Week 2 Flashcards
General linear model is?
Normal linear model is?
If X^TX is singular ?
There is no unique LSE
LSE is chosen to minimise
LSE of β
If XTX is non-singular, else LSE doesn’t exist
Least Square ESTIMATE
Where y is observed value
Important properties of LSE
Gauss Markov Theorem
Residual (error) sum of squares(vector)
Residual (error) mean square
S2 = MSE = SSE/(n-p)
It is an unbiased estimator of σ2
(p is params?)
MLE of σ2 for normal distribution
SSE/n
This is biased
Unbiased is Residual Maximum Likelihood Estimator (REML)
H
Total sum of squares
And also
SSM + SSE = SST
Model/regression sum of squares
Analysis of variance identity
DoF and SS for Treatments
t-1
SSM
DoF and SS for Residual under treatment model
n-t
SSE
Varíance (F) ratio
F = MSM / MSE
F test?
Test all treatment effects are equal
Under H0 ; F ~ Ft-1,n-t
2 main reasons for randomising experiments
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.
SUTVA
Stable unit-treatment value assumption
Treatment and unit effects are additive
Permutation test defined by the randomisation
- 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
Randomisation performed by
Writing down combinatorial design
Randomly allocating units to unit labels
Over the population of randomisations, below model becomes
Sum of e_j =0
When using nonlinear least squares estimation we choose θ to minimise
Where f is a nonlinear function of parameters
What is a p value in a 2 sided t test
The P of observing a result as extreme as the one seen
For low p we reject H_0 : μ_1 = μ_2
Var(ε) =?
From εi = Σjδij ej
Where J is a matrix of 1’s
Randomisation and SUTVA ensure
We get BLUEs of any function of parameters
Var(AX) = ?
For a constant vector A, RV X
