Example

Solution • No, it doesn’t. A is

wrong.• Yes, it does. B looks

good.• No, the Mauchly is a

preliminary test of one of the assumptions of the within subjects ANOVA, namely, that the data have the sphericity property (homogeneity of covariance). So C is wrong.

• We already have a respectable candidate: we choose B.

Example

Solution

• No, it makes that LESS likely. The Bonferroni test is CONSERVATIVE. A is wrong.

• Yes. Conservative tests are less likely to show significance. B looks good.

• No, the ANOVA F test pinpoints a difference only in the special case of an experiment with two treatment conditions. When there are three or more conditions, further analysis is necessary. C is wrong.

• We have a respectable candidate in B, so we can dismiss D.

When to apply the Mauchly test

• The Mauchly test only arises in the context of experiments that have within subjects factors (or factors with REPEATED MEASURES). A Mauchly test is required for all within subjects factors having THREE OR MORE levels.

• Should the within subjects factor have ONLY TWO LEVELS, the Mauchly test is inapplicable.

Lecture 5

WITHIN SUBJECTS FACTORIAL EXPERIMENTS

Last week

• We considered the simplest type of within subjects experiment, in which there is just ONE TREATMENT FACTOR.

• This week, we turn to experiments with two factors, both of which are within subjects. Alternatively, such experiments are said to have REPEATED MEASURES on both factors.

• Within subjects experiments with two or more factors are known as WITHIN SUBJECTS FACTORIAL experiments.

A within subjects factorial experiment

• Three markspersons are tested on shooting accuracy with two differently-marked TARGETS: (1) PLAIN; (2) PATTERNED.

• They shoot at either target from a vehicle moving on 3 different PATHS: (1) STRAIGHT ahead; (2) on a CIRCULAR path; (3) on an irregularly TWISTING path.

• Each participant (subject) shoots at the target under all six combinations of conditions.

• The dependent variable is the number of hits.

• This is a TWO-FACTOR WITHIN SUBJECTS FACTORIAL EXPERIMENT. The factors are Target (2 levels) and Path (3 levels).

Experimental design

• 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 achieved under the six combinations of conditions.

The results

• It’s not easy to discern patterns in raw data sets from such complex experiments.

• Graphic displays can be of great assistance when you are getting to know your data.

Entering the data into SPSS

• Work in Variable View first.

• Avoid cluttering the display that will appear later in Data View – get rid of the decimal point.

• Devise informative labels for the variables. This will pay off at the output stage.

In Variable View

• These Names are for use only within the Data Editor. They can be quite cryptic.

• Notice that the variable Labels are very clear and explicit.

• This is important for the quality of the output.

To view the entire lengths of your labels, create more space by dragging the

boundary between the columns.

→

In Data View

• Our careful work in Variable View has paid off. • Our six combinations of the two treatment

factors Target and Path are clearly labelled. • The removal of the decimals avoids clutter. • The variable names are brief, but phrasing is

preserved by the use of the upper case.

Clustered bar charts or multiple line graphs?

• These are NOT available for this kind of data set.

• For a clustered bar chart you must specify a ‘category’ variable (a grouping variable).

• We have no grouping variables in this data set.

• So far, SPSS will think there are SIX dependent variables: we have no ‘score’ variable to place in ‘Bars Represent’.

Ordering a within subjects ANOVA

The factorial within subjects ANOVA is found in the Repeated Measures submenu.

Defining your factors

• So far, SPSS knows nothing of your two treatment factors, Target and Path.

• When you come to define your factors, make sure that your definitions are compatible with order in which you defined the six combinations of conditions in Variable View.

The Repeated Measures

Define Factor(s) dialog box • Here is your

opportunity to tell SPSS about your factors and identify the dependent variable (Score).

• This information is NOT in Data View.

How to define your factors

• When defining your within factors, enter first the factor whose levels change more slowly as you move from left to right across the table.

• 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’.

Define the Target factor first. • First define your Target factor

as having two levels. • After you have defined the

Target factor, define the Path factor (with three levels).

• After your second factor has been defined as having 3 levels, SPSS will know that, in Data View, 3 of the variables contain scores achieved with one target and 3 contain scores with the other target.

• On the other hand, the SPSS doesn’t yet know where to find the variables.

Name the dependent variable

• In this experiment, the dependent variable is the score a person achieves.

• Type Score into the Measure Name box and the Add button will become live.

• Press Add to transfer the name ‘Score’ into the lower box. This is the name of your dependent variable.

The two factors have now been defined

• The Repeated Measures routine is ready for six variables, but now you must supply the names.

• If you have been systematic in your ordering of the variables, this can be done very quickly.

Filling in the slots

• Hold down the control button and click on the other variables to select the whole lot.

• Click on the arrow in the middle pillar to move the six variables en masse to the panel on the right.

Click here to transfer variable names

The Options button

1. Move all the variables on the left into the panel on the right in the usual way.

2. Mark Descriptive Statistics.

3. Mark Compare Main Effects.

4. From the drop-down menu on the right, select Bonferroni.

Order a PICTURE!

• To interpret the data from a complex experiment, you need a PICTURE of your data.

• We want to compare performance on the two targets when the participants are travelling on the three different paths.

• We need to order Profile Plots.

Ordering Profile Plots

• Click Continue to return to the Repeated Measures dialog.

• Click on Plots. • In the Repeated

Measures: Profile Plots dialog box, put the Path factor on the horizontal axis and ask for Separate Lines (on the same graph) for each target. Click on Plots… to enter the

Repeated Measures: Profile Plots dialog.

The pruning operation

The output is very extensive. Remove the following:

1. All tables with the caption ‘Multivariate Tests’.

2. Table of ‘Within-Subjects Contrasts’.

3. Table of ‘Between-Subjects Effects’.

Communication check

• Make sure that the order here corresponds to the the order in which the variables appear in Data View.

• Keep this table for future reference.

Go to the Profile Plots first!

• To see what happened in the experiment, the first thing to look at is the graph showing the Profile Plots.

• There should be a Plain profile and a Patterned profile.

• Double-click on the figure to enter the graphics editor and substitute meaningful value and line labels for the numbers.

Default profile plot

The edited plot, with meaningful labels

Main effects? • For two of the three

paths, the Plain line is much higher.

• We can expect a main effect of the Target factor.

• The means are highest for the Staight path, and progressively lower for the others.

• We can expect a main effect of the Path factor as well. The means get lower as we move from left to right across the graph.

An interaction?

• The profiles are NOT PARALLEL, so there’s an interaction pattern.

• On the Twisting path only, there is little difference between the mean scores on the two targets.

Edit your tables

• Initially, code numbers appear in the tables, making them difficult to interpret.

• I suggest you edit the default tables and substitute meaningful labels.

• To edit any table, just double-click on it and (having clicked on the cell concerned) type in the label you want.

• I have edited the original output to insert the names of the levels of the Target and Path factors in the tables of marginal means.

Descriptive statistics

• Clearly the scores tend to be higher with the plain target; though the trend is much less evident with the twisting path.

• The marginal means suggest there may be main effects of both the Target and Path factors.

The cell means

• This pattern confirms the appearance of the graph.

• There are big differences between the Plain and Patterned target means with the Straight and Curved paths, but none with the Twisting path.

• There’s an interaction here.

The Mauchly tests • The Mauchly test for homogeneity of covariance

(sphericity) is applicable to WITHIN SUBJECTS factors only.

• The factor must have THREE OR MORE LEVELS.• The strange entries in the TARGET row mean that the

test is INAPPLICABLE to that factor.

These values mean that the Mauchly test is inapplicable here.

Results of the Mauchly test

• The Mauchly test is applicable to the Path factor and the Target × Path interaction.

• The p-values are high, so the null hypothesis of sphericity is accepted.

Implications of the Mauchly results

• We can make the usual ANOVA F-tests without making any adjustment of the degrees of freedom.

• We therefore retain only the rows labelled ‘Sphericity assumed’.

The ANOVA summary table

• This table is far too busy, so you must do some extensive editing.

• Because the Mauchly test gave us no reason to doubt that our data have the sphericity property, we only need the lines labelled ‘Sphericity Assumed’.

Editing the ANOVA table

Editing the table …

• Click on the first item you want to remove.

• Press Control and (holding Control down) click on the other items you want to remove.

• Finally press Delete.

The ANOVA summary table

• I double-clicked on .000, right-clicked to get Cell Properties and increased the number of decimal places to obtain the p-value for the Path factor.

• I double-clicked on some of the Sources and amended the labels.

An F-RATIO

• An F ratio always takes the same general form.• The numerator is a variance estimate for the

effect you are testing.• The denominator, or error term, measures the

noisiness of your data.

MSMS

error

sourceF

Results of the F tests

Table shows the results of three F tests: one for each main effect, the third for the interaction. With alpha = .05, there are

1. a main effect of Target: F(1, 2) = 69.63; p = .01 ;2. a main effect of Path: F(2, 4) = 437.20; p < .013. a significant interaction between Target and Path: F (2,

4) = 46.18; p < .01 • Notice there is a different error term for each F test.

Unplanned multiple comparisons

• We have a significant main effect of the Path factor. But which pairwise differences among the column means are significant?

• This is why we asked for the Bonferroni test.

For the Path factor …

Sensible comparisons only!

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

• In the graph, we only want to compare points in the same column or points on the same curve.

• There is no point in comparing, say,(Plain Target & Straight Path) with (Patterned Target & Curved Path), because you wouldn’t know which factor was responsible for the difference.

Unplanned multiple comparisons among the cell means

• Suppose we want to look at the three Paths with the Plain target only.

• So we want to make comparisons among the three points on the Plain profile.

• Equivalently, we want to make comparisons among the first row of cells in the table of means.

Proceed as follows:• Get into Data View and do a

one-factor within subjects ANOVA on the first three columns, calling your factor ‘Path’.

• In the dialog, specify a ‘Path’ factor with three levels and, in Define Factors, enter the first three variables in your data set.

Bonferroni comparisons

• In the ANOVA dialog, click on the Options button at the bottom.

• Complete the dialog as shown.

Comparisons among the first row of cell means

• All three pairs among the three Path means are statistically significant.

• In the first two columns, the cryptic numbers have been replaced by the names of the conditions.

• All three comparisons show significance.

Multiple-choice example

Second example