Logistic regression is a statistical method that predicts the probability of a binary outcome, such as “yes” or “no.” It models the relationship between one or more predictor variables and this two-category result. The method transforms a linear combination of input variables into a probability value between 0 and 1.
Does Logistic Regression Require Normality?
Logistic regression does not require the assumption of normality for its error terms or independent variables. This is a key difference from linear regression. The reason lies in the nature of the outcome variable: logistic regression deals with a binary dependent variable (0 or 1), not continuous values. Therefore, errors follow a Bernoulli or binomial distribution, not a normal distribution.
The model estimates probability using a logistic (sigmoid) function. This S-shaped function maps any real number to a value between 0 and 1, transforming predictors into a probability. The relationship is modeled on the log-odds (logit) scale, meaning the model is linear in the log-odds. This approach accommodates the binary nature of the dependent variable, making normality assumptions for errors unnecessary.
Core Assumptions of Logistic Regression
While normality is not an assumption, logistic regression relies on several other conditions for valid results. First, observations must be independent, meaning each data point is unrelated to others. This prevents skewed model estimations; for example, repeated measurements from the same individual would violate this.
Another assumption is the linearity of the log-odds with respect to the predictor variables. This means predictors linearly affect the logarithm of the outcome’s odds, not the outcome probability directly. Analysts can check this by examining scatter plots between each continuous predictor and the logit values of the outcome.
The absence of multicollinearity among predictor variables is also assumed. Multicollinearity occurs when two or more independent variables are highly correlated, making it difficult to determine each predictor’s unique contribution. High multicollinearity can lead to inflated standard errors and unreliable coefficient interpretation.
A sufficiently large sample size is also required. A common guideline suggests at least 10 cases with the least frequent outcome for each independent variable to ensure stable parameter estimates. Finally, the model assumes no perfect separation, which happens when a predictor perfectly separates outcome categories, causing infinite coefficient estimates and computational issues.
Understanding the Distinction from Linear Regression
The question of normality often arises because it is a fundamental assumption in linear regression. Linear regression predicts a continuous outcome and assumes residuals (differences between observed and predicted values) are normally distributed. This normality is important for the validity of statistical tests and confidence intervals.
A key difference between the two models lies in the type of dependent variable they handle. Linear regression is designed for continuous outcomes, while logistic regression is for binary or categorical outcomes.
Linear regression assumes a linear relationship between independent variables and the continuous dependent variable. Logistic regression, however, models the linear relationship between independent variables and the log-odds of the binary outcome, not raw probabilities. This distinction means their statistical properties and assumptions diverge significantly, even as both are generalized linear models.
Addressing Assumption Violations
Violating logistic regression assumptions can lead to unreliable model results, including biased coefficient estimates, incorrect standard errors, and invalid p-values. For example, non-independent observations might underestimate standard errors, leading to overly optimistic conclusions. Multicollinearity can make correlated predictor coefficients unstable and difficult to interpret.
Various diagnostic methods check for assumption violations. Linearity of log-odds can be assessed with scatter plots of predictors against the outcome’s logit, or formal tests like the Box-Tidwell test. Multicollinearity is assessed using the Variance Inflation Factor (VIF); high VIF values (e.g., above 5 or 10) indicate problematic correlations. Perfect separation often shows as warning messages in software, indicating fitted probabilities of 0 or 1, or extremely large coefficient standard errors.
If violations are detected, several strategies can be employed. Non-linearity of log-odds might involve transforming variables or adding polynomial terms. For multicollinearity, one can remove highly correlated predictors, combine them, or use dimensionality reduction. Perfect separation issues can be mitigated by collecting more data, combining categories, or using penalized regression. The specific approach depends on the violation’s nature and severity, requiring careful consideration of the data and research question.