Sequential tests

Sequential tests are best explained graphically. As each new observation comes to hand, the value of a function of all observations recorded up to that time is calculated and plotted against the number of observations on a chart such as shown in Fig. 1.12. 

On the chart are two boundary lines, the positions of which depend upon the risks a and p, of errors of type I and type II, the magnitude of the difference it is important to detect, etc. The lines divide the chart into three zones (1) in which the Null Hypothesis is accepted (2) in which the alternative hypothesis is accepted and (3) in which there is no decision. 

The sequential test then consists in plotting the function of the observations f(X) against the number of observations n and continuing to take observations so long as the plotted points f(X) fall within (3). As soon as point falls outside this zone, that is, either in zone (1) (acceptance of the H0), or in zone (2) (acceptance of the H, ), the observations are discontinued and the indicated decision is taken. 

In practice the chart is often changed with the boundary values being calculated in advance for each value of n. The test is then made by successive comparisons of the value f(X) with the appropriate limits. 

One-sided sequential testing-comparison of a mean with a standard value 

---

Fig. 7.10 (Continued) (b) VN-x inhibits response to female urine by male guinea-pig X-2 sequential trials (duration, sec. x s.e.) (from Beauchamp et al., 1982). (c) Inter-strain (domestic vs. wild) discrimination by male domestic guinea pig of female urines [sequential testing sec/4 min, s.e., Ss above each bar intact/sham-VN-x. above]. WF= wild female, F = domestic female (from Beauchamp et al., 1982). (d) Effects of VN-x on maternal chemoinvestigation ewe responses to lambs, including tongue-manipulation of palate [c.f. Fig. 7.6(d)], procaine = MOE inhibition, a sign, versus control (from Booth and Katz, 2000).

This model leads to sequential tests, which generally use three outcomes and which represent the most effective variant of quality control. 

Based on the actual result of every individual measurement, it has to be checked whether a decision may now be taken or whether studies have to be continued. According to the final aim of sequential analysis, one can differentiate between closed sequential test plans that always come to a decision A > B or A < B (possibly with a large test expenditure) and open sequential test plans, which also allow - after a certain test expenditure -the statement A = B. 

Sequential test plans are also suitable for the direct comparison of two products or procedures based on subjective parameters as well as on measured results. This can be done without any computation, but only according to the following three criteria ... 

Fig. 4.13. Sequential test plan for direct characteristic comparison... 

For a sequential test, the number of experiments is not predefined. Rather, experiments are performed sequentially (surprise ), and the series terminated as soon as enough data is available that a decision can be made as to whether the difference is large enough . True, it is theoretically possible for such a sequence of experiments to be indefinitely long in practice, however, it is far more common for the situation to become decidable after fewer experiments than are required for the case of a fixed number of experiments. 

Examination of the matrix in Table XII shows that 6 mechanisms are possible on the basis of the steps proposed by BischofT and Froment. They identified m, m2, m4, m5, and m6, and developed 15 rate equations corresponding to various choices of rate-controlling steps. After a set of experiments involving sequential testing and model discrimination, they retained mechanism m4 with step st as the rate-controlling step. According to the present procedure m3 might be an additional mechanism to consider. 

Also, sequential tests may prove more useful for predicting the availability of nutrients to... 

Sequential testing may be considered during development in addition to Procedure 1. The vibration test durations may be six (6) hours (2 hours per axis) at room temperature and three (3) hours (1 hour per axis) at each temperature extreme for a total test time of 12 hours for the three (3) temperatures. The temperature sequence of room temperature —65°F and 160°F is recommended... 

Applying the sequential test to the differences of observed breaking load in repeat pairs of observations, then ... 

It is also recommended to sample a sufficient quantity of waste so as to be able to undertake Tier I, II and III. Cost-effective conditions of storage must therefore be defined (for example, at ambient temperature, in darkness and inside well-closed containers) to prevent air contact prior to initiating analyses. Chemical and ecotoxicological stability of a waste sample should also be verified via time-sequential testing of sub-samples. 

In sequential testing, a hypothesis is established and limits are set for both Type I and Type II errors. The data are then accumulated sequentially, and, as each new observation is obtained, the information from all the data is tested against the hypothesis with one of three actions possible the hypothesis may be rejected, the hypothesis may be accepted, or no decision may be made and another datum point is obtained. Data are collected until the hypothesis is either rejected or accepted. 

M This aspect of statistics has been developed considerably recently c.f. Sequential. Analysis of Statistical DataApplications Statistical Research Group. Coliunbia University Columbia University Press, and G. A. Barnard "Sequential Tests in Industrial Statistics (Supplement to the Journal of the Royal Statistical Society, Vol. VIII, No. 1, p. 1. 1946). 

Skin tests are first applied as a prick test for safety. In the absence of a local or systemic reaction, an intradermal test is performed and interpreted as described elsewhere (181,182). Experience with skin testing in penicillin allergy has been reviewed (176,183). Properly performed sequential testing is considered a safe procedure, and only an estimated 1% or less of penicillin allergic patients will have systemic symptoms while undergoing skin tests. However, at least three deaths have been reported with both epicutaneous and intradermal testing (184). 

Following the strategy outlined in Table 15-6, the first testis specificity determines the cost of sequential testing. 

---