Sequential Analysis


Also found in: Dictionary, Thesaurus, Medical, Legal, Wikipedia.

sequential analysis

[si′kwen·chəl ə′nal·ə·səs]
(statistics)
The continuous analysis of data, obtained via sampling, performed as the amount of sampling increases.

Sequential Analysis

 

in mathematical statistics, a method for the statistical testing of hypotheses. In this method, the number of observations required is not fixed in advance but is determined during the course of the test. The proper application of a chosen method of sequential analysis often requires considerably fewer observations for the same degree of validity than do methods where the number of observations is fixed in advance. Since the number of observations in sequential analysis is a random variable, this number is smaller only on the average.

Figure 1. Graphical representation of the process of sequential analysis

For example, suppose the problem consists in a choice between the hypotheses H1 and H2 according to the results of independent observations. Hypothesis H1 states that the random variable X has a probability distribution with density f1 (x); hypothesis H2 states that X has density f2 (x). The problem is solved in the following manner. Two numbers A and B are chosen such that 0 < A < B. After the first observation, the ratio λ1 = f2 (x1)/f1 (x1) is computed, where x1 is the result of the first observation. If λ1 < A, then H1 is accepted. If λ1 > B, then H2 is accepted. If A ≤ λ1B, then the process is continued: a second observation is made; the quantity λ2 = f2(x1)f2(x2)/f1(x1)f1(x2), where x2 is the result of the second observation, is analyzed; and appropriate action is taken. The probability is 1 that the process terminates with either the selection of H1 or the selection of H2. The quantities A and B are determined from the condition that the probabilities of errors of the first and second type have the specified values α1 and α2, respectively. An error of the first type is the rejection of hypothesis H1 when it is true, and an error of the second type is the acceptance of H1 when H2 is true.

In practice, it is more convenient to consider instead of λn the logarithms of λn. For example, let hypothesis H1 be that X has a normal distribution

with a= 0 and σ = 1; let hypothesis H2 that X has a normal distribution with a= 0.6 and σ = 1; and let α1 = 0.01 and α2 = 0.03. The corresponding calculations show that in this case A = 1/33, B = 97, and

Therefore, the inequalities λn < 1/33 and λn > 97 are equivalent to the inequalities

and

respectively. The process of sequential analysis in this case admits of a simple graphic representation (see Figure 1). On the xy-plane there are drawn the two straight lines y= 0.3x —5.83 and y = 0.3x + 7.62 and a broken line with vertices at the points

If the broken line first leaves the region bounded by the straight lines through the upper boundary, then H2 is accepted. If the broken line leaves the region through the lower boundary, then H1 is accepted. In this example, the method of sequential analysis requires on the average not more than 25 observations to decide between H1 and H2. More than 49 observations would be required to decide between the hypotheses on the basis of samples of a fixed size.

REFERENCES

Blackwell, D., and M. A. Girshick. Teoriia igr i statisticheskikh reshenii. Moscow, 1958. (Translated from English.)
Wald, A. Posledovatel’nyi analiz. Moscow, 1960. (Translated from English.)
Shiriaev, A. N. Statisticheskii posledovatel’nyi analiz. Moscow, 1969.

IU. V. PROKHOROV

References in periodicals archive ?
Observing interaction: An Introduction to sequential analysis.
An interesting finding of the sequential analysis is the fact that the Simon effect emerged to be significant in Experiment 2 only when the same type of target was repeated in the present and in the preceding trial as reflected by the significant interaction between previous target, present target and present correspondence.
Sequential analysis was chosen based on the statements of Keijsers et al.
Analizing Interaction: Sequential analysis with SDIS and GSEQ.
Quera and Bakeman (2000) explicitly state that sequential analysis can also be used to study the development of social skills and play in children, family relationships, interaction in clinical and educational settings, and communication processes.
One could clearly code a discussion using the IAM and then apply DAT's sequential analysis methodology--thus allowing one to answer questions not only about the types of knowledge construction evidenced in the data but also regarding sequences of interactions.
Simultaneous data collected on teacher directives and the occurrence of target behaviors allowed for a sequential analysis examining the effects of contextual factors on the relationship between adult directives and subsequent target child behaviors (Conroy, Asmus, Ladwig, Sellers, & Boyd, 2004).
We conducted a series of three analyses: 1) data reduction using lag sequential analysis (LSA) and sequence repetition analysis (SRA); 2) correlational analyses of sequential variables with outcome variables; and 3) multiple regression.
Neftci (1982) proposed a method using sequential analysis to calculate the probability of cyclical turning point.
Mother-Child Interaction in the Medical Setting: A Sequential Analysis.
Based on Calypto's patented sequential analysis technology, PowerPro CG reduces power by up to 60% with little or no impact on timing or area.

Full browser ?