Robust estimation in structural equation models using Bregman divergences

Structural equation models seek to find causal relationships between latent variables by analysing the mean and the covariance matrix of some observable indicators of the latent variables. Under a multivariate normality assumption on the distribution of the latent variables and of the errors, maximum likelihood estimators are asymptotically efficient. The estimators are significantly influenced by violation of the normality assumption and hence there is a need to robustify the inference procedures. We propose to minimise the Bregman divergence or its variant, the total Bregman divergence, between a robust estimator of the covariance matrix and the model covariance matrix, with respect to the parameters of interest. Our approach to robustification is different from the standard approaches in that we propose to achieve the robustification on two levels: firstly, choosing a robust estimator of the covariance matrix; and secondly, using a robust divergence measure between the model covariance matrix and its robust estimator. We focus on the (total) von Neumann divergence, a particular Bregman divergence, to estimate the parameters of the structural equation model. Our approach is tested in a simulation study and shows significant advantages in estimating the model parameters in contaminated data sets and seems to perform better than other well known robust inference approaches in structural equation models. References E. G. Baranoff, S. Papadopoulos and T. W. Seger. Capital and Risk Revisited: A Structural Equation Model Approach for Life Insurers. Journal of Risk and Insurance, 74:653–681, 2007. doi:10.1111/j.1539-6975.2007.00229.x K. Bollen. Structural Equations with Latent Variables. Wiley, New York, 1989. I. S. Dhillon and J. A. Tropp. Matrix Nearness Problems with Bregman Divergences. SIAM Journal on Matrix Analysis and Applications, 29:1120–1146, 2008. doi:10.1137/060649021 F. Nielsen and S. Boltz. The Burbea–Rao and Bhattacharyya Centroids. IEEE transactions in information theory, 57:5455–5466, 2011. doi:10.1109/TIT.2011.2159046 B. C. Vemuri, M. Liu, S.-I. Amari and F. Nielsen. Total Bregman Divergence and Its Applications to DTI Analysis. IEEE Transactions on medical imaging, 30:475–483, 2011. doi:10.1109/TMI.2010.2086464 S. Verboven and M. Hubert. LIBRA: a MATLAB library for robust analysis, Chemometrics and intelligent laboratory systems, 75:127–136, 2005. doi:10.1016/j.chemolab.2004.06.003 K.-H. Yuan and P. M. Bentler. Structural equation modeling with robust covariances. Sociological Methodology, 28:363–396, 1998. doi:10.1111/0081-1750.00052 K.-H. Yuan and P. M. Bentler. Robust mean and covariance structure analysis, British Journal of Mathematical and Statistical Psychology, 51:63–88, 1998. doi:10.1111/j.2044-8317.1998.tb00667.x K.-H. Yuan, P. M. Bentler and W. Chan. Structural Equation Modeling with Heavy Tailed Distributions, Psychometrika, 69:421–436, 2004. doi:10.1007/BF02295644 X. Zhong and K.-H. Yuan. Bias and Efficiency in Structural Equation Modeling: Maximum Likelihood Versus Robust Methods. Multivariate Behavioral Research, 46:229–265, 2011. doi:10.1080/00273171.2011.558736 LIBRA: a Matlab Library for Robust Analysis. http://wis.kuleuven.be/stat/robust/LIBRA/LIBRA-hom

Teaching applied statistics courses using computer laboratory final examinations

Courses taught in Applied Statistics, such as regression or multivariate analysis, tend to have the examination component based on a final written paper, either with computer output attached for interpretation or with summary statistics given so the calculator can be used to evaluate test statistics and hence make inference, or a mixture of both. Assignments may be based on students using a statistical package to do the analysis. The authors found this approach unsatisfactory. The first author trialled at the University of Canberra in 2001 both a mid semester computer based examination and final computer based examination with the students being allowed to use a variety of statistical packages. Student feedback was so favourable that she taught the course the following year in the same manner as well as a regression course. When she moved to Macquarie University she split the final examination for the third year regression course she was in charge of in 2004, 2005 and 2006 into a computer laboratory examination and a separate written paper. We have been unable to find more than a small number of other examples of this approach, which seems to have considerable promise as a way of implementing authentic assessment in applied courses. As well as case studies, issues associated with setting, running and marking such examinations are discussed

Client experiences and confidence in nutrition advice delivered by registered exercise professionals

Objectives The aims of this cross-sectional study were to describe the client experiences of registered exercise professionals (REP) in regard to nutrition advice, and to identify factors considered important when selecting a REP. Design A convenience sample of fitness facility-users were recruited. Participants completed a 46 item online questionnaire. Method Frequency distributions were calculated for nutrition advice received from a REP, confidence in the nutrition advice received, and factors considered when selecting a REP. The effect of sex and level of education on participant responses were investigated. Results A total of 455 individuals participated, of which 55.6% had worked with a REP. Participants working with a REP had received nutrition advice on body fat loss (77.4%), muscle gain (78.7%), and clinical nutrition (51.9%). Sex and level of education did not influence confidence in nutrition advice, with 60.7% of participants reporting being ‘very or extremely confident’ in the nutrition advice. Over half of participants (58.3%) prefer to work with a REP who offers nutrition advice. Basic and personal professional conduct were identified as ‘very or extremely important’ by >70% of participants when selecting a REP. Conclusion Clients of REP prefer to work with a REP who provides nutrition advice and are generally confident in this advice. Educating fitness facility-users on the nutrition scope of practice of REP and promoting the role of accredited nutrition professionals should be advocated. Encouraging REP to work collaboratively with nutrition professionals will help ensure clients receive appropriate nutrition care, while better allowing REP to remain within their scope of practice

The Role of Music in Families of Children With Hearing Loss and Normal Hearing in Australia, Finland, and the UK

The primary aim of this current study was to compare the role, importance and value placed on music by families with normally hearing (NH) children, to those who had a child with a hearing loss (HL) who wore either hearing aids and/or cochlear implants. A secondary aim was to see whether this differed between the countries. Parents of children aged 2–6 years living in Australia, Finland, and the United Kingdom were invited to complete the Role of Music in Families Questionnaire (RMFQ). Two groups of participants were recruited from each country: (i) parents of NH children, and (ii) parents of children with a HL. The RMFQ had seven subsections covering topics such as music participation, attitudes to music, importance of music in the family, and future perspectives on music. Three hundred and twenty-two families of NH children, and 56 families of children with HL completed the questionnaire (Australia: 50 NH, 25 HL; Finland: 242 NH, 21 HL; United Kingdom: 30 NH, 10 HL). Analyses compared between NH and HL groups within each country, and between the three countries for the NH group, and the HL group, independently. Overall, there were few significant differences between the participation levels, role, or importance of music in families with NH children compared to those with a child who had a HL, regardless of whether the families lived in Australia, Finland or the United Kingdom. Children first started to respond to music at similar ages, and overall music participation frequency, and music enjoyment were relatively similar. The importance of music in the family was also similar between the NH and HL groups. In comparing between the countries, Finnish children had a tendency to have higher participation rates in musical activities, with few other differences noted. Overall, the results of this study indicate that children, regardless of hearing levels or country of residence, have similar levels of music engagement and enjoyment, and HL is not seen as a contraindication to music participation and involvement by the parents involved in this study.Peer reviewe

Validation of a sports nutrition knowledge questionnaire for athletes in the United Kingdom and Ireland.

Sound general and sports nutrition knowledge in athletes is essential for making appropriate dietary choices. Assessment of nutrition knowledge enables evaluation and tailoring of nutrition education. However, few well-validated tools are available to assess nutrition knowledge in athletes. The objective of the present study was to establish the validity of the Platform to Evaluate Athlete Knowledge Sports - Nutrition Questionnaire (PEAKS-NQ) for use in the United Kingdom and Irish (UK-I) athletes. To confirm content validity, twenty-three sports nutritionists (SNs) from elite, UK-I sports institutes provided feedback on the PEAKS-NQ via a modified Delphi method. After minor changes, the UK-I version of the PEAKS-NQ was administered to UK-I SN from the British Dietetic Association Sport and Exercise Nutrition Register, and elite athletes (EA) training at elite sports institutes in the UK and Ireland. Independent samples -test and independent samples median tests were used to compare PEAKS-NQ total and subsection scores between EA and SN (to assess construct validity). Cronbach's alpha (good ≥ 0⋅7) was used to establish internal consistency. The SN achieved greater overall [SN ( 23) 92⋅3 (9⋅3) EA ( 154): 71⋅4 (10⋅0)%; < 0⋅001] and individual section scores ( < 0⋅001) except Section B, Identification of Food Groups ( = 0⋅07). Largest knowledge differences between SN and EA were in Section D, Applied Sports Nutrition [SN: 88⋅5 (8⋅9) EA: 56⋅7 (14⋅5)%; < 0⋅00]. Overall ES was large (2⋅1), with subsections ranging from 0⋅6 to 2⋅3. Cronbach's alpha was good (0⋅83). The PEAKS-NQ had good content and construct validity, supporting its use to assess nutrition knowledge of UK-I athletes

Applications of l 1 regularisation

The lasso algorithm for variable selection in linear models, intro- duced by Tibshirani, works by imposing an l1 norm bound constraint on the variables in a least squares model and then tuning the model estimation calculation using this bound. This introduction of the bound is interpreted as a form of regularisation step. It leads to a form of quadratic program which is solved by a straight-forward modifica-tion of a standard active set algorithm for each value of this bound. Considerable interest was generated by the discovery that the complete solution trajectory parametrised by this bound is piecewise linear and can be calculated very effciently. Essentially it takes no more work than the solution of either the unconstrained least squares problem or the quadratic program at a single bound value. This has resulted in the study both of the selection problem for different objective and constraint choices and of applications to such areas as data compres- sion and the generation of sparse solutions of very under-determined systems. One important class of generalisation is to quantile regression estimation problems. The original continuation idea extends to these polyhedral objectives in an interesting two phase procedure which involves both the constrained and Lagrangian forms of the problem at each step. However, it is significantly less computationally effective than is the original algorithm for least squares objectives. In contrast, the piecewise linear estimation problem can be solved for each value of the l1 bound by a relatively effcient simplicial descent algorithm, and that this can be used to explore trajectory information in a manner which is at least competitive with the homotopy algorithm in this context. The form of line search used in the descent steps has an important bearing on the effectiveness of the algorithm. A comparison is given between the relative performance of the simplicial descent algorithm used and an interior point method on the piecewise linear estimation problem