## Week 13 - Mixed Anova Assumptions Part 1

Assumptions one and two are methodological and should have been addressed either before or during data collection, the other assumptions can be tested with the statistical package. In this case, we're using SPSS assumptions, three and five are similar to the repeated measures ANOVA. However, when there's a between subject's factor the additional assumptions of homogeneity of variance and homogeneity of covariance apply now, neither of these assumptions are critical when sample sizes are equal or close. To equal so what we're looking at here is a mixed ANOVA or a mixed factorial and over so that's, a combination of between subjects and over and a within subjects and over, so there's, a there's, a between subjects measures and uh within subjects measures. So let's just step through, uh, the data.

So what we've got here is ID, which just and that allows us to identify the actual individual. And then we've got activity the activity of abseiling bowls and meditation, and they're variables that each. Participant does so each high school student does abseil each high school student does bowls. And each high school student does meditation.

And then we've got some measure of personality, which in this case is a measure of introversion or extroversion. So let's just have a look. And what these variables are in more detail. And this you can see here that the personality is either a two or a one. So this is the between subjects, uh, variable, it's, a natural group's design for that personality. So let's have a look at the variable view. So we can see here that when we look at the labels or the label for absolute bowls, meditates the same as a variable name.

So it doesn't give us much information and not as personality. However, when we look at the values, we can see that extrovert is scored as one and introvert is scored as two. So as I said before this that that's a natural groups design variable, you know, the millions of moments that have made up you and the millions of moments that have.

Made up me, we can't assign those people randomly so that you're either an extrovert or you're an introvert and that. So hence, it's a natural group's design, let's, see, okay and go back to data view. So if we have a look at participant, seven ID, seven, this person, they had sailed. They got an enjoyment score of 26. They did lawn bowls. Furthermore, they got an enjoyment score of 16., and they meditated, and they got an enjoyment score of 29. And now hopefully, uh, those activities have been counter balanced, appropriately, To ensure that fatigue effects, don't play out in those scores.

And then we've got a score of personality of one. So we went back to there. We can see one are extroverts. So we can see that participant, seven ID seven, uh.

This person is an extrovert. So the first thing we want to do is evaluate some assumptions. Now one of the assumptions are normality across the variables for each cell. So we're going to have to split we're going to have to split personality into introverts. And extroverts. And look at those separately and check out for normality for each of the variables, abseiling bowls and meditation.

And we want to also look for any univariate outliers in that space. So to do that first thing I want to do is split my file, so data split file. And I want to split it by my between subjects variable, which was personality. So compare groups by personality. So what that's essentially saying, so I'll just go back to there, what that's essentially saying now that I've split my variable, my data. By personality, whatever I do from now on I'm just going to be looking at extroverts and introverts separately. So I want to look at outliers for each of those variables across extroverts and introverts.

And I want to look at normality across those variables of abseiling bowls and meditation across introverts and extroverts. So let's do that let's, look at analyze descriptive statistics explore. And I want to look, uh, my dependent variables are abseiling bowls. Sorry, the variables I want to look at and. I want to pull them into the dependent list.

Those variables there, no, I don't need to worry about personality, because I have split on personality already so let's, look at statistics, I'll leave that sitting there. So I'll just give you. The means, the standard deviations, the measures of central tendency and so forth and variance let's. Look at plots. Now, let's just get rid of stem and lethal, though that's a very useful plot to have, but I'll just get rid of that and let's. Look at box plots with.

Those dependents together so I'm just going to look at abseiling across introverts, and extroverts do a box plot on that and look at bowls across introverts and extroverts and do a box plot on that and so forth and meditation. And I want to do normality plots across each of those variables, let's press, continue options, I'll, leave them as a default press. Okay, the case processing summary table shows how many cases were analyzed, and how many were dropped due to missing data.

So in this example, we. Can see that there are five extroverts and ten introverts in the study and no participants were missing from the data. The table of descriptive contains much information around central tendency range, skewness Kurtis. And in particular in terms of skewness and co Kurtis.

These are measures of normality. So skewness, as the name suggests indicates the extent to which there's either positive, the distributions either positively or negatively skewed and Kurtis relates to how peaked with thin or fat. Tails is the distribution and the more that these depart from zero, the less normal.

The distribution is so dividing skewness and Kurtis by the standard errors, uh of those two skewness and Kurtis figures or those skewness and ketosis figures gives the z scores for both skewness and Kurtis. So here we're looking for seen if those z scores are above or below that parameter of 1.96. And what I can see here is that the range of for all those groups are within the positive negative 1.96.

However, Warning bells are starting to sound for the extrovert group and the activity of meditation, which I'll come to I'll examine in more detail in the following normality tests and outlier analysis. The Shapiro will is a standard test of normality. So both it and the kolmogorov smirk of Tess tests, the null hypothesis that the data come from a normal distribution, and they often lead to the same conclusion. But in the event that there is a disagreement, uh, such as in to meditate, slash, you know, the. Extroverts who meditated, uh, then use the Shapiro will. So if the test is not significant.

So for example, if the p-value or the SIG value is greater than 0.05, then the null hypothesis is retained. And the normality assumption for that cell is met. And the data is assumed to come from a normal distribution. If the test is significant. So in other words, the p value or the SIG value is less than 0.05.

Then the null hypothesis is rejected in favor of the alternative hypothesis. And the normality assumption is not met, and the data are assumed to come from a distribution, that's, not normal. In this case, the significance of all groups of scores for the Shapiro wilt statistic is non-significant. So we can conclude that the assumption of normality is not violated.

So histograms stem and leaf plots box plots they're, always great things to do. And they give you a very good visual interpretation of what's going on. So here I'm showing you box plots are also called box and. Whisker plots. So here the box is a 50 of scores ranging from the 25th quartile up to the 75th quartile. And then you've got your whiskers.

And the whiskers are the extent to which the highest and the lowest points in the distribution are that aren't outliers or extreme scores here you can see that for each box plot, it's roughly symmetrical for extroverts who played bowls and abseiling. But the same cannot be said for meditation, which further confirms the disagreement between the two types of. Normality tests that were talked about previously so outliers and extreme scores are also you can also see in the box plots, and they're indicated in the box plots by the empty or colored circle respectively. And the star represents an extreme value. So if you wanted to confirm any of these suspected outliers, you can obtain z scores for each of the participants score within each cell. Now, any test participant with a z score of more than 3.29 would be an outlier, a univariate outlier. Now, In the meditation group there is an extreme score for case, 4., however, in the interests of brevity, I ran z scores for this cell, and for all other cells and found the z score was not above 3.29.

So as a result, this extreme score has not been modified in any way, and I haven't gone into it in any more detail in this analysis here you can see that each box plot is roughly symmetrical for introverts and confirms the results of the normality tests that were shown previously there, don't appear to be any. Outliers in the introvert group, okay, so we've looked at normality, and we've looked at outliers. And we've found that there is normality in our variables for each cell and there's, also, no univariate outliers. So the next step is to look at homogeneity of variance for the between subjects variable of personality and also mortal's test of spherical for the repeated measures variables and conduct our mixed ANOVA.

So let's, go ahead and do that me. But before actually before we do that, I always. Forget if you've got your data file split, then let's click on data and go to split file, and you may already have your personality from you may have already had that, uh. So let's, take that off let's just go.

Analyze all cases do not create groups. So what that's essentially doing is it's essentially saying, I don't want to split my file by personality anymore. What I want to do is I want to look at all my data so press, ok, let's, go back to the data. So we're going to go analyze general linear model. Let's click on repeated measures. Now we've got that repeated or within subject, factor name called factor 1.

Well, that's a pretty user unfriendly name. So essentially our within subjects factors, they're, they're, abseiling bowls and meditation. So to me that says activity so let's just call it activity.

Now when you look at activity, there are three levels for that repeated measures variable of activity, that's that's, abseiling bowls and meditation. So let's call it three level set and let's go. Ahead so what that's saying is, we've got an activity variable it's got three levels of one of those levels, slap salad bowls and meditation. Let's click define now it's asking where those question marks are on the right-hand side, it's asking us. Well, given that we've identified or defined this variable called activity, tell me what or tell SBS what the levels are.

And those levels are abseil bowls and meditation. Let's pull those over. Now at this point in time, where we're at is, uh, well, we're. At the same spot where we were with a one-way repeated measure and over the only difference now is that we're about to tell SBS there's a between subjects factor of personality, you're, either an introvert or you're, an extrovert you're, not both so let's, grab personality and put it in the between subjects factor. You've got a bunch of options here. And really all we want to use are two of these at this present point time.

The first option we want to use our options. So let's click on options. Now, We're interested when we do a myth stand over we're interested in the main effect for personality and activity and also an interaction effect of personality and activity so let's, grab all of those and let's display memes for those. Now, as soon as I park those over into the display means four you'll, see, compare main effects pops up as a valid thing to dick let's tick that.

And once we tick that what you're going to get is three options, Ellis, Chuck's, LSD, which doesn't correct for family wise error. So. I'd recommend it's rare that you would use that then you've got your conferring or Shiva, let's use Shiva. In this case because it's slightly more powerful and let's texture that now we're interested in a few things here.

We want to get descriptive so let's get the mean and standard deviations let's get this estimates of effect size let's, get the observed power, because we've got a between subject's, uh, variable or personality. Let's get homogeneity test let's do a homogeneity test or Levine's test. To test for equal variances across that between subjects variable, so let's click continue. Now whenever we're doing a mixed ANOVA or looking at any interaction, or even not uh, it's best to get a visual representation of what the data is doing so let's do some plots let's click on plots.

And here we've got the two factors or two independent variables, uh, we've got activity and let's park that in horizontal in the horizontal axis. And the between measures are variable, which is personality. And so let's. Pass that over there and that what that will do is give us activity for introverts as a separate line, and extroverts will be a separate line. Let's click, add and go continue. Now when we press, ok, we're going to run the analysis and be able to check for homogeneity of variance and atrocity and also look for any main effects for activity or personality.

And any interaction effects.