One-way ANOVA (One way analysis of variance)

and MANOVA

(Multivariate Analysis of variance)

Prepared by

Ammara Aftab

ammara.aftab63@gmail.com

Applied Statistics (MSc) 2015

(University of Karachi)

One-way ANOVA (one way analysis of variance)

Independent

Dependent

The Purpose of ONE-WAY ANOVA :

Is to test differences in means (for groups or variables) for statistical significance..

ONE-WAY ANOVA

H0: There are no differences among the mean values of the groups being compared (i.e., the group means are all equal)– H0: µ1 = µ2 = µ3 = …= µk

Ha (Conclusion if H0 rejected)?Not all group means are equal(i.e., at least one group mean is different from the rest).

H0 in ANOVA?

Number of steps involved in ANOVA

2 Groups: A B one-step test :

Step 1: Check weather the mean of two groups are different or not

Scenario 2: If we are comparing 3 or more groups :>3 Groups: A B C

It is a two-step test:

Step 1: Overall test that examines if all groups are equal or not. And, if not all are equal (H0 rejected), then:

Step 2: Pair-wise (post-hoc) comparison tests to see where (i.e., among which groups) the differences exit, and how.

Scenario 1 If we are comparing 2 groups :

Demo

MANOVA(multivariate analysis of variance)

Independent

The Purpose of MANOVA

Is an extension of ANOVA methods to cover cases where there is more than one dependent

variable.

Is to tests for the difference in two or more vectors of means.

What are the Basic requirements?

2 or more continuous DVs

1 or more categorical IVs

Examples

With a single DV you “put all of your eggs in one basket”

Assumptions

Normal distribution.

Linearity.

Homogeneity of variance:

WHY WE NEED MANOVA

It measure >1 dependent variable–Multiple correlated responses

It provides a joint test for any significant effects.

It is use to tests for patterns.

Its…–Power can be reduced by irrelevant variables

–Tests linear combinations of variables

Demo

MANOVA test statistics

 

 Pillai - This is Pillai's Trace, one of the four multivariate criteria test statistics used in manova.  We can calculate Pillai's trace using the generated eigenvalues Divide each eigenvalue by (1 + the eigenvalue).

 Hotellings - This is Lawley-Hotelling's Trace. It is very similar to Pillai's Trace. It is the sum of the eigenvalues and is a direct generalization of the F statistic in ANOVA.

 Wilks - This is Wilk's Lambda. This can be interpreted as the proportion of the variance in the outcomes that is not explained by an effect. To calculate Wilks' Lambda, for each eigenvalue, calculate 1/(1 + the eigenvalue), then find the product of these ratios.

 Roys - This is Roy's Largest Root. We can calculate this value by dividing the largest eigenvalue by (1+largest eigenvalue).

Theoretical and practical issues in MANOVA

• The interpretation of MANOVA results are always taken in the context of the research design.

• Choice of IVs and DVs takes time and a thorough research of the relevant literature

• As with any analysis, theory and hypotheses come first, and these dictate the analysis that will be most appropriate to your situation.

• Choice of DVs also needs to be carefully considered, and very highly correlated DVs weaken the power of the analysis

• Missing data, unequal samples, number of subjects and power are also issue.

Anova vs. Manova

Anova vs. Manova

One way anova just have 1 dependent and 1 independent variable while manova have more than dependent variable.

Measuring several dependent variables in a single experiment, there is a better chance of discovering which factor is truly important. While in one way anova we have to perform every Dv’s indivually again and again.

Thank you