Regression analysis

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❤ Simple linear regression

1. Y = β0 + β1*X + e

where:

Y - dependent variable (response)

X - independent variable (predictor/explanatory)

β0 - intercept

β1 - slope of the regression line

e - random error

2. Y' = b0 + b1*X

where: Y' - predicted value of Y

e = Y - Y'

3. Least squarea regression minizes the sum of the square of the errors and can be used to estimate b0 and b1.

4. Measuring the fit of the estimated model.

- The varibility of Y

SST (Sum of Squared Total): total variability about the mean, SST = sum((Y - mean(Y))^2);

SSE (Sum of Squared Error): variability about the regression line, SSE = sum(e^2) = sum((Y - mean(Y'))^2), SSE is unexplained varibility;

SSR (Sum of Squares due to Regression): variability that is explained, SSR = sum((Y' - mean(Y))^2), SSR is explained varibility.

Note that SST = SSE + SSR.

- Coefficient of determination

r^2: proportion of explained variability by the regression equation.

0 <= r^2 = 1 - SSE/SST = SSR/SST <= 1

- Correlation coefficient

r: strength of the relationship between X and Y.

-1 <= r <= 1

5. Assumptions in the regression model

Errors are independent, normally distributed, with the mean of zero, with a constant variance.

The assumptions can be tested by using residual analysis.

6. MSE (Mean Squared Error)

Estimation of error variance of the regression equation.

s^2 = MSE = SSE / (n - k - 1)

where:

n - number of observations in the sample

k - number of independent variables

Standard deviation of the regression: s = sqrt(MSE) is also frequently used.

❤ Test the model for significance: F-test

Used to statistically test the null hypothesis H0: there is no linear relationship between Y and X (i.e. β1 = 0).

If p value is low, then we regect H0 and conclude there is linear relationship:

F = MSR / MSE

where: MSR = SSR / k

Good regression model should have significant F value and high r^2 value.

Statistical test can be performed on the regression coefficients. H0: the βs are 0.

For a simple linear regression, the test for regression coefficient gives the same information as the ones given by F-test.

❤ ANOVA tables

The general form of the ANOVA table is helpful for understanding the interrelatedness of error terms.

❤ Multiple regression

Similar to the simple regression model, but there are more than one X in the multiple regression models.

Y' = b0 + b1*X1 + b2*X2 + ... + bn*Xn

Note that if indenpendent variables is correlate to each other, colinearity or multicolinearity will happen. This will cause problems when intepreate variables individually although the overall model estimation may still be good.