A Bayesian mixture model for missing data in marine mammal growth analysis

Abstract

Much of what is known about bottle nose dolphin (Tursiops truncatus) anatomy and physiology is based on necropsies from stranding events. Measurements of total body length, total body mass, and age are used to estimate growth. It is more feasible to retrieve and transport smaller animals for total body mass measurement than larger animals, introducing a systematic bias in sampling. Adverse weather events, volunteer availability, and other unforeseen circumstances also contribute to incomplete measurement. We have developed a Bayesian mixture model to describe growth in detected stranded animals using data from both those that are fully measured and those not fully measured. Our approach uses a shared random effect to link the missingness mechanism (i.e. full/partial measurement) to distinct growth curves in the fully and partially measured populations, thereby enabling drawing of strength for estimation. We use simulation to compare our model to complete case analysis and two common multiple imputation methods according to model mean square error. Results indicate that our mixture model provides better fit both when the two populations are present and when they are not. The feasibility and utility of our new method is demonstrated by application to South Carolina strandings data.

Appendices

Appendix

Full conditional posterior distributions for all model parameters.

$$\begin{aligned}&P(\beta _0 |Q_{-\beta _0 } ,\mathbf{{w}},\mathbf{{m}},\mathbf{{z}}) \propto \prod \limits _{i = 1}^n {p_i^{z_i } (1 - p_i )^{1 - z_i } } \prod \limits _{i = 1}^n {p_i^{m_i } (1 - p_i )^{1 - m_i } \frac{1}{{10\sqrt{2\pi } }}\exp \left( { - \frac{{\beta _0^2 }}{{200}}} \right) } \\&P(\beta _1 |Q_{ - \beta _1 } ,\mathbf{{w}},\mathbf{{m}},\mathbf{{z}}) \propto \prod \limits _{i = 1}^n {p_i^{z_i } (1 - p_i )^{1 - z_i } } \prod \limits _{i = 1}^n {p_i^{m_i } (1 - p_i )^{1 - m_i } \frac{1}{{10\sqrt{2\pi } }}\exp \left( { - \frac{{\beta _1^2 }}{{200}}} \right) } \\&P(\delta |Q_{ - a_0 } ,\mathbf{{w}},\mathbf{{m}},\mathbf{{z}}) \propto \frac{1}{{50}}\prod \limits _{i = 1}^n {\phi \left( {w_i |\mu _0 (age_i ),\sigma ^2 } \right) ^{1 - z_i } } \\&P(b_0 |Q_{ - b_0 } ,\mathbf{{w}},\mathbf{{m}},\mathbf{{z}}) \propto \frac{1}{5}\prod \limits _{i = 1}^n {\phi \left( {w_i |\mu _0 (age_i ),\sigma ^2 } \right) ^{1 - z_i } } \\&P(k_0 |Q_{ - k_0 } ,\mathbf{{w}},\mathbf{{m}},\mathbf{{z}}) \propto \frac{1}{2}\prod \limits _{i = 1}^n {\phi \left( {w_i |\mu _0 (age_i ),\sigma ^2 } \right) ^{1 - z_i } } \\&P(a_1 |Q_{ - a_1 } ,\mathbf{{w}},\mathbf{{m}},\mathbf{{z}}) \propto \prod \limits _{i = 1}^n {\phi \left( {w_i |\mu _0 (age_i ),\sigma ^2 } \right) ^{1 - z_i } } \phi \left( {w_i |\mu _1 (age_i ),\sigma ^2 } \right) ^{z_i }\\&\qquad \times \frac{1}{{10a_1 \sqrt{2\pi } }}\exp \left( { - \frac{{(\ln a_1 )^2 }}{{200}}} \right) \\&P(b_1 |Q_{ - b_1 } ,\mathbf{{w}},\mathbf{{m}},\mathbf{{z}}) \propto \frac{1}{5}\prod \limits _{i = 1}^n {\phi \left( {w_i |\mu _1 (age_i ),\sigma ^2 } \right) ^{z_i } } \\&P(k_1 |Q_{ - k_1 } ,\mathbf{{w}},\mathbf{{m}},\mathbf{{z}}) \propto \frac{1}{2}\prod \limits _{i = 1}^n {\phi \left( {w_i |\mu _1 (age_i ),\sigma ^2 } \right) ^{z_i } } \\&P(\sigma |Q_{ - \sigma } ,\mathbf{{w}},\mathbf{{m}},\mathbf{{z}}) \propto \frac{1}{5}\prod \limits _{i = 1}^n {\phi \left( {w_i |\mu _0 (age_i ),\sigma ^2 } \right) ^{1 - z_i } } \phi \left( {w_i |\mu _1 (age_i ),\sigma ^2 } \right) ^{z_i } \\&P(z_i = 1|Q,w_i ,m_i ) = \frac{{\phi \left( {w_i |\mu _1 (age_i ),\sigma ^2 } \right) p_i }}{{\phi \left( {w_i |\mu _0 (age_i ),\sigma ^2 } \right) (1 - p_i ) + \phi \left( {w_i |\mu _1 (age_i ),\sigma ^2 } \right) p_i }} \\&P(w_i |Q,m_i ,z_i ) = \phi \left( {w_i |\mu _0 (age_i ),\sigma ^2 } \right) ^{1 - z_i } \phi \left( {w_i |\mu _1 (age_i ),\sigma ^2 } \right) ^{z_i } \end{aligned}$$

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