To calculate the predicted probabilities \(\widetilde{p_{01}}\) and \(\widetilde{p_{11}}\), we need to use the logistic regression model.
The logistic regression model can be defined as:
\[P(Y_i = 1|X_i = x) = \frac{1}{1 + e^{-\left(\beta_0 + \beta_1 x\right)}}\]
where \( Y_i \) is the binary dependent variable, \( X_i \) is the independent variable, \( \beta_0 \) is the intercept, and \( \beta_1 \) is the slope coefficient.
We are given the maximum likelihood estimators \( \widehat{\beta}_0 \) and \( \widehat{\beta}_1 \), which are the estimated values for \( \beta_0 \) and \( \beta_1 \).
To calculate the predicted probabilities, we substitute the values of \( \widehat{\beta}_0 \) and \( \widehat{\beta}_1 \) into the logistic regression model.
1. For \( \widetilde{p_{01}} \):
\[ \widetilde{p_{01}} = P(Y_i = 1|X_i = 0, \widehat{\beta}_0, \widehat{\beta}_1) = \frac{1}{1 + e^{-\left(\widehat{\beta}_0 + \widehat{\beta}_1 \cdot 0\right)}}\]
Since \( \widehat{\beta}_1 \cdot 0 = 0 \), the equation simplifies to:
\[ \widetilde{p_{01}} = \frac{1}{1 + e^{-\widehat{\beta}_0}}\]
2. For \( \widetilde{p_{11}} \):
\[ \widetilde{p_{11}} = P(Y_i = 1|X_i = 1, \widehat{\beta}_0, \widehat{\beta}_1) = \frac{1}{1 + e^{-\left(\widehat{\beta}_0 + \widehat{\beta}_1 \cdot 1\right)}}\]
Since \( \widehat{\beta}_1 \cdot 1 = \widehat{\beta}_1 \), the equation simplifies to:
\[ \widetilde{p_{11}} = \frac{1}{1 + e^{-\left(\widehat{\beta}_0 + \widehat{\beta}_1\right)}}\]
Therefore, the predicted probabilities in terms of \( f_{kl} \) are:
\(\widetilde{p_{01}} = \frac{1}{1 + e^{-\widehat{\beta}_0}}\) (enter "A")
\(\widetilde{p_{11}} = \frac{1}{1 + e^{-\left(\widehat{\beta}_0 + \widehat{\beta}_1\right)}}\) (enter "D")
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}}
unanswered
\widetilde{p_{11}}
1 answer
1 answer