# Fit a parametric disperison model to thinned counts

### Description

Fit a parametric dispersion model to RNA-Seq counts data
prepared by `prepare.nbp`

. The model parameters
are estimated from the pseudo counts: thinned/down-sampled
counts that have the same effective library size.

### Usage

1 | ```
estimate.disp(obj, model = "NBQ", print.level = 1, ...)
``` |

### Arguments

`obj` |
output from |

`model` |
a string, one of "NBQ" (default), "NBP" or "NB2". |

`print.level` |
a number, controls the amount of messages printed: 0 for suppressing all messages, 1 for basic progress messages, larger values for more detailed messages. |

`...` |
additional parameters controlling the estimation of the parameters. |

### Details

For each individual gene *i*, a negative binomial (NB)
distribution uses a dispersion parameter *φ_i* to
capture the extra-Poisson variation between biological
replicates: the NB model imposes a mean-variance
relationship *σ_i^2 = μ_i + φ_i μ_i^2*. In
many RNA-Seq data sets, the dispersion parameter
*φ_i* tends to vary with the mean *μ_i*. We
proposed to capture the dispersion-mean dependence using
parametric models.

With this function, `estimate.disp`

, users can choose
from three parametric models: NB2, NBP and NBQ (default).

Under the NB2 model, the dispersion parameter is a constant and does not vary with the mean expression levels.

Under the NBP model, the log dispersion is modeled as a
linear function of preliminarily estimated log mean
relative frequencies (`pi.pre`

):

log(phi) = par[1] + par[2] * log(pi.pre/pi.offset),

Under the NBQ model, the log dispersion is modeled as a
quadratic function of preliminarily estimated log mean
relative frequencies (`pi.pre`

):

log(phi) = par[1] + par[2] * log(pi.pre/pi.offset) + par[3] * (log(pi.pre/pi.offset))^2;

The NBQ model is more flexible than the NBP and NB2 models, and is the current default option.

In the NBP and NBQ models, `pi.offset`

is fixed to be
1e-4, so par[1] corresponds to the dispersion level when
the relative mean frequency is 100 reads per million (RPM).

The dispersion parameters are estimated from the pseudo counts (counts adjusted to have the same effective library sizes). The parameters are estimated by maximizing the log conditional likelihood of the model parameters given the row sums. The log conditional likelihood is computed for each gene in each treatment group and then summed over genes and treatment groups.

### Value

The list `obj`

from the input with some added
components summarizing the fitted dispersion model. Users
can print and plot the output to see brief summaries of the
fitted dispersion model. The output is otherwise not
intended for use by end users directly.

### Note

Users should call `prepare.nbp`

before calling
this function. The function `prepare.nbp`

will
normalize the counts and adjust the counts so that the
effective library sizes are approximately the same
(computing the conditional likelihood requires the library
sizes to be the same).

### References

Di Y, Schafer DW, Cumbie JS, and Chang JH (2011): "The NBP Negative Binomial Model for Assessing Differential Gene Expression from RNA-Seq", Statistical Applications in Genetics and Molecular Biology, 10 (1).

### See Also

`nbp.test`

, `exact.nb.test`

### Examples

1 | ```
## See the example for nb.exact.test
``` |

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