Week 5 Flashcards
E of Simple linear regression
E of Quadratic regression model
Least squares estimator for β^ for linear model, and variance of
E of Exponential relationships generally form
Where θ3 is clearly non linear
Reciprocal relationships
covariance Matrix of non linear estimator? What is assumed
Asymptotic dependence is important!!
F
3 reasons to fit a response function to experimental data
1) estimating parameters of functions of; for a scientific interpretation
2) providing smooth version of data to enable prediction
3) as part of a process to identify optimal levels of factors
Optimal design theory,
Focus on what? Why?
(N)LSE are unbiased therefore
Focus on minimising variance
Therefore minimising confidence region (in hyperspace)
D optimal design
Minimising the volume of the joint confidence interval for the set of parameters
A optimality
A stands for average
Minimising average variance of parameter estimators by minimising φA
Generalised D optimal design
A is a matrix intended to remove the intercept
It is p x (p + 1) so as to remove the β0 term
When is a response function relevant
If the treatments in an experiment are levels of a quantitative factor
Extra condition for optimality of non linear models
Φ may depend on values of (some elements of) θ
Manage by either (a) prior estimate of some values of θ or (b) updating values of θ by running experiment sequentially
Comparing designs under D optimality
Workout relative efficiency
Efficiency = (ΦDalt / ΦDoriginal) 1/p
Where p is number of params being estimated
Form D optimal design for SLRM
And n2 = N/2 units assigned to xi = 1
For n odd we choose n1 = (N+1)/2 and n2 = (N-1)/2
Standardised variance at x
