Mauchly's Test of Sphericity with Repeated Measures ANOVA in SPSS

https://www.spss-tutorials.com/spss-repeated-measures-anova-example-2/ 

hello this is dr. Gandhi welcome to my video on testing sphericity using spss

the assumption of C erisa T is used for repeated measures ANOVA and to test the assumption we test the null hypothesis

that the variances of the differences between all groups are equal so taking a

look at these fictitious data have loaded the data view in SPSS you can see

that I have independent variable program with two levels experimental and

treatment as usual and three observations three dependent variables a

pretest a test that occurs six weeks later and a post-test that occurs twelve

weeks later these all represent the same instrument so we have these three dependent variables and looking at this

first case participant 1 0 0 1 these three scores would all be generated by

that one participant so these are within subjects this is within subjects all

three of these dependent variablesncreated by the same participant so when

we talk about the variances of the differences between all possible groups

being equal the groups we're talking about are these three dependent

variables not the independent variable not the levels of the independent

variable but rather these three dependent variables so I'm going to

conduct two repeated measures ANOVA one that has just the three dependent

variables here the three scores and then one that has the between-subjects factor

and I'll show you how we test for sphericity using mock Lee's test so

first I'm going to go to analyze the general linear model and then repeated

measures and you can see this is whatthe first dialog looks like by default

has within subject factor name and then the number of levels so this is only the

within subject actor this is not the independent

variable so in this case let's assume that these tests the pretest and the

test that occurs six weeks after in 12 weeks after let's assume that they're

measuring depression so I'm going to enter depression in there so I'm going

to change factor 1 which is what is there by default to depression and the

number of levels will be 3 because we have 3 dependent variables so go down

here and enter 3 and then click Add so depression and then 3 then I'm going to

go down here and click define the bottom left to fine and then I get the repeated

measures dialog and you can see it's already set up 3 within-subjects

variables but they're blank get 1 2 & 3 so for the first one I'm going to move

over pretest the next the test occurs 6 weeks after and then for the last 4 3

the test that occurs 12 weeks after and for this first example I'm not going to

use a between subjects factor now to generate mock waste test of sarisa T I

don't need to make any changes under these buttons to the right I just need

to click OK and conduct the repeated measures ANOVA and I'm going to move

down to the maquas test of cerissa T and you can see here that we have a p-value

for the statistic of 0.026 now mock Wiis test of cerissa T uses an alpha of 0.05

so this is a statistically significant result this means we have violated the

assumption of sphericityin order to assume that we have Spiro

city we'd have to have a value of greater than point zero five here so

what can we do when we violate the assumption of sarisa T oftentimes when

using parametric statistics we note that some parametric statistics are robust to

some violations of the assumptions associated with them however in the case

of repeated measures ANOVA repeated measures ANOVA are sensitive to

violations of cerissa t so we need to act when we have a statistically

significant value we can't just assume that the statistic is robust to ERISA T

because it's not fortunately SPSS includes a few

Corrections that we can use in the event that we do violate ERISA t one is the

greenhouse Kyser and the other is the wind felt now here we have the values of

epsilon for these statistics not the P values that's down here in the test of

within-subjects effects but we need to first look at epsilon for greenhouse

Geiser and for wind felt and you can see in this case one is 0.85 one and the

other is 0.88 six the number that we want to keep in mind when we're looking

at these two values is 0.75 if we have values here that are less than 0.75

we're going to interpret the greenhouse Geyser correction which is down here if

the value is greater than 0.75 we're going to interpret the wind felt

correction in this case we can see that both of these values are greater than

point five so we would interpret the wind felt so moving down to the test that

within-subjects effects we can see the first row is sphericity assumed we can't

use that value because we violated the assumption of curiosity then we have

green house Keizer again we're not going to use that one because we have an

epsilon value here of greater than 0.75 and then we have wind felt this is the

one we would interpret and you can see it is statistically significant another

option that you have when you have data that is violated the assumption of

cerissa T is to conduct a manova a multivariate analysis of variance as

opposed to repeated measures ANOVA because manova does not have the

assumption of sarisa t so now i'm going to go back and conduct another repeated

measures ANOVA except this time I'm going to add

program as a between-subjects factor that's the only change I'm going to make

a quick okay and you can see for mock waste test erisa T now I have a p-value

of 0.1 for two it's a non statistically significant result so I can assume that

I've met the assumption of sphericity so in this case if we were interested in

depression x program we would use the sphericity assumed row and we have a

p-value here of 0.0 0.2 mock waste test of sarissa t this test has a tendency to

miss violations of cerissa t when working with small samples and it has a

tendency to detect violations of Spira city that aren't actually there in

large samples also mock waste tests of sarissa t is only interpretable if we

have at least three dependent variables i hope you found this video on mock ways

test of cerissa t to be useful as always if you have any questions or concerns, feel free to contact me and i'll be happy to assist you

We usually use 0.8 power in social science. 

Calculator:

http://www.psychologie.hhu.de/arbeitsgruppen/allgemeine-psychologie-und-arbeitspsychologie/gpower.html

Effect Size d:

Cohen suggested that d=0.2 be considered a 'small' effect size, 0.5 represents a 'medium' effect size and 0.8 a 'large' effect size. This means that if two groups' means don't differ by 0.2 standard deviations or more, the difference is trivial, even if it is statistically signficant.

Cohen's d calculator: https://www.socscistatistics.com/effectsize/default3.aspx

SPSS repeated measures ANOVA tests if the means of 3 or more metric variables are all equal in some population. 
The simplest repeated measures ANOVA involves 3 outcome variables, all measured on 1 group of cases (often people). Whatever distinguishes these variables (sometimes just the time of measurement) is the within-subjects factor.

SPSS

Analyze - General Linear Model - Repeated Measuers

1. We may freely choose a name for our within-subjects factor. 

2. We may also choose a name for our measure: whatever each of the four variables is supposed to reflect. 

3. We now select all four variables and move them to the Within-subjects variables box with the arrow to the right.

4. Under Options we'll select Descriptive statistics.

5. Clicking Paste results in the syntax below.

Repeated Measures ANOVA Output - Mauchly’s Test

We now turn to Mauchly's test for the sphericity assumption. As a rule of thumb, sphericity is assumed if Sig. > 0.05. 

Repeated Measures ANOVA Output - Within-Subjects Effects

Since our data seem spherical, we'll ignore the Greenhouse-Geisser, Huynh-Feldt and lower bound results in the table below. We'll simply interpret the uncorrected results denoted as “Sphericity Assumed”.

Reporting the Measures ANOVA Result

When reporting a basic repeated-measures ANOVA, we usually report

• the descriptive statistics table
• the outcome of Mauchly's test and
• the outcome of the within-subjects tests.

Repeated Measures ANOVA

Repeated measures ANOVA is

1. the extension of the dependent t-test

2. the equivalent of the one-way ANOVA, but for related, not independent groups. 

3. a within-subjects ANOVA or

4. a ANOVA for correlated samples.

5. a test to detect any overall differences between related means.

The dependent variable needs to be continuous (interval or ratio) and the independent variable categorical (either nominal or ordinal).

When to use a Repeated Measures ANOVA (Two Types)

We can analyze data using a repeated measures ANOVA for two types of study design.

Studies that investigate either

(1) changes in mean scores over three or more time points

(2) differences in mean scores under three or more different conditions.

The important point with these two study designs is that the same people are being measured more than once on the same dependent variable (i.e., why it is called repeated measures).

In repeated measures ANOVA, the independent variable has categories called levels or related groups.

Where measurements are repeated over time, such as when measuring changes in skill levels due to an exercise-training programme, the independent variable is time. Each level (or related group) is a specific time point. Hence, for the exercise-training study, there would be three time points and each time-point is a level of the independent variable (a schematic of a time-course repeated measures design is shown below)

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