Supplementary materials 2. Descriptions of the prior distributions of the recruitment

compensation metrics using the fitdistrplus package in R

Two main tools of the fitdistrplus package in R are used to describe the prior distributions

of the compensation ratio (κ) and steepness (h) for Atlantic croaker (Micropogonias undulatus),

bluefish (Pomatomus saltatrix), red snapper (Lutjanus campechanus), sheepshead (Archosargus

probatocephalus), and spot (Leiostomus xanthurus) in the U.S. Atlantic Ocean. First is the

Cullen and Frey Plot (CFP), also called the skewness–kurtosis graph, which helps in choosing

the candidate, theoretical probability density functions (pdfs; i.e., normal, uniform, exponential,

logistic, beta, lognormal, and gamma pdfs) to fit to κ and h simulated distributions. Second, the

package fits the candidate pdfs and provides the maximum loglikelihoods and the information

criteria. These statistics help in determining the pdf that matches the unknown distribution best.

CPRs of all species’ κ are similar; they are exemplified by Figure S2.1 for Atlantic

croaker. The CPR results are obtained by comparing the κ data skewness and kurtosis with their

counterparts for predefined theoretical distributions. The κ data skewness and kurtosis show

considerable uncertainty (i.e., are widely dispersed). Because the κ data skewness and kurtosis

are positive (Table 3), the candidate pdfs to fit to κ data of all species are lognormal, gamma, and

Weibull models.

CPRs of the BH–SRR h for Atlantic croaker, bluefish, red snapper, and sheepshead are

similar. They are illustrated with Figure S2.2 for Atlantic croaker. Uncertainty in the BH–SRR h

data skewness and kurtosis is negligible. The candidate pdfs to fit to the BH–SRR h simulated

distributions are the beta and normal models. The beta model is a candidate pdf because the BH–

SRR h skewness are negative, indicating left-tailed distributions (Table 3), and the (skewness,

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kurtosis) pairs are in the beta distribution area (Fig. S2.2). The normal model is a candidate pdf

because the BH–SRR h kurtosis is close to 3.

For spot, the BH–SRR h CRP analysis produces a skewness close to zero and a kurtosis

close to 3 (Fig. S2.3; Table 3). The pair (skewness, kurtosis) is at the boundary of the beta

distribution area. Therefore, the beta and uniform pdfs, and perhaps the normal pdf, are

candidates for fitting the BH–SRR h distributions for spot.

As for CRPs for κ, CPR for the R–SRR h data are similar across species and are

associated with uncertain, positive skewness and kurtosis located between the skewness-kurtosis

lines of the lognormal, gamma, and Weibull models (Fig. S2.4). These models are therefore

candidate pdfs to fit to the R–SRR h distributions.

The identified and fitted pdfs can be represented by

(S1a) f ( x∨μ , σ )= 1σ √2π

e−( x−μ )2

2σ 2 for the normal pdf,

(S1b) f ( x∨μ , σ )= 1xσ √2 π

e−[ ln ( x )− μ¿ ]2

2 σ 2 for the lognormal pdf,

(S1c) f ( x∨p ,θ )= x p−1 e−xθ

Γ ( p )θp for the gamma pdf,

(S1d) f ( x|p , q )=Γ ( p+q )Γ p Γ q

x p−1 (1−x )q−1 for the two-parameter beta pdf and,

(S1e) f ( x|p , θ )= pθp x p−1 e

−( xθ )

p

for the Weibull pdf.

In these pdfs, x represents either κ or h. They are fitted to κ and h data after discarding the

simulated values of κ and h that lead to unviable populations (i.e., 1 < κ < +∞, 0.2 < h < 1 for the

BH–SRR, and 0.2 < h < +∞ for the R–SRR).

2

The parameters of eqs. S1a–S1e are the mean μ and standard deviation σ for the normal

pdf; the mean μ and the standard deviation σ of the log-transformed data (μ and σ are denoted,

respectively, as meanlog and sdlog in the R terminology) for the lognormal distribution; the

shape p and the scale θ for the gamma and Weibull pdfs; and the shape parameters p (shape1)

and q (shape2) for the two-parameter beta pdf.

Figures S2.5 and S2.6 give the density plots of the candidate model fits to various

simulated distributions and Tables S2.1–S2.3 give the parameters and information criteria for the

fitted models. The lognormal pdf matches the κ and R–SRR h distributions best because its

maximum loglikelihood is highest and because its information criteria are lowest. The beta pdf

matches the BH–SRR h best for Atlantic croaker, bluefish, red snapper, and sheepshead because

its maximum loglikelihood is highest and because its information criteria are lowest, but for the

BH–SRR h of spot, either the (symmetric) beta pdf or the uniform pdf is suitable.

3

Fig. S2.1. Skewness-kurtosis plot for the simulated data of the compensation ratio (κ) for

Atlantic croaker in the U.S. Atlantic Ocean. The bootstrapped values of the estimated skewness

and kurtosis (orange points) indicate uncertainty around them (depicted by the blue point and

labelled Observation).

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Fig. S2.2. Skewness-kurtosis plot for the BH–SRR steepness data of Atlantic croaker in the U.S.

Atlantic Ocean. The bootstrapped values (orange points) indicate uncertainty around the

estimated skewness and kurtosis (depicted by the blue point and labelled Observation).

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Fig. S2.3. Skewness-kurtosis plot for the BH–SRR steepness data of spot in the U.S. Atlantic

Ocean. The bootstrapped values (orange points) indicate uncertainty around the estimated

skewness and kurtosis (depicted by the blue point and labelled Observation).

6

Fig. S2.4. Skewness-kurtosis plot for the R–SRR steepness data of Atlantic croaker in the U.S.

Atlantic Ocean. The bootstrapped values (orange points) indicate uncertainty around the

estimated skewness and kurtosis (depicted by the blue point and labelled Observation).

7

Fig. S2.5. Plots of the Weibull, gamma, and lognormal distributions fitted to the simulated

compensation ratio (κ) of Atlantic croaker (ACR), bluefish (BLF), red snapper (RDS),

sheepshead (SPH), and spot (SPT) in the U.S. Atlantic Ocean.

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Fig. S2.6. Plots of various distributions fitted to the simulated BH–SRR steepness (left panels)

and R–SRR steepness (right panels) for Atlantic croaker (ACR), bluefish (BLF), red snapper

(RDS), sheepshead (SPH), and spot (SPT) in the U.S. Atlantic Ocean.

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Table S2.1. Summary statistics from a simulation run for various probability density functions (pdfs: W for Weibull, G for Gamma,

and LN for lognormal) fitted to simulated distributions of the compensation ratio (κ) of Atlantic croaker (ACR), bluefish (BLF), red

snapper, (RDS) sheepshead (SPH), and spot (SPT) in the U.S. Atlantic Ocean (the gray-shaded pdfs match κ distributions best).

Species pdf Parameters and standard error (SE) Goodness-of-fit and model selection b

shape (SE) scale a (SE) meanlog (SE) sdlog (SE) loglike AIC BIC

ACR W 1.048 (0.007) 14.29 (0.15) −35831.61 71667.21 71681.6

G 1.26 (0.0161) 0.09 (0.0014) −35697.39 71398.79 71413.18

LN 2.2 (0.0093) 0.935 (0.0066) −349824.97 69653.94 69668.34

BLF W 1.09 (0.008) 11.7 (0.1156) −33426.39 66856.79 66871.17

G 1.33 (0.017) 0.118 (0.0018) −33265.38 66534.76 66549.13

LN 2.00 (0.009) 0.903 (0.0065) −32467.67 64939.34 64953.72

RDS W 1.04 (0.0074) 24.68 (0.25) −41759.8 83523.61 83538.02

G 1.22 (0.0155) 0.05 (0.008) −41659.46 83322.92 83337.34

LN 2.72 (0.0096) 0.961 (0.007) −40940.69 81885.38 81899.8

SPH W 0.93 (0.0067) 20.74 (0.24) −398869.9 7973.862.69 79758.18

G 0.99 (0.012) 0.046 (0.0007) −39930.27 79864.54 79878.92

LN 2.49 (0.011) 1.079 (0.0077) −39033.82 78071.64 78086.02

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SPT W 0.98 (0.0073) 8.82 (0.017) −28368.53 56741.07 56755.26

G 1.15 (0.0159) 0.129 (0.0021) −28319.51 56643.01 56657.2

LN 1.69 (0.01) 0.95 (0.007) −27228.45 54460.9 54475.09

a For the Weibull and gamma models, the parameters are called slope and scale (the scale parameter for the gamma model is referred

to as rate in the package fitdistrplus).

b The goodness-of-fit is measured through the maximum loglikelihood (loglike); the model selection is based on the Akaike

Information Criterion (AIC) and the Bayesian Information Criterion (BIC).

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Table S2.2. Summary statistics from a simulation run for various probability density functions (pdfs: B for beta, N for Normal, and U

for Uniform) fitted to the BH–SRR steepness simulated distributions for Atlantic croaker (ACR), bluefish (BLF), red snapper (RDS),

sheepshead (SPH), and spot (SPT) in the U.S. Atlantic Ocean (the gray-shaded pdfs match the steepness distributions best). sd is

standard deviation; the range pertains to the uniform pdf, which has no other estimated statistics (NA = Not Applicable).

Species pdf Parameters and SE Goodness-of-fit and model selection b

shape1 (SE) shape2 a (SE) mean (SE) or range sd(SE) loglike AIC BIC

ACR B 4.13 (0.058) 2.01 (0.028) 3653.24 −72302.48 −7288.09

N 0.663 (0.0018) 0.18 (0.001) 2965.46 −5926.92 −5912.52

BLF B 3.88 (0.055) 2.31 (0.031) 3247.27 −6490.54 −6476.16

N 0.63 (0.002) 0.18 (0.001) 2729.05 −5454.09 −5439.71

RDS B 5.23 (0.075) 1.71 (0.022) 5574.18 −11144.36 −11129.9

N 0.76 (0.002) 0.16 (0.001) 4194.77 −8385.54 −8371.13

SPH B 3.64 (0.05) 1.51 (0.02) 3686.17 −7368.33 −7353.95

N 0.71 (0.002) 0.19 (0.001) 2373.47 −4742.94 −4728.56

SPT B 3.04 (0.044) 2.37 (0.034) 2147.50 −4291.01 −4276.82

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N 0.56 (0.002) 0.20 (0.001) 1751.34 −3498.69 −3484.50

U 0.2–0.98 NA NA NA

a For the beta model, the parameters are called shape1 and shape2; for the normal model, the parameters are mean and standard

deviation.

b The goodness-of-fit is measured through the maximum loglikelihood (loglike); the model selection is based on the Akaike

Information Criterion (AIC) and the Bayesian Information Criterion (BIC).

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Table S2.3 Summary statistics from a simulation run for various probability density functions (pdfs: W for Weibull, G for Gamma,

and LN for lognormal) fitted to the R–SRR steepness simulated distributions for Atlantic croaker (ACR), bluefish (BLF), red snapper

(RDS), sheepshead (SPH), and spot (SPT) in the U.S. Atlantic Ocean (the gray-shaded pdfs match the steepness distributions best).

Species pdf Parameters and SE Goodness-of-fit and model selection b

shape (SE) scale a (SE) meanlog (SE) sdlog (SE) loglike AIC BIC

ACR W 1.31 (0.009) 1.68 (0.014) −13447.45 26898.9 26913.3

G 1.92 (0.025) 1.25 (0.019) −13044.16 26092.31 26106.7

LN 0.14 (0.0075) 0.74 (0.0053) −12441.51 24887.03 24901.42

BLF W 1.36 (0.0097) 1.43 (0.01) −11578.7 23161.41 23175.79

G 2.01 (0.03) 1.55 (0.023) −11184.4 22372.81 22387.18

LN −0.012 (0.007) 0.72 (0.005) −10621.02 21246.03 21260.41

RDS W 1.3 (0.0093) 2.6 (0.021) −18039.23 36082.47 36096.88

G 1.83 (0.024) 0.77 (0.012) −17703.76 35411.52 35425.94

LN 0.57 (0.008) 0.767 (0.005) −17220.72 34445.44 34459.85

SPH W 1.16 (0.008) 2.26 (0.021) −17036.62 34077.24 34091.62

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G 1.47 (0.02) 0.69 (0.01) −16820.55 33645.1 33659.48

LN 0.38 (0.009) 0.86 (0.006) −16201.21 32406.43 32420.8

SPT W 1.24 (0.009) 1.14 (0.01) −9026.7 18057.39 18071.58

G 1.74 (0.024) 1.65 (0.03) −8685.89 17375.77 17389.96

LN −0.26 (0.008) 0.76 (0.006) −7887.58 15779.15 15793.34

a For the Weibull and gamma models, the parameters are called slope and scale (the scale parameter for the gamma model is referred

to as rate in the package fitdistrplus).

b The goodness-of-fit is measured through the maximum loglikelihood (loglike); the model selection is based on the Akaike

Information Criterion (AIC) and the Bayesian Information Criterion (BIC).

15