Package {circhelp}

Type:	Package
Title:	Circular Analyses Helper Functions
Version:	1.4.0
Description:	Light-weight functions for computing descriptive statistics in different circular spaces (e.g., 2pi, 180, or 360 degrees), to handle angle-dependent biases, pad circular data, and more. Specifically aimed for psychologists and neuroscientists analyzing circular data. Basic methods are based on Jammalamadaka and SenGupta (2001) <doi:10.1142/4031>, removal of cardinal biases is based on the approach introduced in van Bergen, Ma, Pratte, & Jehee (2015) <doi:10.1038/nn.4150> and Chetverikov and Jehee (2023) <doi:10.1038/s41467-023-43251-w>.
URL:	https://achetverikov.github.io/circhelp/index.html, https://github.com/achetverikov/circhelp
BugReports:	https://github.com/achetverikov/circhelp/issues
License:	CC0
Encoding:	UTF-8
Language:	en-GB
LazyData:	true
Suggests:	testthat (≥ 3.0.0), knitr, rmarkdown, ggridges
Imports:	mathjaxr, data.table, stats, ggplot2 (≥ 3.4.0), MASS, patchwork, rlang
Depends:	R (≥ 4.1.0)
Config/testthat/edition:	3
VignetteBuilder:	knitr, rmarkdown
RdMacros:	mathjaxr
BuildManual:	TRUE
Config/roxygen2/version:	8.0.0
NeedsCompilation:	no
Packaged:	2026-10-09 15:36:14 UTC; anche7026
Author:	Andrey Chetverikov [aut, cre], Eline Van Geert [ctb]
Maintainer:	Andrey Chetverikov <andrey.chetverikov@uib.no>
Repository:	CRAN
Date/Publication:	2026-10-09 16:10:10 UTC

circhelp: Circular Analyses Helper Functions

Description

Light-weight functions for computing descriptive statistics in different circular spaces (e.g., 2pi, 180, or 360 degrees), to handle angle-dependent biases, pad circular data, and more. Specifically aimed for psychologists and neuroscientists analyzing circular data. Basic methods are based on Jammalamadaka and SenGupta (2001) doi:10.1142/4031, removal of cardinal biases is based on the approach introduced in van Bergen, Ma, Pratte, & Jehee (2015) doi:10.1038/nn.4150 and Chetverikov and Jehee (2023) doi:10.1038/s41467-023-43251-w.

Author(s)

Maintainer: Andrey Chetverikov andrey.chetverikov@uib.no (ORCID)

Authors:

• Andrey Chetverikov andrey.chetverikov@uib.no (ORCID)

Other contributors:

• Eline Van Geert eline.vangeert@kuleuven.be (ORCID) [contributor]

See Also

Useful links:

• https://achetverikov.github.io/circhelp/index.html

• https://github.com/achetverikov/circhelp

• Report bugs at https://github.com/achetverikov/circhelp/issues

Data from a motion estimation task

Description

A dataset with the motion estimation results from Bae & Luck (2018) available from https://osf.io/download/4m2kb (some variables are removed, see the link for the full dataset).

Usage

Bae_Luck_2018_data

Format

A data frame with 20480 rows and 8 variables:

subject_Num

observer ID

trial_Num

trial number

TargetDirection

true motion direction

RespAngle

reported motion direction

motionCoh

motion coherence

Block

block number

Session

session number

err

estimation error

Source

https://osf.io/download/4m2kb

References

Bae, G.-Y., & Luck, S. J. (2018). Decoding motion direction using the topography of sustained ERPs and alpha oscillations. NeuroImage, 184(August 2018), 242-255. doi:10.1016/J.NEUROIMAGE.2018.09.029

Data from an orientation estimation task

Description

A dataset with the orientation estimation results from Experiment 2 in Pascucci et al. (2019) available from https://doi.org/10.5281/zenodo.2544946.

Usage

Pascucci_et_al_2019_data

Format

A data frame with 4400 rows and 5 variables:

observer

observer ID

orientation

true orientation

reported

reported orientation

rt

response time

err

estimation error

Source

https://zenodo.org/record/2544946/files/Experiment2_rawdata.csv?download=1

References

Pascucci, D., Mancuso, G., Santandrea, E., Libera, C. D., Plomp, G., & Chelazzi, L. (2019). Laws of concatenated perception: Vision goes for novelty, decisions for perseverance. PLoS Biology, 17(3). doi:10.1371/journal.pbio.3000144

Differences between angles in different circular spaces

Description

Differences between angles in different circular spaces

Usage

angle_diff_rad(a, b)

angle_diff_360(a, b)

angle_diff_180(a, b)

angle_diff_90(a, b)

angle_diff_180_45(a, b)

angle_diff_360_90(a, b)

angle_diff(a, b, period = 360)

Arguments

Details

By default, all functions return values in ± half-range space (e.g., -pi to pi for 2pi radian space used by angle_diff_rad()) but angle_diff_180_45() and angle_diff_360_90() return values in [-1/4 range, 3/4 range] space

Value

difference between a and b

Functions

• angle_diff_rad(): angle difference in radians

• angle_diff_360(): angle difference in 360 degree space

• angle_diff_180(): angle difference in 180 degree space (e.g., line orientation)

• angle_diff_90(): angle difference in 90 degree space

• angle_diff_180_45(): angle difference in 180 degree space from -45 to 135

• angle_diff_360_90(): angle difference in 360 degree space from -90 to 270

• angle_diff(): angle difference in a space with an arbitrary period (e.g., 180 or 360)

Examples

angle_diff_180(5, 175)
angle_diff_360(5, 175)
angle_diff_90(5, 175)
angle_diff_rad(5, 175)

angle_diff_360(300, 0)
angle_diff_360_90(300, 0)

Circular correlation coefficient

Description

Computes a circular correlation coefficient as defined in Jammalamadaka & SenGupta (2001).

Usage

circ_corr(a, b, ill_defined = FALSE, mu = NULL, na.rm = FALSE)

Arguments

Value

correlation coefficient

References

Jammalamadaka, S. R., & SenGupta, A. (2001). Topics in Circular Statistics. WORLD SCIENTIFIC. doi:10.1142/4031

Examples

set.seed(1)
x <- rnorm(1000)
y <- 0.5 * x + sqrt(0.75) * rnorm(1000)
circ_corr(x, y)

A set of descriptive statistics for circular data

Description

A set of descriptive statistics for circular data

Usage

circ_descr(x, w = NULL, d = NULL, na.rm = FALSE)

Arguments

Value

a list with descriptive statistics

• mu - mean

• sigma - standard deviation

• skew_pewsey - skewness as defined by Pewsey

• skew_fischer - skewness as defined by Fischer

• rho - mean resultant length

• skew_rel_to_zero - skewness relative to zero

Examples

x <- c(rnorm(50, 0, 0.5), rnorm(20, 1, 0.5))
circ_descr(x)

Circular-linear correlation

Description

Implementation of the circular-linear correlation measure introduced by Mardia (1976) and Johnson and Wehrly (1977) as cited in Jammalamadaka & Sengupta (2001).

Usage

circ_lin_corr(circ_x, lin_x, na.rm = FALSE)

Arguments

Details

This measure is computed as \[r^2 = (r_{xc}^2+r_{xs}^2-2 r_{xc} r_{xs}r_{cs})/(1-r_{cs}^2)\] where \(r_{xc} = corr(x, cos(\alpha))\), \(r_{xs} = corr(x, sin(\alpha))\), \(r_{cs} = corr(cos(\alpha), sin(\alpha))\), and \(\alpha\) and \(x\) are the circular and linear variables, respectively.

Value

circular-linear correlation measure

References

Jammalamadaka, S. R., & SenGupta, A. (2001). Topics in Circular Statistics. WORLD SCIENTIFIC. doi:10.1142/4031

Examples

set.seed(1)
x <- rnorm(50)
a <- rnorm(50, sd = 0.4)
circ_lin_corr(x + a, x)

An implementation of circular-linear locally-weighted regression (LOESS)

Description

Provides an locally-weighted average when the independent variable is circular and depended variable is linear. Mainly to use with ggplot2.

Usage

circ_loess(
  formula = NULL,
  data = NULL,
  angle = NULL,
  y = NULL,
  xseq = NULL,
  circ_space = NULL,
  span = 0.75,
  ...
)

Arguments

Details

Weights for the regression are computed as \[w = (1-(d/d_{max})^3)^3\] where d is the angular difference between the point at which the estimate is computed and the angles in the data, and \(d_{max}\) is the maximum possible distance. If span is above 1, all points are included and \(d_{max} = {circ\_space}/(4*span)\). Otherwise, a proportion \(\alpha\) of the points included based on their distance to the point at which the estimate is computed and \(d_{max}\) is the corresponding maximal distance.

Value

an object of circ_loess class with the following parameters:

• angle the angles in the data

• y the dependent variable vales in the data

• xseq the grid on which the loess values are estimated

• y_est the estimated loess values

• y_se standard errors

• w weights

• circ_space circular space used

• span span used

See Also

stats::loess()

Examples

library(ggplot2)
p <- ggplot(Pascucci_et_al_2019_data, aes(x = orientation, y = err)) +
  geom_point(alpha = 0.05) +
  labs(x = "Orientation, deg.", y = "Error, deg.")
p1 <- p + geom_smooth(method = "loess") + ggtitle("Standard LOESS")
p2 <- p + geom_smooth(method = "circ_loess", method.args = list(circ_space = 180, span = 0.5)) +
  ggtitle("Circular LOESS, span = 0.5")
p3 <- p + geom_smooth(method = "circ_loess", method.args = list(circ_space = 180, span = 0.2)) +
  ggtitle("Circular LOESS, span = 0.2")
(p1 + p2 + p3)

Circular mean

Description

Circular mean

Usage

circ_mean_rad(x, na.rm = FALSE)

circ_mean_180(x, na.rm = FALSE)

circ_mean_360(x, na.rm = FALSE)

circ_mean(x, period = 360, na.rm = FALSE)

Arguments

Value

mean of values in the vector

Functions

• circ_mean_rad(): circular mean in 2pi space

• circ_mean_180(): circular mean in 180° space (e.g., line orientation)

• circ_mean_360(): circular mean in 360° space

• circ_mean(): circular mean in a space with an arbitrary period (e.g., 180 or 360)

Examples

x <- runif(1000, -pi, pi)
mean(x)
circ_mean_rad(x)

Circular standard deviation

Description

Circular standard deviation

Usage

circ_sd_rad(x, na.rm = FALSE)

circ_sd_360(x, na.rm = FALSE)

circ_sd_180(x, na.rm = FALSE)

circ_sd(x, period = 360, na.rm = FALSE)

Arguments

Value

standard deviation of values in the vector

Functions

• circ_sd_rad(): SD of angles in radians

• circ_sd_360(): SD of angles in 360 degree space

• circ_sd_180(): SD of angles in 180 degree space

• circ_sd(): SD of angles in a space with an arbitrary period (e.g., 180 or 360)

Examples

circ_sd_rad(rnorm(50))
circ_sd_180(rnorm(50))

Get angle value in [-pi, pi] space

Description

Get angle value in [-pi, pi] space

Usage

correct_angle_rad(x)

Arguments

Value

angle in [-pi, pi] space

Examples

correct_angle_rad(4 * pi)

Compute asymmetry in weighted probability density

Description

This function calculates the asymmetry in the probability density of a given variable (usually errors) relative to another variable (usually dissimilarity) using kernel density estimation. The asymmetry is computed for each x-axis value, and the result can be averaged or returned for each value individually.

Usage

density_asymmetry(
  dt,
  circ_space = 180,
  weights_sd = 10,
  kernel_bw = NULL,
  xvar = "abs_td_dist",
  yvar = "bias_to_distr_corr",
  by = c(),
  n = 181,
  average = T,
  return_full_density = F,
  normalize = T,
  wrap = TRUE,
  exclude_antipode = TRUE,
  n_wraps = 1,
  rescale_narrow = TRUE,
  scale_safety = 0.5
)

Arguments

Value

A data.table with the grouping variables, dist - the values of X-axis variable at which the density is computed, and delta - the difference (asymmetry) in probability density for positive and negative values of yvar; or the full density data if return_full_density is TRUE.

Examples

data(Pascucci_et_al_2019_data)
ex_data <- Pascucci_et_al_2019_data
ex_data[, err := angle_diff_180(reported, orientation)] # response errors
ex_data[, prev_ori := shift(orientation), by = observer] # orientation on previous trial

# determine the shift in orientations between trials
ex_data[, diff_in_ori := angle_diff_180(prev_ori, orientation)]
ex_data[, abs_diff_in_ori := abs(diff_in_ori)]
ex_data[, err_rel_to_prev_targ := ifelse(diff_in_ori < 0, -err, err)]

err_dens <- density_asymmetry(ex_data[!is.na(err_rel_to_prev_targ)],
  circ_space = 180, weights_sd = 10, xvar = "abs_diff_in_ori",
  yvar = "err_rel_to_prev_targ", by = c("observer")
)

library(ggplot2)
ggplot(err_dens, aes(x = dist, y = delta)) +
  geom_line(stat = "summary", fun = mean) +
  labs(y = "Asymmetry in error probability density, %", x = "Absolute orientation difference, °")

Compute asymmetry in weighted probability density for discrete data

Description

This function calculates the asymmetry in the probability density of a given variable using kernel density estimation. Unlike density_asymmetry, it does not take an xvar and assumes that the asymmetry is calculated for the whole dataset or subsets defined by by (which could also include a discrete x variable if needed) .

Usage

density_asymmetry_discrete(
  dt,
  yvar = "bias_to_distr_corr",
  circ_space = 180,
  kernel_bw = NULL,
  by = c(),
  n = 181,
  average = T,
  return_full_density = F,
  normalize = T,
  wrap = TRUE,
  exclude_antipode = TRUE,
  n_wraps = 1,
  rescale_narrow = TRUE,
  scale_safety = 0.5
)

Arguments

Value

A data.table with the grouping variables and delta - the difference (asymmetry) in probability density for positive and negative values of yvar; or the full density data if return_full_density is TRUE.

Examples

data(Pascucci_et_al_2019_data)
ex_data <- Pascucci_et_al_2019_data
ex_data[, err := angle_diff_180(reported, orientation)] # response errors
ex_data[, prev_ori := shift(orientation), by = observer] # orientation on previous trial

# determine the shift in orientations between trials
ex_data[, diff_in_ori := angle_diff_180(prev_ori, orientation)]
ex_data[, abs_diff_in_ori := abs(diff_in_ori)]
ex_data[, err_rel_to_prev_targ := ifelse(diff_in_ori < 0, -err, err)]
ex_data[, similarity_discrete := ifelse(abs_diff_in_ori>45, 'Dissimilar', 'Similar')]
err_dens_discrete <- density_asymmetry_discrete(ex_data[!is.na(err_rel_to_prev_targ)],
  circ_space = 180, yvar = "err_rel_to_prev_targ", by = c("observer","similarity_discrete")
)

library(ggplot2)
ggplot(err_dens_discrete, aes(x = similarity_discrete, y = delta))+
geom_violin() + geom_point() +
  labs(y = "Asymmetry in error probability density, %", x = "Previous target")

Get polynomial predictions for values at the boundaries

Description

A helper function for remove_cardinal_biases().

Usage

get_boundary_preds(
  group_i,
  group_label,
  data,
  dist_to_centers_mat,
  space,
  reassign_range,
  poly_deg,
  angle_diff_fun,
  weights = NULL,
  extract_only = FALSE,
  boundary_data = NULL
)

Arguments

Value

a data.table with predicted values

Plots biases using the data from remove_cardinal_biases

Description

Plots biases using the data from remove_cardinal_biases

Usage

make_plots_of_biases(data, poly_deg, sd_val)

Arguments

Value

does not return anything

Pad circular data on both ends

Description

Pad circular data on both ends

Usage

pad_circ(
  data,
  circ_var,
  circ_borders = c(-90, 90),
  circ_part = 1/6,
  verbose = FALSE
)

Arguments

Details

Pads the data by adding a part of the data (default: 1/6th) from one end to another end. Useful to roughly account for circularity when using non-circular methods.

Value

a padded data.table

Examples

dt <- data.table::data.table(x = runif(1000, -90, 90), y = rnorm(1000))
pad_circ(dt, "x", verbose = TRUE)

Compute predictions for circular LOESS

Description

Compute predictions for circular LOESS

Usage

## S3 method for class 'circ_loess'
predict(object, newdata, ...)

Arguments

Value

a data.frame with predictions

Remove cardinal biases

Description

Remove cardinal biases

Usage

remove_cardinal_biases(
  err,
  x,
  space = "180",
  bias_type = "fit",
  plots = "hide",
  poly_deg = 4,
  var_sigma = TRUE,
  var_sigma_poly_deg = 4,
  reassign_at_boundaries = TRUE,
  reassign_range = 2,
  break_points = NULL,
  init_outliers = NULL,
  debug = FALSE,
  do_plots = NULL
)

Arguments

Details

If the bias_type is set to fit, the function computes the cardinal biases in the following way:

1. Create two sets of bins, splitting the stimuli vector into bins centered at cardinal and at oblique directions.

2. For each set of bins, fit a penalised spline (P-spline — a regression spline with a roughness penalty that controls smoothness) for the mean response in each bin. Optionally (see var_sigma), the response variability (SD) is modelled jointly via a log-linear model, allowing the response distribution to vary in width as a function of distance to the nearest cardinal (regardless of whether the bins are centered at the cardinal or at the oblique, the width of the response distribution usually increases as the distance to cardinals increases).

3. Choose the best-fitting model between the one using cardinal and the one using oblique bins.

4. Optionally (see reassign_at_boundaries), reassign observations near bin boundaries to the bin whose fitted model best describes them, iterating until convergence.

5. Compute the residuals of the best-fitting model - that's your bias-corrected error - and the biases (see below).

The bias is computed by flipping the sign of errors when the average predicted error is negative, so, that, for example, if on average the responses are shifted clockwise relative to the true values, the trial-by-trial error would count as bias when it is also shifted clockwise.

If bias_type is set to obl or card, only one set of bins is used, centred at cardinal or oblique angles, respectively.

For additional examples see the help vignette: vignette("cardinal_biases", package = "circhelp")

Value

If plots=='return', returns the three plots showing the biases (combined together with patchwork::wrap_plots()). Otherwise, returns a list with the following elements:

• is_outlier - 0 for outliers (defined as ⁠±3*pred_sigma⁠ for the model with varying sigma or as ⁠±3\*SD⁠ for the simple model)

• pred predicted error

• be_c error corrected for biases (⁠be_c = observed error - pred⁠)

• which_bin the numeric ID of the bin that the stimulus belong to

• bias the bias computed as described above

• bias_typ bias type (cardinal or oblique)

• pred_lin predicted error for a simple linear model for comparison

• pred_sigma predicted SD of the error distribution

• coef_sigma_int, coef_sigma_slope intercept and slope for the sigma prediction

References

• Chetverikov, A., & Jehee, J. F. M. (2023). Motion direction is represented as a bimodal probability distribution in the human visual cortex. Nature Communications, 14(7634). doi:10.1038/s41467-023-43251-w

• van Bergen, R. S., Ma, W. J., Pratte, M. S., & Jehee, J. F. M. (2015). Sensory uncertainty decoded from visual cortex predicts behavior. Nature Neuroscience, 18(12), 1728-1730. doi:10.1038/nn.4150

Examples

# Data in orientation domain from Pascucci et al. (2019, PLOS Bio),
# https://doi.org/10.5281/zenodo.2544946

ex_data <- Pascucci_et_al_2019_data[observer == 4, ]
remove_cardinal_biases(ex_data$err, ex_data$orientation, plots = "show")


# Data in motion domain from Bae & Luck (2018, Neuroimage),
# https://osf.io/2h6w9/
ex_data_bae <- Bae_Luck_2018_data[subject_Num == unique(subject_Num)[5], ]
remove_cardinal_biases(ex_data_bae$err, ex_data_bae$TargetDirection,
  space = "360", plots = "show"
)

# Using a stricter initial outlier boundary

remove_cardinal_biases(ex_data_bae$err, ex_data_bae$TargetDirection,
  space = "360", plots = "show",
  init_outliers = abs(ex_data_bae$err) > 60
)

# We can also use just one bin by setting `bias_type` to custom
# and setting the `break_points` at the ends of the range for x

remove_cardinal_biases(ex_data_bae$err, ex_data_bae$TargetDirection,
  space = "360", bias_type = "custom",
  break_points = c(-180, 180), plots = "show",
  reassign_at_boundaries = FALSE, poly_deg = 8,
  init_outliers = abs(ex_data_bae$err) > 60
)

Remove cardinal biases for data with orientation (color, motion, ...) set in discrete steps

Description

Remove cardinal biases for data with orientation (color, motion, ...) set in discrete steps

Usage

remove_cardinal_biases_discrete(err, x, space, init_outliers = NULL)

Arguments

Value

returns a data.table with the following columns:

• is_outlier - 0 for outliers (defined as ±3*predicted SD, where SD and mean are computed after excluding initially assumed outliers)

• be_c error corrected for biases (⁠be_c = observed error - pred⁠)

Compute smoothed estimate of circular standard deviation

Description

This function calculates smoothed estimates of circular standard deviation using weighted kernel density estimation. It works similarly to density_asymmetry by applying Gaussian weights across an x-axis variable to compute localized circular SD estimates.

Usage

smoothed_circ_sd(
  dt,
  circ_space = 180,
  weights_sd = 10,
  xvar = "abs_td_dist",
  yvar = "err",
  by = c(),
  n_points = 91
)

Arguments

Value

A data.table with the grouping variables, dist - the values of X-axis variable at which the circular SD is computed, and circ_sd - the smoothed circular standard deviation estimate at each point.

Examples

data(Pascucci_et_al_2019_data)
ex_data <- Pascucci_et_al_2019_data
ex_data[, err := angle_diff_180(reported, orientation)] # response errors
ex_data[, prev_ori := shift(orientation), by = observer] # orientation on previous trial

# determine the shift in orientations between trials
ex_data[, diff_in_ori := angle_diff_180(prev_ori, orientation)]
ex_data[, abs_diff_in_ori := abs(diff_in_ori)]

circ_sd_smooth <- smoothed_circ_sd(ex_data[!is.na(diff_in_ori)],
  circ_space = 180, weights_sd = 10, xvar = "abs_diff_in_ori",
  yvar = "err", by = c("observer")
)

library(ggplot2)
ggplot(circ_sd_smooth, aes(x = dist, y = circ_sd)) +
  geom_line(stat = "summary", fun = mean) +
  labs(y = "Circular SD, °", x = "Absolute orientation difference, °")

Conversion between the circular SD and kappa of von Mises

Description

Conversion between the circular SD and kappa of von Mises

Usage

vm_kappa_to_circ_sd(kappa)

vm_kappa_to_circ_sd_deg(kappa)

vm_circ_sd_to_kappa(sd)

vm_circ_sd_deg_to_kappa(sd_deg)

Arguments

Value

vm_kappa_to_circ_sd and vm_kappa_to_circ_sd_deg return circular SD (in radians or degrees, respectively) corresponding to a given kappa. vm_circ_sd_to_kappa and vm_circ_sd_deg_to_kappa return kappa corresponding to a given circular SD (in radians or degrees, respectively).

Functions

• vm_kappa_to_circ_sd_deg(): get circular SD (in degrees) from kappa

• vm_circ_sd_to_kappa(): get kappa from circular SD (in radians)

• vm_circ_sd_deg_to_kappa(): get kappa from circular SD (in degrees)

Examples

vm_kappa <- 5
vm_sd <- vm_kappa_to_circ_sd(vm_kappa)

vm_circ_sd_to_kappa(vm_sd)

vm_kappa_to_circ_sd(vm_circ_sd_to_kappa(vm_sd))

Weighted circular parameters

Description

Weighted circular parameters

Usage

weighted_circ_mean(x, w, period = NULL, na.rm = FALSE)

weighted_circ_mean2(x, w, na.rm = FALSE)

weighted_circ_sd(x, w, na.rm = FALSE)

weighted_circ_rho(x, w, na.rm = FALSE)

Arguments

Value

weighted mean of values in the vector

Functions

• weighted_circ_mean(): weighted circular mean

• weighted_circ_mean2(): an alternative way to compute weighted circular mean (the results are the same)

• weighted_circ_sd(): weighted circular SD

• weighted_circ_rho(): weighted mean resultant length

Examples

x <- rnorm(1000, 0, 0.5)
w <- runif(1000, 0, 1)
weighted.mean(x, w)
weighted_circ_mean(x, w)

Weighted standard error of the mean (SEM_w)

Description

Computes the variance of a weighted mean following the definitions given by Kirchner (2006).

Usage

weighted_sem(x, w, na.rm = FALSE)

Arguments

Details

James Kirchner describes two different cases when the weighted variance is computed. The code here implements Case I where "one wants to give more weight to some points than to others, because they are considered to be more important" and "the weights differ but the uncertainties associated with the individual xi are assumed to be the same" (Kirchner, 2006, p. 1). The formula used is: \[SEM_w = \sqrt{\left(\sum_{i = 1}^{N} (w_{i} x_i^2)-\bar{x}^2\right)\frac{\sum_{i = 1}^{N} w_i^2}{1-\sum_{i = 1}^{N} w_i^2}} \] The expected error is within 5% of the bootstrapped SEM (at larger sample sizes).

Value

weighted standard error of the mean

References

• Kirchner, J. 2006. Data Analysis Toolkit #12: Weighted averages and their uncertainties. https://seismo.berkeley.edu/~kirchner/Toolkits/Toolkit_12.pdf. Retrieved on 04.07.2024.

• Bevington, P. R. 1969. Data Reduction and Error Analysis for the Physical Sciences. McGraw-Hill, 336 pp.

Examples

set.seed(1)
n_obs <- 200
w <- runif(n_obs)
w <- w / sum(w)
x <- rnorm(n_obs, sd = 5)
weighted_sem(x, w)