Probability and odds Suppose we a frequency distribution for the variable “TB status” The probability of an individual having TB is frequencyRelative frequency Have TB Don’t have TB Total680 2

The odds of having TB is Definition of odds Definition of odds : 3

The odds of 0.12 can be interpreted as: For the community in consideration, we expected about one eighth of them to have TB. Or inverting the odds, an individual is eight times as likely not to have TB as having TB. Interpreting odds 4

Association between a dependent variable and an independent variable If an independent variable impacts or has a relationship with the dependent variable, it will change the odds of being in the key dependent variable group( group with the event of interest) For example suppose we have information on HIV status of the individuals whose “TB status” we had earlier: 5

The impact of HIV on TB status can be measured using odds ratio 6 Have TBDon’t have TBTotal HIV+ve HIV-ve Total

The odds ratio is equal to The odds of having TB for HIV +ve individuals is 57/32= The odds of having TB for HIV-ve individuals is 106/505=

Therefore the odds ratio is /0.21= 8.48 The odds ratio can also be calculated directly as Interpretation: HIV +ve individuals are 8.5 times more likely to have TB compared to HIV-ve individuals. 8

Another example….. The table below gives the contingency table of number of women in a study according to use of contraceptive pill and presence/absence of myocardial infarction Myocardial infraction total contraceptiveYesNo using pill not using pill Total

The odds of women using the pill having infraction is 23/49 = And the odds of women not using the pill having infraction is 35/132= Thus the odds ratio having infraction for women using the pill compared to those not using the pill is 0.469/0.265=

Women using the pill are 1.77 times more likely to have myocardial infraction compared to women not using the pill. women using the pill are 77% [ ( ) x 100%] more likely to have myocardial infraction compared to women not using the pill. Interpreting the odds ratio 11

Logistic model It is a mathematical expression used to determine if a relationship exists between a binary dependent variable and a set of independent variables Logistic regression combines the independent variables to estimate the probability that a particular event will occur, i.e. an individual will be a member of one of the groups defined by the binary dependent variable 12

13 If we have only one independent variable, the model is log(odds of event) = a + b  predictor If we have two or more predictors, the model is log(odds of event) = a + b 1  predictor 1 + b 2  predictor 2 +….. + b k  predictor k b 1, b 2, …., b k are known regression coefficients

The independent variables can be either qualitative (categorical) or quantitative (continuous). The independent variables usually include exposure variables, potential confounders and potential effect modifiers 14 Measurements of independent variables:

15 Interpreting output of logistic regression If a coefficient is positive, its transformed log value will be greater than one, meaning that the modeled event is more likely to occur. If a coefficient is negative, its transformed log value will be less than one, and the odds of the event occurring decrease. A coefficient of zero (0) has a transformed log value of 1.0, meaning that this coefficient does not change the odds of the event one way or the other

16 The transformed log value is an odds ratio. For a qualitative independent variable, one level of the variable is selected as an reference and the other levels compared to it. For example, using gender as a variable; then suppose female is chosen as reference then the coefficient corresponding to this variable is interpreted using

17 Another example, suppose age is categorized into five groups : , 30 – 39, 40 – 49, 50 – 59, ; and 20 – 29 group is chosen as the reference group and then have four odds ratio (OR) for this variable

18 OR = odds of having disease for age group 30 – 39 odds of having disease for age group 20 – 29 OR= odds of having disease for age group 40 – 49 odds of having disease for age group 20 – 29

19 OR = odds of having disease for age group 50 – 59 odds of having disease for age group 20 – 29 OR = odds of having disease for age group 60 – 69 odds of having disease for age group 20 – 29

20 For a quantitative independent variable, then we compare two groups with a difference of one unit of measurement of the variable. For example if blood pressure is an independent variable Then odds ratio is OR = odds of having disease for those with (x + 1)mm/Hg odds of having disease for those with x mm/Hg where x is say 120

21 Each independent variable is interpreted adjusting for others. When reporting the results it is advised to report both the unadjusted and adjusted odds ratios

22 A study is designed to assess the association between obesity (defined as BMI > 30) and incident cardiovascular disease. Data were collected from participants who were between the ages of 35 and 65, and free of cardiovascular disease (CVD) at baseline. Each participant was followed for 10 years for the development of cardiovascular disease. A logistic regression analysis is fitted to assess the association between obesity(independent variable ) and CVD (present=1,absent=0) For independent variable, non obese persons is reference group. Example:

23 independent variable regression coefficient (  ) z-valuep-value Exp(  ) (odds ratio) 95% for odds ratio constant obesity The results of the fit were: exp(0.658) = 1.93, is the unadjusted odds ratio. The odds of developing CVD are 1.93 times higher among obese persons as compared to non obese persons. Obese persons are 93% more likely to develop CVD. The association between obesity and incident CVD is statistically significant (p=0.0017).

24 When examining the association between obesity and CVD, age was determined as a confounder. To adjust for it, a logistic regression model is fitted with obesity and age as independent variables and CVD as the dependent variable. Age is categorized as less than 50 years of age and 50 years of age and older. For the analysis, age group of less than 50 years of age is reference group.

25 The fitted model was: Log(odds of developing CVD)= obesity+0.655age Exp(0.415)=1.52; the odds of developing CVD are 1.52 times higher among obese persons as compared to non obese persons, adjusting for age. This is adjusted odd ratio.

26 Example: Researchers examined the relationship between coronary heart disease (CHD) risk and the risk factors: age(in years), cholesterol (mg/dL), systolic blood pressure (mmHg), body mass index (BMI) and smoking status. Using a logistic model, they obtained the results below:

27 independent variable regression coefficient (  ) z-valuep-value Exp(b) Odds ratio 95% for odds ratio constant age < cholestrol < sbp < bmi smokes <

28 Adjusting for cholesterol level, systolic blood pressure, body mass index and smoking status; for every additional year in age, an individual is 1.06 times more likely to have CHD. Alternatively, we can say that an is 6% more likely to have CHD for every additional age while adjusting for other variables. The unadjusted odds ratio for age was 1.08 Age is a significant predictor since p-value is small( <0.001).