linkfüggvény

A linkfüggvény, or link function, is a fundamental concept in generalized linear models (GLMs). It is a transformation that connects the linear predictor of a GLM to the expected value of the response variable. In simpler terms, it's a mathematical function that maps the output of a linear model to the range of possible values for the response variable, which may not be linear.

The general form of a GLM is $g(E(Y)) = X\beta$, where $E(Y)$ is the expected value of the response variable $Y$, $X$ is the matrix of predictor variables, $\beta$ is the vector of regression coefficients, and $g()$ is the link function. The linear predictor, $X\beta$, can take any real value, but the expected value of the response variable often has constraints. For instance, probabilities must lie between 0 and 1, and counts must be non-negative. The link function ensures that the model's predictions are consistent with these constraints.

Common link functions are chosen based on the distribution of the response variable. For a Normal distribution, the identity link function ($g(E(Y)) = E(Y)$) is used, which is the standard linear regression. For a Binomial distribution (e.g., binary outcomes), the logit link function is widely used, where $g(E(Y)) = \log(E(Y) / (1 - E(Y)))$. For a Poisson distribution (e.g., count data), the log link function is common, $g(E(Y)) = \log(E(Y))$. Other link functions include the probit, cloglog, and inverse links, each suited for different response variable distributions and research questions. The choice of link function influences the interpretation of the model's coefficients and the overall model fit.

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