Variety Of Correlation Structures Research Articles

The impact of correlated exposures and missing data on multiple informant models used to identify critical exposure windows.

There has been heightened interest in identifying critical windows of exposure for adverse health outcomes; that is, time points during which exposures have the greatest impact on a person's health. Multiple informant models implemented using generalized estimating equations (MIM GEEs) have been applied to address this research question because they enable statistical comparisons of differences in associations across exposure windows. As interest rises in using MIMs, the feasibility and appropriateness of their application under settings of correlated exposures and partially missing exposure measurements requires further examination. We evaluated the impact of correlation between exposure measurements and missing exposure data on the power and differences in association estimated by the MIM GEE and an inverse probability weighted extension to account for informatively missing exposures. We assessed these operating characteristics under a variety of correlation structures, sample sizes, and missing data mechanisms considering various exposure-outcome scenarios. We showed that applying MIM GEEs maintains higher power when there is a single critical window of exposure and exposure measures are not highly correlated, but may result in low power and bias under other settings. We applied these methods to a study of pregnant women living with HIV to explore differences in association between trimester-specific viral load and infant neurodevelopment.

A Mass-Shifting Phenomenon of Truncated Multivariate Normal Priors

We show that lower-dimensional marginal densities of dependent zero-mean normal distributions truncated to the positive orthant exhibit a mass-shifting phenomenon. Despite the truncated multivariate normal density having a mode at the origin, the marginal density assigns increasingly small mass near the origin as the dimension increases. The phenomenon accentuates with stronger correlation between the random variables. This surprising behavior has serious implications toward Bayesian constrained estimation and inference, where the prior, in addition to having a full support, is required to assign a substantial probability near the origin to capture flat parts of the true function of interest. A precise quantification of the mass-shifting phenomenon for both the prior and the posterior, characterizing the role of the dimension as well as the dependence, is provided under a variety of correlation structures. Without further modification, we show that truncated normal priors are not suitable for modeling flat regions and propose a novel alternative strategy based on shrinking the coordinates using a multiplicative scale parameter. The proposed shrinkage prior is shown to achieve optimal posterior contraction around true functions with potentially flat regions. Synthetic and real data studies demonstrate how the modification guards against the mass shifting phenomenon while retaining computational efficiency. Supplementary materials for this article are available online.

Resampling-based stepwise multiple testing procedures with applications to clinical trial data.

Endpoints in clinical trials are often highly correlated. However, the commonly used multiple testing procedures in clinical trials either do not take into consideration the correlations among test statistics or can only exploit known correlations. Westfall and Young constructed a resampling-based stepdown method that implicitly utilizes the correlation structure of test statistics in situations with unknown correlations. However, their method requires a "subset pivotality" assumption. Romano and Wolf proposed a more general stepdown method, which does not require such an assumption. There is at present little experience with the application of such methods in analyzing clinical trial data. We advocate the application of resampling-based multiple testing procedures to clinical trials data when appropriate. We have conjectured that the resampling-based stepdown methods can be extended to a stepup procedure under appropriate assumptions and examined the performance of both stepdown and stepup methods under a variety of correlation structures and distribution types. Results from our simulation studies support the use of the resampling-based methods under various scenarios, including binary data and small samples, with strong control of Family wise type I error rate (FWER). Under positive dependence and for binary data even under independence, the resampling-based methods are more powerful than the Holm and Hochberg methods. Last, we illustrate the advantage of the resampling-based stepwise methods with two clinical trial data examples: a cardiovascular outcome trial and an oncology trial.

A Road to Classification in High Dimensional Space: The Regularized Optimal Affine Discriminant

SummaryFor high dimensional classification, it is well known that naively performing the Fisher discriminant rule leads to poor results due to diverging spectra and accumulation of noise. Therefore, researchers proposed independence rules to circumvent the diverging spectra, and sparse independence rules to mitigate the issue of accumulation of noise. However, in biological applications, often a group of correlated genes are responsible for clinical outcomes, and the use of the covariance information can significantly reduce misclassification rates. In theory the extent of such error rate reductions is unveiled by comparing the misclassification rates of the Fisher discriminant rule and the independence rule. To materialize the gain on the basis of finite samples, a regularized optimal affine discriminant (ROAD) is proposed. The ROAD selects an increasing number of features as the regularization relaxes. Further benefits can be achieved when a screening method is employed to narrow the feature pool before applying the ROAD method. An efficient constrained co-ordinate descent algorithm is also developed to solve the associated optimization problems. Sampling properties of oracle type are established. Simulation studies and real data analysis support our theoretical results and demonstrate the advantages of the new classification procedure under a variety of correlation structures. A delicate result on continuous piecewise linear solution paths for the ROAD optimization problem at the population level justifies the linear interpolation of the constrained co-ordinate descent algorithm.

Time Variation in Diversification Benefits of Commodity, REITs, and TIPS

Diversification benefits of three “hot” asset classes—Commodity, Real Estate Investment Trusts (REITs), and Treasury Inflation-Protected Securities (TIPS)—are well-studied on an individual basis and in a static setting. Using data from 1970 to 2010, this paper documents both that the three asset classes are in general not substitutes for each other, and that diversification benefits of each asset class change substantially over time. Therefore, all three asset classes ought to be included in investors’ portfolios. Furthermore, we show that the observed time variation in diversification benefits can be explained by time-varying return correlations. To see the implications of these findings for asset allocation in practice, we examine the out-of-sample performance of portfolio strategies, based on a variety of correlation structures. We find that the Dynamic Conditional Correlation (DCC) model (Engle, J Bus Econ Stat 20(3):339–350, 2002) outperforms other correlation structures, such as rolling-window, historical, and constant correlations. Our findings suggest that diversification benefits of the three asset classes should be examined in a dynamic setting, and that investors need to use appropriate correlation estimates to adjust for such time variation.

Why do we test multiple traits in genetic association studies?

In studies of complex disorders such as nicotine dependence, it is common that researchers assess multiple variables related to a disorder as well as other disorders that are potentially correlated with the primary disorder of interest. In this work, we refer to those variables and disorders broadly as multiple traits. The multiple traits may or may not have a common causal genetic variant. Intuitively, it may be more powerful to accommodate multiple traits in genetic traits, but the analysis of multiple traits is generally more complicated than the analysis of a single trait. Furthermore, it is not well documented as to how much power we may potentially gain by considering multiple traits. Our aim is to enhance our understanding on this important and practical issue. We considered a variety of correlation structures between traits and the disease locus. To focus on the effect of accommodating multiple traits, we examined genetic models that are relatively simple so that we can pinpoint the factors affecting the power. We conducted simulation studies to explore the performance of testing multiple traits simultaneously and the performance of testing a single trait at a time in family-based association studies. Our simulation results demonstrated that the performance of testing multiple traits simultaneously is better than that of testing each trait individually for almost models considered. We also found that the power of association tests varies among the underlying models. The advantage of conducting a multiple traits test is minimized when some traits are influenced by the gene only through other traits; and it is maximized when there are causal relations between the traits and the gene, and among the traits themselves or when there are extraneous traits.

Time Variation in Diversification Benefits of Commodity, REITs, and TIPS

Diversification benefits of three “hot” asset classes — Commodity, Real Estate Investment Trusts (REITs), and Treasury Inflation-Protected Securities (TIPS) — are well-studied on an individual basis and in a static setting. Using data from 1970 to 2010, this paper documents both that the three asset classes are in general not substitutes for each other, and that diversification benefits of each asset class change substantially over time. Therefore, all three asset classes ought to be included in investors' portfolios. Furthermore, we show that the observed time variation in diversification benefits can be explained by time-varying return correlations. To see the implications of these findings for asset allocation in practice, we examine the out-of-sample performance of portfolio strategies, based on a variety of correlation structures. We find that the Dynamic Conditional Correlation (DCC) model (Engle 2002) outperforms other correlation structures, such as rolling-window, historical, and constant correlations. Our findings suggest that diversification benefits of the three asset classes should be examined in a dynamic setting, and that investors need to use appropriate correlation estimates to adjust for such time variation.

A semiparametric bootstrap approach to correlated data analysis problems

In this note, we outline a simple to use yet powerful bootstrap algorithm for handling correlated outcome variables in terms of either hypothesis testing or confidence intervals using only the marginal models. This new method can handle combinations of continuous and discrete data and can be used in conjunction with other covariates in a model. The procedure is based upon estimating the family-wise error (FWE) rate and then making a Bonferroni-type correction. A simulation study illustrates the accuracy of the algorithm over a variety of correlation structures.

Strategic Trading When Agents Forecast the Forecasts of Others

ABSTRACTWe analyze a multi‐period model of trading with differentially informed traders, liquidity traders, and a market maker. Each informed trader's initial information is a noisy estimate of the long‐term value of the asset, and the different signals received by informed traders can have a variety of correlation structures. With this setup, informed traders not only compete with each other for trading profits, they also learn about other traders' signals from the observed order flow. Our work suggests that the initial correlation among the informed traders' signals has a significant effect on the informed traders' profits and the informativeness of prices.

Strategic Trading When Agents Forecast the Forecasts of Others

We analyze a multi-period model of trading with differentially informed traders, liquidity traders and a market maker. Each informed traders' initial information is a noisy estimate of the long-term value of the asset, and the different signals received by informed traders can have a variety of correlation structures. With this setup informed traders not only compete with each other for trading profits, they also learn about other traders' signals from the observed order flow. Our work suggests that the initial correlation among the informed traders' signals has a significant effect on the informed traders' profits and the informativeness of prices.

An approximation to the null distribution and power of the Durbin—Watson statistic

SUMMARY It is computationally complex to obtain the exact distribution of the Durbin-Watson statistic for detecting autocorrelation in regression residuals. Moreover, the distribution cannot be tabulated once and for all because of its dependence on the regressors. It is found that the distribution can be well approximated by an appropriate Pearson distribution which is computationally simple, easily programmed and seems to be valid when the regressors are commonly observed economic time series. It is also shown that the power of the statistic is relatively insensitive to such choices of the regressors and to a variety of correlation structures of the regression residuals.