore, the Task factor is within subjects.

• Alternatively, we can say that there are repeated measures on the Task factor.

• The other factor, Gender, is between subjects. Here, then, is a mixture of between subjects and within subjects factors.

• This is the simplest type of mixed factorial experiment.

The experimental design

• This is a hybrid of between subjects and within subjects designs.

• If we forget about the men, the upper half of the table shows a one-factor (Task), within subjects experiment.

• In fact, we have a one-factor within subjects experiment at each level of the Gender factor.

A hybrid design

• Suppose we were to take the mean of each person’s scores under the verbal, the neutral and the spatial conditions.

• We would have a set of data suitable for a one-way ANOVA.

The experimental hypotheses

• The women should do better on the verbal task, comparably well on the neutral task and not so well on the spatial task.

• So there should be a Gender × Task interaction.

• No main effect of either factor is expected.

The results

• These look promising: the women did better on the verbal task, but less well on the spatial task.

• However, we must confirm this pattern with some formal statistical tests.

Entering the data

• Work in Variable View first.

• Avoid clutter – get rid of the decimal point.

• DEVISE INFORMATIVE LABELS FOR THE VARIABLES – THAT WILL PAY OFF AT THE OUTPUT STAGE.

• The variable names, which appear in Data View, can be more cryptic.

In Variable View

• Note the informative variable labels. • We have removed the decimals.• We have labelled the numerical values of our

grouping variable (Gender).

In Data View

• Easy to read, thanks to our preliminary work in Variable View.

• We have clear variable names and no clutter with unnecessary decimals.

• When typing in the data, it can be helpful to view the value labels for the grouping variable (Gender).

• As with the within subjects factorial ANOVA, we choose Repeated Measures.

• The mixed ANOVA is included in Repeated Measures.

Finding the mixed ANOVA

Define the within subjects factor

1. Specify the factor ‘Task’, with three levels.

2. Enter Score in the Measure slot.

3. The Add arrow will come alive.

4. Press Add to transfer the Measure Name.

Assigning the data to the slots

• We slot in the variables at the three levels of the within subjects factor Task en masse in the usual way.

Transferring the data. Identifying the between

subjects factor

• You must select only the variables containing the data.

• Click the arrow in the central pillar.

• Select Gender and click the arrow on the left of the Between Subjects Factor(s) box.

Ordering profile plots

• Click on Plots …, to obtain the Repeated Measures: Profile Plots dialog.

• Transfer Task to the Horizontal Axis slot.

• Transfer Gender to the Separate Lines slot.

• Click on Add to place Task*Gender into the lower panel.

Options

• Transfer all the data variables to the panel on the right.

• Ask for Descriptive statistics.

• Ask for Compare Main Effects and choose Bonferroni.

• Click the Continue button.• In the ANOVA dialog,

click OK to run the procedure.

Edit the output

• Remove Within Subjects Factors and Between Subjects factors tables (inspect them first).

• Remove the Multivariate Tests results. • Remove the table of Within Subjects

Contrasts.• Remove Pairwise Comparisons for

Gender.• Remove Univariate Tests.

Keep

• The Mauchly test.

• Within Subjects Effects.

• Between Subjects Effects.

• The tables of means and standard deviations.

• Pairwise Comparisons for the Task factor.

• The graph.

Edit the graph

• We want to show the entire range on the vertical axis.

• Double-click to enter the Editor.

• Double-click on the vertical axis.

• Cancel Auto for Minimum and set Custom to zero.

• Click Apply to implement the change.

Entire vertical scale visible

Adding the Task names

• Double-click on one of the numbers on the horizontal axis.

• Type in the Task name.

Editing completed

The summary statistics

• Edit the table in the output to make it more readable.

• The CELL means confirm the pattern evident in the graph. There’s an obvious interaction, but the MARGINAL (row and column) MEANS show no main effects.

Expectations

• Our inspection of the graph and the tables suggests that the only pattern worthy of note is an interaction between the Gender and Task factors.

• The marginal means of the tables show no sign of a main effect of either factor.

The Mauchly test

• The assumption of sphericity (homogeneity of covariance) is tenable: p = 0.618 .

• Write this result as : χ2(2) = 0.962; p = .618 .• Therefore we make the usual F tests for the

within subjects factor and the interaction.

The ANOVA summary table

• The table comes in two parts: 1. Between subjects effects: Gender; the

error term for Gender.2. Within subjects effects: Task; Task ×

Gender interaction; the error term for the within subjects effects.

• The same error term is used for both the Task factor and the interaction (which is also a within subjects source).

The ANOVA summary table

The edited ANOVA table

• The F-ratios for the Gender and Task factors have very small values. Neither factor has a main effect.

• The F-ratio for the Gender × Task interaction is very large, with a small p-value. There is a significant interaction.

• The experimental hypotheses are confirmed by these results.

Writing out the results

With the significance level at .05,

• The main effect of Gender is insignificant: F(1,4) = 0.143; p = 0.73.

• The main effect of Task is insignificant: F(2, 8) = 0.175; p = 0.843 .

• There is a significant interaction between the Gender and Task factors: F(2, 8) = 65.425; p < .01 .

Multiple comparisons

We shall want to make multiple pairwise comparisons when …

1. the ANOVA has shown that a factor with more than two levels has a significant main effect.

2. the ANOVA has shown a significant interaction.We need only pursue the interaction, because Task has no main effect.

Meaningful comparisons

• It makes sense to test comparisons only within rows or columns of the table of means.

• In the graph, this corresponds to making comparisons either between genders with each task or among the points on each profile.

Gender • Using Gender as the

grouping variable, use independent-samples t-tests to compare males with females on each of the three tasks.

• Make the test more conservative by multiplying the p-value by 3 (Bonferroni method).

Results• On the Verbal task, the

females scored significantly higher than the males: t(4) = 12.25; p < .01. (Double-click on the given p-value to see more places of decimals. Multiply the given p-value by 3. It’s still less than .01.)

• On the Spatial task, the males scored significantly higher than the females: t(4) = -10.96; p <.01 (even when multiplied by 3).

• There was no significant difference on the neutral task: t(4) = -.50; p = .64.

Tasks within each gender

• We shall also want to make paired comparisons of means among the females and among the males.

• We want to make comparisons among the means on each of the two profiles.

Applying a filter

• Apply a filter, so that only the female cases are active.

• Complete the Select Cases: dialog.

• We are going to select those cases whose Gender value is 1 (i.e. the female participants).

The filter in operation

Any subsequent analysis (such as a one-factor within subjects ANOVA) will now be run only with the data from the female participants.

Selecting Bonferroni tests

• Order a one-factor, within subjects ANOVA.

• We define the Task factor (3 levels).

• We slot in the variable names.

• We complete the Options dialog as usual, asking for Bonferroni conservative tests.

Results of the Bonferroni tests • Among the female

participants there were significant differences between Verbal and Neutral and Neutral and Spatial.

• There’s no significant difference between Neutral and Spatial.

• To make similar comparisons among the men, get into Select Cases… and set Gender = 2.

More complex mixed designs

• This week, the SPSS practical exercise will involve running a mixed ANOVA on data from a two-factor experiment.

• Next week, you will be running a mixed ANOVA on data from more complex experiments with three factors. It will be good practice for you, but these more complex analyses raise no new issues.

• Next Monday, Steve Yule will be taking this slot and telling you about surveys. I shall be back the following week for my last lecture to you.

Two within-subjects factors

Two between subjects factors

Storage × Interference interaction

Simple sums only

Complex sums

A three-factor interaction• With the simple

sums, only Verbal interference has an effect.

• With complex sums, the effect works in opposite directions with the Verbalisers and the Visualisers.

Multiple-choice example

Second example

Second example