Normal distribution and confidence intervals

lecture for COMP6235

DISTRIBUTION FUNCTIONS, MEAN SQUARED ERRORS, AND CONFIDENCE LIMITS FOR RIDGE REGRESSION ESTIMATORS

A large number of ridge regression estimators have been proposed and used with little knowledge of their true distributions. Because of this lack of knowledge, these estimators cannot be used to test hypotheses or to form confidence intervals.^ This paper presents a basic technique for deriving the exact distribution functions for a class of generalized ridge estimators. The technique is applied to five prominent generalized ridge estimators. Graphs of the resulting distribution functions are presented. The actual behavior of these estimators is found to be considerably different than the behavior which is generally assumed for ridge estimators.^ This paper also uses the derived distributions to examine the mean squared error properties of the estimators. A technique for developing confidence intervals based on the generalized ridge estimators is also presented. ^

Conditional Confidence Interval for the Scale Parameter of a Weibull Distribution

2000 Mathematics Subject Classification: 62F25, 62F03.

Small sample confidence intervals for the mean of a positively skewed distribution

This thesis proposes some confidence intervals for the mean of a positively skewed distribution. The following confidence intervals are considered: Student-t, Johnson-t, median-t, mad-t, bootstrap-t, BCA, T1 , T3 and six new confidence intervals, the median bootstrap-t, mad bootstrap-t, median T1, mad T1 , median T3 and the mad T3. A simulation study has been conducted and average widths, coefficient of variation of widths, and coverage probabilities were recorded and compared across confidence intervals. To compare confidence intervals, the width and coverage probabilities were compared so that smaller widths indicated a better confidence interval when coverage probabilities were the same. Results showed that the median T1 and median T3 outperformed other confidence intervals in terms of coverage probability and the mad bootstrap-t, mad-t, and mad T3 outperformed others in terms of width. Some real life data are considered to illustrate the findings of the thesis.

Confidence to cook vegetables and the buying habits of Australian households

Cooking skills are emphasized in nutrition promotion but their distribution among population subgroups and relationship to dietary behavior is researched by few population-based studies. This study examined the relationships between confidence to cook, sociodemographic characteristics, and household vegetable purchasing. This cross-sectional study of 426 randomly selected households in Brisbane, Australia, used a validated questionnaire to assess household vegetable purchasing habits and the confidence to cook of the person who most often prepares food for these households. The mutually adjusted odds ratios (ORs) of lacking confidence to cook were assessed across a range of demographic subgroups using multiple logistic regression models. Similarly, mutually adjusted mean vegetable purchasing scores were calculated using multiple linear regression for different population groups and for respondents with varying confidence levels. Lacking confidence to cook using a variety of techniques was more common among respondents with less education (OR 3.30; 95% confidence interval [CI] 1.01 to 10.75) and was less common among respondents who lived with minors (OR 0.22; 95% CI 0.09 to 0.53) and other adults (OR 0.43; 95% CI 0.24 to 0.78). Lack of confidence to prepare vegetables was associated with being male (OR 2.25; 95% CI 1.24 to 4.08), low education (OR 6.60; 95% CI 2.08 to 20.91), lower household income (OR 2.98; 95% CI 1.02 to 8.72) and living with other adults (OR 0.53; 95% CI 0.29 to 0.98). Households bought a greater variety of vegetables on a regular basis when the main chef was confident to prepare them (difference: 18.60; 95% CI 14.66 to 22.54), older (difference: 8.69; 95% CI 4.92 to 12.47), lived with at least one other adult (difference: 5.47; 95% CI 2.82 to 8.12) or at least one minor (difference: 2.86; 95% CI 0.17 to 5.55). Cooking skills may contribute to socioeconomic dietary differences, and may be a useful strategy for promoting fruit and vegetable consumption, particularly among socioeconomically disadvantaged groups.

Look before you leap : a confidence-based method for selecting species criticality while avoiding negative populations in τ-leaping

The stochastic simulation algorithm was introduced by Gillespie and in a different form by Kurtz. There have been many attempts at accelerating the algorithm without deviating from the behavior of the simulated system. The crux of the explicit τ-leaping procedure is the use of Poisson random variables to approximate the number of occurrences of each type of reaction event during a carefully selected time period, τ. This method is acceptable providing the leap condition, that no propensity function changes “significantly” during any time-step, is met. Using this method there is a possibility that species numbers can, artificially, become negative. Several recent papers have demonstrated methods that avoid this situation. One such method classifies, as critical, those reactions in danger of sending species populations negative. At most, one of these critical reactions is allowed to occur in the next time-step. We argue that the criticality of a reactant species and its dependent reaction channels should be related to the probability of the species number becoming negative. This way only reactions that, if fired, produce a high probability of driving a reactant population negative are labeled critical. The number of firings of more reaction channels can be approximated using Poisson random variables thus speeding up the simulation while maintaining the accuracy. In implementing this revised method of criticality selection we make use of the probability distribution from which the random variable describing the change in species number is drawn. We give several numerical examples to demonstrate the effectiveness of our new method.

Is simulation training effective in increasing podiatrists' confidence in foot ulcer management?

Background: Foot ulcers are a frequent reason for diabetes-related hospitalisation. Clinical training is known to have a beneficial impact on foot ulcer outcomes. Clinical training using simulation techniques has rarely been used in the management of diabetes-related foot complications or chronic wounds. Simulation can be defined as a device or environment that attempts to replicate the real world. The few non-web-based foot-related simulation courses have focused solely on training for a single skill or “part task” (for example, practicing ingrown toenail procedures on models). This pilot study aimed to primarily investigate the effect of a training program using multiple methods of simulation on participants’ clinical confidence in the management of foot ulcers. Methods: Sixteen podiatrists participated in a two-day Foot Ulcer Simulation Training (FUST) course. The course included pre-requisite web-based learning modules, practicing individual foot ulcer management part tasks (for example, debriding a model foot ulcer), and participating in replicated clinical consultation scenarios (for example, treating a standardised patient (actor) with a model foot ulcer). The primary outcome measure of the course was participants’ pre- and post completion of confidence surveys, using a five-point Likert scale (1 = Unacceptable-5 = Proficient). Participants’ knowledge, satisfaction and their perception of the relevance and fidelity (realism) of a range of course elements were also investigated. Parametric statistics were used to analyse the data. Pearson’s r was used for correlation, ANOVA for testing the differences between groups, and a paired-sample t-test to determine the significance between pre- and post-workshop scores. A minimum significance level of p < 0.05 was used. Results: An overall 42% improvement in clinical confidence was observed following completion of FUST (mean scores 3.10 compared to 4.40, p < 0.05). The lack of an overall significant change in knowledge scores reflected the participant populations’ high baseline knowledge and pre-requisite completion of web-based modules. Satisfaction, relevance and fidelity of all course elements were rated highly. Conclusions: This pilot study suggests simulation training programs can improve participants’ clinical confidence in the management of foot ulcers. The approach has the potential to enhance clinical training in diabetes-related foot complications and chronic wounds in general.

Factors influencing research engagement : research interest, confidence and experience in an Australian speech-language pathology workforce

Background Recent initiatives within an Australia public healthcare service have seen a focus on increasing the research capacity of their workforce. One of the key initiatives involves encouraging clinicians to be research generators rather than solely research consumers. As a result, baseline data of current research capacity are essential to determine whether initiatives encouraging clinicians to undertake research have been effective. Speech pathologists have previously been shown to be interested in conducting research within their clinical role; therefore they are well positioned to benefit from such initiatives. The present study examined the current research interest, confidence and experience of speech language pathologists (SLPs) in a public healthcare workforce, as well as factors that predicted clinician research engagement. Methods Data were collected via an online survey emailed to an estimated 330 SLPs working within Queensland, Australia. The survey consisted of 30 questions relating to current levels of interest, confidence and experience performing specific research tasks, as well as how frequently SLPs had performed these tasks in the last 5 years. Results Although 158 SLPs responded to the survey, complete data were available for only 137. Respondents were more confident and experienced with basic research tasks (e.g., finding literature) and less confident and experienced with complex research tasks (e.g., analysing and interpreting results, publishing results). For most tasks, SLPs displayed higher levels of interest in the task than confidence and experience. Research engagement was predicted by highest qualification obtained, current job classification level and overall interest in research. Conclusions Respondents generally reported levels of interest in research higher than their confidence and experience, with many respondents reporting limited experience in most research tasks. Therefore SLPs have potential to benefit from research capacity building activities to increase their research skills in order to meet organisational research engagement objectives. However, these findings must be interpreted with the caveats that a relatively low response rate occurred and participants were recruited from a single state-wide health service, and therefore may not be representative of the wider SLP workforce.

Foot ulcer simulation training (FUST) : are podiatrists FUST with long-term clinical confidence?

Background Foot ulcers are a leading cause of diabetes-related hospitalisations. Clinical training has been shown to be beneficial in foot ulcer management. Recently, improved self-confidence in podiatrists was reported immediately after foot ulcer simulation training (FUST) pilot programs. This study aimed to investigate the longer-term impacts of the FUST program on podiatrists’ self-confidence over 12 months in a larger sample. Methods Participants were podiatrists attending a two-day FUST course comprising web-based interactive learning, low-fidelity part-tasks and high-fidelity full clinical scenarios. Primary outcome measures included participants’ self-confidence measured pre-, (immediately) post-, 6-month post- and 12-month post-course via a purpose designed 21-item survey using a five-point Likert scale (1=Very limited, 5=Highly confident). Participants’ perceptions of knowledge gained, satisfaction, relevance and fidelity were also investigated. ANOVA and post hoc tests were used to test any differences between groups. Results Thirty-four participants completed FUST. Survey response rates were 100% (pre), 82% (post), 74% (6-month post), and 47% (12-month post). Overall mean scores were 3.13 (pre), 4.49 (post), 4.35 (6-month post) and 4.30 (12-month post) (p < 0.05); post hoc tests indicated no differences between the immediately, 6-month and 12-month post group scores (p > 0.05). Satisfaction, knowledge, relevance and fidelity were all rated highly. Conclusion This study suggests that significant short-term improvements in self-confidence to manage foot ulcers via simulation training are retained over the longer term. It is likely that improved self-confidence leads to improved foot ulcer clinical practice and outcomes; although this requires further research.

Exact parametric confidence intervals for Bland-Altman limits of agreement

Purpose The previous literature on Bland-Altman analysis only describes approximate methods for calculating confidence intervals for 95% Limits of Agreement (LoAs). This paper describes exact methods for calculating such confidence intervals, based on the assumption that differences in measurement pairs are normally distributed. Methods Two basic situations are considered for calculating LoA confidence intervals: the first where LoAs are considered individually (i.e. using one-sided tolerance factors for a normal distribution); and the second, where LoAs are considered as a pair (i.e. using two-sided tolerance factors for a normal distribution). Equations underlying the calculation of exact confidence limits are briefly outlined. Results To assist in determining confidence intervals for LoAs (considered individually and as a pair) tables of coefficients have been included for degrees of freedom between 1 and 1000. Numerical examples, showing the use of the tables for calculating confidence limits for Bland-Altman LoAs, have been provided. Conclusions Exact confidence intervals for LoAs can differ considerably from Bland and Altman’s approximate method, especially for sample sizes that are not large. There are better, more precise methods for calculating confidence intervals for LoAs than Bland and Altman’s approximate method, although even an approximate calculation of confidence intervals for LoAs is likely to be better than none at all. Reporting confidence limits for LoAs considered as a pair is appropriate for most situations, however there may be circumstances where it is appropriate to report confidence limits for LoAs considered individually.

Confidence measurement in the light of signal detection theory

We compare three alternative methods for eliciting retrospective confidence in the context of a simple perceptual task: the Simple Confidence Rating (a direct report on a numerical scale), the Quadratic Scoring Rule (a post-wagering procedure), and the Matching Probability (MP; a generalization of the no-loss gambling method). We systematically compare the results obtained with these three rules to the theoretical confidence levels that can be inferred from performance in the perceptual task using Signal Detection Theory (SDT). We find that the MP provides better results in that respect. We conclude that MP is particularly well suited for studies of confidence that use SDT as a theoretical framework.

A survey of radiographers' confidence and self-perceived accuracy in frontline image interpretation and their continuing educational preferences

Introduction The provision of a written comment on traumatic abnormalities of the musculoskeletal system detected by radiographers can assist referrers and may improve patient management, but the practice has not been widely adopted outside the United Kingdom. The purpose of this study was to investigate Australian radiographers’ perceptions of their readiness for practice in a radiographer commenting system and their educational preferences in relation to two different delivery formats of image interpretation education, intensive and non-intensive. Methods A cross-sectional web-based questionnaire was implemented between August and September 2012. Participants included radiographers with experience working in emergency settings at four Australian metropolitan hospitals. Conventional descriptive statistics, frequency histograms, and thematic analysis were undertaken. A Wilcoxon signed-rank test examined whether a difference in preference ratings between intensive and non-intensive education delivery was evident. Results The questionnaire was completed by 73 radiographers (68% response rate). Radiographers reported higher confidence and self-perceived accuracy to detect traumatic abnormalities than to describe traumatic abnormalities of the musculoskeletal system. Radiographers frequently reported high desirability ratings for both the intensive and the non-intensive education delivery, no difference in desirability ratings for these two formats was evident (z = 1.66,P = 0.11). Conclusions Some Australian radiographers perceive they are not ready to practise in a frontline radiographer commenting system. Overall, radiographers indicated mixed preferences for image interpretation education delivered via intensive and non-intensive formats. Further research, preferably randomised trials, investigating the effectiveness of intensive and non-intensive education formats of image interpretation education for radiographers is warranted.

A call for better care: the impact of postnatal contact services on women’s parenting confidence and experiences of postpartum care in Queensland, Australia

Background Universal postnatal contact services are provided in several Australian states, but their impact on women’s postnatal care experience has not been evaluated. Furthermore, there is lack of evidence or consensus about the optimal type and amount of postpartum care after hospital discharge for maternal outcomes. This study aimed to assess the impact of providing Universal Postnatal Contact Service (UPNCS) funding to public birthing facilities in Queensland, Australia on women’s postnatal care experiences, and associations between amount and type (telephone or home visits) of contact on parenting confidence, and perceived sufficiency and quality of postnatal care. Methods Data collected via retrospective survey of postnatal women (N = 3,724) were used to compare women who birthed in UPNCS-funded and non-UPNCS-funded facilities on parenting confidence, sufficiency of postnatal care, and perceived quality of postnatal care. Associations between receiving telephone and home visits and the same outcomes, regardless of UPNCS funding, were also assessed. Results Women who birthed in an UPNCS-funded facility were more likely to receive postnatal contact, but UPNCS funding was not associated with parenting confidence, or perceived sufficiency or perceived quality of care. Telephone contact was not associated with parenting confidence but had a positive dose–response association with perceived sufficiency and quality. Home visits were negatively associated with parenting confidence when 3 or more were received, had a positive dose–response association with perceived sufficiency and were positively associated with perceived quality when at least 6 were received. Conclusions Funding for UPNCS is unlikely to improve population levels of maternal parenting confidence, perceived sufficiency or quality of postpartum care. Where only minimal contact can be provided, telephone may be more effective than home visits for improving women’s perceived sufficiency and quality of care. Additional service initiatives may be needed to improve women’s parenting confidence.

Changing citizen confidence: Orientations towards political and social institutions in Australia, 1983-2010 [Review]

Since the early 1980s, when confidence in institutions was first measured in an Australian academic social survey, Australia - And the world - has faced many political, social and economic changes. From corporate scandals and company collapses, to unprecedented terrorist attacks, to major ongoing international conflicts, to changes in government and all manner of political machinations, to the global financial crisis and its aftermath. One consequence of such developments has been that many major political, social and economic institutions have come under intense pressure. Using survey research data, this paper investigates how public confidence in various Australian institutions and organisations has changed over time. The results are variable and in some instances surprising. Confidence in some institutions has remained high, and in some low, over an extended period of time. In other cases, confidence has varied quite markedly at different time points. As well as looking at trends in the level of public confidence in institutions, the paper examines different dimensions of confidence together with underpinning socio-political factors. It also discusses theoretical and practical implications of the data.