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Passing-Bablok Regression for Method Comparison

Source: R/pb_reg.R
pb_reg.Rd

[Experimental]

A robust, nonparametric method for fitting a straight line to two-dimensional data where both variables (X and Y) are measured with error. Particularly useful for method comparison studies.

Usage

pb_reg(
  formula,
  data,
  id = NULL,
  method = c("scissors", "symmetric", "invariant"),
  conf.level = 0.95,
  weights = NULL,
  error.ratio = 1,
  replicates = 0,
  se_method = c("bootstrap", "jackknife", "dufey"),
  model = TRUE,
  keep_data = TRUE,
  ...
)

Arguments

formula

A formula of the form y ~ x specifying the model.

data

Data frame with all data.

id

Column with subject identifier (optional). If provided, measurement error ratio is calculated from replicate measurements.

method

Method for Passing-Bablok estimation. Options are:

  • "scissors": Scissors estimator (1988) - most robust, scale invariant (default)

  • "symmetric": Original Passing-Bablok (1983) - symmetric around 45-degree line

  • "invariant": Scale-invariant method (1984) - adaptive reference line

conf.level

The confidence level required. Default is 95%.

weights

An optional vector of case weights to be used in the fitting process. Should be NULL or a numeric vector.

error.ratio

Ratio of measurement error variances (var(x)/var(y)). Default is 1. This argument is ignored if subject identifiers are provided via id.

replicates

Number of resamples for confidence intervals and the variance-covariance matrix. For se_method = "bootstrap" this is the number of bootstrap resamples; for se_method = "jackknife" it is the number of random delete-d subsets. If 0 (default), analytical confidence intervals are used and no variance-covariance matrix is returned (unless se_method = "dufey"). Resampling is recommended for weighted data and 'invariant' or 'scissors' methods.

se_method

Method used to estimate the variance-covariance matrix and confidence intervals of the coefficients. Options are:

  • "bootstrap": Nonparametric pairs (case) bootstrap with percentile confidence intervals (default). Used when replicates > 0.

  • "jackknife": Delete-d jackknife with d = floor(n/2), using replicates random subsets (Shao & Wu, 1989). Used when replicates > 0.

  • "dufey": [Experimental] Analytic sandwich estimator of Dufey (2020). Does not require resampling (replicates is ignored). Only available for the "scissors" method without case weights and with error.ratio = 1. See Details before using it for joint tests.

model

Logical. If TRUE (default), the model frame is stored in the returned object. This is needed for methods like plot(), fitted(), residuals(), and predict() to work without supplying data. If FALSE, the model frame is not stored (saves memory for large datasets), but these methods will require a data argument.

keep_data

Logical indicator (TRUE/FALSE). If TRUE, intermediate calculations are returned; default is FALSE.

...

Additional arguments (currently unused).

Value

The function returns a simple_eiv object with the following components:

  • coefficients: Named vector of coefficients (intercept and slope).

  • residuals: Residuals from the fitted model.

  • fitted.values: Predicted Y values.

  • model_table: Data frame presenting the full results from the Passing-Bablok regression.

  • vcov: Variance-covariance matrix for slope and intercept (if resampling or se_method = "dufey" is used; otherwise NULL).

  • df.residual: Residual degrees of freedom.

  • call: The matched call.

  • terms: The terms object used.

  • model: The model frame.

  • x_vals: Original x values used in fitting.

  • y_vals: Original y values used in fitting.

  • weights: Case weights (if provided).

  • error.ratio: Error ratio used in fitting.

  • conf.level: Confidence level used.

  • method: Character string describing the method.

  • method_num: Numeric method identifier (1, 2, or 3).

  • kendall_test: Results of Kendall's tau correlation test.

  • cusum_test: Results of CUSUM linearity test.

  • n_slopes: Number of slopes used in estimation.

  • boot: Resampling results (if replicates > 0 and se_method is "bootstrap" or "jackknife").

  • se_method: Method actually used for standard errors ("analytic", "bootstrap", "jackknife", or "dufey").

Details

Passing-Bablok regression is a robust nonparametric method that estimates the slope as the shifted median of all possible slopes between pairs of points. The intercept is then calculated as the median of y - slope*x. This method is particularly useful when:

  • Both X and Y are measured with error

  • You want a robust method not sensitive to outliers

  • The relationship is assumed to be linear

  • X and Y are highly positively correlated

Methods

Three Passing-Bablok methods are available:

"scissors" (default): The scissors estimator (1988), most robust and scale-invariant. Uses the median of absolute values of angles.

"symmetric": The original method (1983), symmetric about the y = x line. Uses the line y = -x as the reference for partitioning points.

"invariant": Scale-invariant method (1984). First finds the median angle of slopes below the horizontal, then uses this as the reference line.

Measurement Error Handling

If the data are measured in replicates, then the measurement error ratio can be directly derived from the data. This can be accomplished by indicating the subject identifier with the id argument. When replicates are not available in the data, then the ratio of error variances (var(x)/var(y)) can be provided with the error.ratio argument (default = 1, indicating equal measurement errors).

The error ratio affects how pairwise slopes are weighted in the robust median calculation. When error.ratio = 1, all pairs receive equal weight. When error.ratio != 1, pairs are weighted to account for heterogeneous measurement precision.

Weighting

Case weights can be provided via the weights argument. These are distinct from measurement error weighting (controlled by error.ratio). Case weights allow you to down-weight or up-weight specific observations in the analysis.

Standard Errors and Variance-Covariance Matrix

The analytical (Passing & Bablok, 1983) confidence intervals do not provide a covariance between the intercept and slope, so a variance-covariance matrix (needed by joint_test() and plot_joint()) is only returned when one of the following is used:

  • Pairs bootstrap (se_method = "bootstrap", replicates > 0): whole (x, y) observations are resampled with replacement and the model is refit. The variance-covariance matrix is the covariance of the bootstrap estimates and the confidence intervals are percentile intervals.

  • Delete-d jackknife (se_method = "jackknife", replicates > 0): the model is refit on replicates random subsets that each leave out d = floor(n/2) observations. The delete-1 jackknife is inconsistent for median-type estimators such as Passing-Bablok; deleting d observations with \(\sqrt{n}/d \to 0\) restores consistency (Shao & Wu, 1989). Confidence intervals are Wald-type intervals using a t(n-2) quantile.

  • Dufey (2020) (se_method = "dufey"): an analytic, distribution-free estimator for the equivariant ("scissors") Passing-Bablok estimator, based on the U-statistic variance of Kendall's tau and a sandwich estimator for the intercept. The slope interval is formed from order statistics of the pairwise slopes and the intercept interval is Wald-type. This option is experimental. In simulations its marginal standard errors were accurate for n >= 20, but joint tests of intercept and slope (joint_test()) rejected a true null hypothesis about 7-11% of the time at a nominal 5%, suggesting the intercept-slope covariance is not yet reliable. Prefer the pairs bootstrap or delete-d jackknife for joint tests.

Resampling is particularly useful for:

  • Weighted regression (case weights or error.ratio != 1)

  • Methods 'invariant' and 'scissors' (where analytical CI validity is uncertain)

  • Joint tests of intercept and slope

The method automatically:

  • Tests for high positive correlation using Kendall's tau

  • Tests for linearity using a CUSUM test

  • Computes confidence intervals (analytical or bootstrap)

References

Passing, H., & Bablok, W. (1983). A New Biometrical Procedure for Testing the Equality of Measurements from Two Different Analytical Methods. Application of linear regression procedures for method comparison studies in Clinical Chemistry, Part I. Cclm, 21(11), 709-720. doi: 10.1515/cclm.1983.21.11.709

Passing, H., & Bablok, W. (1984). Comparison of Several Regression Procedures for Method Comparison Studies and Determination of Sample Sizes Application of linear regression procedures for method comparison studies in Clinical Chemistry, Part II. Clinical Chemistry and Laboratory Medicine, 22(6). doi: 10.1515/cclm.1984.22.6.431

Bablok, W., Passing, H., Bender, R., & Schneider, B. (1988). A General Regression Procedure for Method Transformation. Application of Linear Regression Procedures for Method Comparison Studies in Clinical Chemistry, Part III. Clinical Chemistry and Laboratory Medicine, 26(11). doi: 10.1515/cclm.1988.26.11.783

Dufey, F. (2020). Derivation of Passing-Bablok regression from Kendall's tau. The International Journal of Biostatistics, 16(2), 20190157. doi: 10.1515/ijb-2019-0157

Sen, P. K. (1968). Estimates of the regression coefficient based on Kendall's tau. Journal of the American Statistical Association, 63(324), 1379-1389. doi: 10.1080/01621459.1968.10480934

Shao, J., & Wu, C. F. J. (1989). A general theory for jackknife variance estimation. The Annals of Statistics, 17(3), 1176-1197. doi: 10.1214/aos/1176347263

Examples

if (FALSE) { # \dontrun{
# Basic Passing-Bablok regression (scissors method, default)
model <- pb_reg(method2 ~ method1, data = mydata)

# With known error ratio
model_er <- pb_reg(method2 ~ method1, data = mydata, error.ratio = 2)

# With replicate measurements
model_rep <- pb_reg(method2 ~ method1, data = mydata, id = "subject_id")

# With bootstrap confidence intervals
model_boot <- pb_reg(method2 ~ method1, data = mydata,
                     error.ratio = 1.5, replicates = 1000)

# Delete-d jackknife or analytic (Dufey 2020) variance-covariance matrix
model_jack <- pb_reg(method2 ~ method1, data = mydata,
                     se_method = "jackknife", replicates = 1000)
model_dufey <- pb_reg(method2 ~ method1, data = mydata, se_method = "dufey")
joint_test(model_dufey)

# Symmetric method
model_sym <- pb_reg(method2 ~ method1, data = mydata, method = "symmetric")

# Scale-invariant method
model_inv <- pb_reg(method2 ~ method1, data = mydata, method = "invariant")

# With case weights
model_wt <- pb_reg(method2 ~ method1, data = mydata,
                   weights = mydata$case_weights)

# View results
print(model)
summary(model)
plot(model)
} # }