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Mese: Dicembre 2018

Measure Dependence and Regression line

In statistics, dependence or association is any statistical relationship, whether casual or not, between two random variables or bivariate data. Formally, random variables are dependent if they do not satisfy a mathematical property of probabilistic independence. However, when used is a technical sense, correlation refers to any of several specific types of relationship between mean values. There are several correlation coefficients, measuring the degree of correlation. The most common of these is the Pearson correlation coefficient, which is sensitive only to a linear relationship between two variables. Other correlation coefficients have been developed to be more robust than the Pearson correlation – that is more sensitive to nonlinear relationships. Mutual information can also be applied to measure dependence between two variables. Correlation and linearity The Pearson correlation coefficient indicates the strength of a linear relationship between two variables, but its value generally does not completely characterize their relationship. In particular, if the conditional mean of Y given X, denoted E(Y | X), is not linear in X, the correlation coefficient will not fully determine the form E(Y | X).   In statistics, linear regression is a linear approach to modelling the relationship between a scalar response (or dependent variable) and one or more explanatory variables (or independent variables). The case of one explanatory variable is called “simple linear regression”. Linear regression models are often fitted using the least squares approach, but they may also be fitted in other ways, such as by minimizing the “lack of it” in some other norm (as with least absolute derivation regression), or by minimizing a penalized version of the least squares cost function as in ridge regression (L2-norm penalty) and lasso (L1-norm penalty). Conversely, the least squares approach can be used to fit models that are not a linear one. Thus, although the terms “least squares” and “linear model” are closely linked, they are not synonymous. A fitted linear regression model can be used to identify the relationship between a single predictor variable Xj and the response variable y when all the other predictor variables in the model are “help fixed”. Specifically, the interpretation of βj is the expected change in y for a one-unit change in Xj when the other covariates are help-fixed. That is the expected value of the partial derivative of y with respect to Xj. This is sometimes called the unique effect of Xj on y. In contrast, the marginal effect of Xj on y can be assessed using a correlation coefficient or simple linear regression model relating only Xj to y;…