The problem of estimating the parameters which determine a mixture density has been the subject of a large, diverse body of literature spanning nearly ninety years. During the last two decades, the...

Mixture densities, maximum likelihood, and the EM algorithm

When populations are cross-classified with respect to two or more classifications or polytomies, questions often arise about the degree of association existing between the several polytomies. Most of the traditional measures or indices of association are based upon the standard chi-square statistic or on an assumption of underlying joint normality. In this paper a number of alternative measures are considered, almost all based upon a probabilistic model for activity to which the cross-classification may typically lead. Only the case in which the population is completely known is considered, so no question of sampling or measurement error appears. We hope, however, to publish before long some approximate distributions for sample estimators of the measures we propose, and approximate tests of hypotheses. Our major theme is that the measures of association used by an empirical investigator should not be blindly chosen because of tradition and convention only, although these factors may properly be g...

Measures of association for cross classifications

BACKGROUND: Postal and electronic questionnaires are widely used for data collection in epidemiological studies but non-response reduces the effective sample size and can introduce bias. Finding ways to increase response to postal and electronic questionnaires would improve the quality of health research. OBJECTIVES: To identify effective strategies to increase response to postal and electronic questionnaires. SEARCH STRATEGY: We searched 14 electronic databases to February 2008 and manually searched the reference lists of relevant trials and reviews, and all issues of two journals. We contacted the authors of all trials or reviews to ask about unpublished trials. Where necessary, we also contacted authors to confirm methods of allocation used and to clarify results presented. We assessed the eligibility of each trial using pre-defined criteria. SELECTION CRITERIA: Randomised controlled trials of methods to increase response to postal or electronic questionnaires. DATA COLLECTION AND ANALYSIS: We extracted data on the trial participants, the intervention, the number randomised to intervention and comparison groups and allocation concealment. For each strategy, we estimated pooled odds ratios (OR) and 95% confidence intervals (CI) in a random-effects model. We assessed evidence for selection bias using Egger's weighted regression method and Begg's rank correlation test and funnel plot. We assessed heterogeneity among trial odds ratios using a Chi(2) test and the degree of inconsistency between trial results was quantified using the I(2) statistic. MAIN RESULTS: PostalWe found 481 eligible trials. The trials evaluated 110 different ways of increasing response to postal questionnaires. We found substantial heterogeneity among trial results in half of the strategies. The odds of response were at least doubled using monetary incentives (odds ratio 1.87; 95% CI 1.73 to 2.04; heterogeneity P < 0.00001, I(2) = 84%), recorded delivery (1.76; 95% CI 1.43 to 2.18; P = 0.0001, I(2) = 71%), a teaser on the envelope - e.g. a comment suggesting to participants that they may benefit if they open it (3.08; 95% CI 1.27 to 7.44) and a more interesting questionnaire topic (2.00; 95% CI 1.32 to 3.04; P = 0.06, I(2) = 80%). The odds of response were substantially higher with pre-notification (1.45; 95% CI 1.29 to 1.63; P < 0.00001, I(2) = 89%), follow-up contact (1.35; 95% CI 1.18 to 1.55; P < 0.00001, I(2) = 76%), unconditional incentives (1.61; 1.36 to 1.89; P < 0.00001, I(2) = 88%), shorter questionnaires (1.64; 95% CI 1.43 to 1.87; P < 0.00001, I(2) = 91%), providing a second copy of the questionnaire at follow up (1.46; 95% CI 1.13 to 1.90; P < 0.00001, I(2) = 82%), mentioning an obligation to respond (1.61; 95% CI 1.16 to 2.22; P = 0.98, I(2) = 0%) and university sponsorship (1.32; 95% CI 1.13 to 1.54; P < 0.00001, I(2) = 83%). The odds of response were also increased with non-monetary incentives (1.15; 95% CI 1.08 to 1.22; P < 0.00001, I(2) = 79%), personalised questionnaires (1.14; 95% CI 1.07 to 1.22; P < 0.00001, I(2) = 63%), use of hand-written addresses (1.25; 95% CI 1.08 to 1.45; P = 0.32, I(2) = 14%), use of stamped return envelopes as opposed to franked return envelopes (1.24; 95% CI 1.14 to 1.35; P < 0.00001, I(2) = 69%), an assurance of confidentiality (1.33; 95% CI 1.24 to 1.42) and first class outward mailing (1.11; 95% CI 1.02 to 1.21; P = 0.78, I(2) = 0%). The odds of response were reduced when the questionnaire included questions of a sensitive nature (0.94; 95% CI 0.88 to 1.00; P = 0.51, I(2) = 0%).ElectronicWe found 32 eligible trials. The trials evaluated 27 different ways of increasing response to electronic questionnaires. We found substantial heterogeneity among trial results in half of the strategies. The odds of response were increased by more than a half using non-monetary incentives (1.72; 95% CI 1.09 to 2.72; heterogeneity P < 0.00001, I(2) = 95%), shorter e-questionnaires (1.73; 1.40 to 2.13; P = 0.08, I(2) = 68%), including a statement that others had responded (1.52; 95% CI 1.36 to 1.70), and a more interesting topic (1.85; 95% CI 1.52 to 2.26). The odds of response increased by a third using a lottery with immediate notification of results (1.37; 95% CI 1.13 to 1.65), an offer of survey results (1.36; 95% CI 1.15 to 1.61), and using a white background (1.31; 95% CI 1.10 to 1.56). The odds of response were also increased with personalised e-questionnaires (1.24; 95% CI 1.17 to 1.32; P = 0.07, I(2) = 41%), using a simple header (1.23; 95% CI 1.03 to 1.48), using textual representation of response categories (1.19; 95% CI 1.05 to 1.36), and giving a deadline (1.18; 95% CI 1.03 to 1.34). The odds of response tripled when a picture was included in an e-mail (3.05; 95% CI 1.84 to 5.06; P = 0.27, I(2) = 19%). The odds of response were reduced when "Survey" was mentioned in the e-mail subject line (0.81; 95% CI 0.67 to 0.97; P = 0.33, I(2) = 0%), and when the e-mail included a male signature (0.55; 95% CI 0.38 to 0.80; P = 0.96, I(2) = 0%). AUTHORS' CONCLUSIONS: Health researchers using postal and electronic questionnaires can increase response using the strategies shown to be effective in this systematic review.

/pdf/methods-to-increase-response-to-postal-and-electronic-4sjz3r84m4.pdf

Methods to increase response to postal and electronic questionnaires

by J. Biggs and C. Tang, Maidenhead, England,
Open University Press, 2007, 360 pp., £29.99, ISBN-13: 978-0-335-22126-4

Teaching for quality learning at university

https://rss.onlinelibrary.wiley.com/doi/pdf/10.2307/2344237

A Review of Classification

In clinical trials, both unequal randomization design and sequential analyses have ethical and economic advantages. In the single-stage-design (SSD), however, if the sample size is not adjusted based on unequal randomization, the power of the trial will decrease, whereas with sequential analysis the power will always remain constant. Our aim was to compare sequential boundaries approach with the SSD when the allocation ratio (R) was not equal. We evaluated the influence of R, the ratio of the patients in experimental group to the standard group, on the statistical properties of two-sided tests, including the two-sided single triangular test (TT), double triangular test (DTT) and SSD by multiple simulations. The average sample size numbers (ASNs) and power (1-β) were evaluated for all tests. Our simulation study showed that choosing R = 2 instead of R = 1 increases the sample size of SSD by 12% and the ASN of the TT and DTT by the same proportion. Moreover, when R = 2, compared to the adjusted SSD, using the TT or DTT allows to retrieve the well known reductions of ASN observed when R = 1, compared to SSD. In addition, when R = 2, compared to SSD, using the TT and DTT allows to obtain smaller reductions of ASN than when R = 1, but maintains the power of the test to its planned value. This study indicates that when the allocation ratio is not equal among the treatment groups, sequential analysis could indeed serve as a compromise between ethicists, economists and statisticians.

/pdf/sequential-boundaries-approach-in-clinical-trials-with-1cz344lcfv.pdf

Sequential boundaries approach in clinical trials with unequal allocation ratios

This paper presents a procedure for answering the question of whether a mixture of two normal distributions, with five known parameters μ1, μ2, σ1, σ2, p, is unimodal or not. The approach of the study is based on a simple transformation, which might be useful for further investigation of the properties of such mixtures. It is shown, by a geometric interpretation, that a mixture of two normal distributions is either unimodal or bimodal. For finding the modes, a simple iterative procedure is given, which always converges. A brief table of the modes is provided for some values of the parameters. Some conditions under which we have a unique mode are studied. It is shown that, for this purpose, a general sufficient condition is |μ1 – μ2| ≤ 2 min (σ1, σ2). For the special case in which μ1 = μ2 = μ, a necessary and sufficient condition is presented, and from that the simple sufficient condition |μ1 – μ2| ≤ 2σ is derived.

https://www.jstor.org/stable/pdfplus/1267357.pdf

On the Modes of a Mixture of Two Normal Distributions

A Bayesian approach to fuzzy hypotheses testing

We consider a generalization of the Azzalini skew–normal distribution. We denote this distribution by SNBn(λ). Some properties of SNBn(λ) are studied. Its moment generating function is derived, and the bivariate case of SNBn(λ) is introduced. Finally, we illustrate a numerical example and we present an application for order statistics.

The Balakrishnan skew–normal density

In this paper a new approach is presented for testing statistical hypotheses when the hypotheses are fuzzy rather than crisp.  In order to establish optimality criteria, we first give new definitions for probability of type I and type II errors. Then, we state and prove the Neyman-Pearson Lemma, on the basis of these new errors, for testing fuzzy hypotheses, and we give a few examples.

Javad Behboodian

Papers

Sequential boundaries approach in clinical trials with unequal allocation ratios

On the Modes of a Mixture of Two Normal Distributions

A Bayesian approach to fuzzy hypotheses testing

The Balakrishnan skew–normal density

Neyman-Pearson Lemma for fuzzy hypotheses testing