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Logistic RegressionLogistic Regression with 1 PredictorSlide 3Example - Rizatriptan for MigraineExample - Rizatriptan for Migraine (SPSS)Odds Ratio95% Confidence Interval for Odds RatioExample - Rizatriptan for MigraineMultiple Logistic RegressionTesting Regression CoefficientsExample - ED in Older Dutch MenExample - ED in Older Dutch MenLoglinear Models with Categorical VariablesLoglinear ModelsLoglinear ModelsExample - Feminine Traits/BehaviorSlide 17Slide 18Slide 19SPSS OutputInterpreting CoefficientsGoodness of Fit StatisticsGoodness of Fit TestsSlide 24Slide 25Adjusted ResidualsSlide 27Comparing Models with G2 StatisticSlide 29Logit Models for Ordinal ResponsesLogistic Regression for Ordinal ResponseExample - Urban Renewal AttitudesSlide 33Fitted EquationInference for Regression CoefficientsSlide 36Ordinal PredictorsLogistic Regression•Logistic Regression - Dichotomous Response variable and numeric and/or categorical explanatory variable(s)–Goal: Model the probability of a particular as a function of the predictor variable(s)–Problem: Probabilities are bounded between 0 and 1•Distribution of Responses: Binomial•Link Function: 1log)(gLogistic Regression with 1 Predictor• Response - Presence/Absence of characteristic • Predictor - Numeric variable observed for each case• Model - (x)  Probability of presence at predictor level xxxeex1)(•  = 0  P(Presence) is the same at each level of x•  > 0  P(Presence) increases as x increases•  < 0  P(Presence) decreases as x increasesLogistic Regression with 1 Predictor- are unknown parameters and must be estimated using statistical software such as SPSS, SAS, or STATA·Primary interest in estimating and testing hypotheses regarding ·Large-Sample test (Wald Test):·H0: = 0 HA:   0)(::..:..2221,22^^2^obsobsobsXPvalPXRRXSTExample - Rizatriptan for Migraine •Response - Complete Pain Relief at 2 hours (Yes/No)•Predictor - Dose (mg): Placebo (0),2.5,5,10Dose # Patients # Relieved % Relieved0 67 2 3.02.5 75 7 9.35 130 29 22.310 145 40 27.6Example - Rizatriptan for Migraine (SPSS)Variables in the Equation.165 .037 19.819 1 .000 1.180-2.490 .285 76.456 1 .000 .083DOSEConstantStep1aB S.E. Wald df Sig. Exp(B)Variable(s) entered on step 1: DOSE.a. xxeex165.0490.2165.0490.2^1)(000.:84.3:819.19037.0165.0:..0:0:21,05.2220valPXRRXSTHHobsobsAOdds Ratio•Interpretation of Regression Coefficient ():–In linear regression, the slope coefficient is the change in the mean response as x increases by 1 unit–In logistic regression, we can show that:)(1)()()()1(xxxoddsexoddsxodds• Thus erepresents the change in the odds of the outcome (multiplicatively) by increasing x by 1 unit• If  = 0, the odds and probability are the same at all x levels (e=1)• If  > 0 , the odds and probability increase as x increases (e>1)• If  < 0 , the odds and probability decrease as x increases (e<1)95% Confidence Interval for Odds Ratio•Step 1: Construct a 95% CI for :^^^^^^^^^96.1,96.196.1• Step 2: Raise e = 2.718 to the lower and upper bounds of the CI:^^^^^^96.196.1,ee• If entire interval is above 1, conclude positive association• If entire interval is below 1, conclude negative association• If interval contains 1, cannot conclude there is an associationExample - Rizatriptan for Migraine)2375.0,0925.0()037.0(96.1165.0:%95037.0165.0^^^CI• 95% CI for  :• 95% CI for population odds ratio: )27.1,10.1(,2375.00925.0ee• Conclude positive association between dose and probability of complete reliefMultiple Logistic Regression•Extension to more than one predictor variable (either numeric or dummy variables).•With k predictors, the model is written:kkkkxxxxee11111• Adjusted Odds ratio for raising xi by 1 unit, holding all other predictors constant:ieORi• Many models have nominal/ordinal predictors, and widely make use of dummy variablesTesting Regression Coefficients•Testing the overall model:)(..))log(2())log(2(..0 allNot :0:222,210210obskobsobsiAkXPPXRRLLXSTHH• L0, L1 are values of the maximized likelihood function, computed by statistical software packages. This logic can also be used to compare full and reduced models based on subsets of predictors. Testing for individual terms is done as in model with a single predictor.Example - ED in Older Dutch Men •Response: Presence/Absence of ED (n=1688)•Predictors: (p=12)–Age stratum (50-54*, 55-59, 60-64, 65-69, 70-78)–Smoking status (Nonsmoker*, Smoker)–BMI stratum (<25*, 25-30, >30)–Lower urinary tract symptoms (None*, Mild, Moderate, Severe)–Under treatment for cardiac symptoms (No*, Yes)–Under treatment for COPD (No*, Yes) * Baseline group for dummy variablesExample - ED in Older Dutch MenPredictor b sbAdjusted OR (95% CI)Age 55-59 (vs 50-54) 0.83 0.42 2.3 (1.0 – 5.2)Age 60-64 (vs 50-54) 1.53 0.40 4.6 (2.1 – 10.1)Age 65-69 (vs 50-54) 2.19 0.40 8.9 (4.1 – 19.5)Age 70-78 (vs 50-54) 2.66 0.41 14.3 (6.4 – 32.1)Smoker (vs nonsmoker) 0.47 0.19 1.6 (1.1 – 2.3)BMI 25-30 (vs <25) 0.41 0.21 1.5 (1.0 – 2.3)BMI >30 (vs <25) 1.10 0.29 3.0 (1.7 – 5.4)LUTS Mild (vs None) 0.59 0.41 1.8 (0.8 – 4.3)LUTS Moderate (vs None) 1.22 0.45 3.4 (1.4 – 8.4)LUTS Severe (vs None) 2.01 0.56 7.5 (2.5 – 22.5)Cardiac symptoms (Yes vs No) 0.92 0.26 2.5 (1.5 – 4.3)COPD (Yes vs No) 0.64 0.28 1.9 (1.1 – 3.6)Interpretations: Risk of ED appears to be:• Increasing with age, BMI, and LUTS strata• Higher among smokers• Higher among men being treated for cardiac or COPDLoglinear Models with Categorical Variables•Logistic regression models when there is a clear response variable (Y), and a set of predictor variables (X1,...,Xk)•In some situations, the variables are all responses, and there are no clear dependent and independent variables•Loglinear models are to correlation analysis as logistic regression is to ordinary linear regressionLoglinear Models•Example: 3


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UF STA 6127 - Logistic Regression

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