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Logistic Regression Fall 2009

Logistic Regression Fall 2009 - McGill University Advanced...

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Logistic Regression aka Binary Regression
The Logistic Function

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Properties of the logistic function The logistic function is useful in modeling many phenomena that require both lower and upper bounds. Note that there is no restriction on the domain (input) of the function but its range (output) is confined to values between 0 and 1. Logistic regression is a useful way of describing the relationship between one or more risk factors (e.g., age, sex, smoking, etc.) and an outcome such as mortality (which only takes two possible values: death or survival).
LOGISTIC REGRESSION Logistic regression is used to model categorical response variables that take on one of only two possible values, such as Agree/Disagree, Prefer Brand A/Prefer Brand B, or Yes/No. In this case it is clear that the response variable violates the normality assumption, so that an alternative to standard simple or multiple regression models must be developed.

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History of Logistic Regression Logistic regression was originally applied to survival data in the health sciences, where the response variable would assume one of the two values Survive or Die. Logistic regression is based on the concept of the Odds Ratio , which represents the probability of an event of interest occurring, compared with the probability that the event of interest does not occur.
Odds Ratio Probability of an event of interest Odds ratio = 1- probability of an event of interest Example: P(E) = 0.5 0.50 Odds ratio = 1 or 1 to 1 1- 0.50 = Example: P(E) = 0.75 0.75 Odds ratio = 3 or 3 to 1 1 0.75 = -

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Definitions The quantity is called the “log-odds ratio”. It is also referred to as the logistic or logit transformation. ln 1 p p ÷ -
Logistic Regression Model The logistic regression model is based on the natural logarithm of the odds ratio: ln(odds ratio) = β 0 + β 1 X 1 + β 2 X 2 + . .. + β k X k If we set P(E) = , then the model may be written as 0 1 1 2 2 logit( ) =ln ... 1 k k p p X X X p β ε = + + + + + ÷ - p

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Alternative formulation of the model An alternative form of the model is obtained by solving the previous equation for p : Definition of p Note that for a binary variable 0 1 1 2 2 0 1 1 2 2 0 1 1 2 2 ... ... ( ... ) 1 1 1 k k k k k k X X X X X X X X X e p e e β + + + + + + - + + + = = + + 1 if condition A occurs 0 if condition A* occurs y = p = P( y = 1), i.e. The probability that condition A occurs.
Logistic graph Linear Logistic

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When we use sample data to estimate the logistic model we obtain the following logistic regression equation : ln(estimated odds ratio) = Once you have found the logistic regression equation, the estimated odds ratio may be computed using the formula: 0 1 1 2 2 ... k
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Logistic Regression Fall 2009 - McGill University Advanced...

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