Linear regression models

Simple Linear Regression

History Developed by Sir Francis Galton (1822-1911) in his article “Regression towards mediocrity in hereditary structure”

Purposes: To describe the linear relationship between two continuous variables, the response variable (y-axis) and a single predictor variable (x-axis) To determine how much of the variation in Y can be explained by the linear relationship with X and how much of this relationship remains unexplained To predict new values of Y from new values of X

The linear regression model is: Xi and Yi are paired observations (i = 1 to n) β0 = population intercept (when Xi =0) β1 = population slope (measures the change in Yi per unit change in Xi) εi = the random or unexplained error associated with the i th observation. The εi are assumed to be independent and distributed as N(0, σ2).

Linear relationship Y ß1 1.0 ß0 X

Linear models approximate non-linear functions over a limited domain extrapolation interpolation extrapolation

For a given value of X, the sampled Y values are independent with normally distributed errors: Yi = βo + β1*Xi + εi ε ~ N(0,σ2)  E(εi) = 0 E(Yi ) = βo + β1*Xi Y E(Y2) E(Y1) X X1 X2

Fitting data to a linear model: Yi Yi – Ŷi = εi (residual) Ŷi Xi

The residual sum of squares

Estimating Regression Parameters The “best fit” estimates for the regression population parameters (β0 and β1) are the values that minimize the residual sum of squares (SSresidual) between each observed value and the predicted value of the model:

Sum of squares Sum of cross products

Least-squares parameter estimates where

Sample variance of X: Sample covariance:

Solving for the intercept: Thus, our estimated regression equation is:

Hypothesis Tests with Regression Null hypothesis is that there is no linear relationship between X and Y: H0: β1 = 0  Yi = β0 + εi HA: β1 ≠ 0  Yi = β0 + β1 Xi + εi We can use an F-ratio (i.e., the ratio of variances) to test these hypotheses

Variance of the error of regression: NOTE: this is also referred to as residual variance, mean squared error (MSE) or residual mean square (MSresidual)

Mean square of regression: The F-ratio is: (MSRegression)/(MSResidual) This ratio follows the F-distribution with (1, n-2) degrees of freedom

Variance components and Coefficient of determination

Coefficient of determination

ANOVA table for regression Source Degrees of freedom Sum of squares Mean square Expected mean square F ratio Regression 1 Residual n-2 Total n-1

Product-moment correlation coefficient

Parametric Confidence Intervals If we assume our parameter of interest has a particular sampling distribution and we have estimated its expected value and variance, we can construct a confidence interval for a given percentile. Example: if we assume Y is a normal random variable with unknown mean μ and variance σ2, then is distributed as a standard normal variable. But, since we don’t know σ, we must divide by the standard error instead: , giving us a t-distribution with (n-1) degrees of freedom. The 100(1-α)% confidence interval for μ is then given by: IMPORTANT: this does not mean “There is a 100(1-α)% chance that the true population mean μ occurs inside this interval.” It means that if we were to repeatedly sample the population in the same way, 100(1-α)% of the confidence intervals would contain the true population mean μ.

Publication form of ANOVA table for regression Source Sum of Squares df Mean Square F Sig. Regression 11.479 1 21.044 0.00035 Residual 8.182 15 .545 Total 19.661 16

Variance of estimated intercept

Variance of the slope estimator

Variance of the fitted value

Variance of the predicted value (Ỹ):

Regression

Assumptions of regression The linear model correctly describes the functional relationship between X and Y The X variable is measured without error For a given value of X, the sampled Y values are independent with normally distributed errors Variances are constant along the regression line

Residual plot for species-area relationship