The analysis of very small samples of repeated measurements II: a modified box correction

There is a need for appropriate methods for the analysis of very small samples of continuous repeated measurements. A key feature of such analyses is the role played by the covariance matrix of the repeated observations. When subjects are few it can be difficult to assess the fit of parsimonious structures for this matrix, while the use of an unstructured form may lead to a serious lack of power. The Kenward-Roger adjustment is now widely adopted as a means of providing an appropriate inferences in small samples, but does not perform adequately in very small samples. Adjusted tests based on the empirical sandwich estimator can be constructed that have good nominal properties, but are seriously underpowered. Further, when such data are incomplete, or unbalanced, or non-saturated mean models are used, exact distributional results do not exist that justify analyses with any sample size. In this paper, a modification of Box's correction applied to a linear model based $F$-statistic is developed for such small sample settings and is shown to have both the required nominal properties and acceptable power across a range of settings for repeated measurements

Robustness of power in analysis of variance for various designs.

Robustness of power of the analysis of variance technique to the departures from the underlying assumptions of homoskedasticity and independence of error has been considered in various designs, including the mixed and non-orthogonal designs. Distribution of the ratio of two independent quadratic forms is modified with arbitrary scale, parameter g and has been used extensively. The choice of g is also discussed. The results, in general., indicate that the power of the test of equal means is seriously affected when the assumption of homoskedasticity is violated, but'for moderate degree of heteroskedasticity, the actual type I error is not seriously affected. Also, the power of the test of homogeneity of means is highly sensitive to the departure from the fixed effects model to the corresponding random effects model. The problem of design of experiments to optimise power of the test under the constraint of cost is discussed with reference to the one-way classification for both cases of homogeneous and heterogeneous group error variances

Efficient Probit Estimation with Partially Missing Covariates

A common approach to dealing with missing data is to estimate the model on the common subset of data, by necessity throwing away potentially useful data. We derive a new probit type estimator for models with missing covariate data where the dependent variable is binary. For the benchmark case of conditional multinormality we show that our estimator is efficient and provide exact formulae for its asymptotic variance. Simulation results show that our estimator outperforms popular alternatives and is robust to departures from the benchmark case. We illustrate our estimator by examining the portfolio allocation decision of Italian households.missing data, probit model, portfolio allocation, risk aversion

On the history and use of some standard statistical models

This paper tries to tell the story of the general linear model, which saw the light of day 200 years ago, and the assumptions underlying it. We distinguish three principal stages (ignoring earlier more isolated instances). The model was first proposed in the context of astronomical and geodesic observations, where the main source of variation was observational error. This was the main use of the model during the 19th century. In the 1920's it was developed in a new direction by R.A. Fisher whose principal applications were in agriculture and biology. Finally, beginning in the 1930's and 40's it became an important tool for the social sciences. As new areas of applications were added, the assumptions underlying the model tended to become more questionable, and the resulting statistical techniques more prone to misuse.Comment: Published in at http://dx.doi.org/10.1214/193940307000000419 the IMS Collections (http://www.imstat.org/publications/imscollections.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

The Observed Growth of Massive Galaxy Clusters III: Testing General Relativity on Cosmological Scales

This is the third of a series of papers in which we derive simultaneous constraints on cosmological parameters and X-ray scaling relations using observations of the growth of massive, X-ray flux-selected galaxy clusters. Our data set consists of 238 clusters drawn from the ROSAT All-Sky Survey, and incorporates extensive follow-up observations using the Chandra X-ray Observatory. Here we present improved constraints on departures from General Relativity (GR) on cosmological scales, using the growth index, gamma, to parameterize the linear growth rate of cosmic structure. Using the method of Mantz et al. (2009a), we simultaneously and self-consistently model the growth of X-ray luminous clusters and their observable-mass scaling relations, accounting for survey biases, parameter degeneracies and systematic uncertainties. We combine the cluster growth data with gas mass fraction, SNIa, BAO and CMB data. This combination leads to a tight correlation between gamma and sigma_8. Consistency with GR requires gamma~0.55. Under the assumption of self-similar evolution and constant scatter in the scaling relations, and for a flat LCDM model, we measure gamma(sigma_8/0.8)^6.8=0.55+0.13-0.10, with 0.79<sigma_8<0.89. Relaxing the assumptions on the scaling relations by introducing two additional parameters to model possible evolution in the normalization and scatter of the luminosity-mass relation, we obtain consistent constraints on gamma that are only ~20% weaker than those above. Allowing the dark energy equation of state, w, to take any constant value, we simultaneously constrain the growth and expansion histories, and find no evidence for departures from either GR or LCDM. Our results represent the most robust consistency test of GR on cosmological scales to date. (Abridged)Comment: Accepted for publication in MNRAS. 11 pages, 5 figures, 1 table. New figure added: Fig. 4 shows the tight constraints on gamma from the cluster growth data alone compared with those from the other data sets combined

HARDY-WEINBERG EQUILIBRIUM ASSUMPTIONS IN CASE-CONTROL TESTS OF GENETIC ASSOCIATION

The case-control study design is commonly used in genetic association study with a binary trait using unrelated individuals from a population. To test association with a binary trait in a case-control or cohort study, the standard analysis is a chi-square test or logistic regression model that test to detect a difference in frequencies of alleles or genotypes. In this thesis, we derive the maximum likelihood estimator, using Chen and Chatterjee's methods, for standard 1 df genetic tests (dominant, recessive, and multiplicative). We then compare these methods that assume HWE with standard Wald tests and chi-squared tests that do not make the HWE assumption. We consider four different HWE scenarios: 1) when HWE holds in both cases and controls, 2) when HWE does not hold in cases and controls follow HWE, 3) when HWE does not hold in controls, and cases follow HWE and 4) when HWE does not hold in either cases or controls. Our results show that the performances of the three statistics (chi-squared, Wald, and Chen and Chatterjee Wald) are equivalent for multiplicative test under all four HWE scenarios. When HWE holds in both cases and controls, the performances of the three statistics are also equivalent, except for variations attributable to type I error issues. When HWE fails to hold in either cases or controls or both, the 2 df version of the Chan and Chatterjee Wald test (and to a lesser extent the dominant and recessive versions) detects this HWE departure and can therefore "find" a case-control difference even if there is not an allele frequency difference or even a genotype frequency difference. Our results will improve the design and analysis of genetic association studies. Such association studies are a crucial step in understanding the genetic components of many diseases that have a large impact on public health. Better understanding of the etiology of these diseases will lead in the long term to better prevention and treatment strategies

The abundance of high-redshift objects as a probe of non-Gaussian initial conditions

The observed abundance of high-redshift galaxies and clusters contains precious information about the properties of the initial perturbations. We present a method to compute analytically the number density of objects as a function of mass and redshift for a range of physically motivated non-Gaussian models. In these models the non-Gaussianity can be dialed from zero and is assumed to be small. We compute the probability density function for the smoothed dark matter density field and we extend the Press and Schechter approach to mildly non-Gaussian density fields. The abundance of high-redshift objects can be directly related to the non-Gaussianity parameter and thus to the physical processes that generated deviations from the Gaussian behaviour. Even a skewness parameter of order 0.1 implies a dramatic change in the predicted abundance of z\gap 1 objects. Observations from NGST and X-ray satellites (XMM) can be used to accurately measure the amount of non-Gaussianity in the primordial density field.Comment: Minor changes to match the accepted ApJ version (ApJ, 539