The NLMIXED Procedure

Example 82.4 Poisson-Normal Model with Count Data

This example uses the pump failure data of Gaver and O’Muircheartaigh (1987). The number of failures and the time of operation are recorded for 10 pumps. Each of the pumps is classified into one of two groups corresponding to either continuous or intermittent operation. The data are as follows:

data pump;
   input y t group;
   pump = _n_;
   logtstd = log(t) - 2.4564900;
   datalines;
 5  94.320 1
 1  15.720 2
 5  62.880 1
14 125.760 1
 3   5.240 2
19  31.440 1
 1   1.048 2
 1   1.048 2
 4   2.096 2
22  10.480 2
;

Each row denotes data for a single pump, and the variable logtstd contains the centered operation times.

Letting $y_{ij}$ denote the number of failures for the jth pump in the ith group, Draper (1996) considers the following hierarchical model for these data:

$\begin{align*} y_{ij} | \lambda _{ij} & \sim \mbox{Poisson}(\lambda _{ij}) \\ \log \lambda _{ij} & = \alpha _ i + \beta _ i(\log t_{ij} - \overline{\log t}) + e_{ij} \\ e_{ij}| \sigma ^2 & \sim \mbox{Normal}(0,\sigma ^2) \end{align*}$

The model specifies different intercepts and slopes for each group, and the random effect is a mechanism for accounting for overdispersion.

The corresponding PROC NLMIXED statements are as follows:

proc nlmixed data=pump;
   parms logsig 0 beta1 1 beta2 1 alpha1 1 alpha2 1;
   if (group = 1) then eta = alpha1 + beta1*logtstd + e;
   else eta = alpha2 + beta2*logtstd + e;
   lambda = exp(eta);
   model y ~ poisson(lambda);
   random e ~ normal(0,exp(2*logsig)) subject=pump;
   estimate 'alpha1-alpha2' alpha1-alpha2;
   estimate 'beta1-beta2' beta1-beta2;
run;

The selected output is as follows.

Output 82.4.1: Dimensions Table for Poisson-Normal Model

The NLMIXED Procedure

Dimensions
Observations Used	10
Observations Not Used	0
Total Observations	10
Subjects	10
Max Obs per Subject	1
Parameters	5
Quadrature Points	5

The "Dimensions" table indicates that data for 10 pumps are used with one observation for each (Output 82.4.1).

Output 82.4.2: Iteration History for Poisson-Normal Model

Iteration History
Iteration	Calls	Negative Log Likelihood	Difference	Maximum Gradient	Slope
1	4	30.6986932	2.162768	5.10725	-91.6020
2	9	30.0255468	0.673146	2.76174	-11.0489
3	12	29.7263250	0.299222	2.99040	-2.36048
4	16	28.7390263	0.987299	2.07443	-3.93678
5	18	28.3161933	0.422833	0.61253	-0.63084
6	21	28.0956400	0.220553	0.46216	-0.52684
7	24	28.0438024	0.051838	0.40505	-0.10018
8	27	28.0357134	0.008089	0.13506	-0.01875
9	30	28.0339250	0.001788	0.026279	-0.00514
10	33	28.0338744	0.000051	0.004020	-0.00012
11	36	28.0338727	1.681E-6	0.002864	-5.09E-6
12	39	28.0338724	3.199E-7	0.000147	-6.87E-7
13	42	28.0338724	2.532E-9	0.000017	-5.75E-9

NOTE: GCONV convergence criterion satisfied.

The "Iteration History" table indicates successful convergence in 13 iterations (Output 82.4.2).

Output 82.4.3: Fit Statistics for Poisson-Normal Model

Fit Statistics
-2 Log Likelihood	56.1
AIC (smaller is better)	66.1
AICC (smaller is better)	81.1
BIC (smaller is better)	67.6

The "Fit Statistics" table lists the final log likelihood and associated information criteria (Output 82.4.3).

Output 82.4.4: Parameter Estimates and Additional Estimates

Parameter Estimates
Parameter	Estimate	Standard Error	DF	t Value	Pr > \|t\|	95% Confidence Limits		Gradient
logsig	-0.3161	0.3213	9	-0.98	0.3508	-1.0429	0.4107	-0.00002
beta1	-0.4256	0.7473	9	-0.57	0.5829	-2.1162	1.2649	-0.00002
beta2	0.6097	0.3814	9	1.60	0.1443	-0.2530	1.4725	-1.61E-6
alpha1	2.9644	1.3826	9	2.14	0.0606	-0.1632	6.0921	-5.25E-6
alpha2	1.7992	0.5492	9	3.28	0.0096	0.5568	3.0415	-5.73E-6

Additional Estimates
Label	Estimate	Standard Error	DF	t Value	Pr > \|t\|	Alpha	Lower	Upper
alpha1-alpha2	1.1653	1.4855	9	0.78	0.4529	0.05	-2.1952	4.5257
beta1-beta2	-1.0354	0.8389	9	-1.23	0.2484	0.05	-2.9331	0.8623

The "Parameter Estimates" and "Additional Estimates" tables list the maximum likelihood estimates for each of the parameters and two differences (Output 82.4.4). The point estimates for the mean parameters agree fairly closely with the Bayesian posterior means reported by Draper (1996); however, the likelihood-based standard errors are roughly half the Bayesian posterior standard deviations. This is most likely due to the fact that the Bayesian standard deviations account for the uncertainty in estimating $\sigma ^2$ , whereas the likelihood values plug in its estimated value. This downward bias can be corrected somewhat by using the $t_9$ distribution shown here.