Stratified Sampling and Representative Frequency Estimates

Why the Sampling Design Matters

An allele-frequency estimate describes a particular population, not simply the set of chromosomes that happened to be observed. If a study samples different population strata at different rates, the sample may contain a different mix of strata from the target population. A pooled frequency then reflects the sample’s mix, and may not represent the population mix of interest.

The remedy, when the strata and their proportions in the target population are known, is to estimate the frequency within each stratum and combine those estimates using the target population proportions. This is stratified estimation. The weights describe how much each stratum contributes to the target population; they are not determined by how many individuals the study happened to recruit from that stratum.

Definition: Suppose the target population is partitioned into \(H\) mutually exclusive and collectively exhaustive strata. Let \(W_h\) be the proportion of the target population in stratum \(h\), where \(W_h \geq 0\) and \(\sum_{h=1}^{H} W_h=1\). Let \(p_h\) be the frequency of the allele of interest in that stratum, and let \(\hat{p}_h\) be its estimate from the sample drawn from that stratum. The stratified estimate of the overall target-population allele frequency is \(\hat{p}_{\mathrm{str}}=\sum_{h=1}^{H} W_h\hat{p}_h\).

From Genotype Counts to a Weighted Estimate

For diploid individuals with complete genotype calls, each individual contributes two allele copies at the locus. Within stratum \(h\), if \(A_h\) copies carry the allele of interest among \(2n_h\) observed copies, the stratum estimate is \(\hat{p}_h=A_h/(2n_h)\). If the observed allele-copy denominator differs because of missing data or other design features, use the actual number of observed copies as the denominator, and ensure that the usable sample represents that stratum.

Once each \(\hat{p}_h\) has been calculated, multiply it by the corresponding target weight \(W_h\), then add the contributions. A stratum with a large target-population share can have substantial influence even if it was sampled less heavily. Conversely, a small stratum does not become a large part of the target merely because the study sampled many of its members.

The unweighted pooled estimate instead gives each observed allele copy equal weight. With complete diploid data, it is \(\hat{p}_{\mathrm{pool}}=\sum_h A_h/\sum_h(2n_h)=\sum_h(n_h/\sum_j n_j)\hat{p}_h\). Its implicit stratum weights are the sample proportions \(n_h/\sum_j n_j\). It therefore estimates the allele frequency for the sampled mixture, unless those sample proportions match the target weights or the stratum frequencies happen to make the difference immaterial.

Worked Example: A Large Sample From a Small Stratum

Suppose a hypothetical target population consists of 80% members of stratum A and 20% members of stratum B. A study samples 20 individuals from A and 80 from B. At a biallelic locus, the sampled individuals have 12 copies of the allele of interest among 40 copies in A, and 112 among 160 in B. Assume the strata are defined before sampling, all genotypes are correctly called, and sampling is representative within each stratum.

The stratum frequencies are \(\hat{p}_A=12/40=0.30\) and \(\hat{p}_B=112/160=0.70\). The target weights are \(W_A=0.80\) and \(W_B=0.20\), so the weighted estimate is

\hat{p}_{\mathrm{str}}=(0.80)(0.30)+(0.20)(0.70)=0.24+0.14=0.38.

Thus the estimated target-population frequency is 0.38, or 38%. The pooled estimate is \((12+112)/(40+160)=124/200=0.62\), or 62%. The pooled value is much higher because 80% of the sample came from B, although B represents only 20% of the target population. Neither calculation has an arithmetic error; they summarize different population mixtures.

Why the Weighted Estimator Targets the Overall Frequency

The target-population frequency is itself a weighted average of the stratum frequencies. To see this, imagine counting the allele copies in the target population. Stratum \(h\) contributes a fraction \(W_h\) of the population, and its allele frequency is \(p_h\). Its contribution to the overall frequency is therefore \(W_hp_h\). Adding the contributions from all strata gives the population quantity of interest:

p_{\mathrm{target}}=\sum_{h=1}^{H}W_hp_h.

If sampling within each stratum is representative, then each \(\hat{p}_h\) estimates \(p_h\). In particular, if \(E(\hat{p}_h)=p_h\), and the weights are fixed and known, linearity of expectation gives

E(\hat{p}_{\mathrm{str}}) =E\left(\sum_{h=1}^{H}W_h\hat{p}_h\right) =\sum_{h=1}^{H}W_hE(\hat{p}_h) =\sum_{h=1}^{H}W_hp_h =p_{\mathrm{target}}.

This establishes that the weighted estimator is unbiased under those conditions. The argument does not require the stratum estimates to have the same sample size. It does require the weights to refer to the target population and the stratum estimates to be appropriate for their respective strata.

Key condition: Weighting corrects a mismatch between the sample’s stratum proportions and the target population’s stratum proportions. It does not correct biased selection within a stratum, incorrect genotype calls, or weights that describe a different target population.

Worked Comparisons

Worked Example: Three Strata With Different Sampling Rates

In a hypothetical study, the target population is 50% stratum A, 30% stratum B, and 20% stratum C. The study samples 50, 20, and 10 individuals from the respective strata. Among the diploid allele copies, A has 40 copies of the allele among 100, B has 24 among 40, and C has 4 among 20. Assume representative sampling within each stratum and complete, accurate genotype calls.

The stratum estimates are \(\hat{p}_A=40/100=0.40\), \(\hat{p}_B=24/40=0.60\), and \(\hat{p}_C=4/20=0.20\). Applying the target-population weights gives

\hat{p}_{\mathrm{str}}=(0.50)(0.40)+(0.30)(0.60)+(0.20)(0.20) =0.20+0.18+0.04=0.42.

The estimated target frequency is 0.42, or 42%. In the sample, the total is 68 allele copies of interest among 160, so the pooled estimate is \(68/160=0.425\), or 42.5%. These estimates are close, but for different reasons: the pooled sample proportions are 62.5%, 25%, and 12.5%, which are not the target proportions. The closeness is a feature of these particular stratum estimates and weights, not a general guarantee that pooling will work.

Worked Example: Calculating Weights From Population Counts

Suppose a hypothetical target population contains 600 individuals in stratum A, 250 in B, and 150 in C, for a total of 1,000. A study samples 30 individuals from each stratum. The observed allele counts are 6 of 60 copies in A, 30 of 60 in B, and 48 of 60 in C. Assume sampling within each stratum is representative and genotype calls are complete and accurate.

First calculate the target weights from the population counts: \(W_A=600/1000=0.60\), \(W_B=250/1000=0.25\), and \(W_C=150/1000=0.15\). The stratum estimates are \(\hat{p}_A=6/60=0.10\), \(\hat{p}_B=30/60=0.50\), and \(\hat{p}_C=48/60=0.80\). Therefore,

\hat{p}_{\mathrm{str}}=(0.60)(0.10)+(0.25)(0.50)+(0.15)(0.80) =0.06+0.125+0.12=0.305.

The weighted estimate is 0.305, or 30.5%. Because the sample has 30 individuals per stratum, its pooled estimate gives each stratum equal weight: \((6+30+48)/(60+60+60)=84/180\approx0.467\), or about 46.7%. Equal sampling counts do not imply equal target-population proportions. The population counts, not the study’s allocation, determine the weights for this target.

Assumptions, Uncertainty, and Interpretation

A representative weighted estimate requires several distinct conditions. First, the strata must partition the target population: each target member belongs to one and only one stratum. Second, the weights must accurately describe the proportions in the population for which the estimate is intended. Third, within every stratum, the sampled individuals and their usable allele copies must represent that stratum. Finally, allele counts must use a consistent allele definition and valid genotype data. These are design and measurement requirements; the weighting formula cannot establish them by itself.

An overall estimate is specific to its target composition. If two studies have different target-population weights, their overall frequencies can differ even when their stratum-specific frequencies are identical. Reporting the stratum estimates alongside the weights makes this distinction visible and helps readers understand what the overall number summarizes.

Weighting also affects uncertainty. Under a simplified model in which the \(m_h\) sampled allele copies in stratum \(h\) are independent Bernoulli observations with frequency \(p_h\), the variance of the weighted estimate is

\operatorname{Var}(\hat{p}_{\mathrm{str}}) =\sum_{h=1}^{H}W_h^2\frac{p_h(1-p_h)}{m_h},

when samples are independent across strata. The squared weights matter: uncertainty in a heavily weighted stratum contributes more to the overall variance. In diploid data, the two allele copies from one individual need not be independent, so treating all \(2n_h\) copies as independent can give an incorrect standard error. Use a variance method that matches the individual-level sampling design and dependence structure; the weighted point estimate alone does not determine its standard error.

Common Mistakes

Using sample proportions as if they were target weights. The pooled estimate weights strata according to how many individuals were sampled. For a target-population estimate, use the target stratum proportions when they are known.
Weighting by the stratum sample size. Multiplying a stratum frequency by \(n_h\) and dividing by total sample size simply recreates the pooled estimate. That is appropriate only when the sample mix is the target mix, or when the resulting value otherwise coincides with the target estimate.
Assuming weighting repairs all sampling bias. A stratum can still be unrepresentative if some members are more likely to be sampled than others within it. Correct stratum proportions do not fix that problem.
Changing the target without changing the weights. Weights describing one population do not automatically produce an estimate for another population or for a different mixture of populations.
Confusing a weighted point estimate with its precision. The formula for the estimate does not justify a pooled-binomial standard error, particularly when allele copies within individuals may be dependent.

Key takeaway: Estimate the allele frequency within each prespecified stratum, then combine those estimates using the stratum proportions in the population you want to describe. The unweighted pooled estimate instead follows the sample’s composition and is representative only when its implicit weights suit the target.

Check Your Understanding

Answer each question by identifying the target population and the weights that apply to it.

A target population is 70% stratum A and 30% stratum B. The study samples equal numbers from both strata. Which weights should be used for the overall target-population estimate, and why?
For two strata, the target weights are 0.4 and 0.6, and the estimated frequencies are 0.2 and 0.5. Calculate the weighted estimate.
When does the unweighted pooled estimate use the same stratum weights as the target-population estimate?
Explain why weighting cannot correct selection bias among individuals within a stratum.
Why might treating the two allele copies from every diploid individual as independent lead to an inappropriate standard error?