Personality is linked to mental illness. The relationship between the seven temperament and character traits (TCIs), (Novelty seeking (NS), Harm Avoidance (HA), Reward Dependence (RD), Persistence (P), Self-directedness (S), Cooperativeness (C) and Self Transcendence (ST)), of Cloninger (1994) and three symptoms of psychological distress, or SCLs (Depression (D), Anxiety (A) and Psychoticism (Psy)) is investigated across gender and shown to have significantly different symptom profiles post treatment. The data used in this study was earlier analysed by Turner, et al. (2003) and comes from patients measured pre and post-treatment from the NZ Christchurch Psychotherapy of Depression Study (Joyce, et al. 2002). In this study we have used the newly developed direct estimation approach (Beh and Davy, 2004 and Zafar et al., 2015) to estimate the linear-by-linear association in two-way tables, within the framework of ordinal log-linear models (OLLMs), with the aim of analysing associations between the TCIs and SCLs. Two non-iterative estimators were considered for this study-the Beh-Davy non-iterative estimator (BDNI) (Beh and Davy, 2004) and the Log non-iterative estimator (LogNI) (Beh and Farver, 2009). The BDNI and LogNI estimation methods provide closed-form estimators which do not require iteration to estimate the linear-by-linear association parameter of OLLMs, unlike their conventional and iterative counter parts, such as the Newton-Raphson and the iterative proportional fitting methods. The estimates obtained from the BDNI and LogNI estimation methods are reported, for pairwise relationships between TCIs and symptoms, along with the standard errors and p-values for males and females for pre and post treatment. Both estimators, BDNI and LogNI, provide estimates which are close to each other. We found significant changing relationships between the seven TCIs and psychological distress symptoms across gender for NS and P post treatment; with both TCIs and SCLs dichotomised by the median. We found statistically significant differences between the BDNI and LogNI estimates for males and females, post-treatment; establishing that higher levels of NS are associated with less D and Psy in males as compared to females. Higher HA is shown to be associated with higher D, A and Psy in males and females, pre and post-treatment. S is found to be negatively related to D, A and Psy for males and in females, pre and posttreatment. P is demonstrated as gender-specific only in the case of D; with less D associated with higher levels of P in males comparison with females post treatment. In addition, we demonstrate the linear-by-linear association between pre-treatment TCI's and change in depression, anxiety and psychoticism (ΔD, ΔA and ΔPsy), where the change is defined as categorised by the median scores of the (post-pre-treatment) levels. We show that pairwise association between three TCI's (HA, P and C) and two of our three symptoms of psychological distress, ΔD and ΔPsy, are gender-specific. These results reported agree, in par...

The dual-system estimator, or estimators with a similar underlying set of assumptions and structure, is a widely used approach to estimate the unknown size of a population. Within official statistics its use is linked with population census, while in health applications it is often used to estimate true levels of incidence from imperfect reporting systems; the classic example being work by Sekar and Deming exploring the estimation of births in India in the 1940s. Critical to the implementation of dual-system estimation are the assumptions that the probability of being counted in a source is homogeneous and that the event of being counted in each source is independent. When either of these assumptions fails, the two by two table will have an odds ratio different to one and the dual-system estimator will be biased.

The analysis of aggregate, or marginal, data for contingency tables is an increasingly important area of statistics, especially in political science and epidemiology. Aggregation often exists due to confidentiality issues or by source of the data itself. Aggregate data alone makes drawing conclusions about the true association between categorical variables difficult, especially in dealing with the aggregate analysis of single or stratified 2x2 contingency tables. These tables are the most fundamental of data structures when dealing with cross-classifying categorical variables hence it is not surprising that the analysis of this type of data has received an enormous amount of attention in the statistical, and related, literature. However, the information, from which the aggregate data can provide for the inference of association between the variables, is still a long standing issue. In order to analyse the association that exists between the variables of a 2x2 table, or stratified 2x2 tables, based only on the aggregate data, numerous approaches that lie within the area of Ecological Inference (EI) have been proposed. As an application of this new development, we shall analyse a unique record of New Zealand gendered election data from 1893 when it was the first selfgoverning country in the world allowing women to vote, this trend quickly spread across the globe. Since the NZ data structure consists of stratified 2x2 tables, where the stratifier is electorate, the issue of analysing a single 2x2 table shall not be discussed. For stratified 2x2 tables, a number of ecological inference techniques exist but these rely on strong, yet untestable assumptions, which are not applicable to a single 2x2 table. To remedy this, one may analyse the association between two dichotomous variables, given only the aggregate data, by using the Aggregate Association Index (AAI). To date, the AAI has been expressed as a function of a conditional probability and been used to test if a statistically significant association is likely to exist given only aggregate data. Nevertheless, the interpretation about the strength and direction of association cannot be obtained through the current AAI. As a result, the purpose of this study is to broaden our understanding of the AAI by establishing its functional link with other classical association measurements, such as the standardised residual, Pearson's ratio, contingency and correlation indices. For brevity, only the standardised residual shall be considered here as a foundational baseline for the other association measures. This work will allow us to confirm the characteristics of the AAI's generalizability and enable analysis of aggregate data in terms of common association measurements. In other words, we show that the analysis of aggregate data of the 2x2 tables can be extended from justifying the existence of an association to that of determining the strength and direction of the association, if it exists, given only aggregate data. The important nature of association between gender and v...