# Background

The total length (mm) and otolith age of Slimy Sculpin (*Cottus cognatus*) captured in the Arctic Long-Term Ecological Research area were recorded in SculpinALTER.^{1}

^{1} See “CSV file” link in “Source” section of linked page.

# Fit Traditional VBGF

- Examine the plot of TL versus age. Make observations regarding the “shape” of the data (do the results look linear or like a von Bertalanffy growth curve, is there an obvious asymptote, are young fish well represented, how variable are lengths within ages).
- Fit the typical parameterization of the von Bertalanffy growth function (VBGF).
- How realistic do the point estimates of \(L_{\infty}\), \(K\), and \(t_{0}\) seem?
- Write the typical VBGF with parameters replaced by their estimated values.
- Carefully interpret the meaning of each parameter.
- Construct 95% bootstrapped confidence intervals for each parameter. Comment on the widths of these confidence intervals. What explains this?
- Predict the mean TL, with 95% confidence interval, for an age-3 fish. Comment on the width of this confidence interval. What explains this?
- Plot TL versus age and superimpose the best-fit VBGF.
^{2}Comment on model fit. - Construct a residual plot. Comment on model fit.
- Compute the correlation between parameter values. Comment.

^{2} This post may be useful.

# Alternative Parameterization

- Fit the von Bertalanffy’s original parameterization.
^{3}- Interpret the interval estimate for the \(L_{0}\) parameter.
- Write the VBGF with parameters replaced by their estimated values.
- Construct 95% bootstrapped confidence intervals for each parameter. Comment on the widths of these confidence intervals. What explains this?
- Predict the mean TL, with 95% confidence interval, for an age-d fish. Comment on the width of this confidence interval. What explains this?
- Plot TL versus age and superimpose the best-fit VBGF. Comment on model fit.
- Compute the correlation between parameter values. Comment
- How does the estimate of \(L_{\infty}\) and \(K\) from fitting this parameterization compare to that from the typical VBGF fit above. Explain your observation.

^{3} See `growthFunShow("vonBertalanffy",param="original",plot=TRUE)`

) and this.

Solution Code:

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