Beale’s estimator and sample size calculation

2021-08-13

1 Introduction

The R package funtimes contains the function beales that can be used to implement Beale’s (Beale 1962) ratio estimator for estimating total value. The function also calculates recommended sample size for desired confidence level and absolute or relative error.

The Beale’s estimator is often used in ecology to compute total pollutant load, \(\widehat{Y}\), given a sample of the loads \(y_i\) and corresponding river flow or discharges, \(x_i\) (\(i = 1,\ldots,n\)): \[ \widehat{Y} =X\frac{\bar{y}}{\bar{x}}\frac{\left( 1+ \theta\frac{s_{xy}}{\bar{x}\bar{y}}\right)}{\left( 1+\theta\frac{s^2_x}{\bar{x}^2} \right)}, \] where \(\theta=n^{-1} - N^{-1}\), \(s_{xy}=(n-1)^{-1}\sum_{i=1}^n(x_i-\bar{x})(y_i-\bar{y})\), and \(s^2_{x}=(n-1)^{-1}\sum_{i=1}^n(x_i-\bar{x})^2\). Total flow, \(X=\sum_{i=1}^Nx_i\), is assumed to be known. If the data set for flow contains only \(n'\) observations (\(n\leqslant n'< N\)), we use an estimate \(\widehat{X}=\frac{N}{n'}\sum_{i=1}^{n'}x_i\) following formula (2.8) in Thompson (2012).

Help file for the function can be opened from R with:

?beales

The function uses the following groups of arguments as its inputs.

2 General case, when all discharge data are known

The ideal case is when all discharge data are know, and only some measurements of loads are missing.

The inputs should be organized in vectors of same length. Consider a toy example where ten measurements cover the whole period of interest (i.e., the population size N = 10):

discharge <- c(60, 50, 90, 100, 80, 90, 100, 90, 80, 70)
loads <- c(33, 22, 44, 48, NA, 44, 49, NA, NA, 36)

NAs stand for missing values.

To estimate the total load for this period, use:

B10 <- beales(x = discharge, y = loads)
# [1] "Beale's estimate of the total (for population size 10) is 399.176 with 95% confidence interval from 391.315 to 407.037."

By default (the setting verbose = TRUE), the function shows text output. All estimates have been saved in the object B10 and can be extracted from there. For example, see the list of elements saved in B10, then extract the standard error of the load estimate, and population size:

ls(B10)
# [1] "CI"       "N"        "estimate" "level"    "n"        "se"
B10$se
# [1] 4.010797
B10$N
# [1] 10

If a different level of confidence (default is 95%) is needed, set it using the argument level:

B11 <- beales(x = discharge, y = loads, level = 0.9)
# [1] "Beale's estimate of the total (for population size 10) is 399.176 with 90% confidence interval from 392.578 to 405.773."

To suppress the text outputs, use verbose = FALSE:

B12 <- beales(x = discharge, y = loads, level = 0.9, verbose = FALSE)

3 Common case, when some discharge data are missing

It is common that some discharge data are missing. The function fills-in the missing discharge measurements with average estimates automatically. For example, now the first discharge value is missing:

discharge2 <- c(NA, 50, 90, 100, 80, 90, 100, 90, 80, 70)
loads2 <- c(33, 22, 44, 48, NA, 44, 49, NA, NA, 36)

The NA in discharge will be replaced by an average value of the non-missing measurements, and the first pair of discharge and load (average discharge and the corresponding load of 33) will be still used in estimating covariance and other quantities. Simply use the function in the same way as above:

B20 <- beales(x = discharge2, y = loads2)
# [1] "Beale's estimate of the total (for population size 10) is 394.35 with 95% confidence interval from 381.569 to 407.131."

In another case, both discharge and load data might be missing. If they are not represented at all in the data vectors (e.g., by NAs), a simple trick is to set the population size, N, which is one of the arguments in the function. For example, if the data above are ten monthly measurements, and an estimate for the whole year (12 months) is required, set N = 12:

B21 <- beales(x = discharge2, y = loads2, N = 12)
# [1] "Beale's estimate of the total (for population size 12) is 473.25 with 95% confidence interval from 455.161 to 491.339."

which is equivalent to adding two missing values to each vector, like this:

discharge22 <- c(discharge2, NA, NA)
loads22 <- c(loads2, NA, NA)
B22 <- beales(x = discharge22, y = loads22)
# [1] "Beale's estimate of the total (for population size 12) is 473.25 with 95% confidence interval from 455.161 to 491.339."

4 Sample size calculation

The other two arguments of the function, p and d, allow the user to set the desired relative error or margin of error, respectively, for sample size calculations. (If both p and d are defined, the calculations will run for p.) The estimated sample size, \(\hat{n}\), is added to the output list as the element nhat, and an additional sentence is printed out at the output.

For example, using our simulated data for 10 months out of 12, estimate the sample size needed to estimate the total yearly load with the relative error up to 5%:

B30 <- beales(x = discharge2, y = loads2, N = 12, p = 0.05)
# [1] "Beale's estimate of the total (for population size 12) is 473.25 with 95% confidence interval from 455.161 to 491.339."
# [1] "To obtain a 95% confidence interval with a relative error of 5%, a sample of size 6 is required."

What if increase the confidence of such interval (notice the differences in the last line of the output):

B31 <- beales(x = discharge2, y = loads2, N = 12, p = 0.05, level = 0.99)
# [1] "Beale's estimate of the total (for population size 12) is 473.25 with 99% confidence interval from 449.477 to 497.024."
# [1] "To obtain a 99% confidence interval with a relative error of 5%, a sample of size 8 is required."

Similarly, when the margin of error is set:

B32 <- beales(x = discharge2, y = loads2, N = 12, d = 15)
# [1] "Beale's estimate of the total (for population size 12) is 473.25 with 95% confidence interval from 455.161 to 491.339."
# [1] "To obtain a 95% confidence interval with a margin of error being 15, a sample of size 9 is required."

The estimated sample size can be extracted as follows:

B32$nhat
# [1] 9

5 Notes

  1. The function will not run if the inputs x and y are of different lengths.
  2. The reported sample size n is the number of non-missing values in y (missing values in x are automatically replaced with an average of non-missing x).
  3. The function will not run if the argument N is set such that N < length(x) (more discharge samples than possible in a given period) or if N <= n (sample size is bigger than or equals the population size). In the case when N = n, no estimation is needed, because the total load can be calculated just by summing up all individual loads.
  4. The form of the Beale’s estimator assumes n > 1 (for estimating the variances and covariance), and \(\bar{x}\neq 0\) and \(\bar{y}\neq 0\).

References

Beale, E. M. L. 1962. “Some Uses of Computers in Operational Research.” Industrielle Organisation 31 (1): 27–28.
Thompson, S. K. 2012. Sampling. 3rd ed. Hoboken: Wiley.