Contingency calculations

Control valve failure is an important scenario in contingency calculation. The following general guidelines are normally used to estimate control valve failure contingency. 

In applying this rule, the capacity of the pressure relief system must also be sized to handle the quantity of fluid released at this pressure (together with other expected loads during this contingency), so that the built-up back pressure will not result in exceeding 1.5 times the design pressure. This additional load need not, however, be considered in calculations of flare and PR valve radiant heat levels. 

A model of lamellae formation In stretched networks is proposed. Approximately one-half of the chains do not fold. Formation of such lamellae Is accompanied by declining stress. Highly folded systems (high crystallinity), however, can cause a stress Increase. In the calculations crosslinks are assigned to their most probable positions through the use of a characteristic vector. A contingent of amorphous chains Is also Included. The calculations suggest that the concept of fibrillar-lamellar transformations may be unnecessary to explain observed stress-temperature profiles In some cases. 

Bartczak et al. s entire track calculation is contingent on the specific detail of track structure. They argue that branch tracks of energy <50 or 100 KeV should be treated as a single entity whereas those of higher energy could be broken into their constituent spurs and tracks. In this manner, Bartczak and Hummel (1993) found that... 

Ghent has developed and proposed a good (though, if performed by hand, laborious) method extending the calculation of exact probabilities to 2x3, 3x3, and RxC contingency tables (Ghent, 1972). 

As long as the equipment is operating, power and treatment chemicals will be consumed. Generally this cost can be calculated on a daily or volume basis. Difficulties can occur when operation is halted on an unplanned occurrence, such as an accidental power disruption or other cause for system stoppage. Restarting and return of the system to a semiequilibrium status can sometimes be as costly as the initial startup. Contingencies should be included for periodic happenings. 

For the most part, the meta-analyses used in this text are based on fourfold contingency tables, which include the number of responders or nonresponders to a given treatment. An advantage of dichotomous data is that information from each individual subject can be abstracted (i.e., the results come from actual patients). In one sense, this is not strictly a meta-analysis, because calculations are not done on summary statistical parameters but on observations of individual subjects. Such an approach has the advantage of directness, however, because the percentage of patients who respond or do not respond to a new treatment, standard treatment, or placebo is intuitively meaningful to clinicians, whereas a change of 0.8 SD units may not be. 

The number of degrees of freedom used with the chi-square distribution associated with the 2-dimensional distribution would be (N- 1)(M- l)-m, where m is the number of independent parameters estimated from the measurements. For the -dimensional case, the degrees of freedom used with the chi-square distribution would be (N1- )(/V, - I)... (Nf - I) - m, where m is the number of independent parameters estimated from the measurements. The steps used in the implementation of the chi-square test of independence are essentially the same as those listed for the chi-square test for goodness of fit. The only difference is that the expected values must be calculated for all NxM cases in the two-dimensional distribution and for all. .. Nk cases in the -dimensional distribution. The expected values for the cells are often arranged in a table that resembles the contingency table or are sometimes included, inside parentheses, within the same cell of the contingency table as the measurement. 

Once the set of trial probabilities has been calculated, the probabilities that are less than or equal to the probability of the measured contingency table are summed. They can be summed in two ways. The first and easiest is to find the sum of all the Ps in the set. This sum gives the probabilities at both extremes, those that are more extreme in the direction of the measured table, and those that are more extreme in the other direction. This will give the two-tail / -value and the test is termed a two-tailed test. The two-tail consideration describes the probability that a measured contingency table as far away from the expected contingency table as was the measured consistency table would occur. If this probability is less than or equal to the two-tail significance level, a, chosen for the study, then the null hypothesis of no effect is rejected otherwise, the null hypothesis is accepted. 

The hypergeometric distribution can be generalized to a multivariable form, the multivariate hypergeometric distribution, which can be used to extend Fisher s Exact Test to contingency tables larger than 2 by 2 and to multidimensional contingency tables. There is statistical software available to perform these calculations however, due to the complexity of the calculations and the large number of trial tables whose probability of occurrence must be calculated, this extension has received limited use. 

The financial appraisal is normally undertaken quite early in the life of the project, well before the final designs are complete, so a great deal of the cost calculation is based on the skill of the estimator, and much of it is done by factoring on known figures (such as how much a similar plant cost in another place or at another time). As it is an estimate, the final figure will prudently include a contingency allowance - the better the estimate is believed to be, the lower this contingency needs to be. 

It follows that the project can be safely recommended (remember there are 5% contingencies in both plant cost and production cost calculations), but that there may well be better ways to invest 84 millions. A lot will then depend on any other reasons that there might be to want to... 

For those commendably simple experiments that result in 2 x 2 contingency tables, some statistical packages include simple routines to calculate necessary sample size. For anything more complex, you are on your own - quite right too The routine will require you to provide values for the following ... 

Figure 16.3 Calculation of necessary sample size for a contingency chi-square test...

Many statistical packages provide a routine that will generate a 95 per cent Cl for the extent of any change in a proportion within a 2 x 2 table and this can be used to check for practical significance. For experiments that can be described in a 2 x 2 contingency table, it is simple to calculate necessary sample sizes. 

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