1 Models

The underlying model takes the form:

$$W_{i} = \alpha L_{i}^{\beta}e^{z_{i}}$$

where $\alpha$ and $\beta$ are intercept and slope parameters (on log scale), $L_{i}$, $W_{i}$ are the length and weight, repectively, of individual i, and $z_{i} \sim N(\mu,\sigma^2)$

This model assumes log normal observational error (normal/gaussian on the log scale). This is important when the fitted model is used to predict weights from length

This model can be extended to :

$$W_{ji} = \alpha L_{ji} ^ {\beta_j} e^{z_i}$$ for season j, j= 1, …, 4 (spring, summer, fall, winter) or sex j, j = 1, …, 3 (0, 1, 2) or extended further to include season:sex combinations. These models are all nested and can therefore be tested using standard statistical methods.

1.1 Relationship between Normal and Log Normal distribution

If

$$Z \sim ~ LN(\mu, \sigma^2)$$ then

$$ln(Z) \sim N(\mu,\sigma^2)$$

Note that each distribution uses the same parameters but their interpretations are different. For example the interpretation of $\mu$ and $\sigma^2$, under the normal distribution, are mean and variance:

$$E(ln(Z)) = \mu$$ $$Var(ln(Z)) = \sigma^2$$

but under the log normal distibution the mean and variance are:

$$E(Z) = e^{\mu + \frac{\sigma^2}{2}}$$ $$Var(Z) = (e^{\sigma^2} - 1)(e ^{2\mu + \sigma^2})$$

This is important when predicting weight from length.

1.2 Fitting

For this type of model it is natural to take logs and fit using ordinary least squares, since $z_{i} \sim N(\mu,\sigma^2)$

$$ln(W_i) = ln(\alpha) + \beta ln(L_i) + z_i$$ where $ln(W_i)$ is regressed on $ln(L_i)$ to estimate $\ln(a)$, $\beta$, and $\sigma^2$

1.3 Prediction

Now to estimate Weight using this fitted model

$$\begin{aligned} E(W_{i}) &= \hat{W_i} \\ &= E({aL_i^\beta}e^{Z_{i}}) \\ &= E({aL_i^\beta}) E(e^{Z_{i}}) \\ &= {aL_i^\beta}e^{\sigma^2/2} \\ &= e^{ln(a) + \beta ln(L_i)}e^{\sigma^2/2} \\ &= e^{a + \beta ln(L_i) + \sigma^2/2} \end{aligned}$$

It is common, although incorrect, to see

$$\hat{W_i} = e^{a + \beta ln(L_i)}$$ omitting the term $\sigma^2/2$.

By omitting this term the resulting fitted value of weight will be biased toward smaller (in weight) individuals.