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Using a rank-based design in estimating prevalence of breast cancer

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Abstract

It is highly important for governments and health organizations to monitor the prevalence of breast cancer as a leading source of cancer-related death among women. However, the accurate diagnosis of this disease is expensive, especially in developing countries. This article concerns a cost-efficient method for estimating prevalence of breast cancer, when diagnosis is based on a comprehensive biopsy procedure. Multistage ranked set sampling (MSRSS) is utilized to develop a proportion estimator. This design employs imprecise rankings based on some visually assessed cytological covariates, so as to provide the experimenter with a more informative sample. Theoretical properties of the proposed estimator are explored. Evidence from numerical studies is reported. The developed procedure can be substantially more efficient than its competitor in simple random sampling (SRS). In some situations, the proportion estimation in MSRSS needs around 76% fewer observations than that in SRS, given a precision level. Thus, using MSRSS may lead to a considerable reduction in cost with respect to SRS. In many medical studies, e.g., diagnosing breast cancer based on a full biopsy procedure, exact quantification is difficult (costly and/or time-consuming), but the potential sample units can be ranked fairly accurately without actual measurements. In this setup, multistage ranked set sampling is an appropriate design for developing cost-efficient statistical methods.

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Notes

  1. https://cran.r-project.org/web/packages/mlbench/index.html.

  2. It is accessible at http://lib.stat.cmu.edu/datasets/bodyfat.

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Acknowledgements

The authors are grateful to the Area Editor and the reviewer for careful reading of an earlier version of this manuscript and providing many constructive comments. Ehsan Zamanzade’s research was carried out in IPM Isfahan branch and was in part supported by a grant from IPM, Iran (No. 1400620422).

Author information

Authors and Affiliations

  1. Department of Statistics, Hakim Sabzevari University, P.O. Box 397, Sabzevar, Iran

    M. Mahdizadeh

  2. Department of Statistics, Faculty of Mathematics and Statistics, University of Isfahan, P.O. Box 81746-73441, Isfahan, Iran

    Ehsan Zamanzade

  3. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395–5746, Tehran, Iran

    Ehsan Zamanzade

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  1. M. Mahdizadeh
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Appendices

Appendix A

Proof of Proposition 1

(a) Let \(F_{[i]}(x)\) (\(i=1,\ldots ,m\)) be the common distribution function of judgment order statistics with rank i, i.e., \(X_{[i]1},\ldots ,X_{[i]n}\). If the ranking scheme is consistent, then we have

$$\begin{aligned} F(x)=\frac{1}{m} \sum _{i=1}^m F_{[i]}(x), \end{aligned}$$
(1)

where F(x) is the population distribution function (see Lemma 1 in Presnell and Bohn (1999) for details). The unbiasedness of \(\hat{p}_{\text {RSS}}\) is immediate from this identity. Also, the variance expression is obtained by noting that elements of \(\{X_{[i]j}: i=1,\ldots ,m\,;j=1,\ldots ,n \}\) are independent, and each \(X_{[i]j}\) is a Bernoulli random variable with the success probability \(p_{[i]}\).

(b) From the first part, one can write

$$\begin{aligned} Var\left( \hat{p}_{\text {RSS}} \right)= & {} \frac{1}{n m^2} \sum _{i=1}^m p_{[i]} \left( 1-p_{[i]} \right) \\= & {} \frac{1}{n m^2} \left[ \sum _{i=1}^m p_{[i]}-\sum _{i=1}^m p_{[i]}^2 \right] \\= & {} \frac{1}{n m^2} \left[ mp-\sum _{i=1}^m \left( p_{[i]}-p+p \right) ^2 \right] \\= & {} \frac{1}{n m^2} \left[ mp-mp^2-\sum _{i=1}^m \left( p_{[i]}-p \right) ^2 \right] \\= & {} \frac{p(1-p)}{n m}-\frac{1}{n m^2} \sum _{i=1}^m \left( p_{[i]}-p \right) ^2 \\\le & {} Var\left( \hat{p}_{\text {SRS}} \right) , \end{aligned}$$

where in the third equality, identity (1) has been used.

(c) It can be seen that

$$\begin{aligned} \hat{p}_{\text {RSS}}=\frac{1}{n}\left( \frac{1}{m}\sum _{i=1}^m X_{[i]1} +\cdots + \frac{1}{m}\sum _{i=1}^m X_{[i]n} \right) . \end{aligned}$$

To put it another way, \(\hat{p}_{\text {RSS}}\) is the sample mean of n independent and identically distributed random variables with mean p and variance \(\sum _{i=1}^m p_{[i]} \left( 1-p_{[i]} \right) /m^2\). The asymptotic normality is then concluded from the central limit theorem. \(\square \)

Proof of Proposition 2

(a) An identity similar to (1) holds in MSRSS (see Proposition 3 in Mahdizadeh and Zamanzade (2020b)). It establishes that

$$\begin{aligned} F(x)=\frac{1}{m} \sum _{i=1}^m F_{[i]}^{(r)}(x), \end{aligned}$$
(2)

where \(F_{[i]}^{(r)}(x)\) (\(i=1,\ldots ,m\)) is the common distribution function of \(X_{[i]1}^{(r)},\ldots ,X_{[i]n}^{(r)}\). An application of identity (2) shows the unbiasedness of \(\hat{p}_{\text {MSRSS}}^{(r)}\). The corresponding variance is derived similar to that of \(\hat{p}_{\text {RSS}}\).

(b) This is proved in line with part (b) of Proposition 1.

(c) Suppose \(\mathcal {X}_{(i)}^{(r-1)}\) (\(i=1,\ldots ,m\)) is the ith order statistic of \(X_{[1]1}^{(r-1)},\ldots ,X_{[m]1}^{(r-1)}\). Then, we have

$$\begin{aligned} Var\left( \hat{p}_{\text {MSRSS}}^{(r-1)} \right)= & {} \frac{1}{n m^2} Var\left( \sum _{i=1}^m X_{[i]1}^{(r-1)} \right) \nonumber \\= & {} \frac{1}{n m^2} Var\left( \sum _{i=1}^m \mathcal {X}_{(i)}^{(r-1)} \right) \nonumber \\= & {} \frac{1}{n m^2} \left[ \sum _{i=1}^m Var\left( \mathcal {X}_{(i)}^{(r-1)} \right) \right. \nonumber \\&+ \left. \sum _{i\ne j=1}^m Cov\left( \mathcal {X}_{(i)}^{(r-1)}, \mathcal {X}_{(j)}^{(r-1)} \right) \right] . \end{aligned}$$
(3)

It can be shown that the covariance terms in (3) are positive. To do so, without loss of generality, it is assumed that \(i<j\). Then, one can write

$$\begin{aligned} Cov\left( \mathcal {X}_{(i)}^{(r-1)}, \mathcal {X}_{(j)}^{(r-1)} \right)= & {} E\left( \mathcal {X}_{(i)}^{(r-1)} \mathcal {X}_{(j)}^{(r-1)} \right) \nonumber \\- & {} E\left( \mathcal {X}_{(i)}^{(r-1)} \right) E\left( \mathcal {X}_{(j)}^{(r-1)} \right) \nonumber \\= & {} P\left( \mathcal {X}_{(i)}^{(r-1)}=1 \right) \nonumber \\&-P\left( \mathcal {X}_{(i)}^{(r-1)}=1 \right) P\left( \mathcal {X}_{(j)}^{(r-1)}=1 \right) \nonumber \\= & {} p_{[i]}^{(r)} \left( 1-p_{[j]}^{(r)} \right) . \end{aligned}$$
(4)

Putting (3) and (4) together, it follows that

$$\begin{aligned} Var\left( \hat{p}_{\text {MSRSS}}^{(r-1)} \right)\ge & {} \frac{1}{n m^2} \sum _{i=1}^m Var\left( \mathcal {X}_{(i)}^{(r-1)} \right) \nonumber \\= & {} \frac{1}{n m^2} \sum _{i=1}^m Var\left( X_{[i]1}^{(r)} \right) \nonumber \\= & {} Var\left( \hat{p}_{\text {MSRSS}}^{(r)} \right) , \end{aligned}$$

where the first equality results from the fact that \(\mathcal {X}_{(i)}^{(r-1)}\) and \(X_{[i]1}^{(r)}\) are identically distributed according to MSRSS procedure.

d) Proof of asymptotic normality of \(\hat{p}_{\text {MSRSS}}^{(r)}\) parallels that of \(\hat{p}_{\text {RSS}}\), and it is omitted. \(\square \)

Appendix B

Suppose that \(m^{r+1}\) units are randomly identified from the population and are assigned labels \(1,\ldots ,m^{r+1}\). Also, the corresponding measured values for the covariate are stored in ”zcon vector. Then, we may use the following function to draw a single cycle of an rth stage ranked set sample using set size m. It returns a vector of length m, consisting of labels \(\ell _i\) (\(i=1,\ldots ,m\)). The final sample is given by \(X_{[1]1}^{(r)},\ldots , X_{[m]1}^{(r)}\), where \(X_{[i]1}^{(r)}\) is measurement of the variable of interest for the unit with label \(\ell _i\).

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Mahdizadeh, M., Zamanzade, E. Using a rank-based design in estimating prevalence of breast cancer. Soft Comput 26, 3161–3170 (2022). https://doi.org/10.1007/s00500-022-06770-0

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