Some Remarks on Sprop

Exact Hypergeometric Confidence Intervals for Proportion Estimates

In survey sampling on a finite population, a simple random sample is typically selected without replacement, in which case a hypergeometric distribution models the observation. A standard construction for the confidence interval is based on a Normal approximation of the proportion with plug-in estimates for proportion and respective variance.

In most scenarios, this strategy results in satisfactory properties. However, if p is close to 0 or 1, it is recommended to use the exact confidence interval based on the hypergeometrical distribution (Kauermann and Kuechenhoff 2010). The Wald-type interval has a coverage probability as low as n/N for any α (Wang 2015). Therefore, there is no guarantee for the interval to capture the true M with the desired confidence level if the sample is much smaller than the population (Wang 2015).

Implementation in samplingbook

The function samplingbook::Sprop() estimates the proportion out of samples either with or without consideration of finite population correction.

Parameters are

• m an optional non-negative integer for number of positive events,
• n an optional positive integer for sample size,
• N positive integer for population size. Default is N=Inf, which means calculations are carried out without finite population correction.

In case of finite population of size N is provided, different methods for calculating confidence intervals are provided

• approx Wald-type interval based on normal approximation (Agresti and Coull 1998), and
• exact based on hypergeometric distribution as described in more detail in this document.

Sprop(m=3, n = 10, N = 50, level = 0.95)
#> 
#> Sprop object: Sample proportion estimate
#> With finite population correction: N = 50 
#> 
#> Proportion estimate:  0.3 
#> Standard error:  0.1366 
#> 
#> 95% approximate confidence interval: 
#>  proportion: [0.0322,0.5678]
#>  number in population: [2,28]
#> 95% exact hypergeometric confidence interval: 
#>  proportion: [0.08,0.64]
#>  number in population: [4,32]

Exact Hypergeometric Confidence Intervals

We observe X = m, the number of sampled units having the characteristic of interest, where X ∼ Hyper(M, N, n), with

• N is the population size,
• M is the number of population units with characteristic of interest, and
• n is the given sample size.

The respective density, i.e. the probability of successes in a sample given M, N, n, is $$\Pr(X=m) = \frac{{M \choose m} {N-M \choose n-m}}{N \choose n}, \text{ with support }m \in \{\max(0,n+M-N), \min(M,n)\} $$

We want to estimate population proportion p = M/N, which is equivalent to estimating M, the total number of population units with some attribute of interest. Then, the boundaries for the exact confidence interval [L, U] can be derived as follows:

$$ \begin{aligned} \Pr(X \leq m) & = \sum_{x=0}^m \frac{{U \choose x} {N-U \choose n-x}}{N \choose n} = \alpha_1 \\ \Pr(X \geq m) & = \sum_{x=m}^n \frac{{L \choose x} {N-L \choose n-x}}{N \choose n} = \alpha_2,\\ & \text{with coverage constraint } \alpha_1 + \alpha_2 \leq \alpha \end{aligned} $$ For sake of simplicity, we assume symmetric confidence intervals, i.e α₁ = α₂ = α/2.

Some Details on the Implementation

The implementation of the exact confidence interval for proportion estimates uses the hypergeometric distribution function phyper(x, M, N-M, n). Note that the parametrization differs slightly from ours.

We search for the optimal confidence boundaries [L, U] that fulfill the requirements as defined in the equations above.

• Given known total population N, sample size n and number of successes in the sample m, we can define some feasibility boundaries for M:
  • Naturally, the smallest possible value is the observed number of successes M_min = m
  • The largest possible value equals the total number N minus negative observations in the sample, i.e. M_max = N − (n − m).
• Upper boundary U
  • Start with largest possible value for M, i.e. U_max = N − (n − m)
  • Then, decrease incrementally while the Pr (X ≤ m) < α/2, so that we find the largest possible value which still fulfills the equation
• Lower boundary L
  • Start with smallest possible value for M, i.e. L_min = m
  • Rewrite Pr (X ≥ m) = 1 − Pr (X ≤ m) = α/2 ⇔ Pr (X ≤ m) = 1 − α/2
  • Then, increase incrementally while the Pr (X ≤ m) ≥ 1 − α/2, so that we find the smallest possible value which still fulfills the equation