 Hello, everyone. Today I'm presenting a package, G-MEDA, which conducts efficient MEDA analysis based on confidence distribution. This is a joint work with Professor Shea from the State Department at Rutgers University and Dr. Yang and Mr. Chen. I also would like to thank National Science Foundation for their support for this work. I will first introduce briefly the concept of confidence distribution and describe the procedures that we can use, the confidence distribution to conduct MEDA analysis. Then I will concentrate in demonstrating the usage of G-MEDA package. As we all know with the advancement of technologies in connecting, storing, and accessing information, we are getting an exploding amount of data every day. As a result, we need efficient processing mechanism for meaningful results. MEDA analysis is one of such tools and it focuses on combining results, analysis results from multiple separate studies. And it has been proven that MEDA analysis can make more reliable inference from combined results and from just by using individual studies. With a large amount of literature, MEDA analysis still remains an active research area in both applications and the theoretical foundation. Among the theoretical foundation, confidence distribution is a function of sample data to present confidence intervals of all levels. With that, with that or confidence distribution, simply we call it as CDN. We can answer many questions in statistical inferences, such as we can use confidence distribution to get point estimate or confidence or interval estimate or density estimate. Here are some typical examples. For example, when we use a normal sample size M, the distribution estimate is a normal distribution. Use the sample mean as the distribution mean and the one over sample size as the distribution variance. With that, we can derive the point estimate as well as the confidence interval at any level that you want. In particular, we are using confidence distribution in conducting MEDA analysis. There are a few reasons for doing that. CD contains much richer information than a single point estimator or interval estimator. And it also covered a variety of scenarios or cases with supporting theories. The key steps to construct MEDA analysis with CD are very straightforward, at least in concept. First, we summarize information using CD in each study, then combine information from different studies from the combination of those CDs. Here is a generic recipe for combining individual independent CDs. And the recipe is in equation number one. Here, each i, for example, h1, h2, hk, is the individual CD for i's study. And hc is the combined CD. And gc is a function. It is a non-dequasing and a continuous function in every coordinate, one for each study. And we discussed some choices of, sorry. So often, we use two choices for the g function. One is a weighted sum of monotonic functions a0. The other way is a much simpler, simplified version of weighted sum of coordinates, which is in here, equation number three. And we will discuss some choices of weights and monotonic function a used in G-MEDA package. In the first case, we use p-value for each study. And the unit weights and unlock functions. Then we can get the combined p-values or case studies right here. And this is exactly the result from Fisher's method. And when we use inverse of cumulative distribution function for normal random variables, when we combine those p-value, the result right here is actually the stop-first method. Well, CD combination framework can also include the results from fixed effects and meta-analysis. Here is the details. And I will skip some details in the interest of time. The deduction of the result is quite straightforward, as long as we specify a combination function. Besides the fixed effect meta-analysis, almost all existing methods of information combination can be achieved in the CD aggregation framework. Here is the list. They describe details by some of the papers listed here. So let's just try to give you an impression that the CD combination framework is quite generic. And beyond those existing method, the CD combination method is able to expand to include robust meta-analysis. So the first meta-analysis, the robust meta-analysis, is for studies with a large number of subjects. And we download studies set far from the majority by using adaptive weights. By doing so, we can obtain a robust estimate of the true parameter. And the second type of robust meta-analysis is with a large number of studies, which can be considered as an extension to the traditional random effects model. Here, the true parameters of study come from a population, which is a mixed population. It has a regular part and a contaminated part. A combined CD is proposed to robust estimation with protection against possible model specifications. And we will see an example of both robust meta-analysis by using CD. All framework in G-Meta also can conduct the exact inference for two-by-two tables with zero cell counts. Without it, we don't need to use the conventional approach to pre-processing the tables, such as adding continuity corrections of half, or using the large sample approximation. So with all the quick introduction of the groundwork that I explained so far, now I'm ready to explain G-Meta package, which actually mimics the structure of the GLM function in R. For example, the argument family in GLM is corresponding to GC in G-Meta. And the link is mapped to weights in G-Meta package. And GC and weights, as we explained a moment ago, they are the key components of the combination function for CD. And the G-Meta can conduct classic P-value combinations, as well as meta-analysis with fixed effects, random effects, robust meta-analysis, and exact analysis. The package, we can download the package from Crane. There are a few key arguments in the G-Meta function, which we will take a close look at method and the link funk in a moment. The other arguments are GMI, which is for the study, which is for the summary statistics. And GMO x grid is for the range and the grid points to evaluate the combined CD. To demonstrate the package, we include the also data set in the G-Meta package. And this data set has 41 randomized trial of treatment for stomach ulcers. Now let me first show the P-value combination example. And to get ready, we first make continuity corrections by imputing 0.5 to 0 events right here. Then calculate the law, odds ratio, and their standard deviations for each study. Then use those results to compute P-values for all studies. Then the normal method and the tipped method in combining P-values can be simply implemented by using the G-Meta function and the specified method with a method of normal or method of tipped. Then we can print out to combine P-values for all the study. And the G-Meta can support Fisher, Stoffer, Tipper, MaxSum, what are five methods in the P-value combination and that the result is shown in here. For model-based meta-analysis such as fixed or rendering effects meta-analysis, we need to specify a few arguments. For example, method. This is for the assumption of meta-analysis model, unmeta-analysis model. For example, DAL estimator or VML estimator. And the second one is link-funk. This is for A0 function in a combination framework. So the default value is inverse normal CDF. And other choices, for example, can be inverse Laplace CDF. And for the fixed effect meta-analysis, we use individual odds ratio and standard deviation and the default link bound of inverse normal CDF. Inverse normal CDF. And we can get the combined CD from which some estimates can be obtained. For example, mean median standard deviation, no and upper bound of happiness intervals. Here is the result from the fixed effect meta-analysis for all the studies. And you can get the combined CD and from the combined CD, which is a distribution function and calculate the mean and median and confidence lower bound, upper bound. So those are all can be used to estimate the common also ratio. Random effects models are also easy to fit. We only need to select an appropriate method argument. For example, we use the simonium and the LOD, the L method and the restricted maximum likelihood method in this example. And the result from those two methods are quite close. The function core in G meta is very simple. And the only thing you need to change is the method and you specify the grid points. And the result can be printed out by the summary function, the usual summary function. For robust meta-analysis, we only need to change the method to be either fixed robust one or random robust two, respectively, who compare the combined CD from a robust and non-robust approach. This is the fixed-frame model. And the one with the dashed line is the robust version of meta-analysis. And for the random effects model, we use the same line type to indicate the CD from robust and non-robust method. And what we can observe for those two plot is the combined CD from robust method is a little wider than the non-robust method. And this is to explain that robust method will trade efficiency for robustness. In here, I'd like to demonstrate an effectiveness of robust meta-analysis. So we construct and contaminated data with all-lier studies. So we basically multiply the odds ratio in study 5, 13, 14, 20, 2, 35, and 41 by 10. Those are the 1, 2, 3, 4, 5, 6. 6 out of the original 5, 6, 41 studies. Only those 5, 6 studies make those odds ratios all-liers. Then we use random effects analysis, regular random effects meta-analysis and robust meta-analysis on the original data and also contaminated data, data with all-liers. And we estimate the odds ratio and it's 95% confidence interval. And here is the result represented in this table. We can observe that the result from robust meta-analysis on the contaminated data is almost the same as the result from the original data so that the result is robust to all-lier. And there are more analysis that the G-Meta is able to conduct. And due to the time constraint, I would like the audience to explore our package documents and try the examples that we have for those cases. And this concludes my talk today. I welcome, thank you for your time. I welcome comments and questions and suggestions. Thank you very much.