1. An Ecological Trap for Ecologists: Zero-Modified Models Western Mensurationists’ Meeting 2009 Tzeng Yih Lam 06.23.2009 Tzeng Yih Lam, OSU Manuela Huso, OSU Doug Maguire, OSU ^ Possible

2. Ecological Trap [ē-k ə - ˈ lä-ji-k ə l ˈ trap] A preference of falsely attractive habitat and a general avoidance of high-quality but less-attractive habitats. Western Mensurationists’ Meeting 2009 Tzeng Yih Lam 06.23.2009 Wikipedia

3. Cues for Zero-Modified Models Possible Trap #1 Possible Trap #2 Discussions Western Mensurationists’ Meeting 2009 Tzeng Yih Lam 06.23.2009

4. Cues for Using Zero-Modified Models Cues for Zero-Modified Models Possible Trap #1 Possible Trap #2 Discussions Count & Normal Theory The Cues & The Solutions Zero-Inflated Models Hurdle Models Expected Count – Observed Count Conclusions ‘The Cues’: For rare species data, the marginal count frequency distribution contains large number of zeros, Poisson and/or NB GLM have poor fit. ‘The Solutions’: Zero-modified Models: A general class of finite mixture models that account for excessive zeros, Zero-Inflated Models (ZI) 1, Hurdle Models (H) 2. Western Mensurationists’ Meeting 2009 Tzeng Yih Lam 06.23.2009 1 Lambert (1992); 2 Mullahy (1986)

5. Cues for Using Zero-Modified Models Cues for Zero-Modified Models Possible Trap #1 Possible Trap #2 Discussions Count & Normal Theory The Cues & The Solutions Zero-Inflated Models Hurdle Models Expected Count – Observed Count Conclusions Zero-Inflated Models (Poisson; ZIP) Western Mensurationists’ Meeting 2009 Tzeng Yih Lam 06.23.2009 Lambert (1992)  Two States: Perfect and Imperfect States,  Finite Mixture Model (FMM) with 1 latent structure: An observation belongs to either state.  Specify it as Zero-Inflated Negative Binomial (ZINB). Probability of Belonging to Perfect State

6. Cues for Using Zero-Modified Models Cues for Zero-Modified Models Possible Trap #1 Possible Trap #2 Discussions Count & Normal Theory The Cues & The Solutions Zero-Inflated Models Hurdle Models Expected Count – Observed Count Conclusions Hurdle Models (Poisson; HPOIS) Western Mensurationists’ Meeting 2009 Tzeng Yih Lam 06.23.2009 Mullahy (1986) 2 Baughman (2007)  Under FMM framework comparable to ZI models, Hurdle Models with 2 latent structures 2 : An observation either cross the ‘hurdle’ or not, All observations are in the Imperfect State.  Specify it as Hurdle Negative Binomial (HNB). Probability of Crossing the Hurdle

7. Cues for Using Zero-Modified Models Cues for Zero-Modified Models Possible Trap #1 Possible Trap #2 Discussions Count & Normal Theory The Questions The Simulation Study The Bias & AICc Other Preliminary Key Findings Some Plausible Explanations Given known data generating process (dgp): (1)Is there any bias when the data is fitted to different ZI and H model specifications? (2)Is/Are there any universally best fit models? Western Mensurationists’ Meeting 2009 Tzeng Yih Lam 06.23.2009

8. Cues for Using Zero-Modified Models Cues for Zero-Modified Models Possible Trap #1 Possible Trap #2 Discussions Count & Normal Theory The Questions The Simulation Study The Bias & AICc Other Preliminary Key Findings Some Plausible Explanation 4 Factors (1)LAMBDA ( λ ): 0.3 1.5 5.0 (2)INFLA ( p ): 0.0 0.25 0.75 (3)RATIO ( Var/Mean ): 1.0 1.5 3.0 (4)SAMPLE : 25 50 75 100 250 Western Mensurationists’ Meeting 2009 Tzeng Yih Lam 06.23.2009  For each of the 27 dgp (LAMBDA × INFLA × RATIO), generate 1000 sets of SAMPLE random count,  Fit each set to six model specifications: POIS, NB, ZIP, ZINB, HPOIS, HNB,  Calculate mean %RBIAS for each parameter: λ, p, π and compute AICc.

9. Western Mensurationists’ Meeting 2009 Tzeng Yih Lam 06.23.2009 SAMPLE = 100

10. Western Mensurationists’ Meeting 2009 Tzeng Yih Lam 06.23.2009 SAMPLE = 100 Bias at LAMBDA = 0.3

11. Western Mensurationists’ Meeting 2009 Tzeng Yih Lam 06.23.2009 SAMPLE = 100 Bias with ZIP & HPOIS

12. Western Mensurationists’ Meeting 2009 Tzeng Yih Lam 06.23.2009 SAMPLE = 100 Bias with ZINB & HNB

13. Western Mensurationists’ Meeting 2009 Tzeng Yih Lam 06.23.2009 POIS NB HNB NB ZINB HNB ZIP ZINB HPOIS HNB ZIP ZINB HPOIS HNB NB ZINB HNB ZIP ZINB HPOIS HNB ZIP ZINB HPOIS HNB NB ZIP ZINB HPOIS HNB SAMPLE = 100 Lowest AICc

14. Cues for Using Zero-Modified Models Cues for Zero-Modified Models Possible Trap #1 Possible Trap #2 Discussions Count & Normal Theory The Questions The Simulation Study The Bias & AICc Other Preliminary Key Findings Some Plausible Explanations (1)Variance of estimated λ is the highest when LAMBDA = 0.3, (2)Variance of estimated λ decreases with increasing LAMBDA but it increases with increasing RATIO and/or INFLA, (3)Probability in Perfect State, p, from ZI models has largest (+ve and –ve) bias and variance at LAMBDA = 0.3, (4)Overdispersion parameter, θ, requires ≥ 250 SAMPLE to achieve negligible bias. Western Mensurationists’ Meeting 2009 Tzeng Yih Lam 06.23.2009

15. Cues for Using Zero-Modified Models Cues for Zero-Modified Models Possible Trap #1 Possible Trap #2 Discussions Count & Normal Theory The Questions The Simulation Study The Bias & AICc Other Preliminary Key Findings Some Plausible Explanations Maximum Likelihood Theory Ingredient = a simulated set of count Optimize the parameter estimates to match the marginal count distribution. Large bias and variance of λ and p at LAMBDA = 0.3, When there are either too many zeros or ones, Binomial GLM seems to be unstable Min and Agresti (2005)  unstable estimates of p  unstable estimates of λ and θ. There might not be enough information when LAMBDA = 0.3. ZI models estimation are based on EM algorithm, H models separately maximize the likelihood functions of π and λ. Western Mensurationists’ Meeting 2009 Tzeng Yih Lam 06.23.2009

16. Cues for Using Zero-Modified Models Cues for Zero-Modified Models Possible Trap #1 Possible Trap #2 Discussions Count & Normal Theory Perfect & Imperfect States A Priori Knowledge Rarity Conclusions Western Mensurationists’ Meeting 2009 Tzeng Yih Lam 06.23.2009 Perfect State in Ecology Context: It is a set of habitat conditions that do not host the interested species, Imperfect State in Ecology Context: It is a set of habitat conditions that host the interested species but one may not find the species there, This does not directly differentiate sink & source, saturated & unsaturated habitat, fundamental & realized niche etc. Zero Structural Random Accidental Stochastic Sampling True False

17. A: “Did you smoke any cigarette last week?” B: “No” ; 0 A: “Are you a smoker?” B: “Yes” Cues for Using Zero-Modified Models Cues for Zero-Modified Models Possible Trap #1 Possible Trap #2 Discussions Count & Normal Theory Perfect & Imperfect States A Priori Knowledge Rarity Conclusions Western Mensurationists’ Meeting 2009 Tzeng Yih Lam 06.23.2009 Zero-Inflated Models have 2 states: Perfect and Imperfect States Main Assumption: You do not know the observation belong to which state. Hurdle models have 1 state: Imperfect State Zero-Inflated Models Hurdle Models

18. Cues for Using Zero-Modified Models Cues for Zero-Modified Models Possible Trap #1 Possible Trap #2 Discussions Count & Normal Theory Perfect & Imperfect States A Priori Knowledge Rarity Conclusions Western Mensurationists’ Meeting 2009 Tzeng Yih Lam 06.23.2009 A priori knowledge such as species habitat range, will likely influence the model choice In ecology, scale matters … Grains Extent POIS NB ZIP ZINB HPOIS HNB

19. Cues for Using Zero-Modified Models Cues for Zero-Modified Models Possible Trap #1 Possible Trap #2 Discussions Count & Normal Theory Perfect & Imperfect States A Priori Knowledge Rarity Conclusions Western Mensurationists’ Meeting 2009 Tzeng Yih Lam 06.23.2009 Modeling of rare species habitat association Cunningham & Lindenmayer (2005) and many others…

20. Cues for Using Zero-Modified Models Cues for Zero-Modified Models Possible Trap #1 Possible Trap #2 Discussions Count & Normal Theory Perfect & Imperfect States A Priori Knowledge Rarity, Extent & Grains Conclusions Western Mensurationists’ Meeting 2009 Tzeng Yih Lam 06.23.2009 What Do the Ecologists Need To Do?

21. Cues for Using Zero-Modified Models Cues for Zero-Modified Models Possible Trap #1 Possible Trap #2 Discussions Count & Normal Theory Perfect & Imperfect States A Priori Knowledge Rarity, Extent & Grains Conclusions Western Mensurationists’ Meeting 2009 Tzeng Yih Lam 06.23.2009 The Great Escape (1963) Capt. Hilts (The Cooler King) 1961 British 650cc Triumphs

22. Cues for Using Zero-Modified Models Cues for Zero-Modified Models Possible Trap #1 Possible Trap #2 Discussions Count & Normal Theory Perfect & Imperfect States A Priori Knowledge Rarity, Extent & Grains Conclusions Western Mensurationists’ Meeting 2009 Tzeng Yih Lam 06.23.2009 Escape from defining the types of zeros: There is no restriction on threshold for mixing & hurdle, Change current threshold from 0  1, Change perfect to near-perfect state (ZI models), changing ecological implication of the models. N-mixture models (Royle 2004) Escape from using ZI & H models : If one is uncomfortable with two-states processes, Small Area Estimation Rao(2003), Extreme Value Model Coles(2001).

23. Cues for Using Zero-Modified Models Cues for Zero-Modified Models Acknowledgement Possible Trap #2 Discussions Count & Normal Theory Perfect & Imperfect States Acknowledgement A Priori Knowledge Rarity, Extent & Grains Conclusions Western Mensurationists’ Meeting 2009 Tzeng Yih Lam 06.23.2009 Hayes Family Foundation Funds for Silviculture Alternatives Dilworth Awards, OSU Doug Maguire

24. Cues for Using Zero-Modified Models Cues for Zero-Modified Models Acknowledgement Possible Trap #2 Discussions Count & Normal Theory Perfect & Imperfect States Acknowledgement A Priori Knowledge Rarity, Extent & Grains Conclusions Western Mensurationists’ Meeting 2009 Tzeng Yih Lam 06.23.2009 Thank You for Listening! Any *Err… Question?

25. Cues for Using Zero-Modified Models Cues for Zero-Modified Models Possible Trap #1 Possible Trap #2 Discussions Count & Normal Theory The Questions The Simulation Study The Bias & AICc Other Preliminary Key Findings Some Plausible Explanations AICc, Information theory. More flexible model parameterization will have better fit. Sample size is an issue for ZINB and HNB models for fitting. Western Mensurationists’ Meeting 2009 Tzeng Yih Lam 06.23.2009

26. Cues for Using Zero-Modified Models Cues for Zero-Modified Models Possible Trap #1 Possible Trap #2 Discussions Count & Normal Theory The Questions The Simulation Study The Bias & AICc Other Preliminary Key Findings Some Plausible Explanations Western Mensurationists’ Meeting 2009 Tzeng Yih Lam 06.23.2009

27. Cues for Using Zero-Modified Models Cues for Zero-Modified Models Possible Trap #1 Possible Trap #2 Discussions Count & Normal Theory Perfect & Imperfect States A Priori Knowledge Rarity Conclusions Western Mensurationists’ Meeting 2009 Tzeng Yih Lam 06.23.2009 A priori knowledge such as species habitat range, and extent and grains will likely influence the model choice.