Freshwater fisheries conservation can increase biodiversity

(see [27] for a discussion of the utility of this approach in this context). As usual in this litera- ture, we base our analysis of the statistcial significance of our estimates on bootstrapped stan- dard errors, as these allow us to account for the additional noise introduced by the initial statistical modelling of the participation decision [28]. The CIA is a strong assumption as it essentially requires us to accept that no relevant con- founder was left out of the estimation of the propensity score, the function that we use to judge the similarity between treated and control units. Rather than claiming that such assumption is held, we follow the ex ante literature and present a form of sensitivity analysis that relies on the estimation of Rosenbaum bounds [29]. Intuitively, we answer the question “how important does the relative influence of unobserved confounders have to be to change our conclusion about the value of the program?”. More formally, we can express this idea by rewriting the probability of participation as ProbðT ¼ 1 j X; uÞ ¼ FðbX þ guÞ ð5Þ where u are the unobserved variables and γ is the effect of these variables on the participation decision; F is the CDF of u. If we assume that F = Λ, the odds-ratio that two individuals i and j will be treated is given by Pi 1 Pi Pj 1 Pj ¼ Pið1 PjÞ Pjð1 PiÞ ¼ expðbXi þ guiÞ expðbXj þ gujÞ ð6Þ But if matching is well done (ie, if CIA holds), this converts to exp(γ(ui − uj)). Unconfoud- edness means that either ui = uj (balance on unobservables, as under randomization) or γ = 0

(ie, u are not important in explaining participation, given we accounted for all relevant con- founders X). If any of these is not true, then two observationally identical individuals will have different probabilities of getting treatment. Rosenbaum [29] shows that these probabilities (in terms of odds ratio) are bounded by: 1 eg � Pi 1 Pi Pj 1 Pj � eg ð7Þ where the interpretation of eγ = 2 tells us that observationally similar units (in terms of X) could differ in terms of their probability of receiving treatment by as much as a factor of 2. The important question then is: even if true, would that change our estimates of the treatment effects? Finally, we consider that differences in propensity score estimates between treated units (ie, differences in likelihood of participating in ComFishIII) can be interpreted as a measure of heterogeneity in initial conditions. We can use this intuition to examine heterogeneous treat- ment effects, using the approach suggested in [30], who define neighborhoods of treated units that are more homogeneous than the whole sample. This splitting of the original sample then allows for the analysis of the relation (trend) between the ATT estimate and the values of the propensity score, which is estimated using local polynomial regression. Differences in those local estimates would be suggestive of the importance of initial conditions on the magnitude of the treatment effect, with potential implications in terms of targeting, although care must also

be taken in interpreting this link between initial conditions and outcomes as causal, given that in the context of programs that are designed locally, treated villages may exhibit other forms of heterogeneity (for example, with respect to the rules that are adopted). PLOS ONE FCZ and biodiversity PLOS ONE | https://doi.org/10.1371/journal.pone.0233775 May 27, 2020 4 / 14