L.2.2 Linear regression (Cont.)

each leach cycle. Linear regression is generally calculated using the method of

least squares and follows the form

Y = aX + b

(L-17)

where

Y = dependent or response variable

X = independent or predictor variable

a = slope

b = Y-intercept

Linear regression assumes the following:

Y values are statistically independent of one another.

Relationship between Y and X is linear.

Variance of Y is the same for any X (homoscedasticity).

For any fixed value of X, Y has a normal distribution.

As in hypothesis testing, satisfying these assumptions (especially the

assumption of linearity) may require using a data transformation.

Linear regression may be performed using any general-purpose statistical

package; many hand calculators also include regression functions. Data should

always be plotted first in a scattergram to visually inspect for a functional

relationship between the two variables. When regression is used to characterize

the relationship between suspended solids and turbidity, it may be necessary to

use a nonlinear regression model, or to calculate a linear regression only for a

lower, linear portion of the data. Investigators should refer to Thackston and

Palermo (2000) (http://www.wes.army.mil/el/dots/doer/pdf/doere8.pdf) for

instructions on performing the regression analysis.

When a statistical package is used to calculate the regression analysis, the

strength and validity of the relationship between Y and X can be evaluated by

examining the regression output for the F statistic and its associated P-value, and

for the R2 statistic. The P-value of F determines the probability that the

regression coefficient (slope) is significantly different from zero, given the above

assumptions. P-values > 0.05 indicate that no significant linear relationship

exists between the two variables. R2 or coefficient of determination is the

proportion or percent of the variability in Y that is explained by X. Like the

correlation coefficient r, strong relationships are indicated by coefficients

approaching 1 (or 100 percent); conversely, low values of R2 signify weak or

nonexistent relationships.

L26

Appendix L

Statistical Methods