 Hello, and thank you for viewing my presentation, where I'll be talking about how we've used sub-metro-analyses to maintain independence among spatio-templare-reprocated demographic data sets. My name is Alex Nicolharper, and this is work done with Professor Patrick Doncaster and Professor Tom Ezad at the University of Southampton, and Dr Kevin Wood and Dr Jeff Hilton at the Wildfowl and Wetlands Trust. I'm going to kick off with some acknowledgments, so this work is part of my PhD, which is funded by a NERC's Bit Fire DTP Award. The lovely photo in the background of the slides is taken by Kate Evans, the behalf of the Wildfowl and Wetlands Trust, and of course a big thank you to my supervised routine. So a little bit of preamble, I'd like to point out at this point that our metro-analyses is slightly unusual in that we are metro-analyzing mean values rather than effect sizes. This might be termed informal and informal metro-analyses in the sense of Marysin Colleagues 2016, in that we are reporting essentially the mean of a distribution of estimated values in a metadata set. However, we are using meta-analysical statistical methods, so perhaps it would be a formal metro-analyses in that sense. The method that we use is an established method of Doncaster and Spake, and it is inverse variance weighting with adjustment for small samples. And the benefit of this method is that it is accounted for those small sample sizes, and it also allows you to use studies which haven't necessarily provided an estimate of variance. So I'll quickly run through what this method entails. So for the random effects aerostructure, which is what we're using, you're calculating your meta-estimate as the sum of the products of weightings and estimates divided by the sum of the weightings. For study-level means, weighted by 1 over Bi plus T squared, where T squared is Crockwind's estimator of between study variance, and the study-level error variances Vi are calculated by the mean variance got all the studies divided by each sample size. So just to reiterate that, taking the mean of the variances is the adjustment for small sample sizes, but what it allows us to do is to include studies which haven't got an associated variance, and the meta-variance is then 1 divided by some of the weightings. So meta-analyses is a useful tool in demography, so study of populations and what makes them up. For example, this study of the spotted owl by boys and colleagues, they state that meta-analysis can be used to combine results from multiple demographic studies replicated in time and space to obtain estimates of vital rates. In this case, the vital rates that are interested in are juvenile survival, adults' survival, and fertility. This example is the black bear by Beston. The meta-analysis considers vital rates for the black bear across North America. This plot shows the mean cub survival, but one of their findings is that overall survival and fertility values differed between eastern and western North America, with an apparent trade-off across these two main sets of vital rates. While this meant that they couldn't really use overall averages for the whole of North America, generalising across the continent, the process since this did uncover avenues for further research and tailored management. But demographic data sets tend to be messier than those that are typically subjected to meta-analysis in other fields. This is the two reasons. Firstly, because the ecological data sets in general tend to encompass a lot of natural variation and the presence of covariates with a large range of scale of replication. And demographic data sets in particular tend to include lots of observational data, so people going out and counting the number of eggs or other types of offspring, for example, rather than the sorts of experiments that might be used to apply meta-analyses in, say, medicine. Hence, demographic meta-analyses need careful application of accepted methodologies, for example, in relation to avoiding non-independence, as we're discussing here. So, why are we looking at doing these sorts of meta-analyses? In my PhD, we used population modelling to investigate breeding ecology and informed conservation for the common idea that you can see here, a well-studied seed-up of the northern hemisphere. And our theoretical models are designed to be relevant for the species as a whole, and hence we're wanting to parameterise them with some sort of global mean values. We have a data paper which describes alkylation of vital rate estimates for this species, and we had over 20 independent estimates for adult annual survival, clutch size of the number of eggs laid per breeding attempt, and hatching success, which is the proportion of eggs producing young. And with this sort of scale of data, we then wanted to meta-analyse each of these vital rates. However, our data sets for these vital rates can take many studies of internal replication and sort of talk you through what that means. So, much data collected by researchers are in the field to gather data to answer a particular question. So, for example, how does predator control affect adult survival? How does clutch size vary with age? Or how does hatching success has it affected by the presence of predators such as mink? And often they will have data collected across multiple years and or locations, so sites with a variable study area. So, for example, Woodattel 2021 has adult survival studied across three colonies, as you can see here. There are studies which look at clutch size across different islands within a data set, so here we have egg island at Stump Island. And there are studies which run over multiple years, in this case hatching success from 1972 to 1974. Overall, there is this sort of internal replication, so different areas, different years, different types of islands, whether they're wooded or open. For 7% of adult survival studies that we found, 33% of clutch size studies and 12% of hatching success studies. And while these values are non-trivial, they also show that any attempts at some sort of full data analysis would be working with much reduced and effectively intractable data sets. So our question was how to make use of as much available information as possible, while maintaining equivalence between inputs to the metronasis. So it's been stated that ideally a metronasis involves a single effect size estimate being derived for each study in order to maintain independence and to avoid pseudo replication. In our case this is a mean rather than the effect size but the case stands and where possible we decided that we could apply our overarching metronasis methodology to each case as a sub metronasis. Ensuring that the overall metronasis is then conducted on independent replica observations per the way. At our point at this stage we're referring to the sub metronasis, similarly to some mentions of metronasis or super syntheses rather than the sub metronasis. We believe that this aligns with the suggestions of Megasyn and colleagues to avoid losing potential data. You can adjust sample size, variance or weighting to represent true information content and had a way in colleagues. The metronasis reduced data set is preferable to vote counting, a larger data set. So to show more practically what we were considering. The example of wood et al with the adult survival of those different colonies presents three estimates of adult survival with associated standard errors and sample sizes. So what we would do is basically apply the overall methodology of our overall metronasis to this study. So we would first convert the standard errors to variances. We would then calculate a mean variance across those three estimates. We would calculate those BIs, so the mean variance that we just calculated divided by the sample size that will each estimate. We would calculate the T squared corpus estimator. This is used to calculate weightings and from those we would calculate a meta estimate and a meta variance at this study level. So what we would then send forward to the main meta analysis. The estimate is that meta estimate we just calculated and the variance is the meta variance multiplied by the number of contributing estimates to allow compatibility with calculated variance for other studies that haven't undergone meta analysis. So to illustrate the forest plot for adult survival, we have coded the cases where we've used the sub meta analysis with an asterisk and applied to them here. So with the accru set out 2012 case, we've conducted a sub meta analysis across two replicates and that gives a value of 0.72 as shown. It would actually be the same value if we'd just taken simple mean across these two replicates, albeit less precise. And if we had made the decision rather than doing some form of sub mean to just send forward one of the estimate. So for example, if we picked the most precise estimate that with the smallest standard error, that would give 0.761, which is quite similar. So if we would tell 21 across three replicates, the sub meta analysis gives not point nine one eight mean we give not point nine one six. But if we picked either the most precise or the most study with the larger sample size, this would give not point nine four. So again, quite substantially larger. It's true that this sub meta analysis is balancing information content bias. So again, with the would tell case, the adults five was a not point eight seven not point nine two two and not point nine four across those three sites. We selected the estimate with the largest sample size and would likely overestimate as I showed in the previous slide. All three estimates forward to meta analysis this will then false equivalence to average estimates from other studies which themselves likely represent some form of mean. So this methodology is where it's taking a mean, in this case, whether unweighted or weighted with our methodology reduces the risk of bias ensures that comparison. So we have valuable information is retained or avoiding overweight team of potentially biased information. It's similar to senior etals 26 second order meta analysis, which I'll show in the next slide of the comparison, but it's theoretically closer to the solutions or independence offered by this. So just to compare our sub meta analysis to the second order meta analysis of senior etal. In that case it's a meta analysis of meta analyses, which is done in quite a few studies. In our case we have three meta analysis on per vital rate of 1166 studies of which some of those studies are providing a single estimate which is itself a mean. You know, my good, and others provide multiple estimates on what we can do with meta analysis, so each value going forward to the main meta analysis represents a mean upon years and sites. So to show the effect of these decisions. So if you just took our data set, and did a mean with all the estimates of each of the sub estimates, and likewise, equivalently, you get a value of 0.256. If you take a simple mean but you have done sub means across those studies that we've talked about, you get 0.257. The mean overall, so the meta analysis methodology, but sending forward simply the most precise single estimate from relevant studies, the overall value is quite significantly higher at 0.861. Whereas the weighted mean, with either simple means or weighted means across sub studies gives 0.857. While these values might not seem particularly large, because this is adults Bible so it ranges from 0 to 1 potentially. But it's very influential on population dynamics of these sorts of species. And so small changes make a big difference. So it's important that we're getting close to some sort of. We did explore meta regression, but we didn't find any obvious relationships. So I'll show you an example we did think that perhaps clutch, which we have the most data might vary geographically, because there are some species of the species which recognize. We did not find an obvious pattern for clutch size with latitude, but there was a hint that clutch size was smaller within the Arctic Circle. But we wondered whether that was because latitude is perhaps too simplistic. Very different climate regimes could be experienced in the last years, for example, I just found approximately 57 degrees north in both Scotland and Hudson Bay, whereas the latter is much more extreme freezing over in winter, for example. Sorry, that was my cat. So continentality and oceanity are indices for the cooling effect of land masses and the warming effect of oceans respectively. And they were both found to be marginally stiffer predictors of clutch size. On the left, you can see that mean clutch size is decreasing with increasing continentality. So that's harsher conditions, short breeding season, and the opposite of oceanity. There's some hint that perhaps the oceanity effect is strongest for those above the 60 degrees north line and not really that important below, however, the interaction was not significant. So this sort of thing can be useful extension, but for us, we have reasons to focus on single estimates because we were trying to parameterize our population models for analysis. So to summarise, sub metro analysis that are non dependence within studies to provide us with a global mean vital rate, where the interest is more in those underlying drivers, or where there's evidence of strong heterogeneity, such as between species, more so than we found for this species. Multi-level modelling will be the obvious method as recently developed by Nacogawa and colleagues. The limited code for that study implies it requires more balanced data than ours. So to reiterate, we have studies providing a single estimate across a single site year, single estimate, so some form of mean across multiple site year combinations, and multiple estimates across multiple site year combinations, which we did our sub metro analysis on. We don't have no sample sizes and we're therefore excluded due to our method. A sample size but no variance, which can be handled by our method, and sample size and variance. So overall, it's like a really big mixture of data, which we think is probably typical of demographic data sets and less so the other sorts of data sets which we tend to be metro analysed. Thank you very much for listening. Hope this has been useful.