Logistic Regression and Odds Ratio

1 Logistic Regression and odds Ratio A. Chang 1 odds Ratio Review Let p1 be the probability of success in row 1 (probability of Brain Tumor in row 1) 1 p1 is the probability of not success in row 1 (probability of no Brain Tumor in row 1) Odd of getting disease for the people who were exposed to the risk factor: (1 p is an estimate of p1) O+ = Let p0 be the probability of success in row 2 (probability of Brain Tumor in row 2) 1 p0 is the probability of not success in row 2 (probability of no Brain Tumor in row 2) Odd of getting disease for the people who were not exposed to the risk factor: (0 p is an estimate of p0) O = The odds Ratio of having brain tumor for people who were exposed to the risk factor versus not exposed: Interpretation.

2 The odds of having brain tumor are times higher for those who exposed to benzene than those who were not exposed to benzene. If > 1, then the odds of success are higher for column 1(risk factor present) than column 2(risk factor not present). If < 1, then the odds of success are lower for column 1(risk factor present) than column 2(risk factor not present). If = 1, then the odds of success are equal for column 1(risk factor present) and column 2(risk factor not present). Outcome Risk Factor Yes (Disease) No (NoDisease) Total Yes (Exposed) a b a + b No (Unexposed) c d c + d Total a + c b + d n ]|[]|[]|[]|[sedexpoundisease noPsedexpoundiseasePsedexpoundiseaseP-1s edexpoundiseaseP= 1 10000===++= = )]/(/[)]/([)]/(/[)]/([ 1 1 1100110011= ===++++= = === +bcaddcbadcddccbabbaappppppppOOOR Outcome (Brain Tumor)

3 Risk Factor (Benzene) Yes (Case) No (Control) Total Yes 50 20 70 No 100 130 230 Total 150 150 300 ]|[]|[]|[1]|[sedexpodisease noPsedexpodiseasePsedexpodiseaseP - sedexpodiseaseP= 1 11111===++= =bababbaappppRisk factor can always be arranged as either column or row variable, but the definition of odds Ratio is still the same.

4 Logistic Regression and odds Ratio A. Chang 2 Confidence interval for odds Ratio : For large sample, the log of odds Ratio , , follows asymptotically a normal distribution. The (1 )100% confidence interval estimate for the Log odds Ratio is The (1 )100% confidence interval estimate for the odds Ratio is ()*) ln(*) ln(2/2/,szszee + where , standard error of ln( ) is s* , and a, b, c and d should not be zero. Example: (Brain tumor) The 95% confidence interval estimate for the odds Ratio is: This confidence interval doesn t cover 1.

5 In fact, it covers a range that is greater than 1. This means that exposed to benzene is more likely to have brain tumor than those who were not exposed to benzene. Continuity Correction: The (1 )100% confidence interval estimate for the odds Ratio is ()*) ln(*) ln(2/2/,szszee + where , standard error of ln( ) is s* . ) ln( *) ln(2/sz dcba1111+++ bcad= ) , (),(),( *, ) () ln( *, , *) ln(*) ln(2 e ee esz s bcad zzszsz = = ===+++ = ====+ + +++++++dcba) )( () )( ( ++++=cbda Logistic Regression and odds Ratio A.

6 Chang 3 Example: Risk Factors for Dengue Epidemics Data: Dantes, Koopman and Addy et al., Dengue Epidemics on the Pacific Coast of Mexico, International Journal of Epidemiology 17 (1988), p. 178-186. Variable Name Description Identification Age Socioeconomic Status Sector Disease Status Saving Account Status 1-196 Age of person (in years) 1=upper, 2=middle, 3=lower Sector in city: 1 = sector 1 2 = sector 2 1= with disease, 0= no disease 1= has acct., 0 = has no acct. With 100% of the data: Disease Status Yes, Y=1 No, Y=0 X=1, sector 1 22 95 117 Sector X=0, sector 2 35 44 79 57 139 196 Sector 1: 1 p = 22/117, 1 1 p = 1 22/117 = 95/117 Sector 2.

7 0 p = 35/79, 1 0 p = 1 35/79 = 44/79 odds Ratio of having disease for sector 1 sector 2: (22/95)/(35/44) = .291 (The odds of having disease living in sector 1 is around 30% of the odds for living in sector 2.) (It is less likely to have disease living in sector 1 than living in sector 2.) odds Ratio of having disease for sector 2 sector 1: ( 35/44)/(22/95) = (The odds of having disease living in sector 2 is more than 3 times of the odds for living in sector 1.) Estimate and compare proportions of persons who contracted the disease by sector, using first 50% of the data: Disease Status Yes, Y=1 No, Y=0 X=1, sector 1 10 49 59 Sector X=0, sector 2 21 18 39 31 67 98 Sector 1.

8 1 p = 10/59, 1 1 p = 49/59 Sector 2: 0 p = 21/39, 1 0 p = 18/39 odds Ratio (sector 1 sector 2) = (10/49)/(21/18)= .175 (The odds of having disease living in sector 1 is around 18% of the odds for living in sector 2.) (It is less likely to have disease living in sector 1 than living in sector 2.) Logistic Regression and odds Ratio A. Chang 4 Use of SPSS for odds Ratio and Confidence Intervals Layout of data sheet in SPSS data editor for the 50% data example above, if data is pre-organized. Step 1: (Go to Step 2 if data is raw data and not organized frequencies as in figure (a).)

9 First, create the data in SPSS Data Editor as in (a), and then weight the cases entered in the Data Editor by click Data and select Weight Cases as in (b). In the Weight Cases dialog box select freq variable for weighting the cases as in figure (c). Weight Cases will allow users to be able to make a contingency table with the joint frequency distribution entered in (a) and each associate with a joint class. For example, 10 is the frequency for Yes and Sector 1 . The disease variable has internal values 1 and 2 (1 is labeled as Yes and 2 is labeled as No).

10 The sector variable has internal values 1 and 2 (1 is labeled as Sector 1 and 2 is labeled as Sector 2). Step 2: Following the procedure below (Analyze/Descriptive Statistics/Crosstabs) to make the contingency table. In SPSS, the row variable is risk factor and column variable is outcome variable. In the Crosstabs dialog box, click Statistics and check the Risk box in the Crosstabs: Statistics dialog window to obtain risk measurement such as odds Ratio and relative risk for obtaining the following Risk Estimate table. Chi-square test is one of the options too.