The many faces of the Negative Binomial variance function

Ecologists and evolutionary biologists often confront statistical models with overdispersed count data. One common strategy for dealing with overdispersed count data is the use of generalized linear models or generalized linear mixed models that implement a negative binomial (NB) error distribution. The typical interpretation of the NB is that it describes the number of failures before r successes in independent trials with a fixed probability of success in each trial. Alternatively, the NB can be derived as a hierarchical gamma-poisson process where the Poisson intensity (the probability of observing an event) itself follows a gamma distribution. This second formulation is perhaps more intuitive for biological analyses, but the two are equivocal.

Part of what makes the NB so useful for dealing with overdispersed data is that its variance function can be formulated several different ways to describe a variety of mean~variance relations. The well known NB1 & NB2 variance functions correspond to
NB1:     Var(x) = µ + θμ
NB2:     Var(x) = µ + θμ^2,
and allow for the analysis of both linear and quadratic mean~variance relations. Linden & Metanyahoo (2011) demonstrated that the variance function can be extended by incorporating a second estimable overdispersion parameter
Linden:  Var(x) =  ωµ + θμ^2.
This formulation provides more flexibility in describing data arising from processes with non-linear mean~variance relations, but reduces to NB2 ω=1.

As part of an analysis of pollinator count data, I was interested in applying this alternative version of the NB variance function, but could not find readily available code in SAS or R. Luckliy, while trolling SAS forums for advice and ideas, I stumbled on a post by Adam Smith from Dept. Nat. Resources University of Rhode Island, from 2011 looking to do the exact same thing. Neither of us appeared to get any satisfying resolution to our questions on the forums, so I took a long shot and emailed Adam to see if he ever figured it out. We ended up passing code back and forth (mostly he gave me code), and together came up with a working implementation for the Linden NB variance function in SAS. Unfortunately, it only appears to work in PROC NLMIXED, which is extremely inefficient for larger models, or models in which any variable selection process has to happen… but anyway, here is what we came up with, along with a sample dataset provided by Adam.

data counts;
   input ni @@;
   sub = _n_;
   do i=1 to ni;
      input x y @@;
      output;
   end;
   datalines;
  1 29 0
  6  2 0 82 5 33 0 15 2 35 0 79 0
 19 81 0 18 0 85 0 99 0 20 0 26 2 29 0 91 2 37 0 39 0  9 1 33 0
     3 0 60 0 87 2 80 0 75 0  3 0 63 1
  9 18 0 64 0 80 0  0 0 58 0  7 0 81 0 22 3 50 0
 15 91 0  2 1 14 0  5 2 27 1  8 1 95 0 76 0 62 0 26 2  9 0 72 1
    98 0 94 0 23 1
  2 34 0 95 0
 18 48 1  5 0 47 0 44 0 27 0 88 0 27 0 68 0 84 0 86 0 44 0 90 0
    63 0 27 0 47 0 25 0 72 0 62 1
 13 28 1 31 0 63 0 14 0 74 0 44 0 75 0 65 0 74 1 84 0 57 0 29 0
    41 0
  9 42 0  8 0 91 0 20 0 23 0 22 0 96 0 83 0 56 0
  3 64 0 64 1 15 0
  4  5 0 73 2 50 1 13 0
  2  0 0 41 0
 20 21 0 58 0  5 0 61 1 28 0 71 0 75 1 94 16 51 4 51 2 74 0  1 1
    34 0  7 0 11 0 60 3 31 0 75 0 62 0 54 1
  2 66 1 13 0
  5 83 7 98 1 11 1 28 0 18 0
 17 29 5 79 0 39 2 47 2 80 1 19 0 37 0 78 1 26 0 72 1  6 0 50 3
    50 4 97 0 37 2 51 0 45 0
 17 47 0 57 0 33 0 47 0  2 0 83 0 74 0 93 0 36 0 53 0 26 0 86 0
     6 0 17 0 30 0 70 1 99 0
  7 91 0 25 1 51 4 20 0 61 1 34 0 33 2
 14 60 0 87 0 94 0 29 0 41 0 78 0 50 0 37 0 15 0 39 0 22 0 82 0
    93 0  3 0
 16 68 0 26 1 19 0 60 1 93 3 65 0 16 0 79 0 14 0  3 1 90 0 28 3
    82 0 34 0 30 0 81 0
 19 48 3 48 1 43 2 54 0 45 9 53 0 14 0 92 5 21 1 20 0 73 0 99 0
    66 0 86 2 63 0 10 0 92 14 44 1 74 0
  8 34 1 44 0 62 0 21 0  7 0 17 0  0 2 49 0
 13 11 0 27 2 16 1 12 3 52 1 55 0  2 6 89 5 31 5 28 3 51 5 54 13
    64 0
  9  3 0 36 0 57 0 77 0 41 0 39 0 55 0 57 0 88 1
  7  2 0 80 0 41 1 20 0  2 0 27 0 40 0
 18 73 1 66 0 10 0 42 0 22 0 59 9 68 0 34 1 96 0 30 0 13 0 35 0
    51 2 47 0 60 1 55 4 83 3 38 0
 17 96 0 40 0 34 0 59 0 12 1 47 0 93 0 50 0 39 0 97 0 19 0 54 0
    11 0 29 0 70 2 87 0 47 0
 13 59 0 96 0 47 1 64 0 18 0 30 0 37 0 36 1 69 0 78 1 47 1 86 0
    88 0
 15 66 0 45 1 96 1 17 0 91 0  4 0 22 0  5 2 47 0 38 0 80 0  7 1
    38 1 33 0 52 0
 12 84 6 60 1 33 1 92 0 38 0  6 0 43 3 13 2 18 0 51 0 50 4 68 0
;
proc sort; by sub i; run;
proc print; run;

TITLE 'Negative binomial';
proc glimmix data=counts method=quad;
    class sub;
    model y = x / link = log s dist=nb;
    random int / subject=sub;
run;

title 'Linden parameterization';
proc nlmixed data=counts; 
*omega = 1;  *constrains model to quadratic mean-variance relationship, NB2;
             *omega starting value in parms statement should by greater than 1 when estimating
              Quasi-Poisson model to avoid execution error due to zero denominator;
*theta = 0;  *constrains model to linear mean-variance relationship, i.e., NB1, quasiPoisson;
*Starting fixed effect parameter estimates are the rounded coefficient estimates from the NB model;
*Starting value for omega should be > 1 (or at least not 1) to prevent dividing by 0 in calculation of r;
parms b_0=-1 b_1=0 omega=4 theta=1;
eta_lambda = b_0 + b_1*x + u;  /* subject level random intercept */
lambda = exp(eta_lambda);
r = lambda / (omega - 1 + theta*lambda);
p = r / (r + lambda);

loglike = lgamma(y+r) - lgamma(y+1) - lgamma(r) + r*log(p) +
          y*log(1-p);

/* fitting the model */ 
model y ~ general(loglike); 
random u ~ normal(0, exp(2*log_sdSUB)) subject=sub;
run;

/* REPLACING r,p, AND loglike WITH  THE FOLLOWING GIVES*/
/* THE EQUIVALENT GAMMA-POISSON FORMULATION */
alpha = linp / (omega - 1 + theta*linp);
beta = 1 / (omega - 1 + theta*linp);
loglike = lgamma(y+alpha) - lgamma(y+1) - lgamma(alpha) + 
          alpha*log(beta/(1+beta)) + y*log(1/(1+beta));
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