Predicted probabilities (worked example)

5.5. Predicted probabilities (worked example)#

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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:

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

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 $β$ 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^{α + β x}}{1 + e^{α + β 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 $0 = α + β x$ and solve for $x$

We then find that $p (y = 1) = 0.5$ when $x = \frac{- α}{β}$ .

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