Question

11. (5 points) Answer the following questions on regression. X versus Y 45 3 15 . 0 0 1.5 4.5 (1) (1 points) For the given data (shown above), draw two possible linear regression lines. (2) (1 point) Between the two regression lines you draw, which one is better? Why? (3) (1 point) Which of the following offsets do we use in linear regression's least square line fit? Circle the one (either vertical offsets or perpendicular offsets). 2.--- m. vertical offsets perpendicular offsets (4) (2 point) What if the data is not linearly shaped/distributed? Discuss what other approaches are available to do regression analysis.

Answer 1

(1)Two possible regression lines are drawn in the figure. In figure A, the regression line passes through three of the points while in figure B, it passes through 2 of them.

X versus Y X versus Y 45 3 3 15 o 1.5 3 45 1.5 45 B A

(2) The better regression line is in figure B, since it minimizes the sum of least squares error across all the data points, even though the line itself passes through fewer of the points.

(3) The vertical offsets are used in least squares fit linear regression. For any given x, we want to minimize the difference in y between the regression line and the actual data point; the difference in y is the vertical offset.

(4) If data is not linearly distributed, it is still possible to use linear regression. Linear regression is one of the form:

y = k1 + k2 * X1 + k3 * X2 +... where the k are constants and the Xs are independent variables; it is possible to define X2 = X1**2, X3 = X1** 3, X4 = log(X) etc  and using these, we can see if we get a better linear regression fit. Alternatively, non linear regression of the form Y = k X**n (where n > 1) or use of trignometric functions (sin X, cos x, tan X) can be used to express the relationship betwee Y and the independent variables.