Stasinopoulos 2007 - Lice, CD4 and Claims Examples

Summary

Three demonstrations: (1) four count distributions on head-lice data (Sichel wins; Poisson is off by ~25,000 AIC points); (2) CD4 count vs age, where jointly smoothing mean and variance beats polynomials; (3) negative-binomial insurance claims, where term selection finds smooth effects in $\mu$ and covariates in $\sigma$. The authors stress these are capability demonstrations, “not to answer substantive questions in the data.”

Overview

Section 3.1 states the aims. Section 3.5 (head circumference) is in Stasinopoulos 2007 - Centile Curves, LMS and Residual Diagnostics.

Main Content

3.2 Lice data (Williams 1944)

Frequencies of 1,083 Hindu male prisoners (Cannamore, South India, 1937-39) by number of head lice $y$ (frequencies used as weights). Four intercept-only fits (head ~ 1):

Model	df	AIC
Sichel (SICHEL, mixed(10,50))	3	4646.198
Negative binomial I (NBI)	2	4653.687
Poisson inverse Gaussian (PIG)	2	4756.275
Poisson (PO)	1	29174.823

Sichel fit: $\hat{\mu }=e^{1.927}=6.869$, $\hat{\sigma }=e^{4.863}=129.4$, $\hat{\nu }=0.00079$ (SE 0.0157, near zero); global deviance 4640.198, SBC 4661.16. The profile deviance for $\nu$ gives a 95% CI $(-0.0853,0.0887)$ (Fig. 1). Interpretation: heavy overdispersion makes the Poisson hopeless; NBI is within 7.5 AIC of the best; $\nu \approx 0$ shows the third (shape) parameter adds little beyond NBI-like behaviour but AIC still favours it modestly.

3.3 CD4 data (Wade and Ades 1994)

CD4 counts of uninfected children born to HIV-1 mothers vs age (about 600 observations: 8 df plus 601 residual df). Three problems: (a) shape of $E(\text{cd4}\mid \text{age})$; (b) variance decreasing with age; (c) non-normality. Transforming $y$ and $x$ (Fig. 4: ${\text{cd4}}^{-0.5}$, $\log (\text{cd4}+0.1)$, ${\text{cd4}}^{0.5}$ against ${\text{age}}^{-0.5}$, $\log \text{age}$, ${\text{age}}^{0.5}$, age) does not fix all issues at once. GAMLSS addresses them one at a time:

Mean model (normal, constant $\sigma$)	Result
orthogonal polynomial, degree 7	best AIC and SBC but “wobbly” end (Fig. 5)
fractional polynomials fp(age, 3)	best by AIC/SBC; all three powers $=-2$: $557.5-599.3+1116.8{\text{age}}^{-2}+17776.2{\text{age}}^{-2}\log \text{age}+698.6{\text{age}}^{-2}(\log \text{age}{)}^2$; global dev. 8950.37, df 8; “unconvincing”
piecewise polynomial bs()	AIC prefers 7 df, SBC 5
cubic smoothing spline cs()	AIC: 10.85 extra df (13 total) overfits; SBC: 1.854 extra df (4 total) misses small ages

Jointly smoothing $\mu$ and $\log \sigma$ with cs() (m42 = AIC, m31 = SBC): extra df $(\mu ,\log \sigma )=(3.72,1.81)$ under AIC and $(2.55,0.93)$ under SBC. “By picking a suitable model for $\sigma$ the model for $\mu$ is less complicated.” Validation-deviance tuning (VGD()) gave $(4.78,1.38)$ — similar. The lesson: modelling the variance lets the mean be simpler.

3.4 Third-party insurance claims (de Jong and Heller 2007)

176 New South Wales local government areas, 1984-86: Claims, Accidents, KI (killed/injured), Population, Pop_density, statistical division SD (13 levels); analysis uses logs of the continuous variables (L_Population, L_Accidents, L_KI, L_Popdensity).

• Poisson vs NBI: deviance drop of 4606.169 for one extra df $\Rightarrow$ use NBI for all later steps.
• Global deviance 6487.73 (Poisson) vs ~1883.9-1881.56 (NBI) quoted in iterations.
• dropterm(): no $\mu$-term can be dropped; addterm() on two-way interactions: none needed (e.g. SD:L_Popdensity ${\chi }^2=14.10$, $p=0.294$; L_Popdensity:L_Accidents $p=0.574$).
• stepGAIC.CH() with gamlss.scope() finds smooth cs() terms for L_Popdensity, L_Accidents and L_Population but not L_KI.
• Next, linear terms for $\log \sigma$ (excluding SD): L_Population, L_KI, L_Accidents were selected under AIC; the authors judge the AIC model “over complicated” and suggest a higher GAIC penalty.

Examples

What GAMLSS buys: variance modelling on CD4

Model cd4 ~ cs(age, df=a), sigma.fo = ~cs(age, df=b) with $(a,b)=(3.72,1.81)$. At small ages both fitted $\mu$ and $\sigma$ are larger, and the SBC fit has “a slightly lower mean and standard deviation” than AIC at early ages. Predictive intervals computed as $\hat{\mu }(x)\pm z\hat{\sigma }(x)$ then widen correctly at young ages — which a constant-$\sigma$ polynomial would not do.

Connections

• Model comparison criteria: Stasinopoulos 2007 - Fitting Algorithms and Model Selection.
• Counts with overdispersion: Generalized Linear Models; for the Bayesian equivalent a negative binomial with $\alpha$ regressed on covariates.

See Also

Stasinopoulos 2007 - GAMLSS in R - Overview