<![CDATA[
A quality control procedure used when a decision on the acceptability of a batch has to be made from a sample.It minimises the number of items tested when the early results show that the batch clearly meets, or fails to meet, the required standards.
<P>
SEQUENTIAL ACCEPTANCE SAMPLING
<P>
Acceptance sampling is a quality control procedure used
when a decision on the acceptability of the batch has to be made from
tests done on a sample of items from the batch. <p>
Sequential acceptance sampling minimises the number of items
tested when the early results show that the batch clearly meets,
or fails to meet, the required standards. <p>
The procedure has the advantage of requiring fewer
observations, on average, than fixed sample size tests for a similar
degree of accuracy. <p>
The method is illustrated here by application to seed viability testing
where the % viable seeds in a sample is assessed in order to determine
the quality of the seed lot from which the sample is drawn.
<P>
APPLICATION : SEED GERMINATION TESTS
<P>
<a href="slide1_1.htm">
<img alt="picture" src="../images/si02.gif" WIDTH=100 HEIGHT=68> </a>
Seeds laid out on damp paper towels and placed in germinating cabinet
<p><br>
<a href="slide1_2.htm">
<img alt="picture" src="../images/si03.gif" WIDTH=100 HEIGHT=69> </a>
After several days towels are removed and the abnormal seedlings are counted
<p><br>
<font color="FF0000">
N.B. To view in more detail please click on image.
</font>
<p>
<P>
The supply of dependable, high quality seed, is an important element in
successful crop production. <p>
Seed quality has many aspects including purity, germination capacity,
vigour and health. <p>
The quality of seed nationally depends on a well-organised testing service,
with skilled seed analysts, applying sound sampling and analytic techniques.
<p>
This presentation introduces a method for minimising the number of seeds that
must be assessed by analysts. <p>
EXAMPLES OF ABNORMAL SEEDLINGS
<P>
<a href="slide1_3.htm">
<img alt="picture" src="../images/si04.gif" WIDTH=100 HEIGHT=68> </a>
No roots
<p><br>
<a href="slide1_4.htm">
<img alt="picture" src="../images/si05.gif" WIDTH=100 HEIGHT=68> </a>
Stunted roots
<p><br>
<a href="slide1_5.htm">
<img alt="picture" src="../images/si06.gif" WIDTH=100 HEIGHT=68> </a>
Splits in coleoptile
<p><br>
EXAMPLES OF ABNORMAL SEEDLINGS
The two seedlings on the right are abnormal, having no roots, unlike the normal
seed on the left. <p>
</ul>
<p>
<IMG SRC = "../images/i04.jpg" align=middle WIDTH=400 HEIGHT=271>
<p>
TETRAZOLIUM TESTING
<P>
<a href="slide1_8.htm">
<img alt="picture" src="../images/si09.gif" WIDTH=100 HEIGHT=68> </a>
Examining seeds for stains under a microscope
<p><br>
<a href="slide1_9.htm">
<img alt="picture" src="../images/si10.gif" WIDTH=100 HEIGHT=68> </a>
A normal seed
<p><br>
<a href="slde1_10.htm">
<img alt="picture" src="../images/si11.gif" WIDTH=100 HEIGHT=68> </a>
An abnormal seed
<p><br>
<P>
An indication of germination capacity can also be got by a biochemical test
using the reagent tetrazolium. Seeds are soaked in a tetrazolium solution.
In the solution living seed tissue turns red, while dead tissue remains
unstained. The embryos of the seed are then inspected under a microscope
and those that show a minimum staining pattern are counted as normal. <p>
WHAT IS SEQUENTIAL ACCEPTANCE SAMPLING?
<P>
<ul>
<li>Samples are taken from a lot and examined individually or in
groups;
<p>
<li>After each examination a decision is made between:
<p>
<ul>
<li>Accepting the lot;
<p>
<li>Rejecting the lot;
<p>
<li>Examining another sample
<p>
</ul>
</ul>
<p>
<P>
One way of reducing effort is to test the seeds in groups. After examining
a group a decision is made to accept the lot as having met the required
standards, to reject the lot, or to examine another group. Such a method is
known as sequential acceptance sampling. <p>
Note, for the purposes of statistical validity each group of seeds must be
selected at random and independently of all other groups. <p>
RESULTS FROM EXAMINING A SEED LOT
<pre>
ORDER OF SEEDS
EXAMINING RESPONDING
SEEDS NO. %
1- 20 20 100
21- 40 39 98
41- 60 58 97
61- 80 76 95
81-100 95 95
101-120 113 94
121-132 132 94
141-161 151 94
161-180 169 94
| | |
| | |
481-500 470 94
</pre>
DEVELOPMENT OF SEQUENTIAL SAMPLING
<P>
<ul>
<li>PRE WORLD WAR 2
<p>Methods based on simple ad-hoc rules
<p>
<li>1943
<p>First formal theory developed by :
<p><ul><li>A.Wald (Statistical Research Group, Columbia University)<br>
G.A.Barnard (Britain)</ul>
<p>
<li>1980s
<p>Computer program for designing sequential trials<br>
<A href="ref.htm">J.Whitehead and P.Marek (1985) </A>
</ul>
<p>
APPLICATIONS OF SEQUENTIAL SAMPLING
<P>
<UL>
<LI>CLINICAL
<p>Trials of new drugs and treatments
<p>
<li>INDUSTRIAL
<p>Acceptance inspection/sampling of manufactured components
<p>
<li>SEED TESTING
<ul>
<li><A href="ref.htm">Banyai (1978)</A> :
Diploid content of red clover seeds
<li><A href="ref.htm">Ellis et al (1980)</A> :
Monitoring genebank seed
<li><A href="ref.htm">Coster (1988)</A>:
Tetrazolium testing of cereal seed
</ul>
</ul>
<p>
<P>
Sequential techniques have been especially valuable in the medical field where
ethical considerations demand that a trial should be stopped as soon as there
is clear evidence that one of the treatments is preferred. <p>
They are also widely used in industry and are particularly useful where testing
involves the destruction of test material. <p>
In seed testing, sequential methods have been used by Hungarian workers, Banyai
and her colleagues in testing for the diploid content in red clover seeds, and
by Ellis in monitoring seeds in a gene bank for viability. <p>
OUTLINE OF THIS PRESENTATION
<P>
<ul>
<li>Basics of simple (fixed-sized) acceptance sampling<br>
<br>
<li>Principles of sequential sampling<br>
<br>
<li>Guidelines for applying sequential sampling to seed testing<br>
</ul>
<p>
<P>
In this introduction to sequential acceptance sampling we begin by outlining
the basics of fixed-sized acceptance sampling. We go on to consider the
principles underlying sequential sampling and then, show how these principles
can be applied in seed testing. <p>
BASICS OF SIMPLE ACCEPTANCE SAMPLING
<P>
<ul>
<li>Fixed - size sample
<p>
<li>Taken at random from a seed lot in which proportion <i> p </i> will respond
if all are tested
<p>
<li>Observe the proportion p that responds in the sample
<p>
<li>Accept the seed lot if p exceeds a test standard p*
<p>
<li>Reject the seed lot otherwise
</ul>
<p>
Let us look first at the characteristics of simple acceptance sampling. The
primary aim is to classify a seed lot as satisfactory or not satisfactory. A
sample of fixed size is taken at random from a seed lot in which an unknown
proportion p will respond if all seeds are tested. In the sample we find that
proportion p hat respond. We accept the seed lot as satisfactory if p hat
exceeds a pre-determined standard. Otherwise we reject the seed lot as
unsatisfactory. There are only two possible decisions -accept or reject. <p>
A SIMPLE ACCEPTANCE SAMPLING SCHEME
<P>
This diagram illustrates some of the characteristics of a simple acceptance
sampling scheme. It shows the distribution of the proportion of seeds
responding in a sample plotted against the actual proportion in the seed lot.
<p>
The horizontal line at the top of the diagram represents the standard which, if
it is exceeded in the sample, will lead to acceptance of the seed lot. The
actual proportions observed in samples are indicated by an X. <p>
Note that due to the uncertainties of sampling it is possible to obtain a
proportion in a sample that is above the test standard even though the
proportion in the lot as a whole is below the standard. <p>
FACTORS IN SETTING TEST STANDARD
<P>
p0 : the minimum acceptable level for the proportion
responding in the seed lot<p>
p1 : a high level for the proportion responding which
if achieved in a seed lot should mean acceptance.<p>
The test standard is set between p0 and p1
<p>
<P>
How do we decide what should be the level of the test standard? <p>
We do this by establishing what we think represents a poor seed lot and what
represents a very good seed lot. <p>
The lower level, called p o, is the minimum proportion that we would accept
in a seed lot. <p>
The higher proportion p, represents the level which, if achieved in a seed
lot, we would like to have the lot accepted. <p>
The test standard is set between the high level and the minimum acceptance
level. <p>
BALANCING THE RISKS
<P>
Test standards are set by attempting to find a balance between two competing
risk : <p>
<ul>
<li>a (alpha) - the tester's risk
<br> i.e. the probability that a seed lot with a low response proportion
(p0) will be accepted <p>
<li>b (beta) - the applicant's risk
<br> i.e. the probability that a seed lot with a high response proportion
(p1) will be rejected <p>
<li>1-b - the <a href="glossary.htm#power"> power </a>
<br> (the applicant's risk is usually expressed as a difference from 1.
i.e. the power of the test to identify seed lots with high response
rates)
</ul>
<p>
The exact point at which we set the standard is determined by how we wish to
balance two competing risks. <p>
One of these risks, known as the tester's risk, in statistical notation alpha,
is the probability that a seed lot with a low response proportion will in fact
be accepted because of random sampling variation. <p>
The other risk, called the applicant's risk, or beta, is the probability that
a seed lot with a high proportion of seed responding, will be rejected due to
random sampling variation. <p>
The risks alpha and beta are usually expressed in probability terms which
range from <br>
0 - no seeds in the lot will respond <br>
1 - all seeds in the lot should respond. <p>
For convenience the applicants risk is usually expressed as a difference from
one. Thus, one minus beta represents the power of the test to identify seed
lots which contain a high proportion of seeds responding. <p>
OPERATING CHARACTERISTIC CURVE
<P>
Fixed sized (n=467) acceptance sampling
<p>
<img src="../images/graph2.gif" WIDTH=411 HEIGHT=348>
<p>
p0 = 0.85 - minimum acceptable level for the seed lot
<p>
<P>
This diagram shows how the risks of accepting a seed lot are related to the
level of the response proportion in the seed lot. <p>
The operating characteristic curve, called 'OC curve' in industrial quality
control terms, tells us how an acceptance sampling scheme will behave when
seed lots of different qualities are submitted for testing. For any specified
quality of seed lot the probability of accepting the lot can be found by
reading from the horizontal axis to the curve and then across to the vertical
axis. <p>
<P>
Three important points on the OC curve are indicated on the diagram. <p>
<ul>
<li> The probability of accepting a seed lot with a low response proportion.
Here we have chosen a probability of 0.05 that a seed lot with 85% of
seeds responding will be accepted. <p>
<li> The second point is the probability that a seed lot with a high proportion
responding will be accepted - we have chosen a probability of 0.95 that a
seed lot, with 90% of seeds responding, will be accepted. <p>
</ul>
These two points, together with information on the number of seeds tested,
determine the shape of the operating characteristic curve. <p>
<ul>
<li> The third point - marked by the dashed line in the middle of the diagram
- represents the test standard level. This is usually set at the point where
there is an even chance - probability equal to 0.50 - that a seed lot will
either be accepted or rejected. <p>
</ul>
Reading across from the probability level of 0.50 to the OC curve and then down
to the horizontal axis gives us a test standard, p., of 0.87. <p>
This is the standard which should be applied if samples of 467 seeds are
examined under a scheme with a tester's risk of 0.50 and a power of the test to
identify high quality samples of 0.95. <p>
TYPES OF SEQUENTIAL TEST METHODS
<P>
<ul>
<li>Truncated sequential probability ratio test
<p>
<li>Triangular
<p>
<P>
Understanding how standards are developed for simple acceptance sampling makes
it easier to follow the same procedure in sequential sampling. <p>
Here, we describe how standards are developed for two type of sequential
acceptance tests. <p>
<ul>
<li> the truncated sequential probability ratio test, which for brevity we
will call the truncated test. <p>
and <p>
<li> the triangular test. <p>
</ul>
<P>
A TRUNCATED SEQUENTIAL SAMPLING SCHEME
<P>
In the truncated test seeds are taken at random from a seed lot in groups of
fixed size. After each group is examined the proportion of seeds responding
so far is compared with the lower limits for this proportion. If the
proportion responding exceeds the upper limit then the lot is accepted without
further testing. If the proportion responding lies below the lower limit then
the seed lot is rejected. When the proportion lies between the upper and lower
limit then another group is examined. <p>
The diagram illustrates this procedure. The vertical axis represents the
proportion of seeds responding in the sample so far, and the horizontal axis
gives the number of groups tested. <p>
The boundaries are set to give the required tester's risk and power of test to
detect high quality seed lots. <p>
To ensure that a test reaches a conclusion a truncation point is introduced.
This can be specified by the tester. It must be at a level greater than the
number of seeds in a fixed size sample test with the same operating
characteristic curve as the sequential test. <p>
If a test actually reaches the vertical boundary then the sample is accepted
if it lies in the upper half of the vertical boundary and rejected otherwise.
<p>
EXAMPLE OF A TRUNCATED SCHEME
<P>
<pre>
ALPHA <a href="glossary.htm#risk">(tester's risk)</a> = 0.05
<a href="glossary.htm#power">POWER</a>(1 - applicant's risk) = 0.95
p0 (minimum acceptable response) = 0.85
p1 (high level response) = 0.90
No. of seeds tested per group = 40
</pre>
<pre>
Number of seeds responding (n)
Group Seeds Reject when Accept when
number tested n < or = n > or =
1 40 29 41
2 80 64 76
3 120 99 111
4 160 134 146
5 200 169 181
| | | |
| | | |
14 560 484 496
15 600 525 526
</pre>
Note : Equivalent fixed sample size (for same risks) is 467
<P>
This is an example of a truncated sequential sampling plan. The main parameters
of the plan are outlined above the table. These parameters are the same as
those used to define the fixed sample-size scheme that we described earlier.
However, in the truncated test we are examining the seeds in groups of 40
instead of a singe fixed-sample size of 467. <p>
The table shows the output from the SEEDS program developed by Richard Ellis
and John Whitehead. We can see after the second inspection when 80 seeds have
been examined we would reject the seed lot if 64 or less reponded and accept if
75 or more responded. When the number responding lies between 64 and 76 then we
would inspect another 40 seeds. This procedure would continue up to a maximum
of 15 groups being tested. <p>
A TRIANGULAR SEQUENTIAL SAMPLING SCHEME
<P>
In a triangular test the boundaries converge to an apex which corresponds to a
sample size larger than the equivalent fixed sample size. As before, the
boundaries can be constructed to satisfy a specified tester's and applicant's
risk. Unlike the truncated scheme where the truncation point is determined
by the tester, in the triangular test the truncation point is determined by the
scheme. <p>
EXAMPLE OF A TRIANGULAR SCHEME
<P>
This is an example of a triangular scheme. The parameters are the same as for
the truncated test shown earlier. <p>
The test operates in a similar way to that for the truncated scheme. The
boundaries are broader initailly but narrow gradually. Note that the maximum
number of groups of seeds to be tested is 18 compared with 15 under the
truncated test. <p>
AN APPLICATION OF THE TRIANGULAR TEST
<P>
<pre>
ORDER OF SEEDS REJECT ACCEPT
EXAMINING RESPONDING WHEN WHEN
SEEDS NO. % no. < or = no. > or =
1- 20 20 100 5 -
21- 40 39 98 23 -
41- 60 58 97 40 -
61- 80 76 95 58 -
81-100 95 95 76 97
101-120 113 94 93 114
121-132 132 94 111 131
141-161 ACCEPT 129 147
161-180 146 164
| | |
481-500 428 429
</pre>
<p>
SIMPLE VS SEQUENTIAL TEST
<P>
If sequential tests are readily available why should we bother to use fixed-
sample size tests? <p>
In practice, fixed-sample sized tests are easier to use. Only one decision is
needed when all the seeds have been examined. Fixed sized sampling is
particulary suitable when the test is inexpensive and the seeds are not
precious. A further advantage is that the proportion of seed which respond in
the sample is an unbiased estimate of the proportion responding in the seed
lot. This is sometimes useful when it is necessary to report the number of
seeds germimating on a certificate. <p>
With sequential testing more calculations and checking with boundary values are
required. Sequential testing is appropriate when tests are rapid but expensive
to carry out, or the seeds are precious. Under sequential testing fewer seeds
will need to be examined on average. <p>
The proportion of seed responding in a sequential test tends to be an overestimate
of the proportion of seed that might respond in the seed lot from which the sample was
drawn. However, it is possible to make a correction which allows for this bias.
<p>
TRIANGULAR VS TRUNCATED TEST
<P>
If a decision is made to use a sequential test which one should it be? <p>
The choice will depend very much on the response proportion that is likely to
be observed. The truncated test is best when we expect to observe extreme
proportions. The boundaries are initially much closer together than the
triangular test and so we would expect a quicker decision in such cases. <p>
The triangular test is appropriate when the proportions are expected to be
between the minimum acceptance level p0 and the high level p1. <p>
COMPARISON OF DECISION REGIONS
<P>
This diagram shows the relative position of the decision regions for each
of the tests. Note that both the truncated and sequential tests may on
occasions require more seeds to be examined than the fixed size sample. <p>
NUMBER OF SEEDS TESTED DEPENDS ON QUALITY OF SEED LOT
<P>
Although sequential tests may require more seeds to be examined on a few
occasions, on average over many tests far fewer seeds will be needed that with
fixed sample size tests. <p>
Here we show the median or average number of groups of 40 seeds that we would
expect to have to examine in the schemes described earlier. For the fixed-
size sample test, 12 groups of 40 seeds, 467 seeds to be precise - are needed.
This number does not change with the proportion of seed responding in the
seed lot being examined. For the truncated and triangular tests the average
number of seeds to be examined depends on the proportion of seed responding
in the seed lot from which the samples are drawn. The largest number of seeds
are tested when the proportion responding lies mid-way between the minimum
acceptance level and high standard level, at about 0.87. <p>
Even at this level of response the sequential tests requires a third fewer seeds
to be tested than the fixed-sample test. <p>
Note that although the truncated test is more economical in testing effort over
much of the range of response proportions, the triangular test is slightly
better at the point where more of the samples are likely to lie - between 0.86
and 0.88. <p>
WHICH TECHNIQUE TO CHOSE
<P>
In what situations are the various tests best applied? <p>
The fixed-sample test is more approriate for the standard paper-towel
germination test. Having to await the results of one germination test before
deciding to prepare a further group of seed for testing will delay conclusion
of the test considerably. <p>
The truncated test is well suited to testing for viability of seeds in gene
banks. If there is a very low proportion responding in a seed lot then testing
stops sooner, since at the early stages the decision boundaries are closer
together for the truncated test. <p>
The triangular test is more appropriate for quality testing where the
proportion responding in seed lots are likely to lie within a relatively narrow
range. <p>
OTHER APPLICATIONS OF SEQUENTIAL METHODS IN SEED TESTING
<P>
<ul>
<li>DETECTION OF PROPORTION OF SIBS/DIPLOIDS <p>
- use truncated test
<p>
<li>TETRAZOLIUM TEST FOR SEED VIABILITY <p>
- use either truncated or triangular test
<p>
<li>LARGE SEED CONSIGNMENTS DELIVERED TO SEED TRADERS <p>
- use truncated test to reduce delay in deciding whether to accept
a consignment
</ul>
<p>
<P>
Apart from standard germination tests there are other areas in seeds work where
sequential tests may be helpful. <p>
One of these areas is the detection of the proportion of sibs/diploids. The
tests are relatively rapid and there is little delay in results. So the
truncated test could be appropriate. <p>
In the tetrazolium test for seed viability, results are produced much more
rapidly than with the germination test but, since each seed is examined
individually under a microscope, there is a limit to the number of seeds that
can be examined by an analyst in a day. Here, either the truncated of
triangular test should be suitable. <p>
For inspection of large seed consignments by traders the truncated test might
be satisfactory since it would allow the trader to quickly reject low quality
or accept high quality seed lots. <p>
SUMMARISING THE BENEFITS
<P><P>
<ul>
<li>Testing stops as soon as it is possible to make a decision
<p>
<li>Saving in number of seeds used
<p>
<li>Quicker testing
<p>
<li>Cheaper testing
</ul>
<p>
To summarise, the main advantages of sequential testing is that testing stops
as soon as it is possible to make a decision; so there is no unnecessary work.
There is also a saving in the number of seeds used for testing and the more
extreme the response proportion in the seed lot then the greater is that saving.
Testing can be completed more quickly with sequential tests and at less expense.
On average, there is a potential saving of 25-30% of the variable costs
associated with the number of seeds that are examined. <p>
]]>