bivariate regression and correlation

bivariate regression and correlation

1. regression and correlation

   • There are numerous examples that can be given when working with two variables that are related: production and consumption. income and expenses; hours worked and accidents at work; hours of study and academic performance, etc
     • The relationship that can exist between two variables can be classified:
       • one-sided causal dependence. This relationship occurs when one of the variables influences the other but not vice versa. interdependence: occurs when the influence between the two variables is reciprocal. It would be a case of bilateral dependency. indirect dependency. Two variables can show a correlation through a third variable. agreement. It is presented in two independent variables to which the correlation that may exist is determined. chance covariation. when the correlation that occurs in two variables is completely random or accidental
   • regression : the term regression was originally used by Galton to indicate certain relationships in the theory of biological inheritance
     • rthe analysis of the regression, gives rise to the mathematical equation that allows us to discover the relationship between the variables
       • that is to say, to obtain an "ideal" line known as the regression line that describes the relationship or dependence between two variables.
         • This line or mathematical function, in the case of a single independent or explanatory variable, can be expressed through:
           • line or linear function: Y = bx + c quadratic parabola: Y = ax2 + bx + c exponential function: Y = cbx potential function = y = cx
   • The correlation analysis describes the degree or strength with which this relationship occurs, for this we will use a measure known as the correlation coefficient or Pearson correlation.
     • Just as the correlation analysis allows to measure the strength of association between two variables, the regression analysis allows the prediction or the estimate or the estimate of a value or average of a variable
       • We must be clear that the regression analysis, in addition to explaining the relationship between two variables, cause and effect, tells us if the mathematical relationship can be linear, parabolic, exponential, etc.
   • Dispersion diagram
     • The two-dimensional distribution data can be represented graphically in a pair of coordinate axes, considering the axis of the abscissa for the first variable (x) and the vertical axis or of the ordinates.
       • in a Cartesian plane, there are as many points as there are pairs of observations, each point corresponding to a pair of observations; This graphic representation is instinctively called: scattering diagram, scattering, scattering or point clouds.
         • regression and linear correlation
           • Generally, in the regression analysis it is used in a straight line due to its simplicity in the mathematical calculation, in addition, in many real life cases it provides us with sufficient approximations to be accepted. the mathematical model that describes a linear relationship when the value of Y is estimated as a functionof x, is represented indistinctly
             • Y =bx + c ó Y = B, x + B0 ó Y = Bx + A
   • Methods of least squares
     • is a preferred method of estimation, by which the quantities to be estimated are determined by minimizing the sum of the squares of the differences between the observed values ​​(y) and the estimated values ​​Y
       • Procedure for calculating the regression coefficients b and c (B and B) According to the general equation of the line Y = bx + c, in which we have as unknowns ( b and c ), it requires two normal equations for its solution: for this we obtain the first equation by putting the sign of the sum before each term. the second equation is obtained by multiplying the general equation by x, that is, yx = bx squared + cx and then adding the sign of the summation.
         • forecast or point estimate
           • It could be said that one of the purposes, when establishing the function that adequately represents the set of pairs of observations that have been taken for the analysis, is to be able to perform interpolations, since it is not indicated to make extrapolations. This means that if the value of Y is going to be predicted or estimated, it is required that the value of x (independent variable) must take a value that is in the range of said variable and not a value that is superior or lower than the limits of the population range.
             • residual variance and error and standard of estimation -> VR = Sv squared
               • the residual variance, unexplained variance or the sum of the error squares are the names used to identify this measure of dispersion, which allows calculating the degree of variation or dispersion.
                 • standard error of estimate is an expression given to the standard deviation of the observed values ​​with respect to a regression line. It could be said that it is an estimate of the variation that will probably accompany the predictions made by means of the regression equation.
                   • prediction or estimation interval
                     • the prediction interval or confidence limits for an estimated value (Y), consists of establishing two points for the estimator, within which the parameter or statistical value of the population must be contained
   • explained variance = VE So far, the variances for the variable x and Y (s2 and s2) have been calculated, called: total x variances and total variance of y. The values ​​for the unexplained variance or residual variance (S2 YX)
     • The explained variance is a measure that allows us to calculate the degree of dispersion of those values ​​that were explained by the regression line, that is, the dispersion of those points on the graph that are confused with the points of the line or estimated values.
       • correlation coefficient or pearson -> R = r
         • The correlation coefficient is a measure of the interdependence of two random variables.
           • In the first place, we refer to the calculation of the correlation coefficient to the table also known as the determination coefficient, symbolized by R squared. The formulas for the calculation of this coefficient are varied, obtaining an equal result by any of the methods.
             • angular coefficient test or regression coefficient

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