Hyperbolic Regression in Biomedical Analysis      (STB-9: sbe7; STB-11: sbe7.1)
--------------------------------------------

                 ^hbolic^ depvar indepvar


Description
-----------

This command provides the least squares solution for the constants in the
general equation describing a hyperbola: (X-Xo)(Y-Yo)=Co

     Xo   for the independent variable, X ("varname")
     Yo   for the dependent variable, Y ("varname")
     Co

It also computes "Yhat", the predicted values for Y from equation

                        Yhat = Co/(X-Xo) + Yo 

A graph of Yhat and Y vs X gives a visual display of the fitted (standard)
curve.  Residuals are also computed and plotted against Yhat.  The graphs are
saved as ^gph1.gph^ and ^gph2.gph^.   They may be seen again by typing:

	. ^gr using gph1^                           (to see gph1 again)
	. ^gr using gph2^                           (to see gph2 again)
	. ^gr using gph1 gph2^                      (to see both again)

^gph1.gph^ and ^gph2.gph^ are automatically deleted when ^hbolic^ is rerun.


Notes:
------

Biomedical analyses frequently involve a hyperbolic relationship. The response
or dependent variable,Y, results from an analytical method applied in dupli-
cate or triplicate to a set of standards, X.

The independent variable, X, is assumed to be without error. Plotting Y vs. X
then provides a standard curve.  From this curve new X-values (answers) are
computed using ^generate^ and the rearranged equation:

                   answer = Co/(responsevar-Yo) + Xo

"responsevar" being the new Y-values given by the unknowns.  Name the answer
and response variables in the above equation as desired.  Alternatively, add
another line to the program:

                ^generate answer=Co/(`3'-Yo)+Xo^

Also include, "responsevar" (which is now `3'), in the ^hbolic^ command after
"indepvar".

^Xo^, ^Yo^, ^Co^, ^Yhat^ and ^resids^ are dropped automatically when the program is run
again, so new variables for X and Y can be used within the same file.  This
usage requires that ^Xo^, ^Yo^, ^Co^, ^Yhat^ and ^resids^ be renamed if you wish to
retain them.  Also, graphs must be renamed or deleted as mentioned above.

Except in the case of bivariate linear regression and homoscedastic Y-values, I
am aware of no method to assign confidence limits to the results from ^hbolic^
(see Snedecor and Cochran 1989, 170).


References
----------


Snedecor, G. W. and W. G. Cochran.  1989.  ^Statistical Methods^, 8th ed. 
     Ames, IA: Iowa State University Press.


Author
------

Paul J. Geiger, USC School of Medicine, Los Angeles, CA.