<PAGE>   1
                                                                    EXHIBIT 10.4

                              AMENDMENT NUMBER 1 TO
                              EMPLOYMENT AGREEMENT

         THIS AMENDMENT NUMBER 1 TO EMPLOYMENT AGREEMENT (this "Amendment") is
made and entered into as of the 4th day of November 1998, by and between
MEDIRISK, INC., a Delaware corporation (the "Company") and KENNETH M.
GOINS, JR. ("Employee").


                              W I T N E S S E T H:

         WHEREAS, the Company and Employee are parties to an Employment
Agreement dated as of 21 August 1998 (the "Agreement") pursuant to which the
Company employs Employee; and

         WHEREAS, the Company and Employee desire to amend the Agreement in
light of organizational changes made at the Company;

         NOW, THEREFORE, in consideration of the compensation payable to
Employee by the Company pursuant to the Agreement, and the mutual promises,
covenants, representations and warranties contained herein, the receipt and
sufficiency of which are hereby acknowledged, the parties hereto do agree as
follows:

         1.       DUTIES, EXTENT OF SERVICES. Section 4 of the Agreement is
amended by deleting the first sentence of such Section 4 and inserting in lieu
thereof the following:

                  Employee is engaged as President and Chief Operating Officer
                  of the Company and shall perform such duties and
                  responsibilities as are typically incident to such positions,
                  and shall perform in a faithful and competent manner such
                  additional duties as may be reasonably assigned from time to
                  time by the Company.

         2.       TERMINATION. Section 5(c) of the Agreement is amended by
deleting clause (i) therefrom and inserting in lieu thereof the following:

                  (i) without the express written consent of Employee, a
                  material diminution of his position, duties, responsibilities
                  and status with the Company as in effect as of the date of
                  Amendment Number 1 to this Agreement, a material reduction of
                  Employee's reporting responsibilities, titles or offices as in
                  effect as of the date of Amendment Number 1 to this Agreement,
                  or the removal of Employee from, or failure to re-elect
                  Employee to, any position referred to in Section 4 of this
                  Agreement, except in connection with promotions to higher
                  officer or except in connection with the termination of his
                  employment for Cause;


              [the remainder of this page intentionally left blank]



<PAGE>   2


         3.       EFFECT. Except as specifically modified and amended by this
Amendment, the parties ratify and affirm all of the covenants and agreements set
forth in the Agreement, which shall continue in full force and effect as
modified hereby.

         IN WITNESS WHEREOF, the Company and Employee have executed this
Agreement on the day and year first above written.

                                    COMPANY:

                                    MEDIRISK, INC.



                                    By:    /s/ Mark A. Kaiser, Jr.
                                           -------------------------------------
                                           MARK A. KAISER, JR.
                                           Chairman of the Board &
                                           Chief Executive Officer

                                    EMPLOYEE:



                                     /s/ Kenneth M. Goins, Jr.
                                    --------------------------------------------
                                     KENNETH M. GOINS, JR.


                                      -2-