## Acceptance Probabilities-Double Sampling

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/*          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---*/

/*---computer term2---*/

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;

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;

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;
hprob1 = hypprob;

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

mhyp = hprob1 * hprob2;

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

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

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 );

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;
f1 = binprob;

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

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;

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;

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;

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

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;

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;

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

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;

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

/*---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;

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

/*---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;

/*---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;
factor1 = binprob;

p_ = pprime;
k_ = z - x;
n_ = nsample - y;
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;

```