ROC ANALYSIS

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       Computation of W and its standard error. (Radiology 143:29-36, April 1982)


 row              Contents               DEFINITELY NORMAL PROBABLY NORMAL                                           D
                                                                                      QUEST ABNORMAL PROBABLY ABNORMAL EFINITELY ABNORMAL                           Total

   1        # of normals rated x                 33                       6                   6                    11                 2                  nN          58

   2       # of abnormals rated >x               48                     46                    44                   33                 0

   3       # of abnormals rated x                 3                       2                   2                    11                33
                                                                                                                                                       n             51

   4        # of normals rated >x                 0                     33                    39                   45                56

   5          (1)*(2)+1/2*(1)*(3)              1633.5                   282                  270                  423.5              33                             2642

   6     (3)*[(4)sq+(4)*(1)+1/3*(1)sq]          1089                   2598                  3534              28163.66667         107228                        142612.6667

   7     (1)*[(2)sq+(2)*(3)+1/3*(3)sq]          80883                 13256                 12152              16415.66667           726                         123432.6667

           W=total row 5 / (#n*#a)           0.893171062             89.32%
          Q2=total row6 / (#a*#nsq)          0.831250534             83.13%
          Q1=total row 7 / (#n*#asq)         0.818204316             81.82%

       SE W = sqrt((W*(1-W)+(#a-1)*(Q1-Wsq)+(#n-1)*(Q2-Wsq))/(#a*#n))                                            0.031990408         3.20%
                                                                                                           W=AREA UNDER CURVE
                z=(W - 0.5)/SE W     12.29027961                                                           X=RADIOGRAPHIC FINDING OR DIAGNOSIS
RADIOLOGY 148:839-843, 1983 COMPARING AREAS UNDER ROC CURVES                                               SE=STANDARD ERROR
FROM THE SAME CASES            Z=CRITICAL RATIO                                                            FILL IN ROWS WITH BOLD LETTERS


       confidence = +-1.96 SE W                   -0.0627012              0.0627012



                                                               TEST RESULT (RATING CATEGORY)

                                         DEFINITELY NORMAL PROBABLY NORMAL                                           D
                                                                                      QUEST ABNORMAL PROBABLY ABNORMAL EFINITELY ABNORMAL                          TOTAL
   NORMAL OR NONDISEASED                         33                       6                   6                    11                 2                              58
       DISEASED OR ABNORMAL                       3                       2                   2                    11                33                              51
                                                                                                                                                                    109
                                                                                                              FALSE POS          TRUE POS
                            NUMBER 1                             DN + PR + QA + PA            DA
                                                   NORMAL               56                     2                 3.45%
                                                  DISEASED              18                    33                                   64.71%
                            NUMBER 2                               DN + PN + QA            PA + DA
                                                   NORMAL               45                    13                 22.41%
                                                  DISEASED               7                    44                                   86.27%
                            NUMBER 3                                 DN + PN             QA + PA +DA
                                                   NORMAL               39                    19                 32.76%
                                                  DISEASED               5                    46                                   90.20%
                            NUMBER 4                                    DN             PN + QA + PA + DA
                                                   NORMAL               33                    25                 43.10%
                                                  DISEASED               3                    48                                   94.12%                                 ROC Analysis


                                                                                                                                                                 120.00%
       DN = DEFINITELY NORMAL PN = PROBABLY NORMAL                 QA =
                                                                                                                                                                 100.00%
       QUESTIONABLY ABNORMAL                                                             ROC CURVE           FALSE POSITIVE
                                                                                            0.00%                 0.00%
                                                                                                                                                 True Positive
                                                                                                                                                                  80.00%
                                                                                            3.45%                64.71%
               TO CHART: SELECT DATA WITH WORKSHEET TOOL
                                                       TRUE POSITIVE                       22.41%                86.27%                                           60.00%
               CHANGE GRAPH TO XY PLOT
                                                                                           32.76%                90.20%
                                                                                           43.10%                94.12%                                           40.00%
               SMOOTH CHART UNDER SERIES SUBMENU
                                                                                          100.00%               100.00%
                                                                                                                                                                  20.00%

                                                                                                                                                                   0.00%
                                                                                                                                                                                                             COMPARING AREAS -ROC
                                                                                                                                                                          0.00%     100.00%        200.00%
                                                                                                           DETECTING DIFFERENCES BETWEEN                                                                     CURVES- FROM SAME CASES -
            HOW BIG A SAMPLE DO I NEED?                                                                    AREAS UNDER TWO ROC CURVES -                                                                      PAIRED DATA RAD 148: 839-843
                                                                                                                                                                                  False Positive
EXPECTED DIAGNOSTIC ACCURACY             0.90           EXPECTED DIAGNOSTIC                                  N (80%) POWER          13.46                                                                       rN kendall tau               0.39      SUPPLY AREA1, SE1,
                          Q1             0.82           ACCURACY = W (IE 85%)                                              V1        0.05                                                                       rA kendall tau                0.6      AREA2, SE2
                          Q2             0.85           NA = #ABNORMALS WHICH                                              V2        1.10                                                                      avg correlation              0.495
                          NA             30.00          SHOULD = THE # OF                                                DIFF        0.39                                                                        AREA 1 W1                 0.8945      GET R FROM TABLE 1
                          NN             30.00          NORMALS (NN)                                        Q1 SECOND CURVE          0.80                                                                                SE 1                0.03      PAGE 841 RAD 148, 1983
                        SE W             0.041681178                                                        Q2 SECOND CURVE          0.80                                                                        AREA 2 W2                 0.9382
                                                        PUT VALUES IN FOR W AND                                                                                                                                          SE 2               0.026      IF Z>= 1.96 "TRUE" ROC
                                                                                                             W SECOND CURVE         0.50                                                                          AVG AREA                0.91635      AREA ARE DIFFERENT
                                                                                                                                                                                                                                                       (STATISTICALLY
                                                                                                           ENTER QI Q2 AND W FROM SECOND ROC                                                         R CORRELATION COEFF                      0.4
                                                                                                                                                                                                                                                       DIFFERENT)
                                                                                                           CURVE                                                                                             DIFF IN AREAS                 0.0437
                                                                                                                                                                                                          SE(AREA1-AREA2)              0.030854497
                                                                                                           N=#'S OF NORMAL AND ABNORMAL CASES                                                                               Z=         1.416325136
                                                                                                           REQUIRED TO ASSURE (80%) REAL
                                                                                                           DIFFERENCES BETWEEN TWO AREAS W1                                                                                             NORMALS
                                                                                                                                                                                                                                                   P
                                                                                                                                                                                                                                                   Q
                                                                                                                                                                                                                                              TAU RN           0
                                                                                                                                                                                                                                       ABNORMALS
                                                                                                                                                                                                                                                   P
                                                                                                                                                                                                                                                   Q
                                                                                                                                                                                                                                              TAU RA           0
                                                                                                                                                                                                                                       RATING
                                                                                                                                                                                                               MODALITY 1     1     2   3 4      5
                                                                                                                                                                                                                NORMAL         -    -   -   -   -
                                                                                                                                                                                                                ABNORMAL        -    -    - -    -
                                                                                                                                                                                                               MODALITY 2
                                                                                                                                                                                                                 NORMAL
                                                                                                                                                                                                                 ABNORMAL
                                                                                                                                                                                                               TO CALCULATE THE RANK CORRELATION COEFFICIENT, FIRST
                                                                                                                                                                                                               REARRANGE THE RATINGS SO THAT THEY ARE RANKED IN ORDER FOR
                                                                                                                                                                                                               ONE OF THE ATTRIBUTES (NORMAL). THE NEXT STEP IS TO CALCULATE
                                                                                                                                                                                                               TWO QUANTITIES P AND Q ON THE SECOND ROW (NORMAL). STARTING
                                                                                                                                                                                                               WITH THE RANK ON THE LEFT OF THE SECOND ROW, FIND HOW MANY
                                                                                                                                                                                                               RANKS TO ITS RIGHT ARE LARGER. REPEAT FOR EACH RANK TO OBTAIN
                                                                                                                                                                                                               THE FOLLOWING SUM P= #>1 + #>2....
                                                                                                                                                                                                               TO CALCULATE Q START AGAIN WITH THE RANK AT THE LEFT OF
                                                                                                                                                                                                               THESECOND ROW AND FIND HOW MANY RANKS TO THE RIGHT OF IT ARE
                                                                                                                                                                                                               SMALLER Q= #<1 + #<2....
                                                                                                                                                                                                               THE NUMBER OF PAIRS OF RANKS, N = 5

                                                                                                                                                                                                                     KENDALL TAU = (P-Q)/0.5N(N-1)

                                                                                                                                                                                                               RN = KENDALL TAU CORRELATION FOR NORMALS (MODALITY 1 VS
                                                                                                                                                                                                               MODALITY 2)
                                                                                                                                                                                                               RA = KENDALL TAU CORRELATION FOR ABNORMALS (MODALITY 1 VS
                                                                                                                                                                                                               MODALITY 2)




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