Malene Højbjerre

The transverse forefoot arch demonstrated by a novel X-ray projection

OBJECTIVE Metatarsalgia is often treated by metatarsal osteotomy. Exact knowledge of the normal anatomy of the forefoot is essential for pre-operative planning. The objective of this study was to investigate the forefoot arch during maximal loading in a randomly selected population sample. METHODS Two hundred subjects randomly selected from a municipality representative of Denmark were invited to interview and forefoot X-ray examination, including a novel horizontal X-ray projection by which the height of each metatarsal from the floor can be measured under maximal loading. RESULTS One hundred and thirty-four subjects (79%) presented themselves for interview and X-ray examination. The study group was representative of the randomly selected population sample in terms of age, sex and incidence of metatarsalgia. The study verified that the interrelated geometry of the metatarsal heads in the AP plane corresponds to a parabola as suggested previously (Le Lièvres parabola). Also in the horizontal plane, the metatarsal heads generally form an arch, the transverse forefoot arch (TFA). Mean height was 3.91mm (S.E.=0.10). The individual height of the TFA varied from -1 to 10mm and was dependent on the width of the forefoot. The relative height of the arch (arch height divided by forefoot width) was independent of age and sex. A non-significant tendency towards a lower arch among subjects with metatarsalgia was observed. CONCLUSION This population study demonstrated that the metatarsal heads constitute arches in both planes (Le Lièvres parabola in the AP plane and the transverse forefoot arch in the horizontal plane). This knowledge is essential for pre-operative planning in metatarsal osteotomy for metatarsalgia. Formulae for calculating the individual location of each metatarsal head were obtained.

Familial tendency to foetal loss analysed with Bayesian graphical models by Gibbs sampling.

This paper presents several models for investigating whether the HLA allogenotypes DR1/Br, DR3 and DR10 are genetic markers for a predisposition of experiencing unexplained recurrent foetal losses. A total of 199 women from 113 families answered questionnaires concerning their pregnancies and 145 of these women were HLA typed. The analysis of the data is complicated as dependencies between pregnancy outcomes are expected. The main purpose of the paper is to illustrate how such analyses can be performed using Bayesian graphical models and Gibbs sampling. The analyses are made using the programs BUGS and CODA. Markov chain Monte Carlo analyses within a Bayesian framework have become easier with the introduction of these programs. However, experience shows that some caution is required so we recommend making some initial analyses using very simple models and perhaps approximative methods, followed by a model development introducing increasing complexity.

The Multivariate Complex Normal Distribution

This chapter presents the multivariate complex normal distribution. It is introduced by Wooding (1956), but it is Goodman (1963) who initiates a more thorough study of this area. Furthermore Eaton (1983) describes the distribution by using vector space approach. In this book we have also used vector space approach and the book is the first to give a systematic and wide-ranging presentation of the multivariate complex normal distribution. The results presented are known from the literature or from the real case. First the univariate case is considered. We define the standard normal distribution on ℂ and by means of this an arbitrary normal distribution on ℂ is defined. For the univariate standard complex normal distribution we study the rotation invariance, which says that the univariate standard complex normal distribution is invariant under multiplication by a complex unit. For an arbitrary complex normal distribution the property of reproductivity is examined. This is the characteristic that the sum of complex numbers and independent complex normally distributed random variables multiplied by complex numbers still is complex normally distributed. The normal distribution on ℂ p is defined and also the reproductivity property for it is studied. For all the distributions the relation to the real normal distribution is determined and the density function and the characteristic function are stated. We also specify independence results in the multivariate complex normal distribution and furthermore marginal and conditional distributions are examined. We investigate some of the results for the complex normal distribution on ℂ p in matrix form, i.e. for the complex normal distribution on ℂn×p.

Conditional Independence and Markov Properties

When we consider graphical models for the multivariate complex normal distribution, we formulate the models in terms of simple undirected graphs, which illustrate conditional independence of complex random vectors. Therefore this chapter concentrates on conditional independence of complex random vectors. Conditional independence is studied formally by Dawid (1979), but has also been explored by others see e.g. Pearl (1988). We have chosen only to consider complex random vectors with continuous density function w.r.t. Lebesgue measure, since we only consider such complex random vectors in this book. Besides an acquaintance with Lebesgue measure no further measure theory is used. We define the conditional density function and the conditional distribution for complex random vectors. Then the conditional distribution of a measurable transformation of a complex random vector is defined. Using these definitions we are able to state the law of total probability and further to define conditional independence of two measurable transformations of a complex random vector given a third complex random vector. We establish some properties, which are equivalent to the definition of conditional independence, and we find properties which can be used to deduce conditional independences from others. Furthermore we study conditional independence in the special case, where a complex random vector is partitioned and we give useful theorems in this case. Next conditional independence in relation to simple undirected graphs is studied. Three different ways are used to specify conditional independence by means of a graph. This means that the distributions fulfilling these different conditional independence criteria may have different properties. These are the so called Markov properties.

Simple Undirected Graphs

When we consider graphical models for the multivariate complex normal distribution we formulate the models in terms of simple undirected graphs. This chapter presents the concept of simple undirected graphs. As the main purpose is to define and introduce the later needed results, the presentation is at times short and compact. The well known results are stated without proof, but references containing further information are given. Results which are not quite well known are treated in more detail. First of all we define a simple undirected graph and associated basic definitions. Afterwards we consider the concepts separation, decomposition and decomposability of simple undirected graphs. This involves investigation of chordless 4-cycles, running intersection property orderings, the maximum cardinality search algorithm and an algorithm to determine decomposability of a simple undirected graph. Then we move on to definition of collapsibility and we treat the concept a regular edge. We observe that a simple undirected decomposable graph and a decomposable subgraph of it with one edge less differ by a regular edge. Finally we state some decompositions of subgraphs in a simple undirected graph containing a regular edge. We begin by defining a simple undirected graph.

The Complex Wishart Distribution and the Complex U-distribution

This chapter contains results on the complex Wishart distribution and the complex U-distribution. Goodman (1963) is the first to consider the complex Wishart distribution. Khatri (1965a), Khatri (1965b) and Giri (1965) make use of Goodmans results in further statistical analysis. The complex U-distribution is used in Gupta (1971). In this book the distributions are used in multivariate linear complex normal models in connection with distributional results for the maximum likelihood estimators and the likelihood ratio test statistics, and they are also useful in complex normal graphical models. This presentation of the complex Wishart distribution and the complex U-distribution is highly based on matrix algebra and the results are known from the literature or the real case. First the attention is focused on the complex Wishart distribution. We consider definition of the distribution and the mean of a complex Wishart distributed random matrix is determined. The correspondence between the complex Wishart distribution and the chi-square distribution is discussed and then we state under certain assumptions that a complex random quadratic form involving a projection matrix is complex Wishart distributed. Under the restriction of equal variance matrices we find the distribution of the sum of two independent complex Wishart distributed random matrices. We state the very useful result that a complex Wishart distributed random matrix is positive definite with probability one, if the variance matrix is positive definite and the degrees of freedom is greater than or equal to the dimension of the considered matrix. Next we turn to consideration of partitioning a complex Wishart distributed random matrix for which we give the distributions of various complex random matrices associated with this partition.

Multivariate Linear Complex Normal Models

In this chapter we consider linear models for the multivariate complex normal distribution. The results of linear models are widely known from the literature. In our presentation we make extensive use of vector space considerations and matrix algebra. First we define complex MANOVA models and then maximum likelihood estimation of the parameters in the complex MANOVA model is considered. We determine the maximum likelihood estimators and their distributions. We find that these estimators are expressed by means of a projection matrix representing the orthogonal projection onto the vector space involved in definition of the complex MANOVA model. Besides independence of the estimators are stated and we also derive the normal equations. Finally likelihood ratio test concerning the mean and test of independence in complex MANOVA models are presented. In both tests we find the likelihood ratio test statistic and its distribution. It turns out that the test statistics both are complex U-distributed.

Complex Normal Graphical Models

Graphical models are used to examine conditional independence among random variables. In this chapter we take graphical models for the multivariate complex normal distribution w.r.t. simple undirected graphs into consideration. This is the first published presentation of these models. Graphical models for the multivariate real normal distribution, also called covariance selection models, have already been studied in the literature. The initial work on covariance selection models is done by Dempster (1972) and Wermuth (1976) and a presentation of these models is given in Eriksen (1992). Graphical models for contingency tables are introduced in statistics by Darroch et al. (1980) and further these are well-described in Lauritzen (1989). Graphical association models are treated in general in Whittaker (1990) and Lauritzen (1993). The complex normal graphical models are quite similar to the covariance selection models. We have chosen to develop this chapter without use of exponential families. We study definition of the model, maximum likelihood estimation and hypothesis testing. To verify some of the results in the chapter we use results from mathematical analysis. These can be found in e.g. Rudin (1987). In graphical models one uses the concentration matrix instead of the variance matrix as it is more advantageous. Therefore we define this matrix and derive a relation which is basic for complex normal graphical models. Afterwards we formally define a complex normal graphical model w.r.t. a simple undirected graph. As these models are used to examine conditional independence of selected pairs of variables given the remaining ones we are mainly interested in inference on the concentration matrix. It is possible to base the maximum likelihood estimation of the concentration matrix on a complex random matrix with mean zero.