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Acceptance Probabilities-Double Sampling

 /****************************************************************/
 /*          S A S   S A M P L E   L I B R A R Y                 */
 /*                                                              */
 /*    NAME: IEDOUBLE                                            */
 /*   TITLE: Acceptance Probabilities-Double Sampling            */
 /* PRODUCT: QC                                                  */
 /*  SYSTEM: ALL                                                 */
 /*    KEYS: Inspection Sampling,                                */
 /*   PROCS: TABULATE                                            */
 /*    DATA:                                                     */
 /*                                                              */
 /*    MISC:                                                     */
 /*                                                              */
 /*   NOTES: This program tabulates the acceptance probability   */
 /*          for double sampling from a single, finite lot under */
 /*          an imperfect inspection model.                      */
 /*                                                              */
 /*          Notation:                                           */
 /*                                                              */
 /*          nlot    = size of lot                               */
 /*          d       = number of nonconforming items in lot      */
 /*                                                              */
 /*          n1      = first sample size                         */
 /*          n2      = second sample size                        */
 /*                                                              */
 /*          a1      = acceptance number at first stage          */
 /*          a1p     = rejection  number at first stage          */
 /*          a2      = acceptance number at second stage         */
 /*                                                              */
 /*          p       = Pr[ nonconforming item is classified      */
 /*                        as nonconforming ]                    */
 /*          pprime  = Pr[ conforming item is classified as      */
 /*                        nonconforming ]                       */
 /*                                                              */
 /*          accprob = Pr[ acceptance ]                          */
 /*                                                              */
 /*                                                              */
 /*          Procedure:                                          */
 /*                                                              */
 /*          Take a random sample of size n1 and record the      */
 /*          apparent number Z1 of defective items.              */
 /*                                                              */
 /*          If Z1 <= a1 then accept.  If Z1 > a1p then reject.  */
 /*                                                              */
 /*          If a1 < Z1 <= a1p then take a second sample of size */
 /*          n2 and record the apparent number Z2 of defective   */
 /*          items.                                              */
 /*                                                              */
 /*          If Z1 + Z2 <= a2 then accept; else reject.          */
 /*                                                              */
 /*                                                              */
 /*     REF: Johnson, N. L., Kotz, S., and Rodriguez, R. N.      */
 /*          (1986), Statistical Effects of Imperfect Inspection */
 /*          Sampling:  II. Double Sampling and Link Sampling,   */
 /*          Journal of Quality Technology 18, 116-138.          */
 /*          See Table 3.                                        */
 /*                                                              */
 /*          Johnson, N. L., Kotz, S., and Wu, X. (1991).        */
 /*          Inspection Errors for Attributes in Quality         */
 /*          Control.  London:  Chapman & Hall.  See Chapter 4.  */
 /*                                                              */
 /****************************************************************/

data table;

   keep nlot d n1 n2 a1 a2 a1p p pprime accprob;

   label nlot    = 'N (lot)'
         d       = 'D'
         n1      = 'n1'
         n2      = 'n2'
         a1      = 'a1'
         a2      = 'a2'
         a1p     = 'a1'''
         p       = 'p'
         pprime  = 'p'''
         accprob = 'Pr[ Accept ]';

   format zprob   6.4
          accprob 6.4 ;

   /*---set main parameters---*/
   nlot    = 100;
   n1      = 8;
   n2      = 8;
   a1      = 0;
   a1p     = 1;
   a2      = 1;

   /*---loop over d values---*/
   do d = 5, 10, 20;

      /*---loop over p values---*/
      do p = 0.75, 0.90, 0.95, 0.98, 1.00;

         /*---loop over pprime values---*/
         do pprime = 0.0, 0.01, 0.02, 0.05, 0.10;

            /*---compute term1---*/
            link first;

            /*---computer term2---*/
            link second;

            accprob = term1 + term2;

            output;

            end;  /* finish loop over pprime values */

         end;  /* finish loop over p values */

      end;  /* finish loop over d values */

   return;  /* finish main program */

   /*------------------------------------------------------------*/
   /*                                                            */
   /* This module computes the probability Pr[ Z2 <= a1 ]        */
   /*                                                            */
   /* The following serve as input parameters:                   */
   /*                                                            */
   /*    a1      = acceptance value for first sample             */
   /*    nlot    = lot size                                      */
   /*    d       = number of defectives in lot                   */
   /*    n1      = first sample size                             */
   /*    p       = Pr[ correctly classifying a defective item ]  */
   /*    pprime  = Pr[ incorrectly classifying a good item ]     */
   /*                                                            */
   /* The following is returned:                                 */
   /*                                                            */
   /*    term1   = Pr[ Z2 <= a1 ]                                */
   /*                                                            */
   /*------------------------------------------------------------*/
   first:

   term1 = 0.0 ;
   do z2 = 0 to a1 by 1;

      /* nlot, d, p, pprime are globally defined */
      z       = z2;
      nsample = n1;
      link uncond;

      term1 = term1 + zprob;

      end;

   return;  /* finish first */

   /*------------------------------------------------------------*/
   /*                                                            */
   /* This module computes the probability                       */
   /*                                                            */
   /*    Pr[ a1 < Z2 <= a1p , Z2 + Z2P <= a2 ]                   */
   /*                                                            */
   /*                                                            */
   /* The following serve as input parameters:                   */
   /*                                                            */
   /*    a1      = acceptance value for first sample             */
   /*    a1p     = acceptance value for first sample             */
   /*    a2      = acceptance value for second sample            */
   /*    nlot    = lot size                                      */
   /*    d       = number of defectives in lot                   */
   /*    n1      = first sample size                             */
   /*    n2      = second sample size                            */
   /*    p       = Pr[ correctly classifying a defective item ]  */
   /*    pprime  = Pr[ incorrectly classifying a good item ]     */
   /*                                                            */
   /* The following is returned:                                 */
   /*                                                            */
   /*    term2   = Pr[ a1 < Z2 <= a1p, Z2 + Z2P <= a2 ]          */
   /*                                                            */
   /*------------------------------------------------------------*/
   second:

   term2 = 0.0 ;
   z2max = max( a1p, a2 );

   do z2 = 0 to z2max by 1;

      do z2p = 0 to z2max by 1;

         if ( a1 < z2 ) & ( z2 <= a1p ) & ( z2 + z2p <= a2 )
         then do;

            link uncond2;
            term2 = term2 + unprb2;
            end;

         end;

      end;

   return;  /* finish second */

   /*------------------------------------------------------------*/
   /*                                                            */
   /* This module computes the unconditional joint distribution  */
   /* of Z2 and Z2P.                                             */
   /*                                                            */
   /* The following serve as input parameters:                   */
   /*                                                            */
   /*    z2      = number of items classified as defective       */
   /*    z2p     = number of items classified as defective       */
   /*    nlot    = lot size                                      */
   /*    d       = number of defectives in lot                   */
   /*    n1      = sample size                                   */
   /*    n2      = sample size                                   */
   /*    p       = Pr[ correctly classifying a defective item ]  */
   /*    pprime  = Pr[ incorrectly classifying a good item ]     */
   /*                                                            */
   /* The following is returned:                                 */
   /*                                                            */
   /*    unprb2  =                                               */
   /*                                                            */
   /*------------------------------------------------------------*/
   uncond2:

   unprb2 = 0.0 ;
   upp1   = min( d, n1 );
   upp1p  = min( d, n2 );
   lsum   = max( 0, n1 + n2 + d - nlot );
   usum   = min( d, n1 + n2 );

   do ylocal = 0 to upp1 by 1;

      do yplocal = 0 to upp1p by 1;

         if ( lsum <= ylocal + yplocal ) &
            ( ylocal + yplocal <= usum )
         then do;

            /*---absolute hypergeometric probability---*/
            bign_ = nlot;
            litn_ = n1 + n2;
            d_    = d;
            y_    = ylocal + yplocal;
            link hypergmt;
            hprob1 = hypprob;

            bign_ = n1 + n2;
            litn_ = n1;
            d_    = ylocal + yplocal;
            y_    = ylocal;
            link hypergmt;
            hprob2 = hypprob;

            mhyp = hprob1 * hprob2;

            /*--conditional probability that Z2 = z2 ---*/
            nval = n1;
            zval = z2;
            yval = ylocal;
            dval = d;
            link cond;
            mhyp = mhyp * cprob;

            /*---conditional probability that Z2P = z2p ---*/
            nval = n2;
            zval = z2p;
            yval = yplocal;
            dval = d;
            link cond;
            mhyp=mhyp * cprob;

            *--add over y and yp--;
            unprb2 = unprb2 + mhyp;

            end;

         end;

      end;

   return;  /* finish uncond2 */


   /*------------------------------------------------------------*/
   /*                                                            */
   /* This module computes the conditional probability           */
   /*                                                            */
   /*    cprob   = Pr[ Z = zval | Y = yval ]                     */
   /*                                                            */
   /* where                                                      */
   /*                                                            */
   /*    zval    = number of items classified as defective       */
   /*    yval    = number of actually defective items in sample  */
   /*                                                            */
   /*    dval    = number of defectives in the lot               */
   /*    nval    = sample size                                   */
   /*    nlot    = lot size                                      */
   /*    p       = Pr[ correctly classifying a defective item ]  */
   /*    pprime  = Pr[ incorrectly classifying a good item ]     */
   /*                                                            */
   /*                                                            */
   /*------------------------------------------------------------*/
   cond:

   /*---initialize result to zero---*/
   cprob = 0.0;

   /*---set limits for subscript---*/
   lolim = max( 0, nval + dval - nlot );
   uplim = min( nval, dval );

   if ( p = 0 ) & ( pprime = 0 ) then do;

      if ( yval >= lolim ) & ( yval <= uplim) then do;

         if zval = 0 then cprob = 1;

         end;

      end;

   else
   if ( p = 0 ) & ( abs( pprime - 1 ) < fuzz ) then do;

      if ( yval >= lolim ) & ( yval <= uplim ) then do;

         if zval = nval - yval then cprob = 1;

         end;

      end;

   else
   if ( abs( p - 1 ) < fuzz ) & ( pprime = 0 ) then do;

      if ( lolim <= yval ) & ( yval <= uplim ) then do;

         if zval = yval then cprob = 1;

         end;

      end;

   else
   if ( abs( p - 1 ) < fuzz ) & ( pprime > 0 ) & ( pprime < 1 )
   then do;

      if ( lolim <= yval ) & ( yval <= uplim ) then do;

         if ( yval <= zval ) & ( zval <= nval ) then do;

            n_ = nval - yval;
            p_ = pprime;
            k_ = zval - yval;

            link binomial;

            cprob = binprob;

            end;

         end;

      end;

   else
   if ( p > 0 ) & ( p < 1 ) & ( abs( pprime-1 ) < fuzz ) then do;

      if ( lolim <= yval ) & ( yval <= uplim ) then do;

         if ( nval - yval <= zval ) & ( zval <= nval ) then do;

            n_ = yval;
            p_ = p;
            k_ = zval - ( nval - yval );
            link binomial;

            cprob = binprob;
            end;

         end;

      end;

   else
   if ( abs( p-1 ) < fuzz ) & ( abs( pprime-1 ) < fuzz ) then do;

      if ( lolim <= yval ) & ( yval <= uplim ) then do;

         if zval =nval then cprob = 1;

         end;

      end;

   else
   if ( 0 < p ) & ( p < 1) & ( pprime > 0 ) & ( pprime < 1 )
   then do;

      if ( lolim <= yval ) & ( yval <= uplim ) then do;

         if ( 0 <= zval ) & ( zval <= nval ) then do;

            xlo = max( 0, yval + zval - nval );
            xup = min( yval, zval );

            /*---convolution of binomial distributions---*/
            do xval = xlo to xup by 1;

               p_ = p;
               k_ = xval;
               n_ = yval;
               link binomial;
               f1 = binprob;

               p_ = pprime;
               k_ = zval - xval;
               n_ = nval - yval;
               link binomial;
               f2 = binprob;

               cprob =cprob + f1 * f2;

               end;

            end;

         end;

      end;

   else
   if ( p = 0 ) & ( 0 < pprime ) & ( pprime < 1 ) then do;

      if ( yval >= lolim ) & ( yval <= uplim ) then do;

         if ( zval >= 0 ) & ( zval <= nval - yval ) then do;

            n_ = nval - yval;
            p_ = pprime;
            k_ = zval;
            link binomial;

            cprob=binprob;

            end;

         end;

      end;


   else
   if ( pprime = 0 ) & ( 0 < p ) & ( p < 1 ) then do;

      if ( lolim <= yval ) &  ( yval <= uplim ) then do;

         if ( zval >= 0 ) & ( zval <= yval ) then do;

            n_ = yval;
            p_ = p;
            k_ = zval;
            link binomial;

            cprob = binprob ;

            end;

         end;

      end;

   return;  /* finish cond */

   /*------------------------------------------------------------*/
   /*                                                            */
   /* This module computes the probability Pr[ Z = z ], where Z  */
   /* is the number of items classified as defective.            */
   /*                                                            */
   /* The following serve as input parameters:                   */
   /*                                                            */
   /*    z       = number of items classified as defective       */
   /*    nlot    = lot size                                      */
   /*    d       = number of defectives in lot                   */
   /*    nsample = sample size                                   */
   /*    p       = Pr[ correctly classifying a defective item ]  */
   /*    pprime  = Pr[ incorrectly classifying a good item ]     */
   /*                                                            */
   /* The following is returned:                                 */
   /*                                                            */
   /*    zprob   = Pr[ Z = z ]                                   */
   /*                                                            */
   /*------------------------------------------------------------*/
   uncond:

   /*---used for roundoff---*/
   fuzz = 0.0001 ;

   /*---lower and upper limits for y---*/
   miny = max( 0, nsample + d - nlot );
   maxy = min( nsample, d );

   /*---initialize probability to zero---*/
   zprob = 0.0 ;

   /*---Case I: p = 0 ---*/
   if p = 0 then do;

      /*---Ia: pprime = 0 ---*/
      if pprime = 0 then do;

         if z = 0 then zprob = 1 ;

         end;  /* finish Ia */

      /*---Ib: pprime = 1 ---*/
      else if abs( pprime - 1 ) < fuzz then do;

         minz = max( 0, nsample - d );
         maxz = min( nsample, nlot - d );

         if ( minz <= z ) & ( z <= maxz ) then do;

            bign_ = nlot;
            litn_ = nsample;
            d_    = d;
            y_    = nsample - z;
            link hypergmt;

            zprob = hypprob;

            end;

         end;  /* finish Ib */

      /*---Ic:  0 < pprime < 1 ---*/
      else do;

         /* Note: minz =  0 */
         maxz = nsample - max( 0, nsample + d - nlot );

         if ( z <= maxz ) then
         do y = miny to maxy by 1;

            /*---obtain Pr[ Y = y ]---*/
            bign_ = nlot;
            litn_ = nsample;
            d_    = d;
            y_    = y;
            link hypergmt;

            /*---obtain Pr[ Z = z | Y = y ]---*/
            n_ = nsample - y;
            k_ = z;
            p_ = pprime;
            link binomial;

            zprob = zprob + binprob * hypprob ;

            end;

         end;  /* finish Ic */

      end;  /* finish Case I */


   /*---Case II:  p = 1 ---*/
   else if ( abs( p - 1 ) < fuzz ) then do;

      /*---IIa:  pprime = 0 (perfect inspection) ---*/
      if pprime = 0 then do;

         minz = max( 0, nsample + d - nlot );
         maxz = min( nsample, d );

         if ( minz <= z ) & ( z <= maxz ) then do;

            bign_ = nlot;
            litn_ = nsample;
            d_    = d;
            y_    = z;
            link hypergmt;

            zprob = hypprob;

            end;

         end;  /* finish IIa */

      /*---IIb:  pprime = 1 ---*/
      else if ( abs( pprime - 1 ) < fuzz ) then do;

         if z = nsample then zprob = 1 ;

         end;  /* finish IIb */

      /*---IIc:  0 < pprime < 1 ---*/
      else do;

         minz = max( 0, nsample + d - nlot );
         maxz = nsample ;

         if ( minz <= z ) & ( z <= maxz ) then
         do y = miny to maxy by 1;

            /*---compute Pr[ Y = y ] ---*/
            bign_ = nlot ;
            litn_ = nsample ;
            d_    = d;
            y_    = y;
            link hypergmt;

            /*---obtain Pr[ Z = z | Y = y ]---*/
            p_ = pprime;
            k_ = z - y;
            n_ = nsample - y;
            link binomial;

            zprob = zprob + hypprob * binprob;

            end;

         end;  /* finish IIb */

      end;  /* finish Case II */


   /*---Case III:  0 < p < 1---*/
   else do;

      /*---IIIa:  pprime = 0 ---*/
      if pprime = 0 then do;

         /* zmin = 0 */
         zmax = min( nsample, d );

         if z <= zmax then
         do y = miny to maxy by 1;

            /*---obtain Pr[ Y = y ]---*/
            bign_ = nlot ;
            litn_ = nsample ;
            d_    = d;
            y_    = y;
            link hypergmt;

            /*---obtain Pr[ Z = z | Y = y ]---*/
            p_ = p;
            k_ = z;
            n_ = y;
            link binomial;

            /*---increment unconditional probability---*/
            zprob = zprob + binprob * hypprob ;

            end;

         end;  /* finish IIIa */

      /*---IIIb:  pprime = 1 ---*/
      else if abs( pprime - 1 ) < fuzz then do;

         zmin = nsample - min( nsample, d );
         /* zmax = nsample */

         if z >= zmin then
         do y = miny to maxy by 1;

            /*---obtain Pr[ Y = y ]---*/
            bign_ = nlot;
            litn_ = nsample;
            d_    = d;
            y_    = y;
            link hypergmt;

            /*---obtain Pr[ Z = z | Y = y ]---*/
            p_ = p;
            k_ = z - ( nsample - y );
            n_ = y;
            link binomial;

            /*---increment unconditional probability---*/
            zprob = zprob + binprob * hypprob ;

            end;

         end;  /* finish Case IIIb */

      /*---IIIc:  0 < pprime < 1 ---*/
      else
      do y = miny to maxy by 1;

         /*---obtain Pr[ Y = y ]---*/
         bign_ = nlot;
         litn_ = nsample;
         d_    = d;
         y_    = y;
         link hypergmt;

         /*---obtain Pr[ Z = z | Y = y ]---*/
         condprob = 0.0 ;
         minx     = max( 0, y + z - nsample );
         maxx     = min( y, z );

         do x = minx to maxx by 1;

            p_ = p;
            k_ = x;
            n_ = y;
            link binomial;
            factor1 = binprob;

            p_ = pprime;
            k_ = z - x;
            n_ = nsample - y;
            link binomial;
            factor2 = binprob;

            condprob = condprob + factor1 * factor2;

            end;

         /*---increment unconditional probability---*/
         zprob = zprob + condprob * hypprob ;

         end;  /* finish IIIc */

      end;  /* finish Case III */

   return;  /* finish uncond */

   /*---Compute Binomial Probability---*/
   binomial:

   binprob=0.0;

   if n_ = 0 then do;

      if k_ = 0 then binprob = 1.0 ;

      end;

   else
   if n_ > 0 then do;

      if ( k_ > 0 ) & ( k_ < n_ ) then
         binprob = probbnml( p_, n_, k_ ) -
                   probbnml( p_, n_, k_-1 );

      else
      if k_ = n_ then do;
         if ( p_> 0.0 ) & ( p_ < 1.0 ) then
            binprob = p_**n_;
         else if p_ = 1.0 then
            binprob = 1.0;
         end;

      else
      if k_ = 0 then do;
         if ( p_ > 0.0 ) & ( p_ < 1.0 ) then
            binprob = (1.0 - p_)**n_;
         else if p_ = 0.0 then
            binprob = 1.0;
         end;

      end;

   /*---finish binomial computation---*/
   return;

   /*---Compute Hypergeometric Probability---*/
   hypergmt:

      hypprob = 0 ;
      minarg  = max( 0, litn_ + d_ - bign_ );
      maxarg  = min( litn_, d_ );

      if y_ = minarg then

         hypprob = probhypr( bign_, d_, litn_, y_ );

      else
      if ( minarg < y_ ) & ( y_ <= maxarg ) then

         hypprob = probhypr( bign_, d_, litn_, y_     ) -
                   probhypr( bign_, d_, litn_, y_ - 1 );

   /*---finish hypergeometric computation---*/
   return;

run;

proc sort data=table;
   by nlot d n1 n2 a1 a1p a2;

proc tabulate data=table noseps;
   by nlot d n1 n2 a1 a1p a2;
   class p pprime;
   var accprob;
   table p, pprime*accprob=' '*sum=' '*f=8.4 / rts=7;
run;