 Hi there, my name is Wolfgang Fichtbauer and I will now give my talk on selection models for publication buys in meta-analysis. So when we conduct a meta-analysis, we collect the studies that have examined some phenomenon of interest and we quantify that phenomenon in terms of some effect size or outcome measure. And then we can analyze these estimates using, for example, a random effects model to estimate the average true effect and the variance in the true effects. A special case of course arises, if there is no variance in the true effects, then the model collapses down to a fixed effects model and we estimate the true effect. Now a major concern in meta-analysis is publication buys. We want to synthesize all relevant studies on the phenomenon, or at least a representative sample of them, but the studies we find mostly come from the published literature. And those studies have undergone a selection process and as a result, they may no longer be representative of all of the research that has been conducted on a phenomenon. And this may lead to bias in our estimates from our meta-analysis. An example of a meta-analysis where this may be an issue is shown here. This is a funnel plot of studies that have examined the risk of lung cancer in those exposed to environmental tobacco smoke. And so here the results are quantified in terms of log odds ratios with positive values indicated in increased risk in those exposed compared to those not exposed. This funnel plot is not symmetric. So here on the left-hand side we see a gap and this may indicate publication bias which may lead then to bias when we meta-analyze these studies. One way to address this is through selection models. So these selection models try to model and account for the process by which studies may have been selected for meta-analysis. Now when I say selected, I don't mean explicit selection by the person conducting the meta-analysis, but some implicit process by which certain types of studies are more or less likely to be included in our meta-analysis. In all kinds of selection models have been proposed in the literature. Here I will focus on models where the relative likelihood of selection is a function of the statistical significance, so the p-value of the studies. Now one model of this type is the beta selection model by Sitkovitz and Veviyar. And this is what the selection function looks like, but it may be easier just to look at a picture of what that function can look like. So here for example if both parameters of this function are equal to one, then the selection function is flat. So here there is no association between the p-value of the study and its relative likelihood of selection, but for other combinations of these parameters we see how the likelihood of selection is quite high for highly significant studies, at least statistically speaking, and much lower for studies where the p-value is higher. Now another group of models of this type are these exponential decay models that were described by Preston and colleagues. So here the relative likelihood of selection is an exponential function of the p-value and so this is what it can look like. These are one parameter models, so here when this parameter is equal to zero then again we get a flat function and for other values of this parameter this is what this selection function can look like. An extension of this model is the negative exponential power selection model quite a mouthful. This is a two-parameter function again and this gives it more flexibility as to what that selection function can look like. Now in all of these examples there is a smooth function that describes the association between the p-value and the selection likelihood, but another group of models are these step function models. Here we define some cut points and within the intervals defined by these cut points the likelihood of selection is constant. Now we set typically these cut points at inherently interesting p-values such as 0.05, so that is of course that point where a significant effect turns into a non-significant one or 0.5 where for one sided p-values the direction of the effect flips. How can you fit these types of models? Well there are several R packages to do so and I recently added the possibility to fit selection models to the metaphor package. So to give an example here I'm fitting a fixed effects model to the data that I showed to you earlier. So here we have a data set that includes the log arts ratios and the corresponding sampling variances and once I have such a model object then I can use the cell model function to fit for example the beta selection model or step function model and several other models. So these are the results from the fixed effects model and then from the beta selection model so here we have the estimates of the two selection parameters and then the adjusted effect and these are the results from the step function model that I fitted and again the adjusted effect. And this is what the selection function looks like for all these different types of models and some of these models suggest less severe forms of selection and other models suggest quite severe forms of selection and so depending on the model the adjusted estimate can actually be quite close to 0. Now this immediately raises the question which of these results we should trust but there is actually more research really needed to say how should we select a selection model. So that is really still an open question. And I should also mention that fitting these models isn't easy. You shouldn't try this if you only have a handful of studies especially in the context of random effects models. So distinguishing heterogeneity from these selection processes is quite difficult. And you can also then run into convergence problems or other numerical difficulties. So you should really only attempt to do this if you have a sufficiently large number of studies which then raises the question what is sufficiently large and again there is no clear answer to that either. Now I want to finish this talk with one last point here namely that hopefully we will get to a point where we won't need these models in the first place. So if our meta-analysis is based on pre-registered studies then essentially we are minimizing or eliminating publication bias. But while we are not quite there yet so in the meantime we can maybe try to address publication bias through some of these selection models. So here are the references and my contact information. So thank you for your attention.