The sequential probability ratio test, or SPRT, can be used as an efficient tool for process tolerance and mean shift determinations. It also provides for simplifying insights into the nature of random mean shifts when process tolerances and Type I/II errors are selected. The cumulative sum design of the test naturally compensates for random errors, and one-sided random shifts easily calculate as one-half of the process tolerance T, corresponding to a worst case error of ½ (a=b=.5). For a selected tolerance of 3 sigma, a long-term 1.5 sigma shift is simply the critical mean asymptotic limit of an indeterminate process for both one-sided and two-sided SPRTs.
Sequential Analysis Theory
Sequential analysis was created by Abraham Wald in 1943 for wartime military equipment development and inspections. Numerous examples of the theory’s practical applications and sampling economies were later published by Columbia University and example table exhibits of the 50 percent sampling economy can be found in the document. The bases are the probability inequality ratios that define the three inspection regions shown in Table 1 below.
Table 1 | ||
Accept H_{} |
Continue to Inspect |
Accept H_{1} |
I: P_{1N} / P_{0N}<b/ (1 – a) |
II: b / (1 – a) < P_{1N} / P_{0N} < (1 -b) /a |
III: P_{1N} / P_{0N}> (1 -b) /a |
Where P_{0N}, P_{1N} are the probability density functions corresponding to the test H_{} and alternate H_{1} hypothesis for the sequential set of observations X_{1}…X_{N} and a, b are the selected Type I (producer’s risk) and Type II (consumer’s risk) errors. The test is continued when inequalityratio (II) applies, and discontinued at the firstoccurrence of the ratios (I) or (III) resulting in the acceptance of hypotheses H_{} or H_{1}, respectively. When deployed during test applications for such popularprobability functions as the Binomial, Poisson and Normal distributions, the probability inequality ratios are usually transformed into the simple straight line forms Y_{} = S N + h_{}, Y_{1} = S N + h_{1}, where S is the common slope, N the sample number, and h_{}, h_{1} are the intercepts of the acceptance and rejection lines which delineate the regions. The cumulative sums S X_{N} of the observations are then compared with the straight line values for decision purposes. The concept is summarized in Table 2.
Table 2 | ||
Accept H_{} |
Continue to Inspect |
Accept H_{1} |
I: S X_{N} < S N + h_{} |
II: S N + h_{} < S X_{N} < S N + h_{1} |
III: S X_{N} > S N + h_{1} |
One-Sided SPRT Process Mean and Shift Tests
As an example of the test’s applicability to mean testing, consider the sequence of observations X_{1}…X_{N} from a normally distributed population with a known standard deviation and unknown mean M.When the probability density ratios for the test hypotheses H_{}: M = M_{}, H_{1}: M = M_{1} are inserted in Continue to Inspect test region of Table 1, thecumulativesum inequality (1) results. And upon inspection it is easily reduced to the straight line form (2).
Dividing (1) by the cumulative sample size N transforms the Continue to Inspect region into the conceptually useful hyperbolic form:
Where Xca = 1 / N, SX_{N} is the cumulative average and Mc = (M_{} + M_{1}) / 2 is the critical mean and asymptotic limit for the test; i.e., cumulative averages Xca converge to Mc when N becomes sufficiently large and both h_{} / N, h_{1} / N ? 0. The limit also occurs at the test’s maximum level of uncertainty or worst case error (a = b = ½) since h_{} / N = h_{1} / N = 0 at these values. However, Wald has shown that all sequential tests eventually terminate with a probability of one which is evident by the geometry of the cumulative average test illustrated in the graph in Figure 1. Group size 4 random data from Shewhart’s normal bowl with the known average of 30 and standard deviation of 10 together with the Type I and Type II errorsa’ = .001 andb’ = .001 were selected for the test. Since sample grouping in a sequential test has the effect of lowering the risks, a prime designation for the errors is used.
In the Shewhart’s bowl example, the test terminated with the acceptance of M_{} = 30 when the cumulative average of the third subgroup fell within the acceptance region.When the critical mean Mc = 45 is chosen as the criteria fora successive mean shift determination, the test required ninesubgroups leading again to the acceptance of the null hypotheses (M_{} = 30) and rejection of the alternate H_{1} (M_{1}= 45) that a shift has occurred.
Sequential Test Sampling Requirements
Since sequential test inspection plan numbers are dependent on the outcome of the observations, average sample numbers (ASN) are used to estimate sampling needed. A useful table of average sample numbers corresponding to probability of acceptance and mean values for the test is provided in Table 3 below.
Table 3 | ||
Mean Value M |
Probability of Acceptance (PA) |
Average Sample Number (ASN) |
M_{} |
1 -a |
((1 – a)(h_{} + h_{1}) – h_{}) / (S – M_{}) |
Mc |
h_{} / (h_{} + h_{1}) |
h_{} h_{1} / s^{2} |
M_{1} |
b |
((b)(h_{} + h_{1}) – h_{}) – h_{}) / (S – M_{1}) |
When used for practical purposes, the Type I and Type II risks are chosen in a manner which preserves sampling economy and protects both the producer and consumer interests. The larger of the ASNs corresponding to hypothesis H_{} or H_{1} is usually chosen for this purpose.
Two-Sided SPRT Process Tolerance and Mean Shift Tests
Wald’s curvilinear (Method C) two-sided sequential decision test for determining whether a process mean M is within Tolerance T, or M +/- T assumes that the process is normally distributed. For a selected mean M, Tolerance T, alpha/beta errors a, band standard deviation s, the test would proceed as follows when the measured random variables X_{N} are transformed into Z_{N} values:
Method C
I – Stop the test and accept the hypothesis H_{} that M = 0 and the process is within tolerance T when the natural log Cosh function absolute values are within the acceptance region.
(5) Ln(Cosh( I ST Z_{N} I )) < (T^{2} / 2) N + h_{}
II – Continue Testing or reserve the decision whenever.
(6) (T^{2} / 2) N + h_{} < Ln(Cosh( I S T Z_{N} I )) < (T^{2} / 2) N + h_{1}
III – Stop the Test and accept the hypothesis H_{1} that the process mean M is out of tolerance or I M I > T whenever the absolute value of the cumulative T Z_{N} sum falls within the rejection region.
(7) Ln(Cosh( I ST Z_{N} I )) > (T^{2} / 2 N) + h_{1}
Where Z_{N} = (M – Xn) / s, N is the test sample number h_{} = Ln (b / (1 – a)), h_{1} = Ln ((1 -b) /a)
The equations of Method C can be greatly simplified and converted to linear form by utilizing the asymptotic property of Y = Ln(Cosh (x)) which is equivalent to Y = x – Ln(2), depending upon the degree of precision required. For x>3, 4 and 5 the functions are equivalent to within 2, 3 and 4 decimals respectively. When x – Ln(2) is substituted for Ln(Cosh (x)) in the inequality expressions of Method C, linear decision line regions of Method L result:
Method L
I – Stop the test and accept the hypothesis H_{} that M = 0 and the process is within tolerance T when the absolute values of the cumulative Z_{N} sums is within the acceptance region.
(8) I SZ_{N} I < S N + h_{}
II – Continue testing or reserve the decision whenever.
(9) S N + h_{} < I S Z_{N} I < S N + h_{1}
III – Stop the Test and accept the hypothesis H_{1} that the process mean M is out of tolerance or I M I > T whenever the absolute value of the cumulative Z_{N} sum falls within the rejection region.
(10) I SZ_{N} I > S N + h_{1}
Where Z_{N} = (M – Xn) / s, S = T / 2 is the common slope of the acceptance and rejection lines, N is the test sample number, h_{} = 1 / T[Ln (b / (1 – a)) + Ln(2)] is the intercept of the acceptance line and h_{1} = 1 / T[Ln ((1 -b) /a) + Ln(2)] is the intercept of the rejection line.
Hence, a Z value selection of T = 3 for a process tolerance test results in the calculated slope value S = 1.5. However, in the case of a process shift test, the ½ tolerance critical value of Tc = 1.5 would be chosen with the corresponding slope of S = .75.
Process Tolerance and Shift Test Example
A chemical manufacturer routinely controls the specific gravity measurement of a critical to quality ingredient to a tolerance of +/- 3 sigma based on collected data. The mixture operation was shut down for routine preventative maintenance. After start-up, two-sided sequential process shift Tc and tolerance T tests are conducted. For the test parameters, the engineering group selected the historical mean M = .84, standard deviations= .021, alpha/beta errorsa= .001,b= .1 and tolerances of Tc = 1.5 and T = 3 respectively. The tabulated analysis (Tables 4 and 5) indicated the presence of a process shift and out-of-tolerance condition which was investigated and corrected.
Table 4: Method L Sequential Analysis – Mixture Process Shift Determination | |||||
AlphaError |
0.001 |
Beta Error |
0.1 |
||
ProcessMean |
0.84 |
Standard Deviation |
0.021 |
||
ProcessTolerance Tc |
1.5 |
Slope S |
0.75 |
||
Intercept h_{ } |
-1.072 |
Intercept h_{1} |
4.997 |
||
Sample |
< SN + h_{} |
> SN + h_{1} |
Observed |
Z |
Abs Value |
1 |
-0.322 |
5.747 |
0.874 |
1.619 |
1.619 |
2 |
0.428 |
6.497 |
0.861 |
1.000 |
2.619 |
3 |
1.178 |
7.247 |
0.890 |
2.381 |
5.000 |
4 |
1.928 |
7.997 |
0.892 |
2.476 |
7.476 |
5 |
2.678 |
8.747 |
0.880 |
1.905 |
9.361 |
Table 5: Method L Sequential Analysis — Mixture Process Tolerance Determination | |||||
AlphaError |
0.001 |
Beta Error |
0.1 |
||
ProcessMean |
0.84 |
Standard Deviation |
0.021 |
||
ProcessTolerance T |
3 |
Slope S |
1.5 |
||
Intercept h_{ } |
-0.536 |
Intercept h_{1} |
2.499 |
||
Sample |
< SN + h_{} |
> SN + h_{1} |
Observed |
Abs Z |
Abs Value |
1 |
0.964 |
3.999 |
0.874 |
1.619 |
1.619 |
2 |
2.464 |
5.499 |
0.861 |
1.000 |
2.619 |
3 |
3.964 |
6.999 |
0.890 |
2.381 |
5.000 |
4 |
5.464 |
8.499 |
0.892 |
2.476 |
7.476 |
5 |
6.964 |
9.999 |
0.880 |
1.905 |
9.381 |
6 |
8.464 |
11.499 |
0.895 |
2.619 |
12.000 |
Procedure Advantages and Limitations
Although sequential sampling procedures are very economical and easily automated, they do not exempt the manufacturer or organization from best practices continuous monitoring since processes do change in time. Also, the tests discussed are based on the assumption of normally distributed processes. Another limitation of the sequential method is that compensations are not made for the effects of sample grouping on Type I, Type II errors, ASN and OC curve values. Sample grouping by increments results in an increase in the number of items inspected and lower Type I and Type II errors. If the process evaluated is either very good or bad, speedy decisions occur. However, these accept-or-reject decisions are dependent upon the closeness of the mean parameters M_{} and M_{1} and the degree of riska,b that the manufacturer or organization is willing to accept.Ambiguous processes often require indefinite sample sizes,and a truncation point leading to an increaseda orb risk has to be agreed upon.