A Design-Based Mood’s Median Test for Complex Survey Data

1. Introduction

Large-scale epidemiological surveys, such as the World Health Organization (WHO) STEPwise approach to surveillance (STEPS), frequently collect continuous biomarker data, including fasting blood glucose and blood pressure. These variables are often characterized by heavy tails, skewness, and the presence of outliers, violating the normality assumptions required for traditional parametric tests. In such contexts, comparing the central tendency across groups (e.g., urban vs. rural) necessitates robust nonparametric approaches.

Mood’s Median Test is a classical alternative under simple random sampling [1]. However, its direct application to complex survey data is invalid, as it ignores sampling weights, clustering, and stratification. These features affect both point estimation and variance, leading to biased inference and inflated Type I error rates. To remedy this problem, we propose a design-based median comparison procedure that combines weighted quantile estimation with Rao–Scott adjustments for contingency tables. The resulting method is not a literal extension of the hypergeometric Mood’s test, but rather a design-based test of independence based on dichotomization around an estimated population median.

1.1. Motivating Example - WHO STEPS Surveys

The WHO STEPS is widely used to generate nationally representative data on non-communicable disease (NCD) risk factors in adult populations. STEPS surveys are typically designed as population-based cross-sectional studies employing stratified multistage probability sampling to ensure representativeness across demographic and geographic subgroups. The standardized STEPS methodology consists of three sequential components. Step 1 involves the collection of socio-demographic and behavioural information through interviewer-administered questionnaires, including tobacco use, alcohol consumption, diet, physical activity, and self-reported history of chronic conditions. Step 2 comprises physical measurements such as height, weight, waist circumference, and blood pressure, obtained using calibrated instruments and standardized protocols. Step 3 includes biochemical assessments based on blood samples, commonly measuring indicators such as fasting blood glucose and total cholesterol [2].

A key feature of STEPS surveys is the collection of continuous health measurements, including blood pressure, blood glucose, cholesterol, and anthropometric indicators. These variables are frequently characterized by skewed distributions, heavy tails, and the presence of outliers, reflecting the underlying heterogeneity in population health. Such distributional features challenge the assumptions of classical parametric methods, particularly when comparing groups. In many applications, interest lies in simple comparisons of central tendency—for example, differences in blood pressure between males and females or blood glucose levels between rural and urban populations. The median provides a natural and robust summary measure under these conditions. This motivates the use of non-parametric methods such as Mood’s median test. However, the standard formulation of the test assumes simple random sampling and does not account for the complex survey design features—such as clustering, stratification, and unequal weighting—that characterize STEPS data, thereby necessitating appropriate design-based extensions.

2. Methods

2.1. Mood’s Median Test

Under simple random sampling, let $Y_{gi}$ be the ith observation in group g, where $g=1,\dots ,G$ and $i=1,\dots ,n_g$. Let $N={\sum }_{g=1}^G{n}_g.$ Define the global sample median $\tilde{\theta }$ and construct a $2\times G$ table $O_{1g}={\sum }_{i=1}^{n_g}I\left(Y_{gi}>\tilde{\theta }\right),$ and $O_{2g}={\sum }_{i=1}^{n_g}I\left(Y_{gi}\le \tilde{\theta }\right).$

Under $H_0$, the counts follow a multiple hypergeometric distribution, and the Pearson statistic is

$${\chi }^2={\displaystyle \sum _{j=1}^2}{\displaystyle \sum _{g=1}^G}{\displaystyle \frac{{\left(O_{jg}-E_{jg}\right)}^2}{E_{jg}}}\sim {\chi }_{G-1}^2,$$

where $O_{1g}={\sum }_{i=1}^{n_g}I\left(Y_{gi}>\tilde{\theta }\right)$ and $O_{2g}={\sum }_{i=1}^{n_g}I\left(Y_{gi}\le \tilde{\theta }\right).$ Here, $\tilde{\theta }$ is the overall sample median and $I(·)$ is the indicator function. $E_{jg}$ denotes the expected frequency under the null hypothesis of independence, given by

$$E_{jg}={\displaystyle \frac{\left({\sum }_{g=1}^G{O}_{jg}\right)\left({\sum }_{j=1}^2{O}_{jg}\right)}{N}},$$

and $N={\sum }_{g=1}^G{n}_g$ is the total sample size.

2.2. Mood’s Median Test for Complex Surveys

We extend this framework to complex surveys by replacing counts with estimated population totals and using design-based variance estimation. Let $w_{ki}$ denote the sampling weight for individual i in PSU k. The design-consistent estimator of the population median $\theta$ satisfies

$${\displaystyle \frac{{\sum }_k{\sum }_i{w}_{ki}I(Y_{ki}\le \theta )}{{\sum }_k{\sum }_i{w}_{ki}}}=0.5.$$

Because $\theta$ is estimated from the data, subsequent statistics must account for its sampling variability. In particular, the indicator functions $I(Y_{ki}\le \theta )$ are non-smooth functionals, and their variance is obtained using Taylor linearization based on the influence function of the weighted median, or alternatively via replicate-weight methods (e.g., jackknife).

Define estimated population totals as $N_{1g}={\sum }_{k\in S_g}{\sum }_i{w}_{ki}I(Y_{ki}>\theta ),$ and $N_{2g}={\sum }_{k\in S_g}{\sum }_i{w}_{ki}I(Y_{ki}\le \theta ).$ Let $N_{++}={\sum }_k{\sum }_i{w}_{ki},$$p_{jg}=N_{jg}/N_{++},$ and $p_{j·}$ and $p_{·g}$ are defined similarly. The design-based Pearson statistic is

$$X_{\mathrm{survey}}^2=N_{++}{\displaystyle \sum _{j=1}^2}{\displaystyle \sum _{g=1}^G}{\displaystyle \frac{{\left(p_{jg}-p_{j·}{p}_{·g}\right)}^2}{p_{j·}{p}_{·g}}}.$$

This formulation uses estimated population totals rather than sample counts, ensuring design consistency.

2.3. Variance Estimation and Rao–Scott Adjustment

Under complex sampling, $X_{\mathrm{survey}}^2$ does not follow a ${\chi }^2$ distribution. Instead, it follows a $X_{\mathrm{survey}}^2\sim {\sum }_{m=1}^{\nu }{\delta }_m{\chi }_1^2,$ where ${\delta }_m$ are eigenvalues derived from the design-based covariance matrix of $p_{jg}$. This covariance matrix must incorporate

• clustering and stratification,
• unequal weights,
• variability induced by estimating $\theta$.

It is obtained using Taylor linearization applied jointly to the weighted proportions and the median-defining indicators, or via replicate-weight variance estimation.

The first-order Rao–Scott correction is given by

$$X_{RS}^2={\displaystyle \frac{X_{\mathrm{survey}}^2}{\overline{\delta }}},$$

where $\overline{\delta }$ is the mean eigenvalue (average design effect).

2.4. Second-Order Rao–Scott (Satterthwaite) Adjustment

Let

$$\nu =G-1.$$

Define

$$a^2={\displaystyle \frac{{\sum }_{m=1}^{\nu }{\delta }_m^2}{\nu {\overline{\delta }}^2}}-1.$$

The effective degrees of freedom is given by

$${\nu }^{*}={\displaystyle \frac{\nu }{1+a^2}}.$$

The Satterthwaite-adjusted statistic is

$$F_{\mathrm{Satt}}={\displaystyle \frac{X_{\mathrm{survey}}^2}{\nu \overline{\delta }}}.$$

We compare $F_{\mathrm{Satt}}$ to an F distribution with ${\nu }^{*}$ numerator degrees of freedom and ${\nu }^{*}(f-L)$ denominator degrees of freedom, where f is the number of PSUs and L is the number of strata.

2.5. Practical Implementation

The proposed method can be implemented through a sequence of straightforward steps that mirror standard practice in design-based survey analysis.

First, the population median is estimated using the survey weights. This step ensures that the threshold used to divide the data reflects the target population rather than the sample alone. Because the data arise from a complex sampling design, this median should be computed using a design-consistent estimator. Next, each observation is classified relative to the estimated median. Specifically, individuals are grouped according to whether their outcome value falls above or below (or equal to) the estimated population median. This creates a binary variable that captures the relative position of each observation within the overall distribution.

The data are then cross-classified by this median-based indicator and the grouping variable of interest (e.g., sex, residence, or exposure status). A design-based Pearson-type statistic is subsequently computed to assess the association between group membership and median classification. Because the data arise from a complex survey, this statistic does not follow the usual chi-square distribution. Therefore, a Rao–Scott adjustment is applied to correct both the scale of the statistic and its reference distribution, accounting for clustering, stratification, and unequal weights.

2.6. Software Implementation

All steps of the proposed procedure can be implemented in the survey package in R. The weighted median, contingency table, and Rao–Scott adjusted test are supported by standard functions.

3. Results

We used data from the 2024 WHO STEPS survey on non-communicable disease risk factors conducted in Trinidad and Tobago to show the application of this proposes approach. The survey was carried out between May and August 2024 as a nationally representative, population-based cross-sectional study of adults aged 18–69 years. A stratified multistage probability sampling design was employed to ensure representativeness of the target population across the country.

The STEPS survey followed the standardized three-step framework. Step 1 consisted of interviewer-administered questionnaires used to collect information on socio-demographic characteristics and behavioural risk factors, including tobacco use, alcohol consumption, diet, physical activity, and prior diagnoses of hypertension and diabetes. Step 2 involved physical measurements, including height, weight, waist circumference, and blood pressure, obtained using standardized procedures and calibrated instruments, with multiple blood pressure readings recorded per participant. Step 3 comprised biochemical assessments based on fasting venous blood samples, including measurements of blood glucose and total cholesterol.

A total of 4,052 adults participated in the survey, corresponding to an overall response rate of 75.0%. For the present analysis, we focused on comparing blood glucose levels between males and females. The analytical sample was restricted to participants with observed blood glucose measurements. Observations with missing blood glucose values were excluded. The final sample ($n=981$) therefore consisted of individuals with complete data required for the analysis.

Figure 1 shows the weighted median between males and females. The figure shows that the weighted medians for males and females are very similar, indicating little difference in the central tendency of the outcome between the two groups. The interquartile ranges (IQRs) also largely overlap, suggesting comparable variability. Overall, the visual evidence points to minimal differences in the distribution of the outcome by sex at the population level.

The weighted contingency table (Table 1) shows a near balance between males and females above and below the estimated population median. The estimated total above the median is slightly higher for males than females, while the reverse is observed for those below or equal to the median. However, these differences are relatively small in magnitude, suggesting no substantial shift in the distribution of the outcome variable by sex at the population level.

The Rao–Scott adjusted test indicates no statistically significant association between sex and the median-based classification of the outcome variable ($F=2.7594$, $df=1,86$, $p=0.1003$). This suggests that the distribution of observations above and below the estimated population median does not differ meaningfully between males and females. The result is consistent with the descriptive findings from the weighted medians and contingency table, both of which indicated minimal differences between the two groups.

4. Discussion

The proposed design-based extension of Mood’s median test provides a simple and robust approach for comparing medians across groups in complex survey data. By combining weighted median estimation with Rao–Scott adjusted tests of independence, the method accommodates unequal sampling weights, clustering, and stratification while retaining the intuitive interpretation of the classical Mood’s median test. Unlike standard applications of the test, which assume simple random sampling, the proposed approach produces inference that is appropriate for complex survey designs [3,4].

Application of the method to the 2024 Trinidad and Tobago WHO STEPS survey demonstrated its practical utility. The weighted medians of blood glucose were similar for males and females, and the weighted contingency table showed only minor differences in the proportions of individuals classified above and below the estimated population median. Consistent with these descriptive findings, the Rao–Scott adjusted test indicated no statistically significant association between sex and median-based classification ($F=2.7594$, $p=0.1003$). These results suggest that blood glucose levels were comparable between males and females at the population level.

The proposed procedure is particularly useful when survey outcomes are skewed, heavy-tailed, or contain influential outliers, situations in which mean-based comparisons may be difficult to interpret. Because the method relies on median classification, it provides a robust alternative that is easily implemented using standard survey-analysis software. The approach may therefore be useful in epidemiological surveys, health examination studies, and other population-based investigations where design-based inference is required.

5. Conclusions

This article proposed a design-based extension of Mood’s median test for use with complex survey data. The method replaces sample counts with weighted population totals, incorporates survey design features through design-based estimation, and applies Rao–Scott adjustments to obtain valid statistical inference. Application to WHO STEPS survey data demonstrated that the procedure is straightforward to implement and provides an intuitive approach for comparing medians across groups when outcome variables are skewed or contain outliers. The proposed method offers a practical nonparametric alternative for epidemiological and public health surveys where conventional median tests are not appropriate because of complex sampling designs.