Recall: Logistic regression allows one to predict a discrete outcome from a set of variables that may be continuous, discrete, dichotomous, or a mix of any of these The dependent or response variable is dichotomous, such as presence/absence or success/failure. It has no assumption about the distribution of the independent variables. Logistic regression gives odds ratios from regression coefficients: exp( ) β Odds ratio sometimes used as an estimate of risk ratio = relative risk When is odds ratio a good estimate of relative risk?

Rare events: the number of events A is very small relative to the number of non-events B When event is rare in both groups, odds ratio is almost equal withrelative risk. But when events are not rare, odds ratios are not good estimates of relative risk

The log-binomial model has been proposed as a useful approach to compute an adjusted relative risk. Comparison with Logistic Regression Both model are used for the analysis of a dichotomous outcome. Both model the probability of the outcome. Both assume that the error terms have a binomial distribution. ● The difference between the logistic model and the log-binomial model is the link between the independent variables and the probability of the outcome. ● In logistic regression, the logit function is used and For the log-binomial model, the log function is used. definition A generalized linear model where the link function is the logarithm of the proportion under study and the distribution of the error is binomial Since log(π) must be in the interval -∞ to 0, restrictions in the estimation process have to be used to avoid predicting probabilities out of the [0,1] interval. uses ●This analysis is always used on experiments when then variable of interest is a binary outcome. ●Log binomial is an easy to use model than logistic regression to analyze cross-sectional (or longitudinal) data with binary outcomes and a confounding variable. ●Those are more interpretable and easier to communicate. However, using continuous variables on the model as independent variable may cause the log binomial regression to not converge. interpretation With log-binomial model, we estimate relative risk, not odds ratio. Change the function that links the mean π to the linear regression β0 + β1 x Regression coefficient for predictor x

β1 = β0 + β1(x + 1) − β0 + β1(x)

= log p(x + 1) − log p(x)

= log ( p(x + 1)/p(x))

= log (relative risk for unit increase in x) Interpretation:

⇒exp(β1) = relative risk for every unit increase in x

example: Why doesn’t everyone use log-binomial instead of logistic regression?

Logistic is numerically more stable: log-binomial does not always converge to produce an answer.

Logistic is conventional approach, software more developed. basada, calimutan, hormaza, transfiguracion Log binomial models there is a cohort study where experts followed 192 women with breast cancer. They classified them by breast-cancer stage (I, II, III) and receptor level (low, high). They are then classified as dead or alive five years after diagnosis. let's use sas to compare the results logistic and logbinomial models. thank you!

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# log binomial models

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