Download presentation
1
Linear regression models
2
Simple Linear Regression
3
History Developed by Sir Francis Galton ( ) in his article “Regression towards mediocrity in hereditary structure”
4
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
5
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).
6
Linear relationship Y ß1 1.0 ß0 X
7
Linear models approximate non-linear functions over a limited domain
extrapolation interpolation extrapolation
8
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
9
Fitting data to a linear model:
Yi Yi – Ŷi = εi (residual) Ŷi Xi
10
The residual sum of squares
11
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:
12
Sum of squares Sum of cross products
13
Least-squares parameter estimates
where
14
Sample variance of X: Sample covariance:
15
Solving for the intercept:
Thus, our estimated regression equation is:
16
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
17
Variance of the error of regression:
NOTE: this is also referred to as residual variance, mean squared error (MSE) or residual mean square (MSresidual)
18
Mean square of regression:
The F-ratio is: (MSRegression)/(MSResidual) This ratio follows the F-distribution with (1, n-2) degrees of freedom
19
Variance components and Coefficient of determination
20
Coefficient of determination
21
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
22
Product-moment correlation coefficient
23
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.
24
Publication form of ANOVA table for regression
Source Sum of Squares df Mean Square F Sig. Regression 11.479 1 21.044 Residual 8.182 15 .545 Total 19.661 16
25
Variance of estimated intercept
26
Variance of the slope estimator
27
Variance of the fitted value
28
Variance of the predicted value (Ỹ):
29
Regression
30
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
31
Residual plot for species-area relationship
Similar presentations
© 2024 SlidePlayer.com Inc.
All rights reserved.