Logistic Regression and Odds RatiosProbability and OddsInterpreting OddsImpact of an Independent VariableOdds for Independent Variable GroupsThe Odds Ratio Measures the EffectSPSS Output for this RelationshipSW318Social Work StatisticsSlide 1Logistic Regression and Odds RatiosExample of Odds Ratio Using Relationship between Death Penalty and RaceSW388R7Data Analysis IISlide 2Probability and OddsWe 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 = 0.5155SW388R7Data Analysis IISlide 3Interpreting OddsThe 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 imprisonmentOr, inverting the odds,Receiving a life imprisonment sentence is twice as likely as receiving the death penalty.SW388R7Data Analysis IISlide 4Impact of an Independent VariableIf 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:SW388R7Data Analysis IISlide 5Odds for Independent Variable GroupsWe 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 = 0.6222The odds of receiving a death sentence if the defendant was not Black = 22/52 = 0.4231SW388R7Data Analysis IISlide 6The Odds Ratio Measures the EffectThe 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 blacksThe risk of receiving a death sentence are 1.47 times greater for blacks than non blacksThe 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.SW388R7Data Analysis IISlide 7SPSS Output for this RelationshipVariable s in the Equation.386 .350 1.213 1 .271 1.471-.860 .254 11.439 1 .001 .423BLACKDConstantStep1aB S.E. Wald df Sig. Exp(B)Variable(s) entered on step 1: BLACKD.a. The Exp(B) output using SPSS is the change in the odds ratio.The odds ratio is output in SPSS in the column labeled
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