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}}
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")
To find the predicted probabilities in terms of f_{kl}, we need to use the logistic regression model to derive the expressions for \widetilde{p_{01}} and \widetilde{p_{11}}. The logistic regression model assumes that the log odds of the probability of Y_i being 1 is a linear function of X_i:
\log \left( \frac{P(Y_i = 1 | X_i = x)}{1 - P(Y_i = 1 | X_i = x)} \right) = \beta_0 + \beta_1 x
where \beta_0 and \beta_1 are the maximum likelihood estimators.
To find \widetilde{p_{01}}, we substitute X_i = 0 into the logistic regression model:
\log \left( \frac{P(Y_i = 1 | X_i = 0)}{1 - P(Y_i = 1 | X_i = 0)} \right) = \widehat{\beta}_0 + \widehat{\beta}_1 \cdot 0
\log \left( \frac{P(Y_i = 1 | X_i = 0)}{1 - P(Y_i = 1 | X_i = 0)} \right) = \widehat{\beta}_0
Exponentiating both sides to solve for P(Y_i = 1 | X_i = 0):
\frac{P(Y_i = 1 | X_i = 0)}{1 - P(Y_i = 1 | X_i = 0)} = e^{\widehat{\beta}_0}
P(Y_i = 1 | X_i = 0) = e^{\widehat{\beta}_0} (1 - P(Y_i = 1 | X_i = 0))
P(Y_i = 1 | X_i = 0) = \frac{e^{\widehat{\beta}_0}}{1 + e^{\widehat{\beta}_0}}
P(Y_i = 1 | X_i = 0) = \frac{1}{1 + e^{-\widehat{\beta}_0}}
Therefore, \widetilde{p_{01}} = \frac{1}{1 + e^{-\widehat{\beta}_0}}.
Similarly, to find \widetilde{p_{11}}, we substitute X_i = 1 into the logistic regression model:
\log \left( \frac{P(Y_i = 1 | X_i = 1)}{1 - P(Y_i = 1 | X_i = 1)} \right) = \widehat{\beta}_0 + \widehat{\beta}_1 \cdot 1
\log \left( \frac{P(Y_i = 1 | X_i = 1)}{1 - P(Y_i = 1 | X_i = 1)} \right) = \widehat{\beta}_0 + \widehat{\beta}_1
\log \left( \frac{P(Y_i = 1 | X_i = 1)}{1 - P(Y_i = 1 | X_i = 1)} \right) - \log \left( \frac{P(Y_i = 1 | X_i = 1)}{1 - P(Y_i = 1 | X_i = 1)} \right) = \widehat{\beta}_1
\log \left( \frac{P(Y_i = 1 | X_i = 1)}{1 - P(Y_i = 1 | X_i = 1)} \right) = \widehat{\beta}_1
Exponentiating both sides to solve for P(Y_i = 1 | X_i = 1):
\frac{P(Y_i = 1 | X_i = 1)}{1 - P(Y_i = 1 | X_i = 1)} = e^{\widehat{\beta}_1}
P(Y_i = 1 | X_i = 1) = e^{\widehat{\beta}_1} (1 - P(Y_i = 1 | X_i = 1))
P(Y_i = 1 | X_i = 1) = \frac{e^{\widehat{\beta}_1}}{1 + e^{\widehat{\beta}_1}}
Therefore, \widetilde{p_{11}} = \frac{e^{\widehat{\beta}_1}}{1 + e^{\widehat{\beta}_1}}.
Hence, the expressions for the predicted probabilities are:
\widetilde{p_{01}} = \frac{1}{1 + e^{-\widehat{\beta}_0}}
\widetilde{p_{11}} = \frac{e^{\widehat{\beta}_1}}{1 + e^{\widehat{\beta}_1}}
Note: The expressions above are in terms of \widehat{\beta}_0 and \widehat{\beta}_1, not in terms of the individual f_{kl} values.