The use of the likelihood ratio for evaluative and investigative purposes in comparative forensic handwriting examination

This paper extends previous research and discussion on the use of multivariate continuous data, which are about to become more prevalent in forensic science. As an illustrative example, attention is drawn here on the area of comparative handwriting examinations. Multivariate continuous data can be obtained in this field by analysing the contour shape of loop characters through Fourier analysis. This methodology, based on existing research in this area, allows one describe in detail the morphology of character contours throughout a set of variables. This paper uses data collected from female and male writers to conduct a comparative analysis of likelihood ratio based evidence assessment procedures in both, evaluative and investigative proceedings. While the use of likelihood ratios in the former situation is now rather well established (typically, in order to discriminate between propositions of authorship of a given individual versus another, unknown individual), focus on the investigative setting still remains rather beyond considerations in practice. This paper seeks to highlight that investigative settings, too, can represent an area of application for which the likelihood ratio can offer a logical support. As an example, the inference of gender of the writer of an incriminated handwritten text is forwarded, analysed and discussed in this paper. The more general viewpoint according to which likelihood ratio analyses can be helpful for investigative proceedings is supported here through various simulations. These offer a characterisation of the robustness of the proposed likelihood ratio methodology.

The use of classification and regression trees to predict the likelihood of seasonal influenza.

Background Individual signs and symptoms are of limited value for the diagnosis of influenza. Objective To develop a decision tree for the diagnosis of influenza based on a classification and regression tree (CART) analysis. Methods Data from two previous similar cohort studies were assembled into a single dataset. The data were randomly divided into a development set (70%) and a validation set (30%). We used CART analysis to develop three models that maximize the number of patients who do not require diagnostic testing prior to treatment decisions. The validation set was used to evaluate overfitting of the model to the training set. Results Model 1 has seven terminal nodes based on temperature, the onset of symptoms and the presence of chills, cough and myalgia. Model 2 was a simpler tree with only two splits based on temperature and the presence of chills. Model 3 was developed with temperature as a dichotomous variable (≥38°C) and had only two splits based on the presence of fever and myalgia. The area under the receiver operating characteristic curves (AUROCC) for the development and validation sets, respectively, were 0.82 and 0.80 for Model 1, 0.75 and 0.76 for Model 2 and 0.76 and 0.77 for Model 3. Model 2 classified 67% of patients in the validation group into a high- or low-risk group compared with only 38% for Model 1 and 54% for Model 3. Conclusions A simple decision tree (Model 2) classified two-thirds of patients as low or high risk and had an AUROCC of 0.76. After further validation in an independent population, this CART model could support clinical decision making regarding influenza, with low-risk patients requiring no further evaluation for influenza and high-risk patients being candidates for empiric symptomatic or drug therapy.

The likelihood approach to compare populations : a study on DNA evidence and pitfalls of intuitions

The paper follows on from earlier work [Taroni F and Aitken CGG. Probabilistic reasoning in the law, Part 1: assessment of probabilities and explanation of the value of DNA evidence. Science & Justice 1998; 38: 165-177]. Different explanations of the value of DNA evidence were presented to students from two schools of forensic science and to members of fifteen laboratories all around the world. The responses were divided into two groups; those which came from a school or laboratory identified as Bayesian and those which came from a school or laboratory identified as non-Bayesian. The paper analyses these responses using a likelihood approach. This approach is more consistent with a Bayesian analysis than one based on a frequentist approach, as was reported by Taroni F and Aitken CGG. [Probabilistic reasoning in the law, Part 1: assessment of probabilities and explanation of the value of DNA evidence] in Science & Justice 1998.

Fourth order pseudo maximum likelihood methods

We extend PML theory to account for information on the conditional moments up to order four, but without assuming a parametric model, to avoid a risk of misspecification of the conditional distribution. The key statistical tool is the quartic exponential family, which allows us to generalize the PML2 and QGPML1 methods proposed in Gourieroux et al. (1984) to PML4 and QGPML2 methods, respectively. An asymptotic theory is developed. The key numerical tool that we use is the Gauss-Freud integration scheme that solves a computational problem that has previously been raised in several fields. Simulation exercises demonstrate the feasibility and robustness of the methods [Authors]

Uncertainty about the true source. A note on the likelihood ratio at the activity level.

This paper focuses on likelihood ratio based evaluations of fibre evidence in cases in which there is uncertainty about whether or not the reference item available for analysis - that is, an item typically taken from the suspect or seized at his home - is the item actually worn at the time of the offence. A likelihood ratio approach is proposed that, for situations in which certain categorical assumptions can be made about additionally introduced parameters, converges to formula described in existing literature. The properties of the proposed likelihood ratio approach are analysed through sensitivity analyses and discussed with respect to possible argumentative implications that arise in practice.

Adaptively weighted maximum likelihood estimation of discrete distributions

SummaryDiscrete data arise in various research fields, typically when the observations are count data.I propose a robust and efficient parametric procedure for estimation of discrete distributions. The estimation is done in two phases. First, a very robust, but possibly inefficient, estimate of the model parameters is computed and used to indentify outliers. Then the outliers are either removed from the sample or given low weights, and a weighted maximum likelihood estimate (WML) is computed.The weights are determined via an adaptive process such that if the data follow the model, then asymptotically no observation is downweighted.I prove that the final estimator inherits the breakdown point of the initial one, and that its influence function at the model is the same as the influence function of the maximum likelihood estimator, which strongly suggests that it is asymptotically fully efficient.The initial estimator is a minimum disparity estimator (MDE). MDEs can be shown to have full asymptotic efficiency, and some MDEs have very high breakdown points and very low bias under contamination. Several initial estimators are considered, and the performances of the WMLs based on each of them are studied.It results that in a great variety of situations the WML substantially improves the initial estimator, both in terms of finite sample mean square error and in terms of bias under contamination. Besides, the performances of the WML are rather stable under a change of the MDE even if the MDEs have very different behaviors.Two examples of application of the WML to real data are considered. In both of them, the necessity for a robust estimator is clear: the maximum likelihood estimator is badly corrupted by the presence of a few outliers.This procedure is particularly natural in the discrete distribution setting, but could be extended to the continuous case, for which a possible procedure is sketched.RésuméLes données discrètes sont présentes dans différents domaines de recherche, en particulier lorsque les observations sont des comptages.Je propose une méthode paramétrique robuste et efficace pour l'estimation de distributions discrètes. L'estimation est faite en deux phases. Tout d'abord, un estimateur très robuste des paramètres du modèle est calculé, et utilisé pour la détection des données aberrantes (outliers). Cet estimateur n'est pas nécessairement efficace. Ensuite, soit les outliers sont retirés de l'échantillon, soit des faibles poids leur sont attribués, et un estimateur du maximum de vraisemblance pondéré (WML) est calculé.Les poids sont déterminés via un processus adaptif, tel qu'asymptotiquement, si les données suivent le modèle, aucune observation n'est dépondérée.Je prouve que le point de rupture de l'estimateur final est au moins aussi élevé que celui de l'estimateur initial, et que sa fonction d'influence au modèle est la même que celle du maximum de vraisemblance, ce qui suggère que cet estimateur est pleinement efficace asymptotiquement.L'estimateur initial est un estimateur de disparité minimale (MDE). Les MDE sont asymptotiquement pleinement efficaces, et certains d'entre eux ont un point de rupture très élevé et un très faible biais sous contamination. J'étudie les performances du WML basé sur différents MDEs.Le résultat est que dans une grande variété de situations le WML améliore largement les performances de l'estimateur initial, autant en terme du carré moyen de l'erreur que du biais sous contamination. De plus, les performances du WML restent assez stables lorsqu'on change l'estimateur initial, même si les différents MDEs ont des comportements très différents.Je considère deux exemples d'application du WML à des données réelles, où la nécessité d'un estimateur robuste est manifeste : l'estimateur du maximum de vraisemblance est fortement corrompu par la présence de quelques outliers.La méthode proposée est particulièrement naturelle dans le cadre des distributions discrètes, mais pourrait être étendue au cas continu.

On the use of the likelihood ratio for forensic evaluation: response to Fenton et al.

This letter to the Editor comments on the article When 'neutral' evidence still has probative value (with implications from the Barry George Case) by N. Fenton et al. [[1], 2014].

How to assign a likelihood ratio in a footwear mark case: an analysis and discussion in the light of R v T

This article analyses and discusses issues that pertain to the choice of relevant databases for assigning values to the components of evaluative likelihood ratio procedures at source level. Although several formal likelihood ratio developments currently exist, both case practitioners and recipients of expert information (such as judiciary) may be reluctant to consider them as a framework for evaluating scientific evidence in context. The recent ruling R v T and ensuing discussions in many forums provide illustrative examples for this. In particular, it is often felt that likelihood ratio-based reasoning amounts to an application that requires extensive quantitative information along with means for dealing with technicalities related to the algebraic formulation of these approaches. With regard to this objection, this article proposes two distinct discussions. In a first part, it is argued that, from a methodological point of view, there are additional levels of qualitative evaluation that are worth considering prior to focusing on particular numerical probability assignments. Analyses will be proposed that intend to show that, under certain assumptions, relative numerical values, as opposed to absolute values, may be sufficient to characterize a likelihood ratio for practical and pragmatic purposes. The feasibility of such qualitative considerations points out that the availability of hard numerical data is not a necessary requirement for implementing a likelihood ratio approach in practice. It is further argued that, even if numerical evaluations can be made, qualitative considerations may be valuable because they can further the understanding of the logical underpinnings of an assessment. In a second part, the article will draw a parallel to R v T by concentrating on a practical footwear mark case received at the authors' institute. This case will serve the purpose of exemplifying the possible usage of data from various sources in casework and help to discuss the difficulty associated with reconciling the depth of theoretical likelihood ratio developments and limitations in the degree to which these developments can actually be applied in practice.

Evidential relevance in scene to offender transfer cases: development and analysis of a likelihood ratio for offence level propositions

The present paper focuses on the analysis and discussion of a likelihood ratio (LR) development for propositions at a hierarchical level known in the context as 'offence level'. Existing literature on the topic has considered LR developments for so-called offender to scene transfer cases. These settings involve-in their simplest form-a single stain found on a crime scene, but with possible uncertainty about the degree to which that stain is relevant (i.e. that it has been left by the offender). Extensions to multiple stains or multiple offenders have also been reported. The purpose of this paper is to discuss a development of a LR for offence level propositions when case settings involve potential transfer in the opposite direction, i.e. victim/scene to offender transfer. This setting has previously not yet been considered. The rationale behind the proposed LR is illustrated through graphical probability models (i.e. Bayesian networks). The role of various uncertain parameters is investigated through sensitivity analyses as well as simulations.

Different likelihood ratio approaches to evaluate the strength of evidence of MDMA tablet comparisons

Two likelihood ratio (LR) approaches are presented to evaluate the strength of evidence of MDMA tablet comparisons. The first one is based on a more 'traditional' comparison of MDMA tablets by using distance measures (e.g., Pearson correlation distance or a Euclidean distance). In this approach, LRs are calculated using the distribution of distances between tablets of the same-batch and that of different-batches. The second approach is based on methods used in some other fields of forensic comparison. Here LRs are calculated based on the distribution of values of MDMA tablet characteristics within a specific batch and from all batches. The data used in this paper must be seen as examples to illustrate both methods. In future research the methods can be applied to other and more complex data. In this paper, the methods and their results are discussed, considering their performance in evidence evaluation and several practical aspects. With respect to evidence in favor of the correct hypothesis, the second method proved to be better than the first one. It is shown that the LRs in same-batch comparisons are generally higher compared to the first method and the LRs in different-batch comparisons are generally lower. On the other hand, for operational purposes (where quick information is needed), the first method may be preferred, because it is less time consuming. With this method a model has to be estimated only once in a while, which means that only a few measurements have to be done, while with the second method more measurements are needed because each time a new model has to be estimated.