Random Number Statistical Test

l Random Number Generation (See Sect. 4.2) B.1.1 Chi-Square Method 

The Chi-Square method checks for uniformity by dividing a range F of uniformly distributed random numbers a into a series of k adjacent intervals as 

Agarwal, Simulation Studies of Recombination Kinetics and Spin Dynamics in Radiation Chemistry, Springer Theses, DOI 10.1007/978-3-319-06272-3, 

In this test the sample empirical cumulative distribution function (cdf) Sn (x) is compared with a reference probability distribution F (x) to assess the goodness-of-fit. For this test it is assumed that the sample cdf is asymptotically normally distributed. The reference cdf for a uniform distribution is known to be 

The strategy to test the random number generator involves (i) ranking the i.i.d variables from smallest to largest (ii) computing the parameters  

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Randomization means that the sequence of preparing experimental units, assigning treatments, miming tests, taking measurements, and so forth, is randomly deterrnined, based, for example, on numbers selected from a random number table. The total effect of the uncontrolled variables is thus lumped together into experimental error as unaccounted variabiUty. The more influential the effect of such uncontrolled variables, the larger the resulting experimental error, and the more imprecise the evaluations of the effects of the primary variables. Sometimes, when the uncontrolled variables can be measured, their effect can be removed from experimental error statistically. 

As with all of our Fourier transforms of real data, we test the statistical significance of the periods so revealed by generating appropriate sets of Markovian data, each datum consisting of a constant, a, plus a random number e., where the random number e. 

Because of the paucity of events, it is necessary to evaluate the probability for chance coincidence. Several statistical tests were made with random numbers, and the probability for chance coincidences ranges between 1 and 3 95. 

In the simplest case of parallel-group study, a group of patients presenting sequentially are randomized to one of two equally sized treatment groups, until a prospectively determined total number of patients has been recruited. All these patients are followed for a predetermined period of time, or until some end point is achieved. The database is quality assured and locked before the randomization code is broken. The patients are then sorted according to their treatment, the end point measurements are subjected to a statistical test and an interpretation of the effect (or absence thereof) of the drug is made. What could possibly go wrong ... 

We have recently developed a library implementing several of the parallel random number generators and statistical tests of them on the most widely available multiprocessor computers. Documentation and software are available at... 

However, a single statistical test is not adequate to verify the randomness of a sequence, because typical MCMC applications can be sensitive to various types of correlations. However, if the generator passes a wide variety of tests, then our confidence in its randomness increases. The tests suggested by Knuth [7] and those implemented in the DIEHARD package by Marsaglia [36] are a standard. Since there are many generators that pass these tests, there is no reason to consider one that is known to fail. (Of course, any generator will eventually fail most tests, so we always must state how many numbers were used in the test level of accuracy). 

The second type of test is to run an application that uses random numbers in a similar manner to your applications, but for which the exact answer is known. For statistical mechanical applications, the two-dimensional Ising model (a simple lattice spin model) is often used since the exact answer is known and it has a phase transition so one expects sensi-... 

For example, the null hypothesis may postulate the randomness of samples in a group of observations. If the null hypothesis, Hq, is rejected, the alternative hypothesis is accepted. Since with practical measurements, only a limited number of samples out of the population will be available, the statistical tests usually cannot be directly based on the Gaussian distribution, but have to be performed on distributions derived from the normal distribution. 

A source of random numbers is required by any Monte Carlo experiment. It is certainly possible, in principle, to produce numbers that are random in that they are the result of some random physical process such as radioactive decay, but such techniques are almost never used today. Instead one uses a mathematical relation that produces a sequence of numbers that will pass a specified battery of statistical tests. The numbers are not random in that their sequence is determined by the generator, but various statistical tests cannot distinguish them from random numbers. To be more specific we want a sequence of numbers / = 1,2,3,... that are uniform in the interval (0,1) and that are not seriously correlated. A possible sequence of statistical tests would examine uniformity of < in the unit interval, of 2i 2i+i in the unit square, of 3h 3i+u 31+2 in the unit cube, and so on until correlation behavior of a sufficient order (for the experiment in question) has been considered. 

An important problem when using statistical tests (Monte Carlo filtering) is to set the P value under which the effect studied is supposed to be statistically significant. Moreover, when a test is performed many times, the multiple test problem arises. In the case of model 1 (135 random input parameters), it is quite likely that, a non-negligible number of input parameters could be identified as important (under classic P values thresholds such as 0.05 or 0.01). In this case, some input parameters are know to be non-influential with no uncertainty, i.e. the solubility limit of the Sn in the near field when the output considered is the dose due to A criterion that has worked from a practical point of view is to take as such a limit the lowest P value associated to anon-influential input parameter. In average, for that appUcation such limiting P value was around 10 . ... 

It was stated in Section 2.3 that homoscedasticity is a fundamental assumption of LS. In many calibrations, measurement of the calibrators (that is, with the same nominal concentrations) are replicated, and therefore, in practical terms, homoscedasticity means that the variances of the measurements at all concentrations are equal. Nevertheless, there will be random variation among independent estimates of variance, especially those based on small numbers of degrees of freedom. It is useful therefore to have a statistical test for heterogeneity of variances. If they are found to be heterogeneous, the LS fit must be replaced by the weighted LS fit. Alternatively, in some circumstances a transformation of the data may stabilize the variance sufficiently to make the ordinary LS fit applicable. In instances where a constant relative standard deviation is a reasonable assumption, a log-transformation will achieve that. 

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