Small Area Complications

• Some difficult issues:
  • Estimated counts can be small
  • Estimated counts can be zero
  • Margin of Error (MOE) estimates can be larger than estimates

Non-zero small-area estimates

• Estimated count
• Model-based estimate of MOE from observed count data

Zero small-area estimates

• Estimated zero count
• Model-based estimate of MOE from ten-year census

A sampling distribution

(When census estimates are over-dispersed)

• \(Pr[y|\alpha, \beta] = \binom{y + \alpha - 1}{\alpha - 1} \left(\frac{\beta}{\beta+1}\right)^\alpha\left(\frac{1}{\beta+1}\right)^y\)
• \(E[y] = \frac{\alpha}{\beta} = \mu\)
• \(Var[y] = \frac{\alpha}{\beta^2}\left(\beta + 1\right) = \sigma^2 \approx \left(\frac{MOE}{1.645}\right)^2\)[1]
• \(\beta = \frac{\mu}{\sigma^2 - \mu}\)
• \(\alpha = \frac{\mu^2}{\sigma^2 - \mu}\)
• Either:
  • Simulate a \(y_k\) for each sample in a larger model; or
  • Sum out \(y_k\) according to \(Pr[y_k|\alpha,\beta]\)

A sampling distribution

(When census estimates are not over-dispersed)

• Typically heavily populated tracts
• \(Pr[y|\mu, \sigma] = \frac{1}{\sigma\sqrt{2\pi}}e^\frac{(x-\mu)^2}{\sigma^2}\)
• \(E[y] = \mu \)
• \(Var[y] = \sigma^2 \approx \left(\frac{MOE}{1.645}\right)^2\)[1]
• Either:
  • Simulate a \(y_k\) for each sample in a larger model; or
  • Sum out \(y_k\) according to \(Pr[y_k|\mu, \sigma]\)

A sampling distribution

(When census estimates are zero)

• \(Pr[y|\lambda] = 1 - e^\frac{-y}{\lambda}\)
• \(Var[y] = \lambda^2 \approx \left(\frac{MOE}{1.645}\right)^2\)[1]
• \(\lambda = \frac{MOE}{1.645}\)
• Either:
  • Simulate a \(y_k\) for each sample in a larger model; or
  • Sum out \(y_k\) according to \(Pr[y_k|\lambda]\)

Conceptual match

• Sampling a larger population we are trying to estimate
• For non-zero estimates the MOE informs us about over-dispersion of the counts
• For zero estimate the MOE tells us how many we might expect to *try* to count before we count one successfully
• In all cases the MOE plus a single observation leave uncertainty about the distribution
• Further uncertainty comes from the sampling of the counts from the distribution

Helpful properties

• All sampled counts are positive integers
• Sampled counts should include observed census counts
• Calibration should match the calibration of census estimates

(Calibration is worth testing via simulation)

Downstream Estimates

• Populations are typically used as denominators in downstream estimates
• When the numerator, \(x\), is known precisely, the uncertainty in the population, \(N\) can be included directly by calculating \(Pr[x/N]\) from \(Pr[N]\)
• When the numerator should be estimated in the context of denominator uncertainty, there is no direct calculation.

Arbitrary discrete distribution

• Using the three models above calculate \(Pr[N_i = n]\) for all plausible \(n\).
• Drop estimates where \(Pr[N_i = n] \approx 0\)
• Pick up to \(K\) equally spaced points in \(N\) to represent the distribution
• Drop other points
• Re-normalize so that \(\sum_k Pr[N_i = n] = 1\)
• Each of the \(K\) points, \(\eta_k\), stands in for a larger set of similar values

Discrete-Poisson Mixture model

• \( Pr\left[m_i|\lambda, \vec{\eta}\right]=\sum_1^K \eta_k\times \frac{(\lambda_{i,k})^m_i \exp{\left(-\lambda_{i,k} \right)}}{m_i!} \)
• \( \lambda_{i,k} = \exp{\left(\beta_0 + \beta_{TRACT} + ... + \log 10^5 - \log n_k \right)} \)
• Scaling:
  • The main dataset has \(n_{units} \times n_{timepoints} \) rows
  • The dimension resulting from population uncertainty can be including using a second step