Traditionally, sphericity (i.e., independence and homoscedasticity for raw data) is put forward as the condition to be satisfied by the variance–covariance matrix of at least one of the two observation vectors analyzed for correlation, for the unmodified t test of significance to be valid under the Gaussian and constant population mean assumptions. In this article, the author proves that the sphericity condition is too strong and a weaker (i.e., more general) sufficient condition for valid unmodified t testing in correlation analysis is circularity (i.e., independence and homoscedasticity after linear transformation by orthonormal contrasts), to be satisfied by the variance–covariance matrix of one of the two observation vectors. Two other conditions (i.e., compound symmetry for one of the two observation vectors; absence of correlation between the components of one observation vector, combined with a particular pattern of joint heteroscedasticity in the two observation vectors) are also considered and discussed. When both observation vectors possess the same variance–covariance matrix up to a positive multiplicative constant, the circularity condition is shown to be necessary and sufficient. “Observation vectors” may designate partial realizations of temporal or spatial stochastic processes as well as profile vectors of repeated measures. From the proof, it follows that an effective sample size appropriately defined can measure the discrepancy from the more general sufficient condition for valid unmodified t testing in correlation analysis with autocorrelated and heteroscedastic sample data. The proof is complemented by a simulation study. Finally, the differences between the role of the circularity condition in the correlation analysis and its role in the repeated measures ANOVA (i.e., where it was first introduced) are scrutinized, and the link between the circular variance–covariance structure and the centering of observations with respect to the sample mean is emphasized. 相似文献
This article considers an approach to estimating and testing a new Kronecker product covariance structure for three-level (multiple time points (p), multiple sites (u), and multiple response variables (q)) multivariate data. Testing of such covariance structure is potentially important for high dimensional multi-level multivariate data. The hypothesis testing procedure developed in this article can not only test the hypothesis for three-level multivariate data, but also can test many different hypotheses, such as blocked compound symmetry, for two-level multivariate data as special cases. The tests are implemented with two real data sets. 相似文献
We study economies with one private good and one pure public good, and consider the following axioms of social choice functions. Strategy-proofness says that no agent can benefit by misrepresenting his preferences, regardless of whether the other agents misrepresent or not, and whatever his preferences are. Symmetry says that if two agents have the same preference, they must be treated equally. Anonymity says that when the preferences of two agents are switched, their consumption bundles are also switched. Individual rationality says that a social choice function never assigns an allocation which makes some agent worse off than he would be by consuming no public good and paying nothing. In Theorem 1, we characterize the class of strategy-proof, budget-balancing, and symmetric social choice functions, assuming convexity of the cost function of the public good. In Theorem 2, we characterize the class of strategy-proof, budget-balancing, and anonymous social choice functions. In Theorem 3, we characterize the class of strategy-proof, budget-balancing, symmetric, and individually rational social choice functions. 相似文献
Permutation tests for symmetry are suggested using data that are subject to right censoring. Such tests are directly relevant to the assumptions that underlie the generalized Wilcoxon test since the symmetric logistic distribution for log-errors has been used to motivate Wilcoxon scores in the censored accelerated failure time model. Its principal competitor is the log-rank (LGR) test motivated by an extreme value error distribution that is positively skewed. The proposed one-sided tests for symmetry against the alternative of positive skewness are directly relevant to the choice between usage of these two tests.
The permutation tests use statistics from the weighted LGR class normally used for making two-sample comparisons. From this class, the test using LGR weights (all weights equal) showed the greatest discriminatory power in simulations that compared the possibility of logistic errors versus extreme value errors.
In the test construction, a median estimate, determined by inverting the Kaplan–Meier estimator, is used to divide the data into a “control” group to its left that is compared with a “treatment” group to its right. As an unavoidable consequence of testing symmetry, data in the control group that have been censored become uninformative in performing this two-sample test. Thus, early heavy censoring of data can reduce the effective sample size of the control group and result in diminished power for discriminating symmetry in the population distribution. 相似文献
In this paper, we introduce a new nonparametric test of symmetry based on the empirical overlap coefficient using kernel density estimation. Our investigation reveals that the new test is more powerful than the runs test of symmetry proposed by McWilliams [31]. Intensive simulation is conducted to examine the power of the proposed test. Data from a level I Trauma center are used to illustrate the procedures developed in this paper. 相似文献