Environmental Data Analysis BC ENV 3017

Statistics 4

Lines

The slope is the rate at which y increases with x, along the line.

slope = rise/run

The intercept of a line is its height at x=0.
The graph of the equation y= ax + b is a straight line, with slope a and intercept b.

Correlation

The relationship between two variables can be represented visually by a scatter diagram. When the scatter diagram is tightly clustered around a line, there is a strong association or correlation between the variables (fig).
A scatter diagram can be summarized by means of five statistics:

average and SD of x-values
average and SD of y values
correlation coefficient r

In a series of scatter diagram with the same SDs, as r gets closer to ± 1 the points cluster more tightly around a line.

r = average of ((x in standard units) * (y in standard units))

The points cluster around the SD line. This line goes through the point of averages. When r is positive or negative, the slope of the line is:

(SD of y)/(SD of x) or -(SD of y)/(SD of x), respectively.

The correlation coefficient r is a pure number between -1 and +1, without units.
r² is often interpreted as the 'fraction of the variability of y that is explained by the fitted equation'.
r is not affected by:

interchanging the two variables
adding the same number to all values of one variable
multiplying all the values of one variable by the same positive number

Regression

The regression line is to a scatter diagram as the average is to a list. The regression line for y on x estimates the average value for y corresponding to each value of x. The regression line is also sometimes called the Least Square Fit (fig). It basically minimizes the square of the vertical difference between the line and the data points.

A residual is the difference between the observed and predicted value for y.

This line is different from the SD line! (fig)

If you switch the axis, the regression line will look differently

the slope of the regression line is:

r ^. (SD of y)/(SD of x)

there are formulas in EXCEL with which to calculate the slope and intercept of the regression line (e.g. ADD TRENDLINE, LINEST).
The LINEST function allows performing a multiple regression analysis (y = a1*x + a2*x + a3*x + ....... + b)

Significance of Regression lines

Could it happen that the correlation occurred by chance only? The correlation coefficient r is not a good measure of this. If there is no correlation between two parameters, then the regression line should have the slope close to 0.

We can determine whether a significant simple relationship between X and Y exists by testing whether the slope could be equal to zero. If this hypothesis is rejected, we can conclude that there is evidence of a simple linear relationship.

Let us write the t statistic for this case:

t - statistic = slope / SE of the slope

degrees of freedom: n-2

'A correlation is significant at the 95% level’, means that the area under the student curve to the left and right of ± t-statistic is less than 5%. We can obtain the SE of the slope from the LINEST function (check teh help file of the linest function if you do not know how to use it). We can of course again calculate the P-value using the TDIST function. We need to use the two-tailed t-distribution.

Example: Hudson River annual mean discharge

Relevant EXCEL functions:

TDIST, CORRELATION, REGRESSION, LINEST

Resources:

Freedman, D., Pisani, R., Purves, R., and Adhikari, A. (1991) Statistics. WW Norton & Company, New York, 2nd ed. 514pp.

Fisher, F.E. (1973) Fundamental Statistics Concepts. Canfield Press, San Francisco, 371 pp.

Berenson, M.L., Levine, D.M., and Rindskopf, D. (1988) Applied statistics - A first course. Prentice Hall, Englewood Cliffs, NJ, 557pp.