Negative binomial

The negative binomial distribution defines the probability of the k occurrence of an outcome occurring on the x trial as... 

Although a variety of probability distributions have been considered for contagious systems, the most successful appears to be the negative binomial. Here a distinguishing characteristic is that is greater than ji. 

For a negative binomial distribution an index of clumping must be incorporated, and Equation 2 becomes... 

The above treatment has implicitly assumed that the experimental design was such that the number of trials was fixed at 12 and the observation was the number of heads. However, an alternative design could have been to continue tossing the coin until 3 tails were obtained, and the observation would be n, the number of tosses required to produce the 3 tails. In this case, the statistic for judging the data is just n. But the distribution of n, the number of tosses to produce 3 tails, is given by the negative binomial ... 

The stationary distribution itself is a negative binomial or Polya distribution. There is no thermodynamic pe, since the system is subject to a continuous input. 

The Schulz-Zimm distribution is called by statisticians the negative binomial distribution see Ref. 99, p. 291... 

TB Whitaker, JW Dickens, RJ Monroe, EH Wiser. Comparison of the observed distribution of aflatoxin in shelled peanuts to the negative binomial distribution. J Am Oil Chem Soc 49 590-593, 1972. 

When the total number of reads per gene has been summarized, methods such as DESeq (50) evaluate DE using a negative binomial distribution. Comparing the relative abundance of different mRNA splice variants as meant before is a more challenging task. Methods such as cuffdiff (from Cufflinks) have been ad-hoc developed to handle testing for differences in alternative splicing variants between conditions. 

A key feature of a Poisson-distributed random variable is that it is completely described by one parameter, A. For the Poisson distribution, the variance is equal to the mean. However, clinical count data often can exhibit overdispersion, where the variance exceeds the mean. In this case, the variance of Y, Var(T), equals [Pg.702]

F.J. Anscombe, The Transformation of Poisson, Binomial and Negative-Binomial Data, Biometrika, 15 (1948), 246-254. 

Senn SJ, Collie G (1988) Accident blackspots and the bivariate negative binomial. Road Traffic Engineering and Control 29 168-169. 

Fig. 1. Relationship between incidence of hypoglycaemia (Confirmed minor and major events, exclnding symptoms only ) in the previons 12 weeks of the stndy and AlC at end point, as modelled nsing negative-binomial distribntion with a log-link function, (copyright 2006 American Diabetes Association from Diabetes Care , Vol. 29, 2006 1269-1274. Reprinted with permission from The American Diabetes Association).

Geometric Distribution The geometric distribution models the number of i.i.d. Bernoulli trials needed to obtain the first success. It is the simplest of the waiting time distributions, that is, the distribution of discrete time to an event, and is a special case of the negative binomial distribution (we will show this distribution in the next section). Here the number of required trials is the r.v. of interest. As an... 

Negative Binomial Distribution The binomial distribution counts the number of successes in a fixed number of Bernoulli trials, each with a probability p of success. Instead, suppose that the number of successes m is fixed in advance, and the random variable X is the number of trials up to and ineluding this /nth success. The random variable X is then said to have the negative binomial distribution. The probability distribution of X is found as follows. 

It is helpful to have standard probabihty models that are useful for analyzing large biological data, in particular bioinformatics. There are six standard distributions for discrete r.v. s, that is, BemouUi for binary r.v. s, (e.g., success or failure), binomial for the number of successes in n independent BemouUi trials with a common success probabihty p, uniform for model simations where aU integer outcomes have the same probabihty over an interval [a, b, geometric for the number of trials required to obtain the first success in a sequence of independent BemouUi trials with a common success probabihty p, Poisson used to model the number of occurrences of rare events, and negative binomial for the number of successes in a fixed number of Bemoulh trials, each with a probability p of success. 

Pascal s distribution (negative binomial distribution) The distribution of the number of independent Bernoulli trials performed up to and including the i success. The probability that the number of trials, x, is equal to fcis given by P x=k) = Cr ip q " ... 

Both a and /S are unknown and must be estimated from data. For unknown ju., Ni has a negative binomial distribution. Pooling all triggers allows the use of all available data, from which MLE for a and /S can be obtained. Maximum likelihood estimation is more computationally demanding, but it has been shown to provide better results than method of moments estimates (Elvik 2008). The MLE equations do not exist in closed form, but numerical methods can be employed to provide estimates. Estimates for a and P were obtained using the Newton-Raphson method with initial values ... 

Data are randomly generated from a negative binomial distribution parameterised by (a, 1). a takes values of. 1 and 10 to represent situations in which reahsations are either less or more frequent. 

Albers W (2010) The optimal choice of negative binomial charts for monitoring high-quality processes. J Statistical Plann Infer 140 214-225... 

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