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Logistic Regression and Odds Ratios. Example of Odds Ratio Using Relationship between Death Penalty and Race. Probability and Odds. We begin with a frequency distribution for the variable “Death Penalty for Crime”. The probability of receiving a death sentence is 0.34 or 34% (50/147)

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logistic regression and odds ratios

Logistic Regression and Odds Ratios

Example of Odds Ratio

Using Relationship between

Death Penalty and Race

probability and odds
Probability and Odds
  • We begin with a frequency distribution for the variable “Death Penalty for Crime”
  • The probability of receiving a death sentence is 0.34 or 34% (50/147)
  • The odds of receiving a death sentence = death sentence/life imprisonment = 50/97 = 0.5155
interpreting odds
Interpreting Odds
  • The odds of 0.5155 can be stated in different ways:
    • Defendants can expect to receive a death sentence 50% as often as they would receive life imprisonment.
    • Receiving a death sentence is half as likely as receiving a sentence of life imprisonment
  • Or, inverting the odds,
    • Receiving a life imprisonment sentence is twice as likely as receiving the death penalty.
impact of an independent variable
Impact of an Independent Variable
  • If an independent variable impacts or has a relationship to a dependent variable, it will change the odds of being in the key dependent variable group, e.g. death sentence.
  • The following table shows the relationship between race and sentence:
odds for independent variable groups
Odds for Independent Variable Groups
  • We can compute the odds of receiving a death penalty for each of the groups:
  • The odds of receiving a death sentence if the defendant was Black = 28/45 = 0.6222
  • The odds of receiving a death sentence if the defendant was not Black = 22/52 = 0.4231
the odds ratio measures the effect
The Odds Ratio Measures the Effect
  • The impact of being black on receiving a death penalty is measured by the odds ratio which equals:

= the odds if black ÷ the odds if not black

= 0.6222 ÷ 0.4231 = 1.47

  • Which we interpret as:
      • Blacks are 1.47 times more likely to receive a death sentence as non blacks
      • The risk of receiving a death sentence are 1.47 times greater for blacks than non blacks
      • The odds of a death sentence for blacks are 47% higher than the odds of a death sentence for non blacks. (1.47 - 1.00)
      • The predicted odds of a death sentence for black defendants are 1.47 times the odds for non black defendants.
      • A one unit change in the independent variable race (nonblack to black) increases the odds of receiving a death penalty by a factor of 1.47.
spss output for this relationship
SPSS Output for this Relationship

The Exp(B) output using SPSS is the change in the odds ratio.

The odds ratio is output in SPSS in the column labeled Exp(B).

from odds back to probabilities
From Odds Back to Probabilities
  • The formula for computing odds from probabilities is:

odds = probability / (1 – probability)

With a little algebra, we can solve for probability in terms of odds:

probability = odds/(1 + odds)

  • We can interpret our findings in terms of probabilities using the formula that probability = odds/(1 + odds)

the odds if black = .6222

the probability if black = .6222/1.6222 = .3835

the odds if not black = .4231

the probability if not black = .4231/1.4231 = .2973

  • We could also say: if the defendant was black rather than white, the probability that he received a death sentence increased from .2973 to .3835
  • Note that the ratio of .3835 / .2973 is 1.29, not the odds ratio of 1.47