Bayesian small area demography
Section 3. Interpolating and forecasting obesity prevalence

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3.1  The estimation problem

In New Zealand, as in most countries, obesity rates are rising. Public health researchers and policy makers monitor and forecast obesity prevalence, to assess the success, or otherwise, of obesity-reduction measures, and to gauge future demand for services.

The main source of data on obesity prevalence in New Zealand is the New Zealand Health Survey, a nationally-representative survey of around 19,000 people (Ministry of Health, 2013). Like most household surveys, it has a complex design, with stratification and clustering. Obesity is measured using body mass index (BMI). A person is defined as being obese if he or she has a BMI of 30 or higher.

Surveys were carried out in 1997, 2003, 2007, 2012, and 2013. We use data for all these years. Our objective is to obtain prevalence estimates for the period 1997-2013, including non-survey years, and then forecast for the period 2014-2023. Our estimates and forecasts are disaggregated into age groups 15-24, 25-34, 35-44, 45-54, 55-64, 65-74, and 75+.

3.2  The model

Our main input data are published estimates for the proportion of New Zealanders aged $a$ at time $t$ who are obese, which we denote $p_{at},$ and the published standard errors for the $p_{at},$ denoted $s_{at}.$ The $p_{at}$ are graphed in Figure 3.1.

When individual-level data are available, the standard Bayesian approach towards accounting for complex survey design is to include as many features of the design as possible in the estimation model (Gelman et al., 2014, Chapter 8). Chen, Wakefield and Lumely (2014) show, however, how the full individual-level approach can be approximated by an aggregate-level approach that starts from design based estimates such as $p_{at}$ and $s_{at}.$ Chen et al. (2014) assume that the design-based estimates are constructed so as to reflect all the important features of the survey design, and show how these estimates can be converted into a form suitable for inclusion in an aggregate-level model.

Applying the approach of Chen et al. (2014), we approximate the individual-level approach using a Binomial likelihood. We obtain counts of individuals with obesity $y_{at}$ and total counts of individuals $n_{at}$ by finding $y_{at}$ and $n_{at}$ such that $\frac{y_{at}}{n_{at}}\approx p_{at}$ and $\frac{y_{at}}{n_{at}}\left(1-\frac{y_{at}}{n_{at}}\right)\approx s_{at}^2.$ The likelihood is

$$y_{at}\sim \text{Binomial}\left(n_{at}\mathrm{,}{\gamma }_{at}\right)\mathrm{.}(3.1)$$

Here ${\gamma }_{at}$ is the super-population probability of obesity: the probability that a person aged $a$ at time $t$ is obese. Our objective is to estimate ${\gamma }_{at}$ for past years, including years without survey data, and to forecast ${\gamma }_{at}$ for future years.

Our prior model for ${\gamma }_{at}$ is

$$\text{logit}\left({\gamma }_{at}\right)\mathrm{<mo>=</mo>}{\beta }^0+{\beta }_a^{\text{age}}+{\beta }_t^{\text{time}}\mathrm{,}(3.2)$$

which includes age and time effects, but not an age-time interaction. We experimented with an age-time interaction, but found that its size was small enough to omit (results not shown).

As with the mortality model of Section 2, we use a local trend model for the age effect, though in the obesity case we do not have an infant covariate. The rationale for using a local trend model is, once again, to capture the correlations between neighbouring age groups. We also use the same prior for the intercept as we do in Section 2, a normal distribution with mean 0 and standard deviation 10.

We use a local trend model for time,

$${\beta }_t^{\text{time}}\mathrm{<mo>=</mo>}{\alpha }_t^{\text{time}}+u_t^{\text{time}}(3.3)$$

$${\alpha }_t^{\text{time}}\mathrm{<mo>=</mo>}{\alpha }_{t-1}^{\text{time}}+{\delta }_{t-1}^{\text{time}}+v_t^{\text{time}}(3.4)$$

$${\delta }_t^{\text{time}}\mathrm{<mo>=</mo>}{\delta }_{t-1}^{\text{time}}+w_t^{\text{time}}\mathrm{,}(3.5)$$

but with two different sets of assumptions about innovation terms $v_t^{\text{time}}$ and $w_t^{\text{time}}.$

In our first version, we assume that $v_t^{\text{time}}$ and $w_t^{\text{time}}$ are always very close to 0, which we implement by using extremely tight priors on the standard deviations for these terms. The standard deviations for both terms have $\text{half}-t$ priors with scales of 0.001. This version of the local trend model essentially fits a straight line through the data. Aside from assuming no change, this is perhaps the most common approach to forecasting future rates in epidemiology and demography. We refer to this model as the “straight line” model.

Our second version is a generalization of the first. Rather than assuming that $v_t^{\text{time}}$ and $w_t^{\text{time}}$ are always close to 0, we allow them to take values that imply year-on-year changes in obesity rates of a few percentage points. We do this by setting the scale of the prior for the standard deviation of $v_t^{\text{time}}$ to 0.05 and setting the scale of the prior for standard deviation of $w_t^{\text{time}}$ to 0.025. We use a larger scale for $v_t^{\text{time}}$ than for $w_t^{\text{time}}$ on the basis that levels change more rapidly than systematic trends. We refer to the model based on this version of the time effect as the “flexible” model.

We carry out the estimation using our package demest, with the same settings for burnin, production, chains, and thinning as for the mortality application.

3.3  Results

Figure 3.2 shows results based on the “straight line” model. Estimates for survey years are shown in red, and estimates and forecasts for the remaining years are shown in blue. As is conventional with forecasting, we use 80% credible intervals, rather than 95%.

Estimates for years with survey data are more precise than those for years without survey data, as we would expect. Estimates for years between surveys are more precise than those for forecasts. The differences in precision between estimates and forecasts are, nevertheless, small. Strong assumptions about linearity lead to precise forecasts.

Figure 3.3 shows results based on the “flexible” model. Point estimates and forecasts from the flexible model are indistinguishable from those of the straight line model. The level of uncertainty, however, is clearly different. Compared with the straight line model, there is a modest increase in uncertainty for years between surveys and a large increase in uncertainty for the forecast period, particularly in later years.

The flexible model, arguably, gives a better representation of knowledge about obesity trends in New Zealand than the linear model. The linear assumption is conventional, but does not have any strong theoretical basis. Over-reliance on the linear assumption can produce over-confidence. The flexible model illustrates the implications of weaker assumptions.

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