Flexible Count Estimation

Introduction

Creel surveys estimate angler effort — the total angler-hours or angler-trips on a waterbody during a survey period. The count data collected in the field can take two forms:

Instantaneous counts (Hoenig et al. 1993): At a random time within the survey period, an observer records the number of anglers visible on a section. Effort is estimated as count × total open hours.
Progressive counts (Pope et al., in press): An observer travels the entire section (one “circuit”) and records the number of anglers encountered along the way. Effort is estimated as count × circuit travel time × section open hours / circuit travel time, i.e., Ê_d = C × τ × (T_d / τ) = C × T_d where τ is circuit travel time and T_d is the total open hours on day d.

In addition, observers may complete multiple counts per day (e.g., morning and afternoon circuits). When multiple counts are recorded per sampling unit (PSU), tidycreel aggregates them automatically and decomposes variance into a between-day component and a within-day component via the Rasmussen (1994) two-stage formula.

This vignette demonstrates all three workflows using inline data.

Single Count per Day (Instantaneous Baseline)

The classic workflow records one instantaneous count per PSU (day × stratum).

library(tidycreel)

# Survey calendar: four days, two strata
calendar <- data.frame(
  date = as.Date(c("2024-06-01", "2024-06-02", "2024-06-03", "2024-06-04")),
  day_type = c("weekday", "weekday", "weekend", "weekend"),
  open_hours = rep(10, 4)
)

design <- creel_design(calendar, date = date, strata = day_type)

# One count per day
counts <- data.frame(
  date      = as.Date(c("2024-06-01", "2024-06-02", "2024-06-03", "2024-06-04")),
  day_type  = c("weekday", "weekday", "weekend", "weekend"),
  n_anglers = c(15L, 23L, 45L, 52L)
)

design <- add_counts(design, counts)
#> Warning in svydesign.default(ids = psu_formula, strata = strata_formula, : No
#> weights or probabilities supplied, assuming equal probability
result <- estimate_effort(design)
#> Warning: 2 strata have fewer than 3 observations:
#> • Stratum weekday: 2 observations
#> • Stratum weekend: 2 observations
#> ! Sparse strata produce unstable variance estimates.
#> ℹ Consider combining sparse strata or collecting more data.
result$estimates
#> # A tibble: 1 × 7
#>   estimate    se se_between se_within ci_lower ci_upper     n
#>      <dbl> <dbl>      <dbl>     <dbl>    <dbl>    <dbl> <int>
#> 1      135  10.6       10.6         0     89.3     181.     4

With a single count per day, se_within is always zero — there is no within-day replication to estimate within-day variance. The se column equals se_between.

Multiple Counts per Day

When observers complete multiple circuits on the same day, each circuit is a sub-sample within the PSU. Supply the circuit identifier column via count_time_col.

# Two circuits per day: morning ("am") and afternoon ("pm")
calendar_m <- data.frame(
  date = as.Date(c("2024-06-01", "2024-06-02", "2024-06-03", "2024-06-04")),
  day_type = c("weekday", "weekday", "weekend", "weekend"),
  open_hours = rep(10, 4)
)

design_m <- creel_design(calendar_m, date = date, strata = day_type)

multi_counts <- data.frame(
  date = as.Date(rep(c(
    "2024-06-01", "2024-06-02",
    "2024-06-03", "2024-06-04"
  ), each = 2)),
  day_type = rep(c("weekday", "weekday", "weekend", "weekend"), each = 2),
  count_time = rep(c("am", "pm"), 4),
  n_anglers = c(12L, 18L, 20L, 26L, 40L, 50L, 48L, 56L)
)

design_m <- add_counts(design_m, multi_counts, count_time_col = count_time)
#> Warning in svydesign.default(ids = psu_formula, strata = strata_formula, : No
#> weights or probabilities supplied, assuming equal probability
result_m <- estimate_effort(design_m)
#> Warning: 2 strata have fewer than 3 observations:
#> • Stratum weekday: 2 observations
#> • Stratum weekend: 2 observations
#> ! Sparse strata produce unstable variance estimates.
#> ℹ Consider combining sparse strata or collecting more data.
result_m$estimates
#> # A tibble: 1 × 7
#>   estimate    se se_between se_within ci_lower ci_upper     n
#>      <dbl> <dbl>      <dbl>     <dbl>    <dbl>    <dbl> <int>
#> 1      135  13.1       10.6      7.68     78.6     191.     4

The output tibble now contains nonzero se_within values. Total variance is se^2 = se_between^2 + se_within^2 (combined in quadrature). When within-day angler counts vary substantially between circuits, se_within can be the dominant variance component.

When does `se_within` matter?

se_within is small when counts are stable throughout the day (e.g., a lake where angler density is roughly constant). It is large when activity patterns are strongly bimodal (e.g., a river with a morning bite and an evening bite but few mid-day anglers). Collecting multiple counts per day is most valuable when such within-day variation is expected and needs to be quantified rather than assumed away.

Progressive Count Type

Progressive surveys replace a spot-check count with a complete traversal of the section. The observer counts every angler encountered during one circuit of duration τ hours. The estimator is:

$\hat{E}_d = C_d \times T_d$

where C_d is the count on day d and T_d is the total open hours on day d (the period_length_col argument). The circuit time τ cancels algebraically; it is only needed internally to compute the expansion factor κ = T_d / τ.

calendar_p <- data.frame(
  date       = as.Date(c("2024-06-01", "2024-06-02", "2024-06-03", "2024-06-04")),
  day_type   = c("weekday", "weekday", "weekend", "weekend"),
  open_hours = rep(8, 4)
)

design_p <- creel_design(calendar_p, date = date, strata = day_type)

prog_counts <- data.frame(
  date = as.Date(c("2024-06-01", "2024-06-02", "2024-06-03", "2024-06-04")),
  day_type = c("weekday", "weekday", "weekend", "weekend"),
  n_anglers = c(15L, 23L, 45L, 52L),
  shift_hours = rep(8, 4)
)

design_p <- add_counts(
  design_p, prog_counts,
  count_type = "progressive",
  circuit_time = 2,
  period_length_col = shift_hours
)
#> Warning in svydesign.default(ids = psu_formula, strata = strata_formula, : No
#> weights or probabilities supplied, assuming equal probability

result_p <- estimate_effort(design_p)
#> Warning: 2 strata have fewer than 3 observations:
#> • Stratum weekday: 2 observations
#> • Stratum weekend: 2 observations
#> ! Sparse strata produce unstable variance estimates.
#> ℹ Consider combining sparse strata or collecting more data.
result_p$estimates
#> # A tibble: 1 × 7
#>   estimate    se se_between se_within ci_lower ci_upper     n
#>      <dbl> <dbl>      <dbl>     <dbl>    <dbl>    <dbl> <int>
#> 1     1080  85.0       85.0         0     714.    1446.     4

Worked example (Pope et al.)

Pope et al. (in press) give a worked example with C = 234 anglers encountered during a 2-hour circuit (τ = 2 h) on a day with 8 open hours (T_d = 8 h):

$\hat{E}_d = 234 \times 8 = 1{,}872 \text{ angler-hours}$

The code below reproduces this per-day calculation for a two-day survey. The processed design’s $counts slot stores the expanded per-day effort (Ê_d), so we can confirm the first-day value equals 1,872 angler-hours directly.

# Reproduce Pope et al. per-day calculation
cal_pope <- data.frame(
  date       = as.Date(c("2024-06-01", "2024-06-02")),
  day_type   = c("weekday", "weekday"),
  open_hours = c(8, 8)
)
design_pope <- creel_design(cal_pope, date = date, strata = day_type)

cnt_pope <- data.frame(
  date        = as.Date(c("2024-06-01", "2024-06-02")),
  day_type    = c("weekday", "weekday"),
  n_anglers   = c(234L, 200L),
  shift_hours = c(8, 8)
)

design_pope <- add_counts(
  design_pope, cnt_pope,
  count_type = "progressive",
  circuit_time = 2,
  period_length_col = shift_hours
)
#> Warning in svydesign.default(ids = psu_formula, strata = strata_formula, : No
#> weights or probabilities supplied, assuming equal probability

# Per-day Ê_d values stored in the design (first day = 1,872 angler-hours)
design_pope$counts
#>         date day_type n_anglers
#> 1 2024-06-01  weekday      1872
#> 2 2024-06-02  weekday      1600

The n_anglers column in design_pope$counts shows 1,872 for June 1st — matching the Pope et al. formula exactly.

Interpreting `se_between` and `se_within`

Column	Source	Zero when
`se_between`	Between-PSU variance via `survey::svytotal()`	Never (always estimated)
`se_within`	Rasmussen (1994) two-stage within-day formula	Single count per PSU
`se`	Combined SE: `sqrt(se_between^2 + se_within^2)`	Never

Practical guidance:

If you collect one count per PSU, se_within = 0 and se = se_between. The classical creel estimator is recovered exactly.
If you collect multiple counts per PSU, monitor the ratio se_within / se. A large ratio (> 0.5) indicates substantial within-day variation and suggests that daily sampling intensity (number of circuits) is a meaningful driver of precision.
For progressive surveys, the estimator variance structure is the same as for instantaneous counts. The key difference is in how the single-PSU effort Ê_d is calculated.

References

Hoenig, J. M., Robson, D. S., Jones, C. M., and Pollock, K. H. (1993). Scheduling counts in the instantaneous and progressive count methods for estimating sportfishing effort. North American Journal of Fisheries Management, 13, 723–736.
Rasmussen, P. W. (1994). Two-stage variance estimation for creel surveys. Cited in Su, Y.-S., and Liu, P. (2025), equations 17–18.
Su, Y.-S., and Liu, P. (2025). Flexible creel survey estimators. Canadian Journal of Fisheries and Aquatic Sciences, 82, 1–27.