Week 5 & 6 MANOVA (Slides)

Question 1

Matrices underpin MANOVA and other analysis, what exactly are Matrices?

Answer 1

A matrix is a grid of numbers arranged in rows and columns.

• NB: The standard for quoting is rows then columns. e.g., 2 X 3 matrix means it has 2 rows & 3 columns.
• A simple Matrix may be one where rows represent a participant and columns represent measures.

Question 2

How many different types of Matrices are there?

Answer 2

There are many, and differences depend on the type of Data and analysis undertaken.

• Data Matrix
• Correlation Matrix
• Sum of Squares & Cross-Products Matrix
• Variance-Covariance Matrix
• Residuals

Question 3

A simple Matrix may be one where rows represent a participant and columns represent measures. How exactly does that work in practice?

Answer 3

A simple matrix shows values on a number of variables for each of several subjects & can be:

1. Discrete variables (0/1) where numbers are codes for group membership: 0 = male, 1 = female, OR
2. Continuous variables (e.g., Total Optimism) with values such as 15, 23, 25 etc. (i.e. assume underlying continuity)

Question 4

What is a Square & an Identity Matrix?

Answer 4

If you have a square matrix, then there would be diagonal (runs top left to right bottom) and off diagonal (runs top right to bottom left) values. This is known as a square matrix
*an identity matrix would have all values equal to 1 on the diagonal (e.g. correlation matrix - when correlate with themselves = identity matrix)

Question 5

Tell me about a correlation matrix?

Answer 5

• A correlation matrix is unit free
• Unit Free means it reflects the relationships between variables but does not provide information about the relative size of the units of measurement in measures (Mastery or Self Esteem)
• Based on Pearson’s r.

Question 6

How do we derive correlations?

Answer 6

• Correlations are derived from the sum of squares (SS) [Diagonal] and cross-products matrix (sum of products : SP) [known as an S Matrix]
• Each SS is divided by itself. Hence the 1.00s in the correlation matrix on the diagonal
• Each cross-product (SP) [off-diagonal] is divided by the square root of the product of the sum of squared deviations around the mean for each variable in the pair

Question 7

What is an S Matrix?

Answer 7

An S Matrix is Sum of squares (SS) [Diagonal] and cross-products matrix (sum of products : SP)
*NB An S Matrix, is one where the number of scores and measurement size determine the size of entries

Question 8

If the scores are measured on a meaningful scale, do we still calculate a correlation matrix?
NB a meaningful scale is one that has a unique and non-arbitrary zero value

Answer 8

In this case we would use a variance-covariance matrix (SIGMA) which gives us information about:
1. Variance of each variable (diagonal) rather than 1 where the variance = averaged squared deviations of each of the scores from the mean of scores
2. Covariance between pairs of variables – how much they covary with a value relative to the scale’s value that is being used.
NB this is Week 8 Moderation & Mediation

Question 9

How do we calculate a variance-covariance matrix (AKA SIGMA) ?

Answer 9

We calculate a variance-covariance matrix (SIGMA) by dividing each element in the S matrix by n-1

Question 10

What important things do we need to remember about the variance-covariance matrix (SIGMA Matrices)?

Answer 10

• With a variance-covariance matrix (SIGMA) the size of the entries is influenced by the measurement: Scores that are measured in large numbers tend to have large variances, small numbers result in small variances
• Deviations are averaged, so the number of scores does not have an impact
• Covariances are similar to correlations but they retain information about the scales used to measure the variables

Question 11

What is an important thing to remember about SIGMA (Variance-Covariance), S & R (Pearson) Matrices?

Answer 11

They are all square matrices so they are symmetrical

i.e. they are mirror images above and below the diagonal

Question 12

Remind me the principle of ANOVA….

Answer 12

When we use ANOVA we are separating the amount of variance in the equation to find F.
*Difference between the TOTAL VARIANCE in the data. *We separate variance into difference BETWEEN the groups and difference WITHIN the groups (Error Variance).

Question 13

What is MANOVA?

Answer 13

• MANOVA is an extension of ANOVA used where there are 2 or more DVs.
• Using MANOVA rather than multiple ANOVAs controls for familywise error across the multiple tests (just as ANOVA is preferred to a series of t-tests).
• If MANOVA is significant, researchers typically proceed to the univariate ANOVAs & then to analytical comparisons where necessary
• If MANOVA is not significant no further tests are conducted

Question 14

So I have multiple DVs, should I always run a MANOVA?

Answer 14

• No, there’s considerable controversy with its use – particularly as ANOVAs are often used after its analysis in interpretation.
• The DVs need to be at least moderately correlated, otherwise it wouldn’t be logically or appropriate to evaluate the linear combination of them.
• MANOVA is argued to have a lower power than multiple ANOVAs using a Bonferroni correction when the DVs are uncorrelated

Question 15

When is a good time to use MANOVA?

Answer 15

• You need a sound theoretical reason for its use.
• MANOVA can be more powerful than a series of ANOVAs when small differences on individual DVs combine to produce an overall significant effect
• A smaller number of DVs is preferable due to the complexity in interpretation in explaining the combination when & if significant.
• To protect against Type 1 error
• To evaluates differences among groups when it is the linear combination of DVs
• use MANOVA to create a linear combination of DVs to maximise mean group differences

Question 16

I understand sometimes ANOVA is preferred to MANOVA, What circumstances would I run an MANOVA rather than an ANOVA?

Answer 16

• MANOVA Improves the chance of discovering what changes as a result of various treatments and their interactions.
• With several DVs there is protection against inflated Type I error due to multiple tests of correlated DVs.
• In rare conditions MANOVA may reveal differences emerging that don’t appear when individual ANOVAs are undertaken
• but it is sometimes the interaction of DVs that has meaningful explanatory power over and above individual ANOVAs

Question 17

What should I be wary of when considering use of MANOVA?

Answer 17

*with each DV there’s added complexity & ambiguity in interpretation with the possibility of redundancy increasing with each additional DV

Question 18

How is the linear combination of DVs important in MANOVA?

Answer 18

It is essential to assess the strength of the correlation between the DVs if they are too closely correlated, may not be worth including both.

Question 19

What are the assumptions of MANOVA?

Answer 19

• Independence of observations
• Missing data, unequal sample sizes & power
• Normality violations
• Linearity & Absence of Multicollinearity & Singularity
• Univariate & multivariate outliers
• Homogeneity of Variance-Covariance – via Box’s M
• Reliability of Covariates & Homogeneity of Regression if covariate or step-down analysis used

Question 20

When do I check Homogeneity of Regression?

Answer 20

• Only if I am are undertaking covariate analysis
• Or I am undertaking Roy Bargman Stepdown Analysis – that is required if the multivariate analysis is significant but your individual ANOVAs are not significant.

Question 21

How do I create new cells for data cleaning?

Answer 21

*Transform, Compute, Recode into Different Variable.
*Choose Firm and transfer into numeric variable box, give it a name (name) in output variable box and hit change.
*Go to IF and choose region = 0 and hit continue.
Then go to ‘Old and New Values’ – place “0” old value and then “2” New Value, hit add, next hit “2” in Old Value and “3” in New Value and hit “add” then hit continue, then PASTE the syntax.

Question 22

How do I test for Independence of observations, missing data, equal cell sizes?

Answer 22

• Visual inspection of each individual case to ensure it only appears in one cell of the design.
• Sample size. 200 cases in 2X2 MANOVA design (2 DVs)
• Split file on the interaction cell column (Name of new variable) and check descriptives – sample size for each cell and
• hit “save standardised scores” to get the z-scores to check for univariate outliers at the same time
• Check there is no missing data and cells are approximately equal

Question 23

How do I check for Moderate correlation & Normality Assumption?

Answer 23

*Do a correlation of the 2 DVs.
*Normality. Multivariate and univariate normality violations. There is no single way of checking multivariate normality, however, achieved by assessing univariate normality.
Normality assumptions – Visual check of histogram, Normal probability plot, Kurtosis/Skewness, and SPSS runs normality checks in explore function.

Question 24

How do I check for Univariate & Multivariate Outliers?

Answer 24

• Univariate Outliers are measured by obtaining the z-scores: ±3.29 SDs (Ensure split file is on cells when you conduct do this).
• Multivariate outliers are best checked through Regression: Analyze, Regression, Linear and place ID in the Dependent variable box. Then place the 2 DVs as the predictors in the Independent box. Go to the SAVE button. Tick the box marked Mahalanobis Distance in Distances section . Click PASTE.

Question 25

How do I check for Multivariate Outliers?

Answer 25

• Turn off split file and run the analysis to check the values for Mahalanobis Distance – critical value for 2 predictors is 13.816; chi-square statistic, checked at the .001 level.
• The critical values for predictors is 13.816 – check the chi-square value at .001 with 2 degrees of freedom.

Question 26

How do I check for Multicollinearity & Singularity and Homogeneity of Variance-Covariance?

Answer 26

Checked through the analysis but given the Correlation is only .442*** multicollinearity is not an issue.

• For singularity, the level of correlation may be an indication, however, it’s also up to you as the researcher to know this from a theoretical underpinning.
  e. g. you would not place the total DASS value and also the values for Depression, Anxiety and Stress values combined value in the same analysis.

Question 27

How do I know which Multivariate Test result to report?

Answer 27

• There are 4 choices in the multivariate statistics output.
• Pillai’s Trace is considered to be the more robust (against violations of assumptions) and to have acceptable power. This is good when sample sizes are equal.
• Wilk’s Lambda is most often cited in the literature and is an appropriate choice when assumptions are met, as it can be more powerful

Question 28

What do we need to remember when we have significance and therefore want to run follow up univariate tests?

Answer 28

*We need to adjust the alpha for familywise error to .025 on a Bonferroni adjusted alpha

Question 29

What are some oft he more advanced formed of MANOVA?

Answer 29

• Doubly Multivariate Design – multiple DVs when you have repeated measures on these and interactions arise within the measures and the time collections.
• Profile Analysis – multiple DVs where you are wanting to ascertain the patterns of change across multiple DVs.
• MANCOVA – inclusion of covariate/s when multiple DVs are included.