Gamma model with gaussian second stage model

The next model is more complicated than the previous models (lnk til model appendix) as no explicit likelihood solution exists for it (2). In general the models of this type can be formulated as: \begin{equation} \boldsymbol{y} = X \boldsymbol{\beta} + Z \boldsymbol{u} + \boldsymbol{\epsilon} \end{equation} where X is the design matrix of the fixed effects, and the Z is the design matrix of the random effects. Where \boldsymbol{y} follows a Gamma distribution and the \boldsymbol{u} follows a gaussian distribution.
The coefficients are stored in the $\boldsymbol{\beta}$ for the fixed effect and the \boldsymbol{u} contains the random effect coefficient. The explicit likelihood formulation and Laplace approximation solution to this type of the model is seen in the (link til likelihood appendix).

Using the same link function (log link) as presented in the gamma model (link til model appendix), the main effect model of this type can be written as: \begin{equation} \eta_{i}=log(\mu_[i}) where \mu_{i}=\betax_{i}+\Psi_{i} \end{equation} This translates into: \begin{equation} log(Y_{i})=\mu+\alpha(burn.time_{i})+\beta(burn.temp_{i})+\gamma(mix_{i})+\delta(incu.time_{i})+\zeta(sub_{i})+eta(W_{i})+ \A(id_{i})+ \varepsilon_{i} \\ where \ \varepsilon_{i} \sim G(\alpha,\beta) \ and \ id_{i} \sim N(0,\Psi^{2}) \end{equation} The response variable of the model is:

• $Y_{i}$ refers to the load measured in pascal for the i’th fungal cube

The fixed factor effects of the model are:

• $\mu$ is the overall mean
• $\alpha$ is the term describing the different burning times at the level of i=1..2
• $\beta$ is the term describing the different burning temperatures at the level of i=1..2
• $\gamma$ is the term describing the different mixing ratios at the level of i=1..3
• $\delta$ is the term describing the different incubation times at the level of i=1..2
• $\zeta$ is the term describing the different substrates at the level of i=1..2

Each term is to be understood as deviations from the overall mean

The numerical effect of the model is

• $\eta$ which is the weight of each cube $W_{i}$ corresponding to the $Y_{i}$ measurement

• $\A(id_{i})$ which is the individual cubes tested on each tray.

(table above text): It seems the random cube to cube variation does contribute to the model as the LRT is significant and the AIC goes up if the term is removed. Thus the following model is identified: \begin{equation} Y_{i}=\mu+\beta(burn.temp_{i})+\delta(incu.time_{i})+\zeta(sub_{i})+ \A(id_{i})+ \varepsilon_{i} \\ where \ \varepsilon_{i} \sim G(\alpha,\beta) \ and \ id_{i} \sim N(0,\Psi^{2}) \end{equation}

Inserting the 2 factor interactions resulted in a final model where no interaction terms were significant both for random and fixed terms. So, the final model of the gamma mixed model is the one presented above.

The residual analysis of the model is carried out the same way as the gaussian mixed model.

From the model it does not seem as if no autocorrelation is present in residuals or random effects, thus the assumption of independence is fulfilled. The normality assumption seems to hold for the random effect, but could be better for the residuals. The residual vs fitted shows that the model reasonably models out most variation in the data, but the variance is not constant as it is seen for the higher predicted values and the higher variance.

Model comparison/selection

As mentioned in the beginning, to choose the best models several parameters comes into play. The direct comparison via the AIC, BIC and goodness of fit the table is shown here:

From the tabel is it seen that the gamma mixed effect model is the better choice due to goodness of fit of the data and the lower BIC and AIC. With the model choice in place, it is now time to do the post hoc analysis of the gamma mixed model.