New Sampling Design Boosts Efficiency in Breast Cancer Data Estimates

Measurement is often the most expensive step in any scientific study. In medicine, agriculture and environmental monitoring, obtaining an accurate value for a quantity of interest can be slow, costly or even destructive, while placing a handful of units in rough order—by eye, or through a cheaply measured related variable—costs almost nothing. Ranked set sampling, a statistical idea dating back to work on estimating pasture yields in the 1950s, exploits precisely that imbalance. Small sets of units are ranked by judgment, only one unit from each set is actually measured, and the resulting observations are spread across the population in a way that makes every measurement carry more information than it would under simple random sampling. A new study published in Frontiers in Applied Mathematics and Statistics pushes this idea further with a design called truncation-based median ranked set sampling, or TBMRSS, and demonstrates it on real breast cancer diagnostic data.

The new design, proposed by Mohammed Obeidat of the Department of Statistics at Yarmouk University in Jordan, is built around a simple observation: different statistical targets reward different sampling strategies. When estimating a population mean, it can pay to spread the measured observations toward the tails of the distribution. When estimating a median, however, the extremes of a sample are the noisiest, least reliable order statistics, and a design that concentrates on the center tends to win. Existing schemes in the ranked set sampling family pull in one direction or the other—some measure only the extremes, some only the median, some a shifted diagonal that avoids the extremes—but none had anchored the sample on the median while also trimming the tails gently with a single adjustable coefficient.

TBMRSS works as follows. Instead of ranking sets of size m, the researcher draws m comparison sets, each containing m+2 units, and ranks them. A truncation coefficient k, which can range from zero up to half the set size, then controls the shape of the measured sample. The first k sets contribute the second-smallest unit, the last k sets contribute the second-largest unit, and the remaining central sets contribute the median—one central rank when the set size is odd, and two central ranks split equally when it is even. The design always measures m units per cycle, the same as ordinary ranked set sampling, so all schemes can be compared on an equal measurement budget. At k = 0 the design is purely median-based, measuring the median of every enlarged set; at its largest admissible value, k concentrates the sample at the second-smallest and second-largest ranks, a form of symmetric, mild tail trimming. Notably, at small set sizes the k = 1 member exactly reproduces an existing design called except-extreme ranked set sampling, making that scheme a boundary case of the new family.

The theoretical analysis establishes that the mean estimator is unbiased for any symmetric distribution, for both odd and even set sizes, and provides closed-form variances and efficiencies relative to simple random sampling for the uniform and exponential distributions, together with integral expressions for the normal. For the median estimator, the paper proves a striking result: its asymptotic relative efficiency is completely free of the parent distribution. This distribution-free property arises because the density of an order statistic evaluated at the population median takes a form involving only binomial coefficients, and the density factor cancels between the design and simple random sampling. The efficiency depends only on the set size and the truncation coefficient, and it is strictly decreasing in k—so the median-only setting, k = 0, is always asymptotically optimal for estimating the median, regardless of the shape of the underlying distribution.

Monte Carlo simulations, each based on 120,000 replications, compared TBMRSS against simple random sampling, ordinary ranked set sampling, except-extreme ranked set sampling, and the earlier truncation-based design of Al-Omari and Raqab across eight symmetric and skewed parent distributions and set sizes from three to eight. For the population mean, moderate truncation made TBMRSS the most efficient scheme for every parent distribution, by margins of 5 to 32 percent over the strongest existing competitors. The contrast with the older truncation-based design is especially sharp for heavy-tailed parents: for the exponential distribution at set size eight, the earlier scheme’s efficiency collapsed to 0.88—worse than simple random sampling—when its truncation coefficient grew, whereas TBMRSS reached an efficiency of 4.14, more than a fourfold gain over simple random sampling and roughly 40 percent above ordinary ranked set sampling.

The study is unusually candid about where the gains come from. Because TBMRSS uses enlarged sets of size m+2 while most competitors use sets of size m, part of its advantage over simple ranked set sampling comes simply from ranking more units, not from the selection rule itself. Using the except-extreme design as a control—which shares the enlarged sets but spreads its measured ranks along a shifted diagonal—the author decomposed the improvement into two components. For bell-shaped and mildly skewed parents, the enlargement accounts for the larger share, reaching 91 percent for the gamma distribution at one set size. But for the short-tailed uniform distribution and for the exponential at large set sizes, the selection rule itself contributes 70 to 85 percent of the gain, and at identical set size, sampling effort and ranking effort, TBMRSS beat the except-extreme design for every parent and every set size examined, by between 3 and 39 percent.

Choosing the truncation coefficient in practice matters, and the paper offers a concrete rule. For the median, no choice is needed: the distribution-free result makes k = 0 optimal for every continuous parent. For the mean, the best value depends on tail weight and skewness—moderate or heavy tails favor the median-only setting, short tails favor the largest admissible coefficient, and strong skewness favors an intermediate value. Fixing a single coefficient blindly can cost up to 61 percent of the attainable efficiency, but a rule based on the sample skewness and kurtosis of a modest pilot sample selected the optimal coefficient in 38 of 40 test cases. When no pilot information is available at all, the author recommends k = 1 for the mean, which is never the worst choice for any distribution studied.

Robustness to imperfect ranking—a real-world inevitability, since judgment rankings are typically made by eye or from a correlated auxiliary variable—was tested under a standard model in which each ranking is correct with probability p and random otherwise. Two results stand out. First, the mean estimator remains unbiased for symmetric parents at every ranking quality, and for skewed parents the bias is simply attenuated in proportion to the ranking quality. Second, unlike designs that measure raw extremes, TBMRSS is never worse than simple random sampling: over all distributions, coefficients and ranking qualities examined, its lowest observed efficiency was 0.996, and only for the uniform parent with the largest truncation. The older truncation-based scheme, by contrast, fell below simple random sampling for all eight parents when rankings were poor, dipping to 0.93 for the exponential.

The practical payoff is demonstrated on the Wisconsin Diagnostic Breast Cancer data, a public set of 569 fine-needle-aspirate samples, each described by 30 features of cell nuclei. The variable of interest is mean concave points, the most diagnostically powerful of the mean features with an area under the ROC curve of 0.96 for malignancy, but one that requires laborious tracing of nuclear boundaries. Ranking was performed instead on mean radius, a nucleus size measure that is cheap to judge and correlates with the target at 0.82—a strong but genuinely imperfect association. In this realistic setting, TBMRSS with moderate truncation was the most efficient scheme for the mean at every set size, reaching just over twice the efficiency of simple random sampling, while the median-centered version was best for the median at every set size, reaching 3.14 times simple random sampling and exceeding classical median ranked set sampling by about 10 percent.

The author also examines the extra cost honestly. The enlarged sets require ranking m(m+2) units per cycle rather than m squared, a relative increase of 2/m that is largest at small set sizes. Yet because one cycle of the new design with sets of size m+2 ranks exactly one unit fewer than a cycle of ordinary ranked set sampling with sets of size m+1, the comparison can be made at matched ranking effort—and the new design still wins for every parent and every set size, most decisively for heavy-tailed distributions. Under explicit cost accounting, the ordering holds for all cost ratios in every case except the light-tailed uniform parent at the smallest set size, where ordinary ranked set sampling becomes preferable once ranking a unit costs about two-fifths of measuring one. Taken together, the results suggest a design particularly well suited to biomedical research, where assays are expensive, distributions are skewed, human rankings are imperfect, and a single tunable coefficient can adapt the sampling scheme to whichever quantity—mean or median—a study needs to estimate.

Subject of Research: Truncation-based median ranked set sampling for efficient estimation of population means and medians, applied to breast cancer diagnostic data

Article Title: Truncation-based median ranked set sampling with applications to breast cancer data

Article References: Truncation-based median ranked set sampling with applications to breast cancer data. (n.d.). https://doi.org/10.3389/fams.2026.1924702

Image Credits: AI Generated

DOI: 10.3389/fams.2026.1924702

Keywords: ranked set sampling, median ranked set sampling, truncation, population mean, population median, relative efficiency, imperfect ranking, order statistics, Monte Carlo simulation, breast cancer data, statistical sampling design, unbiased estimation