Question

Even though logistic regression is formulated with continuous input data in mind, one can also try to apply it to categorical inputs. For example, consider the following set-up: We observe \, n \, samples \, Y_ i \in \{ 0, 1\} \,, \, i = 1, \dots , n \,, and covariates \, X_ i \in \{ 0, 1\} \,, \, i = 1, \dots , n \,. Moreover, assume that given \, X_ i \,, the \, Y_ i \, are independent.

First, let us apply regular maximum likelihood estimation. To this end, write

\displaystyle f_{00} = {} \displaystyle \frac{1}{n} \# \{ i : X_ i = 0 \text { and } Y_ i = 0 \}
\displaystyle f_{01} = {} \displaystyle \frac{1}{n} \# \{ i : X_ i = 0 \text { and } Y_ i = 1 \}
\displaystyle f_{10} = {} \displaystyle \frac{1}{n} \# \{ i : X_ i = 1 \text { and } Y_ i = 0 \}
\displaystyle f_{11} = {} \displaystyle \frac{1}{n} \# \{ i : X_ i = 1 \text { and } Y_ i = 1 \}
and assume that \, f_{00}, f_{01}, f_{10}, f_{11} > 0 \,. We can parametrize this model in terms of

\displaystyle p_{01} = {} \displaystyle P(Y_ i = 1 | X_ i = 0)
\displaystyle p_{11} = {} \displaystyle P(Y_ i = 1 | X_ i = 1)
Compute the maximum likelihood estimators \, \widehat{p}_{01} \, and \, \widehat{p}_{11} \, for \, p_{01} \, and \, p_{11} \,, respectively. Express your answer in terms of f_{00} (enter “A"), f_{01} (enter “B"), f_{10} (enter “C"), f_{11} (enter “D") and n.

\widehat{p}_{01}
B/(A+B)
correct

\widehat{p}_{11}
D/(C+D)
correct

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(b)
2 points possible (graded)
Although the \, X_ i \, are discrete, we can also use a logistic regression model to analyze the data. That is, now we assume

Y_ i | X_ i \sim \textsf{Ber}\left( \frac{1}{1 + \mathbf e^{-(X_ i \beta _1 + \beta _0})} \right),

for \, \beta _0, \beta _1 \in \mathbb {R} \,, and that given \, X_ i \,, the \, Y_ i \, are independent.

Calculate the maximum likelihood estimator \, \widehat{\beta }_0 \,, \, \widehat{\beta }_1 \, for \, \beta _0 \, and \, \beta _1 \,, where we again assume that all \, f_{kl} > 0 \,. Express your answer in terms of f_{00} (enter “A"), f_{01} (enter “B"), f_{10} (enter “C"), f_{11} (enter “D") and n.

\widehat{\beta }_{0}
unanswered

\widehat{\beta }_{1}
unanswered

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(c)
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Given the maximum likelihood estimators \, \widehat{\beta }_0 \,, \, \widehat{\beta }_1 \,, what are the associated predicted probabilities

\displaystyle \widetilde{p_{01}} = {} \displaystyle P(Y_ i = 1 | X_ i = 0, \widehat{\beta }_0, \widehat{\beta }_1)
\displaystyle \widetilde{p_{11}} = {} \displaystyle P(Y_ i = 1 | X_ i = 1, \widehat{\beta }_0, \widehat{\beta }_1)
in terms of f_{kl}, for k, l \in \{ 0, 1\}?

Express your answer in terms of f_{00} (enter “A"), f_{01} (enter “B"), f_{10} (enter “C"), f_{11} (enter “D") and n.

\widetilde{p_{01}}
1/A
incorrect

\widetilde{p_{11}}