Logistic Regression and Odds Ratios

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Presentation transcript:

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) The odds of receiving a death sentence = death sentence/not death sentence = 50/97 = 0.5155

Interpreting Odds The odds of 0.5155 can be stated in different ways: Defendants can expect to receive a death sentence instead of life imprisonment in about half of their trials 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 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 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 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 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 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).