mle_bnb {depower} | R Documentation |
MLE for BNB
Description
Maximum likelihood estimates (MLE) for bivariate negative binomial outcomes.
Usage
mle_bnb_null(data, ratio_null = 1, method = "nlm_constrained", ...)
mle_bnb_alt(data, method = "nlm_constrained", ...)
Arguments
data |
(list) |
ratio_null |
(Scalar numeric: |
method |
(string: |
... |
Optional arguments passed to the optimization method. |
Details
These functions are primarily designed for speed in simulation. Missing values are silently excluded.
Suppose X_1 \mid G = g \sim \text{Poisson}(\mu g)
and
X_2 \mid G = g \sim \text{Poisson}(r \mu g)
where
G \sim \text{Gamma}(\theta, \theta^{-1})
is the random item (subject) effect.
Then X_1, X_2 \sim \text{BNB}(\mu, r, \theta)
is the joint distribution where
X_1
and X_2
are dependent (though conditionally independent),
X_1
is the count outcome for sample 1 of the items (subjects),
X_2
is the count outcome for sample 2 of the items (subjects),
\mu
is the conditional mean of sample 1, r
is the ratio of the
conditional means of sample 2 with respect to sample 1, and \theta
is
the gamma distribution shape parameter which controls the dispersion and the
correlation between sample 1 and 2.
The MLEs of r
and \mu
are \hat{r} = \frac{\bar{x}_2}{\bar{x}_1}
and \hat{\mu} = \bar{x}_1
. The MLE of \theta
is found by
maximizing the profile log-likelihood
l(\hat{r}, \hat{\mu}, \theta)
with respect to \theta
. When
r = r_{null}
is known, the MLE of \mu
is
\tilde{\mu} = \frac{\bar{x}_1 + \bar{x}_2}{1 + r_{null}}
and
\tilde{\theta}
is obtained by maximizing the profile log-likelihood
l(r_{null}, \tilde{\mu}, \theta)
with respect to \theta
.
The backend method for numerical optimization is controlled by argument
method
which refers to stats::nlm()
, stats::nlminb()
, or
stats::optim()
. If you would like to see warnings from the optimizer,
include argument warnings = TRUE
.
Value
For
mle_bnb_alt
, a list with the following elements:Slot Name Description 1 mean1
MLE for mean of sample 1. 2 mean2
MLE for mean of sample 2. 3 ratio
MLE for ratio of means. 4 dispersion
MLE for BNB dispersion. 5 nll
Minimum of negative log-likelihood. 6 nparams
Number of estimated parameters. 7 n1
Sample size of sample 1. 8 n2
Sample size of sample 2. 9 method
Method used for the results. 10 mle_method
Method used for optimization. 11 mle_code
Integer indicating why the optimization process terminated. 12 mle_message
Additional information from the optimizer. For
mle_bnb_null
, a list with the following elements:Slot Name Description 1 mean1
MLE for mean of sample 1. 2 mean2
MLE for mean of sample 2. 3 ratio_null
Population ratio of means assumed for null hypothesis. mean2 = mean1 * ratio_null
.4 dispersion
MLE for BNB dispersion. 5 nll
Minimum of negative log-likelihood. 6 nparams
Number of estimated parameters. 7 n1
Sample size of sample 1. 8 n2
Sample size of sample 2. 9 method
Method used for the results. 10 mle_method
Method used for optimization. 11 mle_code
Integer indicating why the optimization process terminated. 12 mle_message
Additional information from the optimizer.
References
Rettiganti M, Nagaraja HN (2012). “Power Analyses for Negative Binomial Models with Application to Multiple Sclerosis Clinical Trials.” Journal of Biopharmaceutical Statistics, 22(2), 237–259. ISSN 1054-3406, 1520-5711, doi:10.1080/10543406.2010.528105.
Aban IB, Cutter GR, Mavinga N (2009). “Inferences and power analysis concerning two negative binomial distributions with an application to MRI lesion counts data.” Computational Statistics & Data Analysis, 53(3), 820–833. ISSN 01679473, doi:10.1016/j.csda.2008.07.034.
See Also
Examples
#----------------------------------------------------------------------------
# mle_bnb() examples
#----------------------------------------------------------------------------
library(depower)
set.seed(1234)
d <- sim_bnb(
n = 40,
mean1 = 10,
ratio = 1.2,
dispersion = 2
)
mle_alt <- d |>
mle_bnb_alt()
mle_null <- d |>
mle_bnb_null()
mle_alt
mle_null