Estimate the Shape Parameter of the Gamma Distribution in a GLM Fit

Find the maximum likelihood estimate of the shape parameter of the gamma distribution after fitting a Gamma generalized linear model.

# S3 method for glm
gamma.shape(object, it.lim = 10,
            eps.max = .Machine$double.eps^0.25, verbose = FALSE, ...)

Arguments

object	Fitted model object from a Gamma family or quasi family with variance = "mu^2".
it.lim	Upper limit on the number of iterations.
eps.max	Maximum discrepancy between approximations for the iteration process to continue.
verbose	If TRUE, causes successive iterations to be printed out. The initial estimate is taken from the deviance.
...	further arguments passed to or from other methods.

Value

List of two components

alpha

the maximum likelihood estimate

SE

the approximate standard error, the square-root of the reciprocal of the observed information.

Details

A glm fit for a Gamma family correctly calculates the maximum likelihood estimate of the mean parameters but provides only a crude estimate of the dispersion parameter. This function takes the results of the glm fit and solves the maximum likelihood equation for the reciprocal of the dispersion parameter, which is usually called the shape (or exponent) parameter.

References

Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S. Fourth edition. Springer.

See also

gamma.dispersion

Examples

clotting <- data.frame(
    u = c(5,10,15,20,30,40,60,80,100),
    lot1 = c(118,58,42,35,27,25,21,19,18),
    lot2 = c(69,35,26,21,18,16,13,12,12))
clot1 <- glm(lot1 ~ log(u), data = clotting, family = Gamma)
gamma.shape(clot1)
#>                
#> Alpha: 538.1315
#> SE:    253.5991

gm <- glm(Days + 0.1 ~ Age*Eth*Sex*Lrn,
          quasi(link=log, variance="mu^2"), quine,
          start = c(3, rep(0,31)))
gamma.shape(gm, verbose = TRUE)
#> Initial estimate: 1.060344
#> Iter. 1 Alpha: 1.238408
#> Iter. 2 Alpha: 1.276997
#> Iter. 3 Alpha: 1.278343
#> Iter. 4 Alpha: 1.278345
#>                 
#> Alpha: 1.2783449
#> SE:    0.1345175
summary(gm, dispersion = gamma.dispersion(gm))  # better summary
#> 
#> Call:
#> glm(formula = Days + 0.1 ~ Age * Eth * Sex * Lrn, family = quasi(link = log, 
#>     variance = "mu^2"), data = quine, start = c(3, rep(0, 31)))
#> 
#> Deviance Residuals: 
#>     Min       1Q   Median       3Q      Max  
#> -3.0385  -0.7164  -0.1532   0.3863   1.3087  
#> 
#> Coefficients: (4 not defined because of singularities)
#>                       Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)            3.06105    0.44223   6.922 4.46e-12 ***
#> AgeF1                 -0.61870    0.59331  -1.043 0.297041    
#> AgeF2                 -2.31911    0.98885  -2.345 0.019014 *  
#> AgeF3                 -0.37623    0.53149  -0.708 0.479020    
#> EthN                  -0.13789    0.62540  -0.220 0.825496    
#> SexM                  -0.48844    0.59331  -0.823 0.410369    
#> LrnSL                 -1.92965    0.98885  -1.951 0.051009 .  
#> AgeF1:EthN             0.10249    0.82338   0.124 0.900942    
#> AgeF2:EthN            -0.50874    1.39845  -0.364 0.716017    
#> AgeF3:EthN             0.06314    0.74584   0.085 0.932534    
#> AgeF1:SexM             0.40695    0.94847   0.429 0.667884    
#> AgeF2:SexM             3.06173    1.11626   2.743 0.006091 ** 
#> AgeF3:SexM             1.10841    0.74208   1.494 0.135267    
#> EthN:SexM             -0.74217    0.82338  -0.901 0.367394    
#> AgeF1:LrnSL            2.60967    1.10114   2.370 0.017789 *  
#> AgeF2:LrnSL            4.78434    1.36304   3.510 0.000448 ***
#> AgeF3:LrnSL                 NA         NA      NA       NA    
#> EthN:LrnSL             2.22936    1.39845   1.594 0.110899    
#> SexM:LrnSL             1.56531    1.18112   1.325 0.185077    
#> AgeF1:EthN:SexM       -0.30235    1.32176  -0.229 0.819065    
#> AgeF2:EthN:SexM        0.29742    1.57035   0.189 0.849780    
#> AgeF3:EthN:SexM        0.82215    1.03277   0.796 0.425995    
#> AgeF1:EthN:LrnSL      -3.50803    1.54655  -2.268 0.023311 *  
#> AgeF2:EthN:LrnSL      -3.33529    1.92481  -1.733 0.083133 .  
#> AgeF3:EthN:LrnSL            NA         NA      NA       NA    
#> AgeF1:SexM:LrnSL      -2.39791    1.51050  -1.587 0.112400    
#> AgeF2:SexM:LrnSL      -4.12161    1.60698  -2.565 0.010323 *  
#> AgeF3:SexM:LrnSL            NA         NA      NA       NA    
#> EthN:SexM:LrnSL       -0.15305    1.66253  -0.092 0.926653    
#> AgeF1:EthN:SexM:LrnSL  2.13480    2.08685   1.023 0.306317    
#> AgeF2:EthN:SexM:LrnSL  2.11886    2.27882   0.930 0.352473    
#> AgeF3:EthN:SexM:LrnSL       NA         NA      NA       NA    
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> (Dispersion parameter for quasi family taken to be 0.7822615)
#> 
#>     Null deviance: 190.40  on 145  degrees of freedom
#> Residual deviance: 128.36  on 118  degrees of freedom
#> AIC: NA
#> 
#> Number of Fisher Scoring iterations: 7
#>