Statistics VI - PCA + Meta Exam Questions + Vectors/Matrices

Question 1

What should the PCA be based on if all variables have the same units?

Answer 1

the covariance matrix

Question 2

What should the PCA be based on if variables have different units?

Answer 2

the correlation matrix

Question 3

In a PCA of a bivariate distributions: The coordination system is rotated so that …

Answer 3

… the first axis - the first PC - has the largest possible variance.

Question 4

How come we are allowed to rotate the coordination system in the PCA?

Answer 4

We are allowed to do that, because the total variance is invariant against rotation of data.

Question 5

The sum of squared distances of a distribution = 3.224.

What is the sum of squared distances after rotating the data in the course of a PCA?

Answer 5

the same: 3.224

Question 6

The second PC has the ___________ largest variance and is _________ to the first PC.

Answer 6

The second PC has the SECOND largest variance and is ORTHOGONAL to the first PC.

Question 7

What’s the total variance in the following (graphically akward) covariance matrix?
[0.950 0.647]
[0.647 0.820]

Answer 7

0.950 + 0.820 = 1.770

Question 8

What multivariate tests should we know?

Answer 8

MANCOVA
MANOVA
Hotelling’s T²

Question 9

What univariate tests should we know?

Answer 9

student’s t-test
ANOVA
ANCOVA

Question 10

What is an eigenvector and an eigenvalue?

Answer 10

If A is a square matrix, v != 0 is an eigenvector of A, if
A * v = λ * v

The number λ is called the eigenvalue of A.

Question 11

List at least 6 tests!

Answer 11

student's t-test
Hotelling's T²
ANOVA
ANCOVA
MANOVA
MANCOVA

Question 12

What can the following tests be useful for:
t-test
ANOVA
ANCOVA

Answer 12

t-test: difference btw 2 groups and 1 variable
ANOVA: difference btw several groups, one dependent variable
ANCOVA: difference btw several groups, one dependent variable + controls

Question 13

What can the following tests be useful for:
Hotelling’s T²
MANOVA
MANCOVA

Answer 13

Hotelling’s T²: difference btw 2 groups and several variables
MANOVA: several dependent variables, groups, interaction among variables
MANCOVA: several dependent variables, groups, interaction among variables + controls

Question 14

How is the name of a vector usually written in statistics?

Answer 14

bold lowercase letter

Question 15

How many columns does a vector have?

Answer 15

1

Question 16

What’s the difference between the vector v and the vector v’?

Answer 16

v is a column vector and

v’ is the transposed version of vector v

Question 17

In a matrix, what is represented through rows and columns?

Answer 17

rows: cases
columns: variables

Question 18

How does M’ look like if M =
(2 4 1)
(3 8 4)

Answer 18

(2 3)
(4 8)
(1 4)

Question 19

What’s the distinct property of a square matrix?

Answer 19

Same amount of rows and columns.

Question 20

What’s the distinct property of a symmetric matrix?

Answer 20

S = S’

Question 21

What’s the (main) diagonal of a matrix?

Answer 21

The “row” of numbers from top left to bottom right.

Question 22

What is an identity matrix (aka. unit matrix)?

Answer 22

An identity matrix is a square matrix with ones on the main diagonal.

Question 23

Cor(a,b) = 0.8

How large is the explained variance?

Answer 23

0.8² = 0.64 -> 64 %

Question 24

Cor(a,b) = - 0.3

How large is the explained variance?

Answer 24

• 0.3² = 0.09 -> 9 %

Question 25

What’s the definition of discrete data?

Answer 25

Discrete Data can only take certain values.

Question 26

What’s the definition of continuous data?

Answer 26

Continuous Data is data that can take any value (within a range)

Question 27

What kind of a scale is Kelvin?

Answer 27

ratio scale

Question 28

What kind of a scale is Celsius?

Answer 28

interval scale

Question 29

Define precision!

Answer 29

Degree to which repeated measurements show the same results

Question 30

Define accuracy!

Answer 30

Closeness of measurements of a quantity to the quantity’s actual (true) value.

Question 31

Define bias!

Answer 31

Difference between the average of the measurements and the reference value.