1. Multivariate Statistics Introduction W. M. van der Veld University of Amsterdam

2. Course Overview Matrix Algebra Multivariate Methods and Models Seminar

3. Matrix Algebra Why Matrix Algebra? –The mathematics in which multivariate analysis is cast is matrix algebra. –Basic knowledge of matrix algebra is also essential for those students going on to the advanced courses in structural equation modeling or regression analysis. What is Matrix Algebra? –Matrix algebra is a branch of mathematics –We will look on it simply as a tool that enables us to work with matrices. –Matrices, for our purposes, are nothing but systematic collections of observed data. –Generally, in multivariate analysis, we have observations on a certain number of variables.

4. Multivariate Methods and Models y = a + bx Linear equation If written in matrix form: Y = Xb+e, then it is known as the general linear model (GLIM), of which most multivariate models are a special case. Some examples: (0,a) (b,0)

5. Multivariate Methods and Models Simple Regression: y= α+βx+ε Or in deviation scores: y= βx+ε And normally the error term is not included in the presentation of the formula: y= βx

6. Multivariate Methods and Models Multiple Regression: y = α + β 1 x 1 + β 2 x 2 + …+ β n x n + ε Or in deviation scores: y = β 1 x 1 + β 2 x 2 + …+ β n x n + ε And normally the error term is not included in the presentation of the formula: y = β 1 x 1 + β 2 x 2 + …+ β n x n

7. Multivariate Methods and Models Multivariate Regression: y 1 = α + β 1 x 1 + β 2 x 2 + β 3 x 3 + ε y 2 = α + β 2 x 2 + ε …… y n = α + β 1 x 1 + β 3 x 3 + ε Or in deviation scores: without the alpha’s. Here we normally do include the error term.

8. Multivariate Methods and Models Principal component analysis: ξ = α + β 1 x 1 + β 2 x 2 + …+ β n x n Or in deviation scores: ξ = β 1 x 1 + β 2 x 2 + …+ β n x n Note, that this is a form of multiple regression with an unmeasured y variable.

9. Multivariate Methods and Models Factor analysis: x = α 1 + λ 1 ξ 1 + ε 1 x = α 2 + λ 2 ξ 1 + ε 2 …. x= α n + λ n ξ 1 + ε n Or in deviation scores: without the alpha’s Note, that this is a form of multivariate regression with an unmeasured x variable.

10. Multivariate Methods and Models These multivariate models have two common assumptions? –The dependent variable(s) are not nominal. –The relation between the variables is linear! –So these models can be solved using linear matrix algebra. There are also situations where the dependent variable is nominal. What to do? –Use different models (eg. correspondence analysis). There are however situations in which linearity does not hold. What to do? –Sometimes it is possible to transform the relationship so that the transformed relation is linear (eg. loglinear analysis).

11. Multivariate Methods and Models Please be aware that this description and the suggestions are far from exhaustive, but merely serve as an illustration of multivariate approaches discussed in this course.

12. Seminar During the course you will write a paper (in a group of 3 to 4 people) on a subject of your own choice. However using the tools handed in this course. This paper will be presented in the last meeting, when you will organize a seminar. When possible, I will invite scholars on the topics that you have studied, and ask them to also give a presentation. Otherwise, I will present some of my own work, and ask some colleagues to present some of their own work.