5.5. Predicted probabilities (worked example)

A logistic regression model describes whether the probability of voting for Candidate X in an election depends on \(x\) = the voter’s total family income (in thousands of dollars) the previous year. The prediction equation is:

\(\log{ \left[ \frac{p(y=1)}{1-p(y=1)} \right]} = -2.00 + 0.03x \)

Before we get to python, this exercise is a chance to practice converting logistic regression output into predicted probabilities by hand (i.e., by calculator or in Excel!) to help you see what is going on.

• Identify \(\beta\) and interpret its sign

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0.03. The estimated probability of voting for Candidate X increases as income increases.

• Find the estimated probability of voting for the candidate when income = 10,000.

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When we plug in a value of \(x = 10\), we find logit(p) = -1.7.

The alternative equation for logistic regression (derived from the equation above, see lecture slides) that expresses the probability directly is:

\[ p(y=1) = \frac{e^{\alpha + \beta x}}{1 + e^{\alpha + \beta x}} \]

Thus

\[ p(y=1) = \frac{e^{-1.7}}{1+e^{-1.7}} = \frac{0.182684}{1.182684} = 0.154465 \]

The probability of voting for Candidate X when income = 10,000 (i.e., x = 10) is 0.15, or 15%.

• At which income level is the estimated probability for the candidate equal to 0.50?

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When \(p(y=1) = 0.5\), the odds \(\frac{p(y=1)}{1-p(y=1)} = 0.\)

So, we take \(0 = \alpha + \beta x\) and solve for \(x\)

We then find that \(p(y=1)=0.5\) when \(x=\frac{-\alpha}{\beta}\).

Thus, for this example, \(p(y=1) = \frac{2}{0.03}\) = 66.67 thousand dollars