Hierarchical models Biotieteellinen tiedekunta / Henkilön nimi / Esityksen nimi 15.9.2018

”Hierarchical meta-analysis” ”Hierarchical”: at least three layers of variables Data – Parameters – Hyperparameters Does not necessarily contain multiple exchangeable units, populations, stocks or studies ”Meta-analysis”: joint analysis of multiple analyses More than one study Not necessarily hierarchical ”Hierarchical meta-analysis” Analysis of multiple studies using a hierarchical model Biotieteellinen tiedekunta / Henkilön nimi / Esityksen nimi 15.9.2018

Example: length at first maturity for a Baltic herring stock? Lm : length were 50% of the population are mature FishBase: 18 studies of Lm for Baltic herring already exist What can we say about the 19th study? Biotieteellinen tiedekunta / Henkilön nimi / Esityksen nimi 15.9.2018

Exchangeability : a state of mind Inability to rank the studies before seeing the data If knowledge about the location of the study would help to rank the studies -> not exchangeable Exchangeability may arise conditionally, all studies at same locations may be seen as exchangeable Consequence of exchangeability All studies can be treated as if they were independent draws from the same probability distribution Biotieteellinen tiedekunta / Henkilön nimi / Esityksen nimi 15.9.2018

Distribution of all potential studies Observed studies Distribution of all potential studies Biotieteellinen tiedekunta / Henkilön nimi / Esityksen nimi 15.9.2018

Graphical model sd~d() mean~d() sd mean Lm[i]~N(mean,sd) Lm[1] Lm[2] 0.25 0.25 0.2 0.2 mean~d() 0.15 0.15 sd mean 0.1 0.1 0.05 0.05 5 10 15 20 25 5 10 15 20 25 Lm[i]~N(mean,sd) Lm[1] Lm[2] Lm[19] 0.25 0.25 0.25 0.2 0.2 0.2 0.15 0.15 0.15 0.1 0.1 0.1 0.05 0.05 0.05 5 10 15 20 25 5 10 15 20 25 5 10 15 20 25 Biotieteellinen tiedekunta / Henkilön nimi / Esityksen nimi 15.9.2018

sd~d() mean~d() sd mean Lm[i]~N(mean,sd) Lm[1] Lm[2] Lm[19] Om[1] 0.25 0.25 0.2 0.2 mean~d() 0.15 0.15 sd mean 0.1 0.1 0.05 0.05 5 10 15 20 25 5 10 15 20 25 Lm[i]~N(mean,sd) Lm[1] Lm[2] Lm[19] 0.25 0.25 0.25 0.2 0.2 0.2 0.15 0.15 0.15 0.1 0.1 0.1 0.05 0.05 0.05 5 10 5 10 5 10 Om[1] Om[2] Om[19] 7 Om[i]~N(L[mi],1) 0.25 0.25 0.25 0.2 0.2 0.2 0.15 0.15 0.15 0.1 0.1 0.1 0.05 0.05 0.05 5 10 15 20 25 5 10 15 20 25 5 10 15 20 25

Benefits of hierarchical models Transfer of information between exchangeable cases More precise case specific estimates ”Shrinkage” reduces sampling error -> case specific estimates are pulled towards the common mean Prior distribution for a new case becomes formed in a consistent and transparent way Can be expanded to include explanatory variables Random effect logistic regression! Biotieteellinen tiedekunta / Henkilön nimi / Esityksen nimi 15.9.2018

Challenges of hierarchical models Too many layers of parameters may make the interpretation of parameters very difficult Computational problems may arise: slow convergence of MCMC Biotieteellinen tiedekunta / Henkilön nimi / Esityksen nimi 15.9.2018