Download 1.14 Polynomial regression

16
Exercises
1.14
Polynomial regression
A quite flexible class of models for the mean of a real valued random variable X
given a real valued covariate y is
EX = β0 + β1 y + β2 y 2 + . . . + βd y d ,
thus the mean is a d’th order polynomial in the covariate y.
Let y1 , . . . , yn be given, real numbers – the covariates – and
Xi = β 0 + β 1 y + β 2 y 2 + . . . + β d y d + ε i
where the εi ’s are iid with the N (0, σ 2 )-distribution. Then we can estimate the d + 1
parameters β0 , . . . , βd by least squares linear regression.
Exercise 1.14.1. Download the dataset for this exercise and load it into R using
read.table. You have a data frame with an x column and an y column. Fit polynomial
regression models to the dataset, and find out how large d should be for the model
to fit the data. Report the estimates for the final model including the estimate of σ 2 .
Support your conclusion with graphs etc.
Advice on lm: You can either add additional columns to the data frame with the
values of y 2 , y 3 etc. before you do the linear regression estimation, or you can directly
in the formula specify that you want to regress upon y 2 , y 3 etc. In the last case you
need to write something like lm(x ∼ y+ I(y^2)...). The use of I() tells R that you
want this to be taken literally as an arithmetic operation on y before regression. The
formula x∼y + y^2 has a different interpretation (actually being the same as x∼y for
reasons we are not going to explain here).
Likelihood functions, genetics and MLE
1.15
17
Likelihood functions, genetics and MLE
In genetics one is often interested in estimating the recombination fraction between
two loci (genes or markers, say) on the genome. If the loci both are found in two
alleles, denoted A and a and B and b, respectively, then if we cross two individuals
with allele combinations AaBb and aabb 1 , respectively, the progeny can only get
two allele combinations, AaBb and aabb, if there is no recombination (why). We
will always assume that they are equally likely, that is, the probability of either of
the allele combinations without recombination is 1/2.
We introduce the recombination fraction as a parameter θ ∈ Θ = [0, 1], such that
the probability of the gamete allele combination Ab from the first individual (which
can only occur, if we have recombination) is θ/2.
Exercise 1.15.1. Find the probability of all the four possible progeny allele combinations AaBb, Aabb, aaBb, and aabb when the recombination fraction is θ. Write down
the likelihood function, the minus-log-likelihood function, and the likelihood equation
for estimating θ when observing the allele combinations for n crosses of individuals
with combination AaBb and aabb,
AaBb
n1
Aabb
n2
aaBb
n3
aabb
n4
Total
.
n = n1 + n2 + n3 + n4
Find the maximum likelihood estimator, find its mean and variance.
Exercise 1.15.2. If we observe
AaBb
18
Aabb
4
aaBb
6
aabb
27
Total
,
55
plot the minus-log-likelihood function and compute the maximum-likelihood estimate.
In the F2-cross, we cross AaBb with AaBb 2 . With recombination rate θ ∈ [0, 1] the
probability of gamete allele combination Ab is θ/2. Gamete allele combination aB
has, likewise, probability θ/2.
Exercise 1.15.3. Argue that all 9 (distinguishable) allele combinations are possible
when we cross AaBb with AaBb. Assuming the gamete allele combinations are independent in the two gamete cells that fuse, compute the corresponding probabilities
when the recombination fraction is θ. Write down the likelihood function, the minuslog-likelihood function, and the likelihood equation for estimating θ when observing
the allele combinations for n crosses of individuals with allele combination AaBb and
AaBb.
Hint: For the computation of the 9 point probabilities, you may start by computing
the 16 point probabilities corresponding to the 16 possible combinations of gamete
alleles and then sum out over the indistinguishable ones.
1
This is a backcross – starting from two homozygotes AABB and aabb, the progeny will always
be a heterozygote, AaBb, for both loci, and then we cross this heterozygote back with its parent
homozygote aabb
2
Starting from homozygotes, AABB and aabb, we cross the progeny AaBb with itself
18
Exercises
Exercise 1.15.4. Implement a Newton-Raphson algorithm for estimating θ in the
F2-cross. If we observe
AABB
21
AABb
10
AAbb
2
AaBB
11
AaBb
42
Aabb
6
aaBB
3
aaBb
5
aabb
12
Total
,
112
plot the minus-log-likelihood function and compute the maximum-likelihood estimator
of θ.
Exercise 1.15.5. Using the estimated recombination rate from the previous exercise,
simulate B = 200 replications of the F2-cross and (re)estimate the recombination
rate for each of the simulated crosses. Investigate, empirically, the distribution of the
maximum-likelihood estimator for θ.