A statistical model for validating the periodicity of vertebral growth band formation in elasmobranch fishes


We developed a simple statistical model for centrum edge analysis (CEA) of elasmobranch species. This page provides you with the program along with a brief explanation of the method. To use the age validation program, you need to install "R" from the CRAN homepage.
 Of a variety of age validation methods for fishes, centrum edge analysis (CEA) is frequently employed for elasmobranchs. The underlying principle of CEA is that the outermost increment will exhibit a yearly sinusoidal cycle when plotted against the month of capture if the growth band pairs are formed annually. CEA usually compares the opacity and translucency of the centrum edge.  Excellent reviews of various age validation methods have been published (Campana 2001; Cailliet et al. 2006).

Okamura and Semba (2009) combined a statistical model for binary data with a statistical model for circular data (Fisher 1993) to deal with the periodicity of the CEA data and the difference of the monthly sample size. The periodicity of growth band formation was categorized as no cycle, an annual cycle or a biannual cycle. When a growth band pair is formed annually, a simple von Mises distribution is used in the model. When a growth band pair is formed biannually, a finite mixture of von Mises distributions is used. When the growth bands have no cycle, the corresponding model is a uniform distribution. The Akaike information criterion (AIC) is used to determine which model is the best. The details of the method are given in Okamura and Semba (2009).  The general performance of the method was evaluated using the simulated data. The simulation trials showed that the performance of the method was satisfactory with moderate sample sizes. This method should improve the accuracy of age determination and could be applied to all species that have periodic growth band pairs.

The R program can be downloaded free (agevalid.r). The example codes are:

n <- c(31, 23, 4, 12, 16, 16, 22, 7, 8, 45, 37, 6)  # from Semba et al. (2009)
x <- c(24, 10, 1, 3, 3, 2, 3, 1, 1, 25, 21, 4)  # the number of convex (hypomineralized) edges
res0 <- agevalid(n,x,peak=0,model="vm",method="BFGS",p0=0)  # no cycle
res1 <- agevalid(n,x,peak=1,model="vm",method="BFGS",p0=c(1,0,0)) # an annual cycle
res2 <- agevalid(n,x,peak=2,model="vm",method="BFGS") # a biannual cycle
res1$estimate # the parameter estimates (see Okamura and Semba 2009)
c(res0$AIC, res1$AIC, res2$AIC)  #  the model with the lowest AIC is the best one.
predProb(res1,plot=T)  # prediction for res1

n <- c(1, 1, 3, 0, 4, 2, 0, 0, 5, 2, 3, 4)  # from Smith (1984)
x <- c(1, 1, 3, 0, 2, 1, 0, 0, 3, 2, 3, 4)  # the number of hyaline (hypomineralized) edges
res0 <- agevalid(n,x,peak=0,model="wc",method="BFGS")  
res1 <- agevalid(n,x,peak=1,model="wc",method="Nelder-Mead")
res2 <- agevalid(n,x,peak=2,model="wc",method="Nelder-Mead")
c(res0$AICc, res1$AICc, res2$AICc)
predProb(res2,plot=T)  # prediction for res2

We recommend you try various initial values and multiple optimization methods for your data. If you find any problems or have any questions
on the program, please contact to Hiroshi Okamura (email address: okamura<at>affrc.go.jp  (replace <at> with @). We hope the program is useful for your study.

  1. Cailliet, G.M., Smith, W.D., Mollet, H.F., and Goldman, K.J. 2006. Age and growth studies of chondrichtyan fishes: the need for consistency in  terminology, verification, validation, and growth function fitting. Environmental Biology of Fishes 77: 211-228.
  2. Campana, S.E. 2001. Accuracy, precision and quality control in age determination, including a review of the use and abuse of age validation methods. Journal of  Fish Biology 59: 197-242.
  3. Fisher, N.I. 1993. Statistical analysis of circular data. Cambridge University Press, Cambridge, UK.
  4. Okamura, H. and Semba, Y. 2009. A novel statistical method for validating the periodicity of vertebral growth band formation in elasmobranch fishes. Canadian Journal of Fisheries and Aquatic Sciences 66(5): 771-780.
  5. Semba, Y., Nakano, H., and Aoki, I. 2009. Age and growth analysis of the shortfin mako, Isurus oxyrinchus, in the western and central North Pacific Ocean. Environmental Biology of Fishes 84: 377-391.
  6. Smith, S.E. 1984. Timing of vertebral-band deposition in tetracycline-injected leopard sharks. Transactions of the American Fisheries Society 113: 308-313.