What is Logistic Regression?

Logistic regression is the appropriate regression analysis to conduct when the dependent variable is dichotomous (binary).  Like all regression analyses, the logistic regression is a predictive analysis.  Logistic regression is used to describe data and to explain the relationship between one dependent binary variable and one or more nominal, ordinal, interval or ratio-level independent variables.

Sometimes logistic regressions are difficult to interpret; the Intellectus Statistics tool easily allows you to conduct the analysis, then in plain English interprets the output.

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Type of questions that a logistics regression can examine.

How does the probability of getting lung cancer (yes vs. no) change for every additional pound of overweight and for every pack of cigarettes smoked per day?

Do body weight calorie intake, fat intake, and participant age have an influence on heart attacks (yes vs. no)?

Logistics Regression major assumptions:

  1. That the outcome must be discrete, otherwise explained as, the dependent variable should be dichotomous in nature (e.g., presence vs. absent);
  2. There should be no outliers in the data, which can be assessed by converting the continuous predictors to standardized, or z scores, and remove values below -3.29 or greater than 3.29.
  3. There should be no high intercorrelations (multicollinearity) among the predictors.  This can be assessed by a correlation matrix among the predictors. Tabachnick and Fidell (2012) suggest that as long correlation coefficients among independent variables are less than 0.90 the assumption is met.

Logistic regression assumes that the dependent variable is a stochastic event.  For example, if we analyze a pesticides kill rate, the outcome event is either killed or alive.  Since even the most resistant bug can only be either of these two states, logistic regression thinks in likelihoods of the bug getting killed.  If the likelihood of killing the bug is > 0.5 it is assumed dead, if it is < 0.5 it is assumed alive.

The outcome variable – which must be coded as 0 and 1 – is placed in the first box labeled Dependent, while all predictors are entered into the Covariates box (categorical variables should be appropriately dummy coded).  Sometimes instead of a logit model for logistic regression a probit model is used.  The following graph shows the difference for a logit and a probit model for different values (-4,4).  Both models are commonly used in logistic regression, and in most cases, a model is fitted with both functions and the function with the better fit is chosen.  However, probit assumes normal distribution of the probability of the event, when logit assumes the log distribution.  Thus the difference between logit and probit is typically seen in small samples.

At the center of the logistic regression analysis is the task estimating the log odds of an event.  Mathematically, logistic regression estimates a multiple linear regression function defined as:

logit(p)

for i = 1…n .

Overfitting.  When selecting the model for the logistic regression analysis, another important consideration is the model fit.  Adding independent variables to a logistic regression model will always increase its statistical validity, because it will always explain a bit more variance of the log odds (typically expressed as R²).  However, adding more and more variables to the model makes it inefficient and over fitting occurs.

Reporting the R2 .  Nevertheless, many people want an equivalent way of describing how good a particular model is, and numerous pseudo-R2 values have been developed.  These should be interpreted with extreme caution as they have many computational issues which cause them to be artificially high or low.  A better approach is to present any of the goodness of fit tests available; Hosmer-Lemeshow is a commonly used measure of goodness of fit based on the Chi-square test (which makes sense given that logistic regression is related to crosstabulation).

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Related Pages:

Conduct and Interpret a Logistic Regression

Assumptions of Logistic Regression