I.  Introduction

     This document describes the procedures for and the results of the regression analysis described in item #6 of the document entitled “High Plains Aquifer Evaluation (Ogallala Decline Study), Kansas Water Office Revised Portions of Contract # 99-132”. Descriptions of the procedures, associated graphs, and files follow.

II.  Procedures

     Depth to water data was assembled as described in item #5 of the contract mentioned in the introduction. From these data the (spatial) average depth to water for each year in the period 1989-1999 was determined for each of the regions labeled a, b, and c in item #1 of the contract. The regions are listed below:

         The Ogallala aquifer
         Groundwater Management District #1
         Groundwater Management District #3
         Groundwater Management District #4

 and the portions of the following KWO basins overlying the Ogallala aquifer:

         Cimarron
         Smoky Hill-Saline
         Solomon
         Upper Arkansas River
         Upper Republican River

     Regression equations were developed for each region by fitting the average depth to water data to a polynomial by the least squares method.  In each case the polynomial which resulted in the best fit was chosen as the final regression equation for the region.  In all cases either a linear polynomial or constant function gave the best fit, depending on the region.  The model regression equation for a linear fit is given below:

                    D = b + mT     (1)

where D is the estimated average depth to water in year T (1989 <= T <= 1999),  b is the intercept regression parameter and m is the slope regression parameter.  The actual model equation used for the regressions is a modified version of (1) obtained by substituting the expression for intercept b obtained from the least squares minimization requirement into the model equation above (Draper, 1981, pp. 14-15):

                    D = Dav + m(T -Tav)  (2)

where Dav is the (temporal) mean of the regional average depths to water for the period 1989-1999 inclusive and Tav is the average year for this period (1994).  It was necessary to perform the least squares fits using this form of the model equation because use of equation (2) for fitting resulted in numerical precision errors.  There is an added benefit, however.  The intercept, Dav in eq. (2) has far more meaning than regression the parameter b, which is the estimated average depth to water in year 0.

	The model regression equation for a constant term fit is obtained form (2) by setting m = 0.  Hence, the regression estimate for a constant term fit is simply equal to Dav.  

III. Results

      The results of the regression analysis are presented in the form of nine
graphs and nine accompanying summary files, one for each region under study.  In each graph the average depth to water values are plotted for each year (points) along with the best regression line (solid line).  The 95% confidence limits for the mean average depth to water given the years since 1994 are also plotted (dashed lines) for linear fits only.  The least squares estimate is the estimate of this mean (see ref. 1). Note that the vertical axis is scaled in reverse (depth to water increasing downward.
      The summary files contain a regression summary in the form of an ANOVA table.  The most important output items from these files are explained below:

Degrees of Freedom (DF):	The number of independent pieces of
 	information needed to compute the statistic.

Error Sum of Squares:	The sum of the squares of the residuals (differences between average DTW and estimated average DTW). The smaller that
This is, the more precise the fit.  The mean error sum of squares is obtained from the
error sum of squares by dividing it by the number of degrees of freedom.

F value and p value (Prob.>F)     	These statistics test the significance of
the regression. For linear fits, they test
the null hypothesis that the slope parameter
m is equal to zero. The smaller the p value is, the higher the significance level at which the null hypothesis can be rejected, hence the greater the linearity.

R squared					Coefficient of determination. Indicates the 
fraction of the variation about the overall
mean that can be explained by the regression
model (Draper, 1981, p.19).  The closer that this is to 1, the better the fit (regression polynomial degree being the same).

INTERCEP                         	The intercept regression parameter, Dav.  The standard error is the standard deviation
of this parameter.

DYEAR                            	The slope regression parameter, m (labeled
by its corresponding independent variable
name).

T statistic                    	This is a two-tailed test for the significance of the parameters.  For
	The slope parameter, this is a two-tailed version of the F test for significance.

Durbin-Watson D                  	Test for auto-correlation of average DTW
 						values. The closer this value is to 2,the
 						less auto-correlation between the residuals.

1st order Auto-correlation        	The closer to zero, the less auto-correlation between the residuals.

At the bottom of this file is a summary which contains the following
quantities:

Slope and Intercept (rounded)

Estimated average DTW in 2010        This is computed from the regression
       equation.

Average DTW in 1999                  Latest value in average DTW vs. YEAR data.

Estimated Rate of Decline.           Computed by subtracting the Average DTW
 						 1999 from that estimated for 2010 and
 						 dividing by 11.

	The latter four quantities are included in an accompanying spreadsheet file.  For constant term fits, the slope is defined to be zero, thus the significance tests described above for the slope parameter are not defined for these fits.

REGION	SLOPE	INTERCEPT	EST2010	AVG1999	RATE2010	
Ogallala	0.80715	132.277	145.19	135.29	0.9	
gmd1	0.61965	132.372	142.29	133.09	0.836	
gmd3	1.35408	151.45	173.12	155.59	1.594	
gmd4	0	129.492	129.49	131.37	-0.171	
Cimarron	1.00989	181.344	197.5	182.77	1.34	
Smoky Hill-Saline	0	128	128	126.43	0.142	
Solomon	0	111.154	111.15	111.12	0.003	
Upper Arkansas	1.09854	107.502	125.08	112.09	1.181	
Upper Republican	0.51663	128.131	136.4	132.81	0.326	

IV. References.
 Smith, H. and Draper, N.R. (1981) Applied Regression Analysis.
  New York:Wiley and Sons.