Problem Set: Logit Model for Binary Outcomes

Concept Comprehension

  1. The probability of an event happening is \(.6\). What are the odds of that event happening? [1]

  2. The odds of an event happening are \(2.5\). What is the probabilty of that event happening? [1]

  3. An event is equally likely to happen as it is to not happen. What are the log odds of that event happening? [1]

  4. In a sentence, explain what the expression \(\Pr(y=1|x=0) = .25\) means using words and not any mathematical symbols. [1]

  5. Say that \(\Pr(y=1|x=0) = .25\) and \(\Pr(y=1|x=1) = .75\). Calculate the difference in the log odds that \(y=1\) when \(x=1\) vs. \(x=0\). [2]

  6. In the logit model, the difference you calculated in the prior question would be the coefficient \(\beta_x\). Given that value of \(\beta_x\), what is the odds ratio? [1]

  7. Consider the example of the relationship between a couple’s household income and how likely it is that the couple owns a home.

    • In 3-5 sentences, as if you were talking to an intelligent person unfamiliar with the materials of this course – perhaps yourself at the dawn of your statistical training – explain the difference between how the linear probability model and the logit model would characterize this relationship.

    • (As it happens, home ownership as an example for a different question below. The question here is intended as conceptual and so don’t use or refer to the later results in answering this question.) [3]

  8. In a sentence, describe the difference between the odds and the odds ratio. [1]

I used the 2010-2018 General Social Survey to fit a model in which the outcome is whether the respondent owns/is-buying their dwelling (vs. renting or some other arrangement). The explanatory variables are household income (measured in 10000s of dollars, inflation-adjusted to 2015) and whether one lives in a city. I got the following odds ratios:

  • For household income, the odds ratio estimate is 1.208
  • For the city variable, the odds ratio estimate is .487

  1. In a sentence, interpret the odds ratio for household income. [1]

  2. In a sentence, interpret the odds ratio for the city variable. [1]

  3. How does the relative risk change as the baseline probability increases? [1]