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Age and Variant Creutzfeldt-Jakob Disease

Peter Bacchetti*
*University of California, San Francisco, California, USA

Appendix (Online Only)

The relation that forms the basis for estimation of epidemic parameters is that the observed number of deaths at time j among persons aged a has a Poisson distribution with expected value

    _j

  Σ N_ija h_ija ,

  ⁱ⁼¹

where N_ija is the number of persons infected at time i and still alive and aged a at time j, h_ija is the risk for death from variant Creutzfeldt-Jakob disease (vCJD) at time j for someone of age a who was infected at time i and is still alive at time j (the vCJD death hazard), time and age are measured on the same discrete time scale (quarter-years are used here), and i = 1 is the earliest possible time of infection.   For the model with no influence of age on risk for infection, the N_ija are determined by the formula:

                         _{i-1                                                       j-1}

N_ija = P_i,a-j+i { Π  (1 - g_kσ)} (g_iσ) S_ija {Π (1 – h_im,a-j+m)}.                                     (1)

                    ^{k = max(1, j-a)                                          m = i}

P_i,a-j+i is the population aged a-j+i at time i (known from census data, available from: URL: www.statistics.gov.uk); the g_i define the shape of the infection hazard and are assumed known from studies of the BSE epidemic in cattle (1) and to equal zero after measures taken at the end of 1989 to keep infected beef out of the human food supply, σ is the scale parameter for the infection hazard, (g_iσ) is the infection hazard at time i, S_ija is the background survival probability that someone aged a-j+i at time i will live to time j (known from census data), and the final product in braces is the probability that a person infected at time i and age a-j+i will avoid death from vCJD up to time j.  The population multiplied by the first product in brackets is the number of uninfected persons in the age cohort up to time i, multiplying this by the infection hazard (g_iσ) gives the number infected at time I; and the final two terms convert this to the number of those who survive to time j.  The scale parameter σ, which determines total number infected, must be specified instead of estimated because for any choice of σ with corresponding N_ija and h_ija that fit the observed deaths well, a similar fit can be obtained from σ* = cσ with corresponding N_ija* and  h_ija* = h_ija/c for a constant c.  We focus here on scale choices that produce numbers of infections much larger than the number of observed deaths to date.  This focus ensures that the second product in brackets is close to one, so that the dependence of the N_ija on the h_ija can be ignored.  This allows estimation of the h_ija by using nonlinear Poisson regression, as performed by the SAS NLMIXED procedure (SAS Institute, Cary, NC).

A parametric proportional hazards model (2) for the risk for vCJD death as a function of time since infection, t = j – i, and age is defined by

log(h_ija) = log(h_ta) = f_t (t) + f_a (a)

in terms of current age, or by

log(h_ta) = f_t (t) + f_a (a-t)                                                   (2)

in terms of age at infection, a-t.  Because the N_ija and h_ija are always multiplied together, an influence of age at infection on the risk for death is indistinguishable from an influence of age on risk for infection for the cases considered here.  When both products in brackets in equation (1) are close to one, we can define age-specific infection scale factors
σ_a-t* = σ exp{ f_a (a-t)}.  The resulting N_ija* are then combined with age-independent incubation hazards defined by log(h_ta*) = log(h_t*) = f_t (t), producing the same fit as the above formulation with age at infection instead influencing the h_ta as specified by equation (2).  We therefore used equation (2) to estimate the influence of age on risk for infection. (Note that this nonidentifiability also holds for any other fixed covariate, such as sex.)

For the functions f_t and f_a, we evaluated polynomials and parametric splines (3), using the Akaike information criterion (4) for model selection, a criterion that requires an improvement of 2.0 in the model’s deviance (–2 times the log likelihood) to justify an additional parameter. Parametric splines may be preferable to polynomials in some cases because they can fit a wider variety of shapes (e.g., the asymmetric shapes for age effects shown in the Figure in the main text).  A quadratic polynomial appeared adequate for the influence of time since infection.  For the model of Figure b, the estimated effect of time since infection was f_t (t) = -34.2 + 0.639*t – 0.00673*t².  Assuming a linear form produced a deviance that was worse by 9.3.  A parametric spline with the same degrees of freedom produced the same deviance, and increasing to 3 degrees of freedom improved it by only 1.4.  For the model of Figure a, we obtained  f_t (t) = -31.7 + 0.795*t – 0.00848*t².  Parametric splines with 3 parameters appeared optimal for the influence of age on both risk for infection and risk for death.  For risk for death, 2 degrees of freedom had a deviance that was worse by 21.0, while 4 degrees of freedom improved the deviance by only 0.9.  A cubic polynomial for age had a deviance that was worse by 9.4. We calculated pointwise confidence intervals from the estimated covariance matrix of the three age parameters.

We assessed goodness-of-fit of the models by calculating both Pearson and deviance overdispersion (5), using data cross-classified into 7 time periods (1994–96, 1997, 1998, 1999, 2000, 2001, 2002) by 5 age groups (<20, 20–24, 25–29, 30–39, 40+). Both the Pearson and deviance estimates of overdispersion for the model of Figure b were below the null value of 1.0, indicating no apparent lack of fit. We tested the proportional hazards assumption for the influence of age by adding a term equal to the product of age and time since infection. The proportional hazards assumption in Figure b appeared viable, with virtually no improvement resulting from addition of this nonproportionality term (p = 0.96).

We compared the models of Figure a and b using log likelihoods, specifically the difference between the value obtained assuming that age influenced risk for infection versus the value obtained assuming it instead influenced risk for disease after infection. Because the models are not nested and have the same number of parameters, an ordinary likelihood ratio test is not possible. To obtain a p value, we created 2,000 simulated datasets under the model with age influencing risk for infection, fitted models to each one, assuming an influence of age on risk for infection only and again assuming an influence on risk for death only, and tallied how many times the simulated difference exceeded the observed difference.

References

1. Donnelly CA, Ferguson NM.  Statistical aspects of BSE and vCJD.  London: Chapman & Hall/CRC; 2000. p. 170.
2. Cox DR, Oakes D.  Analysis of survival data. London: Chapman & Hall; 1984. p. 23–4.
3. de Boor C. A practical guide to splines.  Berlin: Springer Verlag; 1978.
4. Akaike H.  Information theory and an extension of the maximum likelihood principle.  In: Petrov B, Csaki F, editors. Second International Symposium on Information Theory.  Budapest: Akademia Kiado; 1973.  p. 267–81.
5. McCullagh P, Nelder JA. Generalized linear models, 2nd  ed. London: Chapman & Hall; 1989. p. 120–128, 198–200.

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This page posted November 20, 2003
This page last reviewed April 13, 2004

Emerging Infectious Diseases Journal
National Center for Infectious Diseases
Centers for Disease Control and Prevention