metaquant: An R package to estimate means, standard deviations and visualising distributions using quantiles
2025-02-08
Chapter 1 Introduction
The metaquant R package provides functions for estimating means, standard deviations, and visualising distributions using quantile summary data introduced by De Livera et al. (2024). This package focuses meta-analysing continuous outcomes, when only the quantile information (e.g., medians, quartiles, or extremes) is available. By using flexible density distributions, metaquant enables researchers to use quantile summary measures to support the implementation of a comprehensive meta-analysis.
The package addresses the following three common scenarios of reported quantile data:
- S\(_1\): { \(a\), \(m\), \(b\) }
- S\(_2\): { \(q_1\), \(m\), \(q_3\) }
- S\(_3\): { \(a\), \(q_1\), \(m\), \(q_3\), \(b\) },
where { \(a\), \(q_1\), \(m\), \(q_3\), \(b\) } denote the minimum, first quartile, median, third quartile and maximum, respectively.
When the 5 number summary is available (S\(_3\)), the package uses the Generalized Lambda Distribution (GLD), a flexible family of distributions capable of approximating several distributions (e.g., normal, logistic, and log-normal). FKML parameterisation (Freimer et al., 1988) which is defined by its quantile function is used in the package. When only 3-point summaries are available, such as in the Scenario 1 (S\(_1\)) and 2 (S\(_2\)) above, the package uses the quantile-based skew logistic distribution (SLD) (van Staden and King, 2015). Using these GLD and SLD density-based approaches, as well as other methods described by Wan et al. (2014), Luo et al. (2018), Shi et al. (2020) and McGrath et al. (2020), the package provides functions to estimate summary measures such as the sample means and standard deviations using the quantile summaries and sample sizes. In addition to the estimation of sample means and standard deviations, metaquant includes functions for visualising the estimated distributions of the sample data, for all 3 scenarios above. The visualisation functions allow users to create density plots using only quantile summaries, enabling exploration of group differences, skewness, and heterogeneity across studies.
References
Alysha De Livera, Luke Prendergast, and Udara Kumaranathunga. A novel density-based approach for estimating unknown means, distribution visualisations, and meta-analyses of quantiles. Submitted for Review, 2024, pre-print available here: https://arxiv.org/abs/2411.10971
Marshall Freimer, Georgia Kollia, Govind S. Mudholkar, and C. Thomas Lin. A study of the generalized Tukey lambda family. Communications in Statistics-Theory and Methods, 17(10):3547–3567, 1988.
P. J. van Staden and R. A. R. King. The quantile-based skew logistic distribution. Statistics & Probability Letters, 96:109–116, 2015.
King R, Dean B, Klinke S, van Staden P (2022). gld: Estimation and Use of the Generalised (Tukey) Lambda Distribution. R package version 2.6.6, https://CRAN.R-project.org/package=gld.
King R, van Staden P (2022). sld: Estimation and Use of the Quantile-Based Skew Logistic Distribution. R package version 1.0.1, https://CRAN.R-project.org/package=sld.
Chapter 2 Installing metaquant
metaquant can be download via CRAN as follows:
Alternatively, the development version can be downloaded using GitHub. To install this version, the user needs to make sure that Rtools has been installed and integrated prior.
Chapter 3 Estimate Summary Statistics using Quantiles
3.1 Estimate Mean
The function ‘est.mean’ estimates the sample mean of a study given either one of the 3-point summaries (S\(_1\) or S\(_2\)) or 5-point summaries (S\(_3\)).
The ‘est.mean’ implements a flexible quantile-based distribution methods for estimating sample means proposed by De Livera et al. (2024) as well as several other methods for estimating sample means as described by Wan et al. (2014),Luo et al. (2018) and McGrath et al. (2020).
The estimation methods implemented in the function are the following:
- ‘gld/sld’: The method proposed by De Livera et al. (2024); i.e., estimation using the Generalized Lambda Distribution (GLD) for 5-number summaries (S\(_3\)), and the Skew Logistic Distribution (SLD) for 3-number summaries (S\(_1\) and S\(_2\)). This is set as the default estimation option in the function.
- ‘luo’: The method proposed by Luo et al. (2018).
- ‘hozo/wan/bland’: The method proposed by Wan et al. (2014); i.e., the method of Hozo et al. (2005) for S\(_1\), method of Wan et al. (2014) for S\(_2\), and method of Bland (2015) for S\(_3\).}
- ‘bc’: The Box-Cox method proposed by McGrath et al. (2020).
- ‘qe’: The quantile matching estimation method proposed by McGrath et al. (2020).
Estimating mean using S\(_3\): { \(a\), \(q_1\), \(m\), \(q_3\), \(b\) }
To illustrate the usage of the function, we first generate example 5-point summary data from the exponential distribution with parameter \(\lambda\).
#Generate quantile summary data
set.seed(123)
n <- 1000
x <- rexp(n=n, rate=10)
quants <- c(min(x), quantile(x, probs = c(0.25, 0.5, 0.75)), max(x))
quants## 25% 50% 75%
## 8.259739e-05 3.066927e-02 7.311667e-02 1.426635e-01 7.211008e-01The (unreported) sample mean is then
## [1] 0.1029979Next, assuming ‘quants’ represents a 5-point summary from a study where we need to estimate the sample mean, the default ‘gld/sld’ method is used for the estimation.
#Estimate sample mean of S3 using 'gld/sld'
estmean_gl <- est.mean(min = quants[1],
q1 = quants[2],
med = quants[3],
q3 = quants[4],
max = quants[5],
n=n)
estmean_gl## $mean
## [1] 0.100545If one needs to estimate the sample mean using a defined alternative method, simply specify the method as follows.
#Estimate sample mean of S3 using the method 'luo'
estmean_luo <- est.mean(min = quants[1],
q1 = quants[2],
med = quants[3],
q3 = quants[4],
max = quants[5],
n=n,
method = "luo")
estmean_luo## $mean
## [1] 0.08589611Estimating mean using S\(_1\): { \(a\), \(m\), \(b\) }
Suppose only the minimum, median, and maximum are available as your quantile summaries.
## 50%
## 8.259739e-05 7.311667e-02 7.211008e-01Then, instead of providing all five quantile inputs to the function, you can use the arguments ‘min’, ‘med’, and ‘max’ along with the sample size ‘n’ to estimate the sample mean. To estimate the sample mean of S\(_1\) using ‘gld/sld’ method,
#Estimate sample mean for S1
estmean_sl_1 <- est.mean(min = quants1[1],
med = quants1[2],
max = quants1[3],
n=n,
method = "gld/sld")
estmean_sl_1## $mean
## [1] 0.1015034Similarly, the ‘method’ argument can be adjusted to use a different estimation method, as described above.
Estimating mean using S\(_2\): { \(q_1\), \(m\), \(q_3\) }
Similarly, if only the first quartile, median, and third quartile are available, use the ‘q\(_1\)’, ‘med’ and ‘q\(_3\)’ arguments of the function.
## 25% 50% 75%
## 0.03066927 0.07311667 0.14266352#Estimate sample mean for S2
estmean_sl_2 <- est.mean(q1 = quants2[1],
med = quants2[2],
q3 = quants2[3],
method = "gld/sld")
estmean_sl_2## $mean
## [1] 0.102022Note that, the method ‘gld/sld’ under S\(_2\) does not require the sample size to estimate the sample mean, so the argument ‘n’ can be omitted in this case. However, all other methods require the sample size for the estimation.
3.2 Estimate Standard Deviation
For completeness, the package provides the function ‘est.sd’ which estimates the sample standard deviation for a study reporting quantile summary data. This includes 3-point summaries (S\(_1\) or S\(_2\)) and 5-point summary (S\(_3\)). While the function operates similarly to ‘est.mean’, it incorporates distinct estimation methods specific to standard deviation calculation..
The following methods for estimating the standard deviation are implemented in the function:
- ‘shi/wan’: The method proposed by Shi et al. (2020).
- ‘gld/sld’: The method of De Livera et al. (2024); i.e., estimation using the Generalized Lambda Distribution (GLD) for 5-number summaries (S\(_3\)), and the Skew Logistic Distribution (SLD) for 3-number summaries (S\(_1\) and S\(_2\)).
- ‘wan’: The method proposed by Wan et al. (2014).
- ‘bc’: The Box-Cox method proposed by McGrath et al. (2020).
- ‘qe’: The quantile matching estimation method proposed by McGrath et al. (2020).
The method by Shi et al. (2020) is set as the default estimation option in the ‘est.sd’ function as we found that it performs well across a variety of scenarios (see De Livera et al. (2024)).
For example, to estimate the sample standard deviation using a given 5-point summary, the ‘est.sd’ can be applied by providing the quantiles and the sample size as inputs. We use the same data ‘quants’ used in section 3.1 with the default option ‘shi/wan’ as the estimation method.
## [1] 0.1004404#Estimate sample SD of S3 using 'shi/wan' method
estsd_shi <- est.sd(min = quants[1],
q1 = quants[2],
med = quants[3],
q3 = quants[4],
max = quants[5],
n=n)
estsd_shi## $sd
## [1] 0.08841593.3 Estimation for Two Groups
In addition to the functions ‘est.mean’ and ‘est.sd’, the package also provides two functions ‘est.mean.2g’ and est.sd.2g’ for estimating the sample mean and standard deviation in two-group studies based on quantile summary measures. These functions implement the methods proposed by De Livera et al. (2024) for two-group cases, as the other estimation methods do not support variations for two-group cases. The approach uses the Generalized Lambda Distribution (GLD) for 5-number summaries (S\(_3\)), and the Skew Logistic Distribution (SLD) for 3-number summaries (S\(_1\) and S\(_2\)) to estimate sample statistics using quantiles by incorporating shared information across the two groups to improve the accuracy of the estimates.
As a result, these two functions do not require a ‘method’ argument to be specified. However, the functions include additional arguments to input the summary measures for the second group.
For instance, consider the following quantile summaries for two groups. In this case, the ‘rexp’ function from the ‘stats’ package is used to generate example samples with exponential distributions. You may need to load the necessary libraries if they are not already loaded.
#Generate 5-point summary data for two groups
set.seed(123)
n_t <- 100
n_c <- 120
x_t <- rexp(n_t, 5)
x_c <- rexp(n_c, 10)
q_t <- c(min(x_t), quantile(x_t, probs = c(0.25, 0.5, 0.75)), max(x_t))
q_c <- c(min(x_c), quantile(x_c, probs = c(0.25, 0.5, 0.75)), max(x_c))Similarly to the single group case, the ‘est.mean.2g’ and est.sd.2g’ functions can be applied as below.
#Estimate sample mean of S3
estmean_2g <- est.mean.2g(q_t[1],q_t[2],q_t[3],q_t[4],q_t[5],
q_c[1],q_c[2],q_c[3],q_c[4],q_c[5],
n.g1 = n_t,
n.g2 = n_c)
estmean_2g## $mean.g1
## [1] 0.2330208
##
## $mean.g2
## [1] 0.08490334#Estimate sample SD of S3
estsd_2g <- est.sd.2g(q_t[1],q_t[2],q_t[3],q_t[4],q_t[5],
q_c[1],q_c[2],q_c[3],q_c[4],q_c[5],
n.g1 = n_t,
n.g2 = n_c)
estsd_2g## $sd.g1
## [1] 0.2587828
##
## $sd.g2
## [1] 0.07264839The above functions can be used similarly, in the cases where only the three number summaries (S\(_1\) and S\(_2\)) are available.
References
Alysha De Livera, Luke Prendergast, and Udara Kumaranathunga. A novel density-based approach for estimating unknown means, distribution visualisations, and meta-analyses of quantiles. Submitted for Review, 2024, pre-print available here: https://arxiv.org/abs/2411.10971
Xiang Wan, Wenqian Wang, Jiming Liu, and Tiejun Tong. Estimating the sample mean and standard deviation from the sample size, median, range and/or interquartile range. , 14:1–13, 2014.
Dehui Luo, Xiang Wan, Jiming Liu, and Tiejun Tong. Optimally estimating the sample mean from the sample size, median, mid-range, and/or mid-quartile range. Statistical methods in medical research, 27(6):1785–1805,2018.
Sean McGrath, XiaoFei Zhao, Russell Steele, Brett D Thombs, Andrea Benedetti, and DEPRESsion Screening Data (DEPRESSD) Collaboration. Estimating the sample mean and standard deviation from commonly reported quantiles in meta-analysis. Statistical methods in medical research, 29(9):2520–2537, 2020.
Jiandong Shi, Dehui Luo, Hong Weng, Xian-Tao Zeng, Lu Lin, Haitao Chu, and Tiejun Tong. Optimally estimating the sample standard deviation from the five-number summary. Research synthesis methods, 11(5):641–654, 2020.
Chapter 4 Visualise Densities using Quantiles
The ‘plotdist’ function estimates and visualizes the density curves of one or two groups of samples using either the 3-point summaries (S\(_1\) or S\(_2\)) or the 5-point summary (S\(_3\)). It returns a customizable and interactive plotly object visualizing the estimated density curve(s) of individual studies as well as the pooled densities.
4.1 Prepare the Dataset
The input data to ‘plotdist’ should be a data frame containing the quantile summary data. For one-group studies, the input data frame can include the following columns:
- study.index : study index or name
- min.g1 : minimum value.
- q1.g1 : first quartile.
- med.g1 : median.
- q3.g1 : third quartile.
- max.g1 : maximum value.
- n.g1 : sample size.
For two-group studies, the data frame can also contain the following columns for the summary data of the second group: min.g2, q1.g2, med.g2, q3.g2, max.g2 and n.g2.
If only 3-number summaries are available, only the respective columns for the 3-point summary should be included in the data frame.
4.2 Visualise Densities of One-Group Studies
Consider the following dataset which includes three one-group studies, each reporting 5-point summary along with the respective sample size.
# Dataset of 5-point summaries for 1 group
data_s3 <- data.frame(
study.index = c("Study1", "Study2", "Study3"),
min.g1 = c(18, 19, 15),
q1.g1 = c(66, 71, 69),
med.g1 = c(73, 82, 81),
q3.g1 = c(80, 93, 89),
max.g1 = c(110, 115, 100),
n.g1 = c(226, 230, 200)
)
data_s3## study.index min.g1 q1.g1 med.g1 q3.g1 max.g1 n.g1
## 1 Study1 18 66 73 80 110 226
## 2 Study2 19 71 82 93 115 230
## 3 Study3 15 69 81 89 100 200Then, using the data above, the densities of the three stduies can be visualised in the same plot using ‘plotdist’ as illustrated below.
# Plot densities
plot_s3 <- plotdist(
data_s3,
xmin = 10,
xmax = 125,
title = "Example Density Plot of S3",
xlab = "x data",
title.size = 11,
lab.size = 10,
color.g1 = "blue",
display.index = FALSE,
display.legend = FALSE
)
plot_s3The function parameters ‘xmin’ and ‘xmax’ must be specified by the user, where ‘xmin’ is a numeric value for the lower limit of the x-axis for density calculation and ‘xmax’ is a numeric value for the upper limit. To ensure the density curve is captured, it is recommended to set ‘xmin’ to a value smaller than the smallest minimum value across studies in the dataset, while setting ‘xmax’ to a value larger than the largest maximum value across the studies. If specific values are not provided for the above parameters, the function itself uses the minimum value of the ‘min.’ columns and maximum value of the ‘max.’ columns, for scenario S\(_1\) or S\(_3\). Note that for scenario S\(_2\) , no default calculation is performed and an error occurs since there are no ‘min.’ and ‘max.’ columns.
Consider a dataset of quantile summaries of 3 studies, each reporting 3-point summary of S\(_1\) along with the sample size.
# Dataset of 3-point summaries for 1 group
data_s1 <- data.frame(
study.index = c("Study1", "Study2", "Study3"),
min.g1 = c(18, 19, 15),
med.g1 = c(73, 82, 81),
max.g1 = c(110, 115, 100),
n.g1 = c(226, 230, 200)
)
data_s1## study.index min.g1 med.g1 max.g1 n.g1
## 1 Study1 18 73 110 226
## 2 Study2 19 82 115 230
## 3 Study3 15 81 100 200In this case, the data frame consists only of the columns representing the minimum, median and maximum values. The density curves of three studies can be visualised using ‘plotdist’ as illustrated below.
4.3 Visualise Densities of Two-Group Studies
Consider the following dataset of two-group studies, each reporting 5-point summaries for both group 1 and group 2 along with their respective sample sizes.
# Dataset of 5-point summaries for 2 groups
data_2g <- data.frame(
study.index = c("Study1", "Study2", "Study3"),
min.g1 = c(18, 19, 15),
q1.g1 = c(66, 71, 69),
med.g1 = c(73, 82, 81),
q3.g1 = c(80, 93, 89),
max.g1 = c(110, 115, 100),
n.g1 = c(226, 230, 200),
min.g2 = c(15, 15, 13),
q1.g2 = c(57, 59, 55),
med.g2 = c(66, 68, 60),
q3.g2 = c(74, 72, 69),
max.g2 = c(108, 101, 100),
n.g2 = c(201, 223, 198)
)
data_2g## study.index min.g1 q1.g1 med.g1 q3.g1 max.g1 n.g1 min.g2 q1.g2 med.g2 q3.g2 max.g2
## 1 Study1 18 66 73 80 110 226 15 57 66 74 108
## 2 Study2 19 71 82 93 115 230 15 59 68 72 101
## 3 Study3 15 69 81 89 100 200 13 55 60 69 100
## n.g2
## 1 201
## 2 223
## 3 198With the following input into the ‘plotdist’ function, the user can obtain the density curves of two groups, displayed in different colors the user provides, within the same plot. Here, we use the default colors defined by the function.
# Plot densities
plot_2g <- plotdist(
data_2g,
xmin = 10,
xmax = 125,
title = "Example Density Plot of Two Groups",
xlab = "x data",
title.size = 11,
label.g1 = "Treatment",
label.g2 = "Control",
display.index = FALSE,
display.legend = TRUE
)
plot_2gTo display the legend labels, the user needs to provide the names of the groups for the ‘label.g1’ and ‘label.g2’ arguments and set ‘display.legend = TRUE’.
4.4 Visualise Pooled Densities
Similar to a standard meta-analysis of numeric data, where pooled estimates of effects are obtained, De Livera et al. (2024) considered pooled densities as a way to combine estimated distributional information across studies. The pooled densities represent weighted average of individual study densities, with weights determined by sample sizes. By combining information across studies, pooled densities provide a comprehensive visualization of group differences in terms of key characteristics such as location, dispersion, and shape.
If the user needs to generate pooled density plots using ‘plotdist’ function, the user can set the ‘pooled.dist’ or ‘pooled.only’ arguments to ‘TRUE’. By default these arguments are set to ‘FALSE’. When ‘pooled.dist = TRUE’, the pooled density curves will be displayed along with the individual density curves and when ‘pooled.only = TRUE’, only the pooled density curves will be plotted, excluding the individual curves.
For example, pooled curves can be added to the previous plot ‘plot_2g’ as follows. Here, the default colors assigned to the two groups are used. The user can customize the colors by using the ‘color.g1’, ‘color.g2’, ‘color.g1.pooled’, and ‘color.g2.pooled’ arguments.
Chapter 5 Contact and Session Information
5.1 Contact Information
For any queries, contact Alysha De Livera a.delivera@latrobe.edu.au or Udara Kumaranathunga u.kumaranathunga@latrobe.edu.au.
5.2 Session Information
## R version 4.4.1 (2024-06-14 ucrt)
## Platform: x86_64-w64-mingw32/x64
## Running under: Windows 10 x64 (build 19045)
##
## Matrix products: default
##
##
## locale:
## [1] LC_COLLATE=English_Australia.utf8 LC_CTYPE=English_Australia.utf8
## [3] LC_MONETARY=English_Australia.utf8 LC_NUMERIC=C
## [5] LC_TIME=English_Australia.utf8
##
## time zone: Australia/Sydney
## tzcode source: internal
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] bookdown_0.42 devtools_2.4.5 usethis_3.0.0 metaquant_0.1.0
##
## loaded via a namespace (and not attached):
## [1] gld_2.6.6 gtable_0.3.6 xfun_0.50 bslib_0.9.0
## [5] ggplot2_3.5.1 htmlwidgets_1.6.4 remotes_2.5.0 processx_3.8.4
## [9] callr_3.7.6 crosstalk_1.2.1 ps_1.7.7 vctrs_0.6.5
## [13] tools_4.4.1 generics_0.1.3 curl_5.2.2 tibble_3.2.1
## [17] proxy_0.4-27 fansi_1.0.6 pkgconfig_2.0.3 data.table_1.16.2
## [21] desc_1.4.3 PKI_0.1-14 lifecycle_1.0.4 compiler_4.4.1
## [25] stringr_1.5.1 munsell_0.5.1 httpuv_1.6.15 htmltools_0.5.8.1
## [29] class_7.3-22 sass_0.4.9 yaml_2.3.10 lazyeval_0.2.2
## [33] plotly_4.10.4 urlchecker_1.0.1 pillar_1.10.1 later_1.3.2
## [37] jquerylib_0.1.4 tidyr_1.3.1 ellipsis_0.3.2 openssl_2.3.2
## [41] rsconnect_1.3.3 sld_1.0.1 cachem_1.1.0 sessioninfo_1.2.2
## [45] mime_0.12 tidyselect_1.2.1 digest_0.6.37 stringi_1.8.4
## [49] dplyr_1.1.4 purrr_1.0.2 labeling_0.4.3 fastmap_1.2.0
## [53] grid_4.4.1 colorspace_2.1-1 lmom_3.0 cli_3.6.3
## [57] magrittr_2.0.3 base64enc_0.1-3 pkgbuild_1.4.4 utf8_1.2.4
## [61] e1071_1.7-16 scales_1.3.0 promises_1.3.0 estmeansd_1.0.1
## [65] rmarkdown_2.29 httr_1.4.7 askpass_1.2.1 memoise_2.0.1
## [69] shiny_1.9.1 evaluate_1.0.3 knitr_1.49 miniUI_0.1.1.1
## [73] viridisLite_0.4.2 profvis_0.3.8 rlang_1.1.4 Rcpp_1.0.13
## [77] xtable_1.8-4 glue_1.7.0 pkgload_1.4.0 rstudioapi_0.16.0
## [81] jsonlite_1.8.8 R6_2.5.1 fs_1.6.4References
Alysha De Livera, Luke Prendergast, and Udara Kumaranathunga. A novel density-based approach for estimating unknown means, distribution visualisations, and meta-analyses of quantiles. Submitted for Review, 2024, pre-print available here: https://arxiv.org/abs/2411.10971
Marshall Freimer, Georgia Kollia, Govind S. Mudholkar, and C. Thomas Lin. A study of the generalized Tukey lambda family. Communications in Statistics-Theory and Methods, 17(10):3547–3567, 1988.
P. J. van Staden and R. A. R. King. The quantile-based skew logistic distribution. Statistics & Probability Letters, 96:109–116, 2015.
King R, Dean B, Klinke S, van Staden P (2022). gld: Estimation and Use of the Generalised (Tukey) Lambda Distribution. R package version 2.6.6, https://CRAN.R-project.org/package=gld.
King R, van Staden P (2022). sld: Estimation and Use of the Quantile-Based Skew Logistic Distribution. R package version 1.0.1, https://CRAN.R-project.org/package=sld.
Dehui Luo, Xiang Wan, Jiming Liu, and Tiejun Tong. Optimally estimating the sample mean from the sample size, median, mid-range, and/or mid-quartile range. Statistical methods in medical research, 27(6):1785–1805,2018.
Sean McGrath, XiaoFei Zhao, Russell Steele, Brett D Thombs, Andrea Benedetti, and DEPRESsion Screening Data (DEPRESSD) Collaboration. Estimating the sample mean and standard deviation from commonly reported quantiles in meta-analysis. Statistical methods in medical research, 29(9):2520–2537, 2020.
Jiandong Shi, Dehui Luo, Hong Weng, Xian-Tao Zeng, Lu Lin, Haitao Chu, and Tiejun Tong. Optimally estimating the sample standard deviation from the five-number summary. Research synthesis methods, 11(5):641–654, 2020.