Estimate theta of the Negative Binomial

Given the estimated mean vector, estimate theta of the Negative Binomial Distribution.

theta.md(y, mu, dfr, weights, limit = 20, eps = .Machine$double.eps^0.25)

theta.ml(y, mu, n, weights, limit = 10, eps = .Machine$double.eps^0.25,
         trace = FALSE)

theta.mm(y, mu, dfr, weights, limit = 10, eps = .Machine$double.eps^0.25)

Arguments

y	Vector of observed values from the Negative Binomial.
mu	Estimated mean vector.
n	Number of data points (defaults to the sum of weights)
dfr	Residual degrees of freedom (assuming theta known). For a weighted fit this is the sum of the weights minus the number of fitted parameters.
weights	Case weights. If missing, taken as 1.
limit	Limit on the number of iterations.
eps	Tolerance to determine convergence.
trace	logical: should iteration progress be printed?

Details

theta.md estimates by equating the deviance to the residual degrees of freedom, an analogue of a moment estimator.

theta.ml uses maximum likelihood.

theta.mm calculates the moment estimator of theta by equating the Pearson chi-square \(\sum (y-\mu)^2/(\mu+\mu^2/\theta)\) to the residual degrees of freedom.

Value

The required estimate of theta, as a scalar. For theta.ml, the standard error is given as attribute "SE".

See also

glm.nb

Examples

quine.nb <- glm.nb(Days ~ .^2, data = quine)
theta.md(quine$Days, fitted(quine.nb), dfr = df.residual(quine.nb))
#> [1] 1.135441
theta.ml(quine$Days, fitted(quine.nb))
#> [1] 1.603641
#> attr(,"SE")
#> [1] 0.2138379
theta.mm(quine$Days, fitted(quine.nb), dfr = df.residual(quine.nb))
#> [1] 1.562879

## weighted example
yeast <- data.frame(cbind(numbers = 0:5, fr = c(213, 128, 37, 18, 3, 1)))
fit <- glm.nb(numbers ~ 1, weights = fr, data = yeast)
summary(fit)
#> 
#> Call:
#> glm.nb(formula = numbers ~ 1, data = yeast, weights = fr, init.theta = 3.586087428, 
#>     link = log)
#> 
#> Deviance Residuals: 
#>       1        2        3        4        5        6  
#> -16.314    3.682    6.923    7.555    4.033    2.813  
#> 
#> Coefficients:
#>             Estimate Std. Error z value Pr(>|z|)    
#> (Intercept) -0.38199    0.06603  -5.785 7.25e-09 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> (Dispersion parameter for Negative Binomial(3.5861) family taken to be 1)
#> 
#>     Null deviance: 408.9  on 5  degrees of freedom
#> Residual deviance: 408.9  on 5  degrees of freedom
#> AIC: 897.06
#> 
#> Number of Fisher Scoring iterations: 1
#> 
#> 
#>               Theta:  3.59 
#>           Std. Err.:  1.75 
#> 
#>  2 x log-likelihood:  -893.063 
mu <- fitted(fit)
theta.md(yeast$numbers, mu, dfr = 399, weights = yeast$fr)
#> [1] 3.027079
theta.ml(yeast$numbers, mu, limit = 15, weights = yeast$fr)
#> [1] 3.586087
#> attr(,"SE")
#> [1] 1.749609
theta.mm(yeast$numbers, mu, dfr = 399, weights = yeast$fr)
#> [1] 3.549593