8 Existence and uniqueness
This chapter is currently undergoing conversion and substantial changes will be made without notice.
Exercises
Exercise 8.1 This exercise is about an additive genetic effects model. The response \(Y\) is an indicator of a phenotype, e.g., whether an individual has brown eyes or whether the individual has a particular disease. Some phenotypes are determined completely by well known genotypes, while others have a less well understood genetic component.
A genotype is an unordered pair of possible alleles (at a given genomic location) and can usually take three different values – here denoted bb, Bb and BB. The brown eye color is known to be dominant over the blue, so if B denotes the brown eye allele, then BB as well as Bb will result in brown eyes whereas only bb will result in blue eyes1.
1 In reality, the genetics of eye color is somewhat more complicated, but this simple version of the story illustrates the principle.
For the purpose of this exercise you don’t need to understand the genetic details, but genetics provides a context where the model considered is relevant. We recode the genotype as a numeric variable \(X\) as follows:
| genotype | bb | Bb | BB |
|---|---|---|---|
| \(X\) | 0 | 1 | 2 |
Thus \(X\) denotes the number of \(B\) alleles, and we will use it to construct a probabilistic model of the phenotype response \(Y\) given the genotype predictor \(X\). The additive genetic effects model is given by \[ \textrm{logit}(P(Y = 1 \mid X = x)) = \beta_0 + \beta_1 x. \tag{8.1}\] Note that this additive model will not be a good model of eye color, but it can be a useful model of the genetic component of a disease.
Having observed \(n\) individuals we can summarize the data into a table of the following form:
| \(X\) | \(0\) | \(1\) | \(2\) | Total |
|---|---|---|---|---|
| number of \(1\)-s | \(m_0\) | \(m_1\) | \(m_2\) | \(m\) |
| number of \(0\)-s | \(n_0 - m_0\) | \(n_1- m_1\) | \(n_2- m_2\) | \(n - m\) |
| Total | \(n_0\) | \(n_1\) | \(n_2\) | \(n\) |
It is assumed that at least two of \(n_0, n_1\) and \(n_2\) are nonzero.
Introduce also the vectors \[ \mathbf{a} = \left(\begin{array}{c} 1 \\ 0 \end{array}\right), \quad \mathbf{b} = \left(\begin{array}{c} 1 \\ 1 \end{array}\right) \quad \text{and} \quad \mathbf{c} = \left(\begin{array}{c} 1 \\ 2 \end{array}\right). \]
The first two questions deal with existence and uniqueness of the MLE in this model. Define \[ C = \{ \mu_0 \mathbf{a} + \mu_1 \mathbf{b} + \mu_2 \mathbf{c} \mid \mu_0 \in (0, n_0), \mu_1 \in (0, n_1), \mu_2 \in (0, n_2) \} \]
- Show that the MLE in the model given by Equation 8.1 exists and is unique if and only if \[ m_0 \mathbf{a} + m_1 \mathbf{b} + m_2 \mathbf{c} \in C. \]
Consider the following concrete data set:
| \(X\) | \(0\) | \(1\) | \(2\) | Total |
|---|---|---|---|---|
| number of \(1\)-s | \(0\) | \(1\) | \(5\) | \(6\) |
| number of \(0\)-s | \(100\) | \(99\) | \(95\) | \(294\) |
| Total | \(100\) | \(100\) | \(100\) | \(300\) |
In this data set there are \(n = 300\) individuals and \(m = 6\) of them had the disease. Of these, 5 had the BB allele, 1 had the Bb allele and 0 had the bb allele.
Make a figure illustrating the set \(C\) for the concrete data set in the table above. Show that in this case the MLE exists and is unique.
Find a general criterion in terms of \(n_0, n_1, n_2\) and \(m_0, m_1, m_2\) for the existence of the MLE.
Hint: You can simplify the problem to the case \(n_0 = n_1 = n_2 = 100\).
Fit the model given by Equation 8.1 to the concrete data set in the table above.
Hint: You can fit the model using
glm()based on the summary data in the table. This requires a specification of a two-column response in the formula. See also the remark about fitting binomial and quasibinomial models in the Details section in the help page forglm().