Author: Mary Gregg, Somnath Datta, Douglas Lorenz

# htestClust: a package for marginal inference of clustered data under informative cluster size

#### Mary Gregg, Somnath Datta, Douglas Lorenz

    library(htestClust)

Note: This document is abridged from a longer article published in R Journal.

# Abstract

When observations are collected in/organized into observation units, within which observations may be dependent, those observational units are often referred to as “clusters” and the data as “clustered data”. Examples of clustered data include repeated measures or hierarchical shared association (e.g., individuals within families).This paper provides an overview of the R package htestClust, a tool for the marginal analysis of such clustered data with potentially informative cluster and/or group sizes. Contained in htestClust are clustered data analogues to the following classical hypothesis tests: rank-sum, signed rank, t-, one-way ANOVA, F, Levene, Pearson/Spearman/Kendall correlation, proportion, goodness-of-fit, independence, and McNemar. Additional functions allow users to visualize and test for informative cluster size. This package has an easy-to-use interface mimicking that of classical hypothesis-testing functions in the R environment. Various features of this package are illustrated through simple examples.

# Introduction

Observations often occur or can be organized into units called clusters, within which those observations may be dependent. For example, individuals may be repeatedly assessed or naturally belong to some hierarchical structure like a family unit. Potential correlation among intra-cluster observations clearly invalidates the use of classical hypothesis tests for the analysis of such data. Instead, inference is generally performed using model-based methods that capture intracluster relationships through parametric or semi-parametric assumptions. Generalized estimating equations (GEEs) are one such approach that fit marginal generalized linear models to clustered data while making a working assumption on the correlation structure. GEE models are appealing for their flexible and robust nature, and several packages in the R environment, such as gee (Carey, Lumley, and Ripley 2019) and geepack (Halekoh et al. 2006), offer an implementation of this method. However, GEEs and other standard methods for analysis of clustered data operate under an assumption that the number of observations within the clusters (defined as the cluster size) is ignorable. In practice, this assumption may not hold and cluster size may vary systematically in a way that carries information related to the response of interest. When this occurs data are said to have informative cluster size (ICS). Examples of ICS can be found in data related to dental health (Williamson, Datta, and Satten 2003), pregnancy studies (Chaurasia, Liu, and Albert 2018), and longitudinal rehabilitation (Lorenz, Datta, and Harkema 2011), among others. For data with ICS, standard model-based methods can produce biased inference as their estimates may be overweighted in favor of larger clusters.

A related but distinct type of informativeness occurs when the distribution of group-defining covariates varies in a way that carries information on the response. Such phenomenon has been called informative within-cluster group size (IWCGS), as well as informative covariate structure (Pavlou 2012), sub-cluster covariate informativeness (Lorenz, Levy, and Datta 2018), and informative intra-cluster group size (Dutta and Datta 2016). This additional informativeness may occur simultaneously with or separately from ICS, and similarly can result in the failure of standard methods to maintain appropriate nominal size (Huang and Leroux 2011; Dutta and Datta 2016).

Williamson, Datta, and Satten (2003) developed a reweighting methodology that corrects for potential bias from cluster- or group-size informativeness. This reweighting originates from a Monte Carlo resampling process, and leads to weighting observations proportional to their inverse cluster or within-cluster group size. Correction for ICS/IWCGS was originally proposed in the context of modeling, and a number of extensions to this application have been established (Bible, Beck, and Datta 2016; Iosif and Sampson 2014; Mitani, Kaye, and Nelson 2019, 2020). However, when adjustment for covariates is not of interest, this reweighting can be directly applied in the estimation of marginal parameters. Under mild conditions, such estimates are asymptotically normal, permitting Wald-type intervals and tests. This methodology has been applied to develop rank-based tests (Datta and Satten 2005, datta08, dutta16), and tests of correlation (Lorenz, Datta, and Harkema 2011), proportions (Gregg, Datta, and Lorenz 2020), means and variances (Gregg 2020). This collection of reweighted non-model-based hypothesis tests includes clustered data analogues of the following classical tests: rank-sum, signed rank, t-, one-way ANOVA, F, Levene, Pearson/Spearman/Kendall correlation, proportion, goodness-of-fit, independence, and McNemar.

These clustered data analogues to standard hypothesis tests provide simple and intuitive means of performing exploratory and preliminary analysis of clustered data in which the cluster and/or group size varies and is potentially informative. However, many of these tests are recent developments that are not available in a software environment. We address this deficiency through the package htestClust, the first R package designed as a comprehensive collection of direct, non-model-based inferential methods for analysis of clustered data with potential ICS and/or IWCGS. Within htestClust are implimented the collection of methods by Datta and Satten (2005), Datta and Satten (2008), Dutta and Datta (2016), Lorenz, Datta, and Harkema (2011), Gregg, Datta, and Lorenz (2020) and Gregg (2020), as well as a method by Nevalainen, Oja, and Datta (2017) that tests for the presence of informative cluster size. The syntax and output of functions contained in htestClust are intentionally modeled after their corresponding analogous classical function, allowing researchers to assess various marginal analyses through intuitive and user-friendly means. We point the interested reader to the original citations for details on the testing theory. The rest of this document is devoted to an overview of the htestClust package, with a description of the features and structure of the provided functions, along with various illustrations.

# Overview of htestClust

htestClust includes ten functions for conducting different hypothesis tests under ICS, one function for visualizing informativeness in cluster size, and a simulated hypothetical data set to illustrate the use of the functions. With the exception of the test of informative cluster size, each of the hypothesis testing functions implemented in htestClust has a well-known analogue test for i.i.d. data. Table 1 contains a summary of the hypothesis testing functions contained in htestClust and their analogue i.i.d. function.

htestClust function | Reweighted test(s) | Classical analogue function
--- | --- | ---
chisqtestClust() | Chi squared goodness of fit, independence | chisq.test()
cortestClust() | Correlation | cor.test()
icstestClust() | Test of ICS | NA
levenetestClust() | K-group test of variance | leveneTest()
mcnemartestClust() | Homogeneity | mcnemar.test()
onewaytestClust() | K-group mean equality | oneway.test()
proptestClust() | Proportion | prop.test()
ttestClust() | Test of means (one/two group, paired) | t.test()
vartestClust() | 2-group test of variance | var.test()
wilcoxtestClust() | Rank sum, signed rank | wilcox.test()

Table 1. Hypothesis testing functions available in the htestClust package. Each row gives the name of a htestClust function, the reweighted test the function performs, and the R function that executes the corresponding classical analogue test. All classical analogue functions are available in R through the stats package, except for leveneTest(), which is included in the car package.

## Package functions, syntax, and output

The syntax and output of the functions in htestClust are designed to conform with that of the analogous i.i.d. functions from the R stats library. A notable but necessary departure from this correspondence is that the htestClust functions require as input (1) a variable identifying the clusters as an argument in the data set or (2) a cluster-level summary of the data. As an example, consider the syntax for the stats and htestClust functions for conducting a test of a single proportion:

    ## syntax for *stats* function
        prop.test(x, n, p = NULL, alternative = c("two.sided", "less", "greater"),
        conf.level = 0.95, correct = TRUE)

    ## syntax for *htestClust* function
        proptestClust(x, id, p = NULL, alternative = c("two.sided", "less",
        "greater"), variance = c("sand.null", "sand.est", "emp", "MoM"),
        conf.level = 0.95)

The stats library function prop.test does not operate on variables in a data frame, but instead takes summary counts as its input. Argument x can be a scalar representing the number of binomial successes, whence n is required as the number of binomial trials. Alternatively, x can be a one-dimensional table or matrix with two entries, whence n is omitted. The remaining arguments customize the test in ways familiar to most users.

The function proptestClust from htestClust operates on binary variables in a data frame or on cluster-level summary counts. In this function, x may be a binary variable measured over clusters, wherein id is required as a vector of cluster identifiers. Alternatively, x may instead be a two-dimensional table of within-cluster counts of failures and successes, wherein id is omitted. As previously noted, several options are available for variance estimation; these may be selected by the user through the variance argument. Additional customization of the test is as in prop.test.

Each of the testing functions in htestClust has been constructed in this vein – parallel to the analogous stats function with contingencies necessary for clustered data. htestClust functions accept vector input that designates the response, grouping (if necessary), and clustering variables. However, for convenience, many functions are designed with a secondary interface accepting tables or formulas. Like their stats package analogues, htestClust testing functions produce list objects of class htest for which the print method behaves in the usual way.

The function icsPlot provides a simple method for illustrating informative cluster size, providing a visual supplement to the results of the test of ICS, performed through the function icstestClust. Briefly, icsPlot plots a within-cluster summary statistic of a variable, such as a mean, against the size of each cluster. For quantitative variables, icsPlot produces a scatterplot of a within-cluster measure of location (mean, median) or variation (SD, variance, IQR, range) against cluster size. For a categorical variable, a barplot of within-cluster proportions is produced.

## Simulated example data set

htestClust includes a simulated data set named screen8 of clustered observations with informativeness, created under a hypothetical scenario we briefly describe here. A large school district has conducted a voluntary comprehensive exit survey for students graduating elementary school, collecting demographic, biometric, and academic performance data. The clustering mechanism for these data are the schools, with students comprising the observations within clusters.

The school district has offered an incentive program to boost participation, wherein schools having higher participation rates are rewarded with priority status for classroom and technology upgrades for the new academic year. This incentive introduces the potential for ICS – resource-poor schools may exhibit greater participation (larger cluster sizes) but also tend to have students with poorer health metrics and standardized test scores.

screen8 contains data from 2224 students from 73 schools in this district. Cluster sizes – the number of students participating in the exit survey at each school – ranged from 17 to 50, with a median of 30. The first few lines of the data are printed below, followed by the tabulated number of partcipants from each school and a summary of the cluster sizes.

    library(htestClust)
    data(screen8)
    head(screen8)
    #>   sch.id stud.id age gender height weight math read phq2 qfit qfit.s activity
    #> 1      1       1  15      M     65    136   69   75    3   Q2     Q2    other
    #> 2      1       2  14      M     66    135   80   57    2   Q4     Q3    other
    #> 3      1       3  15      M     65    146   60   85    0   Q2     Q3   sports
    #> 4      1       4  15      M     68    156   70   83    1   Q3     Q2    other
    #> 5      1       5  15      M     68    170   66   60    1   Q2     Q2   sports
    #> 6      1       6  14      M     63    109   84   62    0   Q1     Q1 academic

    (tab <- table(screen8$sch.id))
    #>
    #>  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
    #> 35 32 26 33 23 25 27 21 39 28 32 38 35 24 29 27 36 29 38 39 25 30 36 29 46 27
    #> 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52
    #> 30 26 28 40 50 40 28 38 40 25 30 34 18 24 30 21 30 31 23 18 44 29 34 39 23 41
    #> 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73
    #> 31 31 40 41 28 20 24 24 22 17 32 21 21 35 34 27 37 27 36 18 35
    summary(as.vector(tab))
    #>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
    #>   17.00   25.00   30.00   30.47   36.00   50.00

Table 2 provides details on the variables in the data set.

Variable | Description
--- | ---
sch.id | School identification variable
stud.id | Student identification variable within school
age | Student age in years
gender | Student gender
height | Student height in inches
weight | Student weight in lbs
math | Student score on standardized math test
read | Student score on standardized reading test
phq2 | Ordinal (0-6) score from a mental health screening. Higher scores correspond to higher levels of depression
qfit | Age-adjusted fitness quartile from physical health assessment taken at end of school year
qfit.s | Age-adjusted fitness quartile from physical health assessment taken at beginning of school year
activity | Student after-school activity

Table 2. Variables in screen8 data set. Each row gives the name of a variable included in the screen8 data set and its associated description.

# Examples

In this section, we demonstrate usage of the functions in htestClust using the screen8 data set. Our illustration is not comprehensive, but users can learn more about functions not covered here by browsing the associated help files. To motivate the demonstration, we’ll investigate the following questions:

1. Is the proportion of students having ``proficient’’ standardized math test scores (65 or greater) more than 0.75?
2. Are participation in extracurricular activity and gender independent?
3. Are mean standardized math test scores different between male and female students?
4. Are mean standardized reading test scores different among groups defined by extracurricular activities?

## Evaluating informative cluster size

Before addressing these questions, we illustrate how to assess the potential informativeness of cluster size in the data set, starting by visualizing ICS through the icsPlot function. The arguments to icsPlot specify the variable of interest, a cluster-identifying variable, and a summary function to be applied to the variable within each cluster. This summary can be any of obs, mean, median, var, IQR, range, prop, producing plots of the observations themselves, measure of location, or measures of variation against cluster size. Option prop can only be used when the variable of interest is a factor, so numerically coded categorical variables must be converted to factors. Standard R graphical parameters can also be specified when calling icsPlot().

    ### Figure 1
    par(mfrow = c(1,2))
    icsPlot(x = screen8$math, id = screen8$sch.id, FUN = "mean", pch = 20)
    icsPlot(x = screen8$read, id = screen8$sch.id, FUN = "mean", pch = 20)

    ### Figure 2
    par(mfrow = c(1,2))
    icsPlot(x = screen8$gender, id = screen8$sch.id, FUN = "prop", ylab = "P(Female)", pch = 20)
    icsPlot(x = screen8$activity, id = screen8$sch.id, FUN = "prop")

Figures 1 and 2 show potential informativeness in cluster size for the screen8 data. Cluster size appears to be negatively associated with average standardized test scores but positively associated with the proportion of male students and the proportion participating in sports-related extracurricular activities.

These empirical results can be verified using the test for ICS, implemented through the function icstestClust, as illustrated below. The result of this test suggests that cluster size is informative for standardized math test scores. Cluster size is also informative for standardized reading test scores, gender, and sports as an extracurricular activity (p < .001, results not shown).

    ## example code to perform test for ICS (not run due to computational time)
    set.seed(100)
    ics.math <- icstestClust(screen8$math, screen8$sch.id, B = 1000, print.it = FALSE)

    ics.math
    Test of informative cluster size (TF)
    data:  screen8$math
    TF = 0.029686, p-value < 2.2e-16

Within the icstestClust function, the type of test statistic, TF or TCM, is specified using the test.method argument. It is suggested to use TCM when there is a small number of distinct cluster sizes, as it tends to be more powerful. TF is preferred when the number of distinct cluster sizes is large and the number of clusters with those sizes is small, as TCM tends to be too liberal. Readers are pointed to Nevalainen, Oja, and Datta (2017) for more details. The number of bootstrap loops is specified by argument B, while argument print.it is a logical indicating whether to print the progress of the bootstrap procedure. We note that the need for bootstrap resampling in icstestClust can make its implementation computationally expensive.

## Testing a marginal proportion

The first question of interest suggests a one-sample test of a proportion via proptestClust. We specify a one-sided alternative and use the default sandwich variance estimator evaluated at the null value of the proportion (variance = “sand.null”), shown to perform best for this test (Gregg, Datta, and Lorenz 2020).

    screen8$math.p <- 1*(screen8$math >= 65)
    proptestClust(screen8$math.p, screen8$sch.id, p = .75, alternative = "great")
    #>
    #>  Cluster-weighted proportion test with variance est: sand.null
    #>
    #> data:  screen8$math.p, M = 73
    #> z = 0.70159, p-value = 0.2415
    #> alternative hypothesis: true p is greater than 0.75
    #> 95 percent confidence interval:
    #>  0.7311459 1.0000000
    #> sample estimates:
    #> Cluster-weighted proportion
    #>                   0.7640235

As noted previously, htestClust functions produce objects of class htest, producing familiar output through the print method for such objects. We conclude that the proportion of students with proficient math test scores is not greater than 0.75.

## Independence test

The second question suggests a test of independence of extracurricular activity and gender. We start by producing cluster-weighted estimates of the proportion of students participating in each activity within each gender.

    tab <- table(screen8$gender, screen8$activity, screen8$sch.id)
    ptab <- prop.table(tab, c(1,3))
    apply(ptab, c(1,2), mean)
    #>
    #>      academic     other    sports
    #>   F 0.3952102 0.2968473 0.3079425
    #>   M 0.3790267 0.3186699 0.3023035

The cluster-weighted proportions appear roughly similar, and we can test using chisqtestClust. Here, the default method of variance estimation is method of moments (variance = “MoM”), demonstrated to be best for the test of independence (Gregg, Datta, and Lorenz 2020).

    chisqtestClust(screen8$gender, screen8$activity, screen8$sch.id)
    #>
    #>  Cluster-weighted Chi-squared test of independence with variance est:
    #>  MoM
    #>
    #> data:  screen8$gender and screen8$activity, M = 73
    #> X-squared = 1.6131, df = 2, p-value = 0.4464

Before proceeding to the next analysis, we note that further evidence of ICS in the screen8 data can be demonstrated by implementing the standard chi-squared test for this question, which suggests that females were more likely to participate in academic extracurricular activities and males in sports.

    prop.table(table(screen8$gender, screen8$activity), 1)
    #>
    #>      academic     other    sports
    #>   F 0.3891323 0.2979842 0.3128834
    #>   M 0.3370268 0.3120960 0.3508772

## Tests of quantitative variables for two or more groups

We compare math test scores between males and females using the ttestClust function. We conclude that mean standardized test scores are equivalent between males and females, a departure from the conclusion reached by the standard t test (p < .001, results not shown).

    ttestClust(math ~ gender, id = sch.id, data = screen8)
    #>
    #>  Two sample group-weighted test of means
    #>
    #> data:  math by gender, M = 73
    #> z = 1.3495, p-value = 0.1772
    #> alternative hypothesis: true difference in means is not equal to 0
    #> 95 percent confidence interval:
    #>  -0.2234259  1.2111344
    #> sample estimates:
    #> weighted mean in group F weighted mean in group M
    #>                 70.75124                 70.25739

Even though this test does not make use of the \(t\) distribution, we have named it as such to parallel the standard \(t\)-test means (t.test in R). Multi-group tests of quantitative parameters in htestClust necessarily implement jackknife variance estimation, so specification of variance estimation method is not necessary. In addition to the formula implementation used above, we note that ttestClust can also accept vectors of data and cluster identifiers for each of the two groups.

An alternative approach to this comparison, particularly if test scores were skewed in any way, would be a rank-based test. wilcoxtestClust implements the group-weighted analogue of the Wilcoxon test, which we use as an alternative method for the comparison of math test scores between males and females.

    ## code to run group-weighted Wilcoxon test analogue (not run due to computational time)
    wilcoxtestClust(math ~ gender, id = sch.id, data = screen8, method = "group")

Our conclusion is the same as with the reweighted test of means. We note that this test requires estimation of the cluster-weighted empirical cumulative distribution (Dutta and Datta 2016) as well as jackknife variance estimation, so there is an added measure of computational expense in using wilcoxtestClust.

Finally, we compare reading test scores among the three groups defined by extracurricular activity, using onewaytestClust. Mean standardized reading test scores are not appreciably different among extracurricular activity groups.

    onewaytestClust(read ~ activity, id = sch.id, data = screen8)
    #>
    #>  Reweighted one-way analysis of means for clustered data
    #>
    #> data:  read and activity, M = 73
    #> X-squared = 1.3191, df = 2, p-value = 0.5171
    #> sample estimates:
    #> academic    other   sports
    #> 60.11498 60.40785 59.69659

# Discussion

Standard model-based inference of clustered data can be biased when cluster or group size is informative. Reweighting methods that correct for this bias have been established and a number of authors have applied such weighting to develop direct hypothesis tests of marginal parameters in clustered data. Such tests can be interpreted as clustered analogues to common classical statistical tests, and include methods related to ranks, correlation, proportions, means and variances. While these methods are effective and intuitive, all but a few of these tests have remained inaccessible to many researchers due to an absence of convenient software.

In this vignette we summarized htestClust, which is the first R package designed as a comprehensive library of inferential methods appropriate for clustered data with ICS/IWCGS. Most functions in htestClust perform hypothesis tests for clustered data that have an analogous classical form, and the interface of the package has been designed to reflect this relationship. Function syntax has been purposefully structured to resemble that of functions available in the native R environment that perform the analogous classical tests. Many functions have been designed with a secondary interface that operates through table or formula input, allowing flexibility in data structure. In addition to the hypothesis tests of marginal parameters, htestClust also includes functions to visualize potential informativeness and test for ICS. These tools allow analysts to explore the effect and degree of informativeness in their data.

With the exception of the test for ICS, the hypothesis tests performed by htestClust are derived through the asymptotic normality of reweighted parameters, and their asymptotic convergence is indexed by the number of clusters. As such, their use should only be considered when the number of clusters is sufficiently large (at least 30). Additionally, these methods retain a cluster-based marginal interpretation, making them appropriate when clusters, rather than intra-cluster observations, are the unit of interest. The marginal nature of these tests provides researchers with an analysis corresponding to a snapshot in time. If analysis of temporal aspects or effects of additional covariates is desired, readers might instead consider reweighted model-based methods such as those by Bible, Beck, and Datta (2016), Neuhaus and McCulloch (2011), and Wang, Kong, and Datta (2011).

htestClust is a tool to facilitate the analysis of clustered data, and we have designed its use to be accessible and intuitive. While we have not shown the full functionality of all of the htestClust functions, the syntax and usage is similar and fully documented with examples in the help files. While the inferntial methods performed by this package have been developed to correct for the biasing effects of ICS/IWCGS, they remain applicable when fluctuations of cluster or group size are unrelated to the outcome of interest. As such, this package is an effective resource for researchers addressing marginal analyses in clustered data with any variation in the cluster and/or group sizes.

# Citations

Bible, Joe, James D Beck, and Somnath Datta. 2016. “Cluster Adjusted Regression for Displaced Subject Data (Cards): Marginal Inference Under Potentially Informative Temporal Cluster Size Profiles.” Biometrics 72 (2): 441–51. https://doi.org/10.1111/biom.12456.

Carey, Vincent J, Thomas Lumley, and Brian Ripley. 2019. Gee: Generalized Estimation Equation Solver. https://CRAN.R-project.org/package=gee.

Chaurasia, Ashok, Danping Liu, and Paul S Albert. 2018. “Pattern-Mixture Models with Incomplete Informative Cluster Size: Application to a Repeated Pregnancy Study.” Journal of the Royal Statistical Society C 67 (1): 255. https://doi.org/10.1111/rssc.12226.

Datta, Somnath, and Glen A Satten. 2005. “Rank-Sum Tests for Clustered Data.” Journal of the American Statistical Association 100 (471): 908–15. https://doi.org/10.1198/016214504000001583.

———. 2008. “A Signed-Rank Test for Clustered Data.” Biometrics 64 (2): 501–7. https://doi.org/10.1111/j.1541-0420.2007.00923.x.

Dutta, Sandipan, and Somnath Datta. 2016. “A Rank-Sum Test for Clustered Data When the Number of Subjects in a Group Within a Cluster Is Informative.” Biometrics 72 (2): 432–40. https://doi.org/10.1111/biom.12447.

Gregg, Mary. 2020. “Marginal Methods and Software for Clustered Data with Cluster- and Group-Size Informativeness.” PhD thesis, UL (University of Louisville).

Gregg, Mary, Somnath Datta, and Doug Lorenz. 2020. “Variance Estimation in Tests of Clustered Categorical Data with Informative Cluster Size.” Statistical Methods in Medical Research 29 (11): 3396–3408. https://doi.org/10.1177/0962280220928572.

Halekoh, Ulrich, Søren Højsgaard, Jun Yan, and others. 2006. “The R Package Geepack for Generalized Estimating Equations.” Journal of Statistical Software 15 (2): 1–11.

Huang, Ying, and Brian Leroux. 2011. “Informative Cluster Sizes for Subcluster-Level Covariates and Weighted Generalized Estimating Equations.” Biometrics 67 (3): 843–51. https://doi.org/10.1111/j.1541-0420.2010.01542.x.

Iosif, Ana-Maria, and Allan R Sampson. 2014. “A Model for Repeated Clustered Data with Informative Cluster Sizes.” Statistics in Medicine 33 (5): 738–59. https://doi.org/10.1002/sim.5988.

Lorenz, Douglas J, Somnath Datta, and Susan J Harkema. 2011. “Marginal Association Measures for Clustered Data.” Statistics in Medicine 30 (27): 3181–91. https://doi.org/10.1002/sim.4368.

Lorenz, Douglas J, Steven Levy, and Somnath Datta. 2018. “Inferring Marginal Association with Paired and Unpaired Clustered Data.” Statistical Methods in Medical Research 27 (6): 1806–17. https://doi.org/10.1177/0962280216669184.

Mitani, AA, EK Kaye, and KP Nelson. 2020. “Marginal Analysis of Multiple Outcomes with Informative Cluster Size.” Biometrics. https://doi.org/10.1111/biom.13241.

Mitani, Aya A, Elizabeth K Kaye, and Kerrie P Nelson. 2019. “Marginal Analysis of Ordinal Clustered Longitudinal Data with Informative Cluster Size.” Biometrics 75 (3): 938–49. https://doi.org/10.1111/biom.13050.

Neuhaus, John M, and Charles E McCulloch. 2011. “Estimation of Covariate Effects in Generalized Linear Mixed Models with Informative Cluster Sizes.” Biometrika 98 (1): 147–62. https://doi.org/10.1093/biomet/asq066.

Nevalainen, Jaakko, Hannu Oja, and Somnath Datta. 2017. “Tests for Informative Cluster Size Using a Novel Balanced Bootstrap Scheme.” Statistics in Medicine 36 (16): 2630–40. https://doi.org/10.1002/sim.7288.

Pavlou, Menelaos. 2012. “Analysis of Clustered Data When the Cluster Size Is Informative.” PhD thesis, UCL (University College London).

Wang, Ming, Maiying Kong, and Somnath Datta. 2011. “Inference for Marginal Linear Models for Clustered Longitudinal Data with Potentially Informative Cluster Sizes.” Statistical Methods in Medical Research 20 (4): 347–67. https://doi.org/10.1177/0962280209347043.

Williamson, John M, Somnath Datta, and Glen A Satten. 2003. “Marginal Analyses of Clustered Data When Cluster Size Is Informative.” Biometrics 59 (1): 36–42. https://doi.org/10.1111/1541-0420.00005.