6 manova Flashcards
What is MANOVA?
not a single outcome, but multiple outcomes
interactions between outcome variables
contrasting to see which groups differ
univariate = one outcome variable
multivarate = many outcome variables
What is discriminant function analysis?
if you carry out multiple tests on the same data, type 1 errors mount up
→ shouldnt do separate linear analyses
whether groups differ along a combination of dimensions (interrelations are accounted for)
What can be said about the power of MANOVA?
greater power than ANOVA in theory
however, contradictory
some studies show: diminishing power as correlation between outcome variables increases
others show that power with high correlations between outcome variables is higher
if you are expecting to find a large effect → MANOVA
measures need to be somewhat different
group differences are same direction for each measure
e.g. if one group has large difference, the other has no difference
→ power will be increased if those are highly correlated
What is a threat of MANOVA?
choosing outcomes
you need a good theoretical basis
p-hacking → do hundreds of combinations and only choose the relevant ones
For all subsequent cards, consider this example of a MANOVA.
OCD - intrusive images/thoughts
CBT on OCD
three groups:
1. CBT
2. behaviour therapy
3. no treatment
two outcomes:
1. occurence of behaviours
2. occurrence of cognitions
What are matrices?
matrix = grid of numbers arranged in columns and rows
specify its dimensions using numbers
values within a matrix = components or elements
columns and rows = vectors
square matrix = equal number of c and r
identity matrix
diagonal elements are 1
off-diagonal elements are 0
spreadsheet
What is the statistical computation of MANOVA considering matrices?
we are interested in computing an F statistic, that represents how much variance can be explained by the fact that certain scores appear in certain groups
here: multivariate
sum of squares due to model variable
→ sum of squares and cross-products matrices
matrix that represents the systematic variance = H
hypothesis sum of squares and cross-products matrix
hypothesis SSCP
unsystematic variance = E
error SSCP
total amount of variance = T
total SSCP
What is a cross product?
cross product = total value for the combined error between two variables
Example: If you’re looking at study time and test scores, the cross product will tell you if students who study more than the average tend to score above average and if students who study less than the average tend to score below average, indicating a relationship between study time and test scores.
What is the sum of squares?
sum of squares = total squared difference between the observed balues and the mean value
Example: For test scores alone, the sum of squares tells you how much students’ scores vary around the average score. If all students scored the same, the sum of squares would be zero, indicating no variability.
What are key differences between cross product and sum squares?
- The cross product is about the relationship and combined variability between two different variables, assessing how they co-vary.
- The sum of squares focuses on the variability within a single variable, measuring how much the observations deviate from the mean of that variable.
How is univariate F for one outcome calculated?
three sums of squares
- within the outcome SSt
- difference between each of the scores and the mean, sqaring them, adding them
- or variance for action scores and multiply this by df (29)
- explained by model SSm
- difference of each mean and the grand mean, sqaring it, multiplying the number of scores in the group, adding the values together
- explained by error SSr
- taking difference between each score and the mean of the group
square, add them - or multiply each group variance by N-1 and add
- taking difference between each score and the mean of the group
calculate the average sum of squares
→ divide the SS by their degrees of freedom
F statistic is the mean squares for the model divided by the mean squares for the error
How are cross products calculated?
calculate each difference between mean and scores for each group
multiply these differences
D1 x D2
model cross product = multiply above with the number of observations
What is the difference between grand mean and group mean?
grand mean = average of all data points
group mean = average within a specific group
What is the total SSCP matrix?
The total SSCP matrix, T, contains the total sums of squares for each outcome variable and the total cross-product between the outcome variables. You can think of the first column and first row as representing one outcome variable and the second column and row as representing the second outcome variable
actions thoughts actions SSt CPt thoughts CPt SSt
Are matrices additive?
Yes!
you can do
T = H + E
What is HE-1?
univariate F is ratio of systematic variance to unsystematic variance
SSm/SSr
divide matrix H by matrix E
H - model
E - error
→ multiply by the inverse of a matrix
mutliply H by inverse E (E-1)
tricky to calculate so do not bother
HE-1 is conceptually the same as the univariate F statistic
What are the main principles of discriminant function variates?
- predicting an independent variable = predict a categorical variable based on a set of outcome variables, which are typically continuous
- variate or component = linear combinations of the outcome variables
- discriminant functions = distinguish between groups
e.g. Let’s say you’re studying the effectiveness of different therapies on depression and collect data on patients’ improvement across various metrics. Discriminant analysis would allow you to create a model that can predict, based on these improvements, which therapy a future patient might find most beneficial.
What is the concept of maximisation in the context of discriminant function analysis?
maximisation
first discriminant function V1 is the linear combination of outcome variables maximizing the difference between groups
ratio SSm/SSR will be maximized for this first variate, but smaller values in subsequent variates
What is the equation used in discriminant function analysis?
yi = b0 + b1X1i + b2X2i
V1i = b0 + b1Outcome11i + b2Outcome22i
b values are weights - obtained from eigenvectors of HE-1
eigenvectors are the vectors associated with a give matrix, unchanged by transformation of that matrix to a diagonal matrix
→ fewer values