Week 2

Question 1

General linear model is?

Answer 1

Question 2

Normal linear model is?

Answer 2

Question 3

If X^TX is singular ?

Answer 3

There is no unique LSE

Question 4

LSE is chosen to minimise

Answer 4

Question 5

LSE of β

Answer 5

If X^TX is non-singular, else LSE doesn’t exist

Question 6

Least Square ESTIMATE

Answer 6

Where y is observed value

Question 7

Important properties of LSE

Answer 7

Question 8

Gauss Markov Theorem

Answer 8

Question 9

Residual (error) sum of squares(vector)

Answer 9

Question 10

Residual (error) mean square

Answer 10

S² = MS_E = SS_E/(n-p)

It is an unbiased estimator of σ²

(p is params?)

Question 11

MLE of σ² for normal distribution

Answer 11

SS_E/n

This is biased

Unbiased is Residual Maximum Likelihood Estimator (REML)

Question 12

H

Answer 12

Question 13

Total sum of squares

Answer 13

And also

SS_M + SS_E = SS_T

Question 14

Model/regression sum of squares

Answer 14

Question 15

Analysis of variance identity

Answer 15

Question 16

DoF and SS for Treatments

Answer 16

t-1

SS_M

Question 17

DoF and SS for Residual under treatment model

Answer 17

n-t

SS_E

Question 18

Varíance (F) ratio

Answer 18

F = MS_M / MS_E

Question 19

F test?

Answer 19

Test all treatment effects are equal

Under H₀ ; F ~ F_t-1,n-t

Question 20

2 main reasons for randomising experiments

Answer 20

Removes any subjective element from allocation

Justifies simple analysis using linear model. Without doing so, we would have to assume measurements made are a random sample from some large population.

Question 21

SUTVA

Answer 21

Stable unit-treatment value assumption

Treatment and unit effects are additive

Question 22

Permutation test defined by the randomisation

Answer 22

• consider all possible randomisations of treatments of units which could have occurred
• assume H₀ , i.e. no response in each unit
• for each randomisation calculate Variance ratio
• p-val is the proportion of randomisations which would have given as large a variance ratio as actual data

Question 23

Randomisation performed by

Answer 23

Writing down combinatorial design

Randomly allocating units to unit labels

Question 24

Over the population of randomisations, below model becomes

Answer 24

Sum of e_j =0

Question 25

Answer 25

Question 26

Answer 26

Question 27

Answer 27

Question 28

When using nonlinear least squares estimation we choose θ to minimise

Answer 28

Where f is a nonlinear function of parameters

Question 29

What is a p value in a 2 sided t test

Answer 29

The P of observing a result as extreme as the one seen

For low p we reject H_0 : μ_1 = μ_2

Question 30

Var(ε) =?

From ε_i = Σ_jδ_ij e_j

Answer 30

Where J is a matrix of 1’s

Question 31

Randomisation and SUTVA ensure

Answer 31

We get BLUEs of any function of parameters

Question 32

Var(AX) = ?

For a constant vector A, RV X

Answer 32