Computes the sample weighted variance of a numeric vector using precision weights. This function implements the standard unbiased weighted variance estimator commonly used in statistical applications where observations have different precisions or levels of confidence.
Arguments
- x
numeric vector; the data values for which to calculate weighted variance. Missing values are allowed if
na.rm = TRUE- w
numeric vector; the precision weights corresponding to each observation in
x. These represent how much confidence to place in each measurement (higher = more trusted). Must be the same length asx. Should be non-negative- na.rm
logical; if
TRUE, missing values (bothNAandNaN) are removed before computation. IfFALSE(default), missing values will cause the function to returnNA
Value
numeric; the weighted sample variance. Returns NA if insufficient
data or if na.rm = FALSE and missing values are present
Details
The weighted variance is calculated using the formula: $$s^2_w = \frac{\sum w_i}{\sum w_i^2 - (\sum w_i)^2} \sum w_i (x_i - \bar{x}_w)^2$$
where \(\bar{x}_w\) is the weighted mean and \(w_i\) are the weights.
This function uses precision weights (also called reliability weights), which represent how much confidence or trust to place in each observation, rather than frequency weights that represent how many times to count each observation.
The denominator correction (\(\sum w_i^2 - (\sum w_i)^2\)) provides an unbiased estimator for precision weights. Examples of precision weights include probabilities (0-1), measurement confidence scores, or inverse error variances.
See also
weighted.mean for weighted mean calculation,
var for unweighted sample variance
Examples
# Example 1: Basic weighted variance calculation
values <- c(2.1, 3.5, 1.8, 4.2, 2.9)
weights <- c(0.8, 0.3, 0.9, 0.5, 0.7)
weighted_var(values, weights)
#> [1] 0.9463317
# Example 2: Compare with unweighted variance
x <- 1:10
equal_weights <- rep(1, 10)
unweighted_var <- var(x)
weighted_var_equal <- weighted_var(x, equal_weights)
# Example 3: Using precision weights
# Measurements with different levels of confidence
measurements <- c(10.2, 9.8, 10.5, 9.9, 10.1)
confidence_levels <- c(0.9, 0.6, 0.8, 0.95, 0.7) # How much we trust each measurement
precision_weighted_var <- weighted_var(measurements, confidence_levels)
# Example 4: Handling missing values
x_with_na <- c(1, 2, NA, 4, 5)
weights_with_na <- c(0.2, 0.3, 0.1, 0.8, 0.4)
# This will return NA
result_na <- weighted_var(x_with_na, weights_with_na, na.rm = FALSE)
# This will calculate after removing NA
result_removed <- weighted_var(x_with_na, weights_with_na, na.rm = TRUE)
# Example 5: Different weight patterns and their effects
data_points <- c(10, 15, 20, 25, 30)
# Equal weights (should approximate unweighted variance)
equal_wts <- rep(1, 5)
var_equal <- weighted_var(data_points, equal_wts)
# Emphasizing central values
central_wts <- c(0.1, 0.3, 1.0, 0.3, 0.1)
var_central <- weighted_var(data_points, central_wts)
# Emphasizing extreme values
extreme_wts <- c(1.0, 0.1, 0.1, 0.1, 1.0)
var_extreme <- weighted_var(data_points, extreme_wts)