Variance Methods: Taylor, Bootstrap, and Jackknife

tidycreel exposes three variance estimation methods through the variance argument of estimate_effort(), estimate_catch_rate(), and related functions:

`variance =`	Description	Requires
`"taylor"`	Taylor linearization (default)	≥ 2 PSUs per stratum recommended
`"bootstrap"`	Bootstrap resampling (500 replicates)	≥ 2 PSUs per stratum
`"jackknife"`	Delete-one jackknife	≥ 2 PSUs per stratum

library(tidycreel)

1 Building a design for the examples

All three methods work on the same creel_design object. Here we use the bundled example_counts and example_interviews datasets, which have 10 sampled weekdays and 4 sampled weekends.

data("example_counts")
data("example_interviews")

cal <- unique(example_counts[, c("date", "day_type")])
design <- creel_design(cal, date = date, strata = day_type)
design <- add_counts(design, example_counts)
design <- add_interviews(
  design, example_interviews,
  catch = catch_total,
  effort = hours_fished,
  trip_status = trip_status
)
#> ℹ Added 22 interviews: 17 complete (77%), 5 incomplete (23%)

2 Comparing the three variance methods

Pass variance = "..." to any estimator. The point estimate is identical across methods; only the SE and CI change.

eff_taylor <- estimate_effort(design, variance = "taylor")
eff_bootstrap <- estimate_effort(design, variance = "bootstrap")
eff_jackknife <- estimate_effort(design, variance = "jackknife")

# Summarise all three for comparison
rbind(
  as.data.frame(summary(eff_taylor)),
  as.data.frame(summary(eff_bootstrap)),
  as.data.frame(summary(eff_jackknife))
)
#>   Estimate    SE CI Lower CI Upper  N
#> 1    372.5 13.18    343.8    401.2 14
#> 2    372.5 13.20    343.7    401.3 14
#> 3    372.5 13.18    343.8    401.2 14

With 10–14 PSUs per stratum the three methods produce very similar standard errors. This convergence is expected: bootstrap and jackknife are asymptotically equivalent to Taylor linearization for simple probability samples.

3 When to use each method

Taylor linearization (`"taylor"`, default)

Use Taylor linearization when between-day variability dominates within strata and the design is a straightforward stratified random sample of days. It is fast, uses a closed-form variance formula, and introduces no additional randomness. It relies on large-sample approximations, however, so it can be less reliable for highly skewed counts or very small strata.

Bootstrap (`"bootstrap"`)

Bootstrap is useful for highly skewed count distributions or complex designs, and when you want variance estimates that do not depend on a distributional assumption. It captures nonlinear variance propagation, but is stochastic: set a seed for reproducibility. It is also slower (500 replicates by default) and requires at least two PSUs per stratum.

Jackknife (`"jackknife"`)

Choose the jackknife when you prefer a deterministic delete-one estimate that is slightly more conservative than Taylor for small samples. It is often more stable than bootstrap with few PSUs, but still needs at least two PSUs per stratum. For unequal inclusion probabilities, use the appropriate jackknife weights.

4 Grouped estimates

All three methods work with the by argument.

eff_grp_taylor <- estimate_effort(design, by = day_type, variance = "taylor")
eff_grp_bootstrap <- estimate_effort(design, by = day_type, variance = "bootstrap")

rbind(
  cbind(method = "taylor", as.data.frame(summary(eff_grp_taylor))),
  cbind(method = "bootstrap", as.data.frame(summary(eff_grp_bootstrap)))
)
#>      method day_type Estimate     SE CI Lower CI Upper  N
#> 1    taylor  weekday    170.6  9.674    149.5    191.7 10
#> 2    taylor  weekend    201.9  8.946    182.4    221.4  4
#> 3 bootstrap  weekday    170.6 10.230    148.3    192.9 10
#> 4 bootstrap  weekend    201.9  8.604    183.2    220.6  4

5 Single-PSU diagnostic

All three variance methods require at least 2 PSUs (sampled days) per stratum. When a stratum has only 1 sampled day, tidycreel raises a structured error naming the stratum:

# Build a minimal design with 1 sampled day per stratum
dates_1psu <- c(as.Date("2024-06-03"), as.Date("2024-06-08"))
cal_1psu <- data.frame(date = dates_1psu, day_type = c("weekday", "weekend"))
d_1psu <- creel_design(cal_1psu, date = date, strata = day_type)
counts_1psu <- data.frame(
  date     = dates_1psu,
  day_type = c("weekday", "weekend"),
  count    = c(10L, 15L)
)
d_1psu <- add_counts(d_1psu, counts_1psu)

# Taylor linearization raises an error when variance cannot be estimated
estimate_effort(d_1psu, variance = "jackknife")
#> Error in `value[[3L]]()`:
#> ! Stratum "weekday" has only 1 PSU — jackknife variance cannot be
#>   estimated.
#> ✖ A stratum must have at least 2 PSUs for jackknife resampling.
#> ℹ Increase the sampling rate for stratum "weekday", or use `variance =
#>   'taylor'`.

The error message names the problematic stratum ("weekday") and suggests either increasing the sampling rate or switching to variance = "taylor" which can still produce a point estimate via the lonely-PSU adjustment.

6 Summary

Variance method comparison
Method	Deterministic	Min PSUs	Speed	When to prefer
taylor	Yes	1 (with lonely-PSU adjust)	Fast	Default; simple designs
bootstrap	No (stochastic)	2	Slow (500 replicates)	Skewed counts; complex designs
jackknife	Yes	2	Moderate	Small samples; deterministic alternative to bootstrap