Multivariate Analysis of Variance(MANOVA) | Realcode4you

Extensions of univariate methods to multivariate methods

• Many of the univariate (single dependent variable) procedures are based on assessing approximate univariate normality of the errors/residuals, such as a t-test, anova and regression.

• These techniques can be extended to multiple dependent variables with an assumption of multivariate normality.

Univariate

• Normal distribution

• Independent t-test

• ANOVA

Multivariate

• Multivariate normal distribution

• Hotellings’ T

• MANOVA

Key properties

A standard normal distribution has mean 0 and variance 1.

• A standardised multivariate normal distribution has mean vector 0, and covariance matrix where all the variances are 1 (diagonal elements), and all covariances are 0 (off-diagonal elements).

• This is the identity matrix.

• The sum of squares of p independent standard normal random variables follows a chi-squared distribution with p degrees of freedom: χ2(p).

Assessing multivariate normality

1. Univariate normality

2. Bivariate scatterplots

3. Multivariate Q-Q plot

4. Formal tests (quite strict): Mardia, Roysten, Henze-Zirkler and Shapiro-Wilk

1. Univariate normality

• IF a set of variables has multivariate normality

  • then each of the individual variables will have a normal distribution

  • any linear combination of individual variables will also have a multivariate normal distribution.

• This is often used as a quick check for multivariate normality.

  • If univariate distributions are normal, multivariate normality is possible (not guaranteed).

  • If univariate distribution are not normal, multivariate distribution cannot be normal.

2. Bivariate scatterplots

• This is an extension of looking at the normality of each individual variable, by looking at every pair together.

• If two variables have bivariate normality they should form an elliptical point cloud in a scatterplot:

3. Multivariate Q-Q plot

• The standardised distance of each point from the mean can be calculated in the multivariate space:

Where S is the sample covariance matrix. This is also called a (squared) Mahalanobis distance because it scales the distances by the covariance matrix.

If you just calculated the distance from a point to the mean vector without this it would be a Euclidean distance (same as calculating distance by triangles in high school).

Mahalanobis distances are compared with a Chi-square distribution with p degrees of freedom

• Where p is the number of variables

Describing the multivariate distribution

• Mean vector

• Var-cov matrix

Python:

dataframe.cov()

R:

cov(dataframe)

Covariance vs correlation

• We can see in the covariance matrix that the numbers depend on the scale of the original measurements.

• In some procedures this may cause an issue and it would be best to standardise each variable first.

• We will also find that some procedures can be done on either the correlation or covariance matrix, so it is important to be aware of the differences.

• Correlations are always scaled between -1 and 1 whereas covariances depend on the original scale of the variables.

Univariate normality

From the plots we can see that there are only small departures from normality: y3 has ‘heavy tails’ and y4 and y6 have a small skew to the right (and potentially an outlier each).

Bivariate scatterplots

The bivariate scatterplots would ideally show elliptical patterns, which some do not.

However, it is possible most of the problems are cause by the two outliers visible in other plots.

Multivariate Q-Q plot

This is assessing the multivariate distribution, not residuals.

If you run a linear model and choose a Q-Q plot as an assumption check, it will be of the residuals against a normal distribution.

If you run a multivariate technique and ask for a Q-Q plot as an assumption check, it will be of the distances against a theoretical distribution.

Question:

Why do the outliers only show on one end of the graph?

Multivariate methods

• The essence of ‘true’ multivariate methods is that we are concerned with the joint distribution of several variables.

• There may not be an independent/dependent structure.

• In the previous slide I created a fake factor just to make the mancova procedure run so that it would calculate the Mahalanobis distances and multivariate Q-Q plot.

• (But the factor is irrelevant. In other software you can ask for the Mahalanobis distances directly.)

SPSS

Manova

Multivariate Analysis of Variance

• In the univariate case we were able to perform tests on 1 or 2 means using the t-distribution.

• If we had more than 2 means the comparison was made on the within and between groups variance using an F-distribution and a F-test.

• Similarly, even though the we use Hotelling’s T2 distribution when performing tests on 2 mean vectors, the process for comparing more than 2 mean vectors uses a different procedure. 

MANOVA has 4 test options

The equivalent of the between and within groups sums of squares are now matrices, and in the text they are called the H and E matrices.

1. Pillai’s Trace

2. Lawley-Hotelling’s Trace (also called Hotelling’s Trace)

3. Wilks’ Lambda

4. Roy’s Largest Root

Each test is based on the eigenvalues of E⁻¹H, but the calculations vary.

Realcode4you experts for assistance with advanced machine learning modeling and analysis projects. The Realcode4you team specializes exclusively in coding and research work. We have been operating for the past 7 years and have successfully delivered over 10,000 projects and homework assignments with complete satisfaction.

For more details or to get help contact us( WhatsApp +91 8267813869) or send your requirement details:

realcode4you@gmail.com