Bivariate Statistical Measures of Regression

Bivariate Statistical Measures of Regression

1. Simple linear regression examines the linear relationship between two continuous variables: a response (Y) and a predictor (X). when the two variables are related, it is possible to predict a response value from a predictor value with greater accuracy than the associated only with probabilities.

   • Linear regression
     • Before beginning this section it is essential to understand what is meant by linear, since there are two possible interpretations: linearity in the variables and linearity in the parameters.
       • A function y = f(x) is said to be linear on X if the variable X appears with unit power (hence, terms like x2, x3, 1/x, √x, for example) are excluded) and is not multiplied or divided by another variable. For example, yj = a + bxi + cxi2 is not a linear function on the variables since the variable X is squared
         • A function is said to be linear in the parameters if they appear with unitary frequency and not They are not multiplied or divided by any other parameter. As an example, yj = a + √bxi is not a linear function on the parameters. However, yj = a + bxi + cxi2 is.
           • Of the two interpretations of linearity, the linearity in the parameters is the more relevant in the context of regression and correlation theory. However, in what follows, both the linearity in the parameters as well as in the variables for the regression to be qualified as line
             • ESTIMATION OF THE PARAMETERS OF THE LINEAR REGRESSION
               • In the case in which it is assumed that the dependence relationship of Y on X is linear, yj = a + bxi + eij, where eij represents the error that is committed as a consequence of the fact that there could be other variables that influence the behavior of variable Y, the question is: How to estimate the parameters a and b of it?
                 • LINEAR COEFFICIENT OF DETERMINATION
                   • Once the rectilinear function has been chosen to represent the dependency relationship of Y on X and estimated its parameters a and b, then we proceed to calculate the coefficient of linear determination in order to measure the degree of dependence of Y on X under the estimated linear regression function

2. Multiple regression: when more than one independent variable is used to evaluate a dependent variable

   • modified nomenclature
     • modified and more formal notational framework is valuable for commenting on multiple regression analysis. Consider the model of general regression with three predictor variables. The modified equation:
       • Y = α + β1 X1 + β3X3 + ∊
         • to which is a simplified state of the more elaborate and precise equation.
           • • Multicollinearity assumption
               • Existing condition in a multiple regression analysis, which consists in the fact that the predictor variables are not independent of each other, as required, but they are correlated.
                 • Partial regression coefficient
                   • Partial regression coefficients is when there is multicollinearity, it just doesn't the "normal" interpretation of those coefficients, such as “the average change of the criterion variable related to the unit change of the predictor variable appropriate when held constant the other predictor variables. The equation would still be useful for purposes of prediction, assuming that the conditions are stable. However, they must not use the partial regression coefficients as a basis for decision making strategic marketing when it is multicollinearity significant
                     • Multiple Correlation and Multiple Determination Coefficients
                       • In multiple determination analysis, the proportion of variation in the criterion variable that is explained by the covariation of the predictive variables.
                         • Partial correlation coefficients
                           • Quantity that results from the multiple regression analysis and indicates the proportion of variation of the criterion variable that is not explained with one or more previous variables and yes with the inclusion of a new variable to the regression equation.
                             • binary variables
                               • One that is assigned one of two values, 0 or 1, and is used to represent the values ​​in numerical form. attributes or characteristics that are not essentially quantitative.
                                 • Variable transformations
                                   • A variable transformation is simply a change in the scale with which a given variable is expressed.

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