Length at maternity analysis

Background

A maturity ogive describes the proportion of females that are sexually mature at a given length or age. For most teleosts that is sufficient for calculating reproductive capacity, since mature females spawn every year. Chondrichthyans differ. Many species reproduce biennially or triennially, so that in any one year only a fraction of mature females are gravid, and gestation may run for longer than a year. The quantity a population model needs is therefore not the proportion mature but the proportion in maternal condition, meaning females that will give birth or lay eggs by the end of the year and so contribute to recruitment at the start of the next (Walker 2005).

Walker (2005) modelled this with a three-parameter logistic function (3PLF) whose upper asymptote is free to sit below one:

\[E[Y_i] = P_{Max}\left(1 + e^{-\ln(19)\,(x_i - x_{50}) / (x_{95} - x_{50})}\right)^{-1}\]

Here \(Y_i\) is a Bernoulli variable indicating whether female \(i\) is in maternal condition, \(x_{50}\) and \(x_{95}\) are the lengths or ages at which 50% and 95% of the maximum are maternal, and \(P_{Max}\) is that maximum: the proportion of the largest females in maternal condition in a given year. For an annual cycle \(P_{Max} \approx 1\), for a biennial cycle \(P_{Max} \approx 0.5\), and for a triennial cycle \(P_{Max} \approx 1/3\). When \(P_{Max} = 1\) the model reduces to the ordinary two-parameter maturity ogive fitted by maturity(), and since the parameterisation is shared the two are directly comparable.

Harry et al. (2024) compared two ways of using this model. In the 3PLF-fixed approach of Walker (2005), \(P_{Max}\) is chosen from knowledge of the ovarian and uterine cycles and only \(x_{50}\) and \(x_{95}\) are estimated. In the 3PLF-estimated approach, \(P_{Max}\) is estimated along with the other parameters. Simulation showed that estimating \(P_{Max}\) is feasible, often improves accuracy and interval coverage relative to a fixed value, and allows reproductive periodicity to be inferred from the data rather than assumed. The cost is sample size: 100–200 maternal females were typically needed for accurate estimation.

maternity() implements both approaches. The model is written in RTMB and fitted by maximum likelihood, with bootstrap confidence intervals.

The data

The example data are the empirical case study from Harry et al. (2024): maturity and maternity at length for female sandbar sharks, Carcharhinus plumbeus, from the Gulf of Mexico and western North Atlantic. It combines the reproductive studies of Baremore and Hale (2012) and Piercy et al. (2016), with maternal condition assigned to each individual.

library(mustelus)
data(sandbar)

str(sandbar)
#> 'data.frame':    1087 obs. of  4 variables:
#>  $ FL             : num  145 146 149 182 148 151 132 168 160 155 ...
#>  $ maturity_stage : int  1 0 0 0 1 1 0 1 0 1 ...
#>  $ maternity_stage: int  0 0 0 0 0 1 0 0 0 0 ...
#>  $ source         : chr  "Baremore" "Baremore" "Baremore" "Baremore" ...

c(
  females  = nrow(sandbar),
  mature   = sum(sandbar$maturity_stage, na.rm = TRUE),
  maternal = sum(sandbar$maternity_stage)
)
#>  females   mature maternal 
#>     1087      640      206

Fork length is in centimetres. Note that maternal condition is recorded for every female, while maturity_stage has a small number of missing values.

Basic usage

By default \(P_{Max}\) is estimated from the data:

mt <- maternity(maternity_stage, FL, data = sandbar, times = 500)
summary(mt)
#> # A tibble: 1 × 18
#>   method      m50 m50_lower m50_upper   m95 m95_lower m95_upper  pmax pmax_lower
#>   <chr>     <dbl>     <dbl>     <dbl> <dbl>     <dbl>     <dbl> <dbl>      <dbl>
#> 1 3PLF-est…  160.      157.       164  174.      168.      183. 0.480      0.396
#> # ℹ 9 more variables: pmax_upper <dbl>, m50_se <dbl>, m95_se <dbl>,
#> #   pmax_se <dbl>, n <int>, N <int>, nll <dbl>, AIC <dbl>, convergence <lgl>

The summary reports the three parameters with bootstrap 95% confidence intervals, the number of females (n) and the number in maternal condition (N), the negative log-likelihood and AIC, and whether the optimiser converged with a positive-definite Hessian.

\(P_{Max}\) is estimated at 0.48 (0.4–0.61), so roughly half of the largest females are in maternal condition in any year. That is what a biennial cycle looks like. Half of \(P_{Max}\) is reached at 160.1 cm.

Plotting

plot() returns a ggplot object. The default shows binned observed proportions scaled by sample size, with the fitted curve and a bootstrap 95% confidence ribbon:

plot(mt) + xlab("Fork length (cm)") + ylab("Proportion in maternal condition")

Figure 1. Proportion of female sandbar sharks in maternal condition against fork length, with the three-parameter logistic function fitted with \(P_{Max}\) estimated. Points are observed proportions in length bins, sized by the number of females in each bin, and the grey ribbon is the bootstrap 95% confidence interval.

The curve in Figure 1 levels off well short of one, which is the whole point of the third parameter. A two-parameter ogive forced through an asymptote of one could not describe these data.

The same raw_data options as maturity() are available: "proportions", "point", "rug", "bootstrap" (a subsample of the resampled curves in place of the ribbon) and "none". The rug option is shown in Figure 2.

plot(mt, raw_data = "rug") + xlab("Fork length (cm)") + ylab("Proportion in maternal condition")

Figure 2. The fit in Figure 1 with individual females shown as a rug, maternal females along the top axis and non-maternal females along the bottom.

Fixing the asymptote and testing reproductive periodicity

Supplying pmax fixes the asymptote and estimates only \(x_{50}\) and \(x_{95}\). This is appropriate when the reproductive cycle is known from other evidence, and it is the only workable option when maternal females are too few to estimate \(P_{Max}\). It also provides a way of testing competing hypotheses about periodicity: fit the model under each and compare AIC.

Reproductive periodicity in sandbar sharks has been reported as biennial by some studies and as longer by others, so both are worth fitting:

mt_biennial  <- maternity(maternity_stage, FL, data = sandbar, pmax = 0.5, times = 500)
mt_triennial <- maternity(maternity_stage, FL, data = sandbar, pmax = 1/3, times = 500)

fits <- rbind(summary(mt), summary(mt_biennial), summary(mt_triennial))
fits$model <- c("3PLF-estimated", "3PLF-fixed, biennial", "3PLF-fixed, triennial")
fits$dAIC  <- round(fits$AIC - min(fits$AIC), 2)

fits[, c("model", "m50", "m95", "pmax", "AIC", "dAIC")]
#> # A tibble: 3 × 6
#>   model                   m50   m95  pmax   AIC  dAIC
#>   <chr>                 <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 3PLF-estimated         160.  174. 0.480  836.   1.9
#> 2 3PLF-fixed, biennial   161.  176. 0.5    834.   0  
#> 3 3PLF-fixed, triennial  156.  167  0.333  848.  14.2

A biennial cycle is strongly supported over a triennial one, by 14.2 AIC units. The fixed biennial model also edges out the estimated model by 1.9 units, but since it has one fewer estimated parameter the two have essentially the same support given the data. This reproduces the result reported by Harry et al. (2024).

The estimated model is doing useful work even where a fixed model fits equally well, because the confidence interval on \(P_{Max}\) shows what the data can and cannot resolve. Here it excludes 1/3 but comfortably includes 0.5.

plot(mt_triennial) + xlab("Fork length (cm)") + ylab("Proportion in maternal condition")

Figure 3. Maternity ogive for female sandbar sharks with \(P_{Max}\) fixed at 1/3, corresponding to a triennial cycle. Points and ribbon as in Figure 1.

Forcing the asymptote down to 1/3 (Figure 3) pulls the curve below the observed proportions at large sizes, which is the source of the AIC penalty.

Comparison with maturity

Both variables are recorded for these sharks, so the maturity and maternity ogives can be fitted to the same animals and compared directly. The parameterisations match, so \(L_{50}\) and \(x_{50}\) mean the same thing on the same scale:

mat <- maturity(maturity_stage, FL, data = sandbar, times = 500)

data.frame(
  ogive = c("Maturity", "Maternity"),
  x50   = c(summary(mat)$L50, summary(mt)$m50),
  x95   = c(summary(mat)$L95, summary(mt)$m95),
  n     = c(summary(mat)$n,   summary(mt)$n)
)
#>       ogive   x50   x95    n
#> 1  Maturity 151.3 168.0 1065
#> 2 Maternity 160.1 174.4 1087

Half of females are mature at 151.3 cm but half of the maternal maximum is not reached until 160.1 cm, so the maternity curve sits to the right of the maturity curve as well as below it. Harry et al. (2024) note that \(L_{50}\) effectively provides a lower bound for \(x_{50}\), which is a useful check on a fit and a source of prior information where maternal data are sparse.

This comparison also shows what is lost by substituting one for the other. Using the maturity ogive in place of the maternity ogive would credit every mature female with annual reproduction, overstating reproductive output by roughly a factor of two here, and would place the onset of reproduction some 9 cm too early.

Using age as the predictor

Any continuous predictor can be used. The sandbar data contain no ages, but spottail does:

data(spottail)

mt_age <- maternity(maternity_stage, age_agree, data = spottail, times = 500)
summary(mt_age)[, c("m50", "m50_lower", "m50_upper", "m95", "pmax", "n", "N")]
#> # A tibble: 1 × 7
#>     m50 m50_lower m50_upper   m95  pmax     n     N
#>   <dbl>     <dbl>     <dbl> <dbl> <dbl> <int> <int>
#> 1  3.36      2.89      3.78  5.07     1    85    57

plot(mt_age, binwidth = 1) + xlab("Age (years)") + ylab("Proportion in maternal condition")

Figure 4. Proportion of female spot-tail sharks in maternal condition against age, in 1 year bins, with \(P_{Max}\) estimated. Points and ribbon as in Figure 1.

Spot-tail sharks reproduce annually, so \(P_{Max}\) is estimated close to one (Figure 4) and the maternity curve is nearly the same as a maturity ogive.

Sample size

Harry et al. (2024) found that estimating \(P_{Max}\) accurately generally required 100–200 maternal females. The sandbar data contain 206, which is why the interval on \(P_{Max}\) above is reasonably tight. With smaller samples the estimate becomes unstable, and the bootstrap will report replicates that failed to converge or contained no maternal females.

Where maternal data are sparse, fixing pmax from independent evidence on the ovarian and uterine cycles is the more defensible choice. The AIC comparison above is then the way to check that the chosen value is at least consistent with the data.

Output structure

The result is a one row tibble with list columns. mods holds the RTMB objective function, the nlminb result, and the sdreport, which gives asymptotic standard errors as an alternative to the bootstrap:

summary(mt$mods[[1]]$sdreport, "fixed")
#>         Estimate Std. Error
#> m50  160.1332890 1.64364813
#> m95  174.4099208 3.58536832
#> pmax   0.4800549 0.05084888

# Bootstrap parameter estimates, one row per replicate
head(mt$boot_coefs[[1]])
#>        m50      m95      pmax
#> 1 159.2499 172.4716 0.4642783
#> 2 161.0113 176.1331 0.5127472
#> 3 162.2913 176.3678 0.5852862
#> 4 161.9157 178.5714 0.5409618
#> 5 158.6454 170.1597 0.4268599
#> 6 161.0868 177.4966 0.5451289

References

Baremore, I.E. and Hale, L.F. (2012) Reproduction of the sandbar shark in the western North Atlantic Ocean and Gulf of Mexico. Marine and Coastal Fisheries 4(1), 560–572. doi:10.1080/19425120.2012.700904

Harry, A.V., Baremore, I.E. and Piercy, A.N. (2024) Quantifying maternal reproductive output of chondrichthyan fishes. Canadian Journal of Fisheries and Aquatic Sciences 81(10), 1481–1494. doi:10.1139/cjfas-2024-0031

Piercy, A.N., Murie, D.J. and Gelsleichter, J.J. (2016) Histological and morphological aspects of reproduction in the sandbar shark Carcharhinus plumbeus in the U.S. south-eastern Atlantic Ocean and Gulf of Mexico. Journal of Fish Biology 88(5), 1708–1730. doi:10.1111/jfb.12945

Walker, T.I. (2005) Reproduction in fisheries science. In: Hamlett, W.C. (ed) Reproductive Biology and Phylogeny of Chondrichthyes: Sharks, Batoids and Chimaeras. Science Publishers, Enfield, NH, pp. 81–127.