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Linear regression models

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Linear regression models

Simple Linear Regression

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

- 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

- 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).

Y

ß1

1.0

ß0

X

extrapolation

extrapolation

interpolation

- 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

- 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

where

Sample variance of X:

Sample covariance:

Thus, our estimated regression equation is:

- 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

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

The F-ratio is: (MSRegression)/(MSResidual)

This ratio follows the F-distribution with (1, n-2) degrees of freedom

- 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.

- 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