Multivariate Analysis

Multiple Linear Regression

Multiple Linear Regression

Regression Equation

Prediction Equation

The Two Predictor Case

Squared Multiple Correlation

  • When r12 = 0

  • When r12 does not equal 0

    Regression Coefficients

    Sums of Squares

    Raw Regression Coefficient vs Standardized Regression Coefficient

    b vs β

  • Use b with raw scores.
  • Use β with standard scores.

    Note

  • When r12 = 0 then β1 = ry1 & β2 = ry2

    Prediction Equation in Standardized Form

    Beta Coefficients

    More on Betas

    Multiple Correlation Coefficient

    Squared Multiple Correlations

    More on Squared Multiple Correlation

    Variance of Estimate/Standard Error of Estimate

    The variance of estimate is also called the mean square error in the ANOVA summary table of the regression analysis.

    The standard error of estimate gives an indication of how far, on the average, observations fall from the regression line.

    Testing the Model

    The Overall F-test

  • Tests R2 equal to zero.
  • Tests the regression equation.
  • Tests that all b's are simultaneously zero.

    Tests of Regression Coefficients

  • Tests b1 = 0 when all other variables in the equation are held constant.

    About Tests of Regression Coefficients

  • Tests a single coefficient with all the others in the regression equation held constant.
  • The larger the r12 the larger the standard error of b, and thus, the lower the power of the t-test.

    Testing Increments in R-squared

  • Let k1 > k2.

    Note:

  • R2y.123 = R2y.321
  • When independent variables are correlated, the incremental proportion of variance accounted for by a single variable depends on, among other things, when the variable enters into the regression equation.
  • Further, assuming all variables are positively correlated, the later the entry point of the variable, the smaller the incremental proportion of variance accounted for.

    Comparing Variables

  • Due to different scales of measurement, b's cannot, generally, be used to compare variables.
  • Care must be taken in using β's to compare variables, since β's are affected by, among other things, the variability of the variables with which they are associated.

    Multivariate Course Page

    Phil Ender, 29Jan98