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Open access peer-reviewed chapter

# Low-Dose Computed Tomography Screening for Lung Cancer

Written By

Trevor Keith Rogers

Submitted: May 24th, 2016 Reviewed: October 17th, 2016 Published: March 1st, 2017

DOI: 10.5772/66358

From the Edited Volume

## Prevention, Diagnosis, and Treatment of Lung Cancer

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## Abstract

In the landmark American National Lung Cancer Screening Trial (NLST), low-dose CT (LDCT) screening produced a relative mortality reduction of 20%. These results have not been replicated in any of the European studies, although these are of limited statistical power. Besides doubt about the general applicability of the NLST findings, if LDCT screening is to be successfully implemented, a number of developments are still required, including better characterisation of entry criteria and refinement of screening and nodule management protocols. The high incidence of false-positive findings increases costs and morbidity. Even when histologically malignant tumours are identified, frequently these would not have manifested as disease, i.e. they are “overdiagnosed”. These patients are liable to receive unnecessary treatment. LDCT screening is relatively expensive in comparison with other cancer screening modalities. Whilst cost-effectiveness can be improved by integration with smoking cessation programmes, how this would be done in practice remains unclear. Furthermore, individuals at high-risk of lung cancer are virtually by definition risk prone, raising concerns about how attractive participation in a screening programme would be, especially given the very small reported absolute risk reduction in the NLST.

### Keywords

• lung cancer
• screening
• low-dose computed tomography
• early diagnosis
• cost-effectiveness
• overdiagnosis

## 1. Introduction

Lung cancer is the commonest cause of cancer death in both men and women across the developed world, due to a combination of its high incidence and relatively short average survival after diagnosis. Most lung cancer patients present symptomatically and most already have incurable disease at presentation. These considerations have resulted in attempts to improve outcomes through screening.

Screening, though, is a challenging strategy for harm reduction. Screening programmes have been introduced for many cancers, often as a result of political pressures rather than on sound evidence. Indeed, the harms of many screening programmes have only become evident well after widespread implementation, and their utility has often become more rather than less controversial with time. Self-evidently, for screening to be effective, earlier disease identification needs to lead to improved treatment outcomes. This calls into question what we know about the natural history of early stage, asymptomatic tumours, which turns out to be surprisingly little. It is now becoming increasingly certain though that, whilst some tumours will progress and ultimately cause premature death, others may never cause harm. This leads to a bias known as overdiagnosis [1]. The other way in which overdiagnosis can occur is when a competing cause of death prevents clinical manifestations of a tumour that would otherwise have proved lethal [2]. Screening is particularly liable to overdiagnosis, because more aggressive tumours have short volume-doubling times and progress rapidly. Thus, the interval between the onset of a radiological abnormality and the emergence of symptoms is relatively short, and the opportunity for presymptomatic detection in a screening programme is small. Conversely, tumours that grow slowly will have a long phase when they exist without symptoms and are particularly liable to be identified through screening.

There is now convincing evidence that overdiagnosis does occur as an inherent harm in most, and probably all, cancer screening programmes, including mammography [3]. It is always harmful because it is unknown which of the tumours identified are the ones overdiagnosed, meaning that some patients will have treatment for a disease that would never have materialised. The resulting costs include financial (unnecessary investigations/treatments), physical (side effects and complications of treatments) and psychological and are all serious, irrespective of any benefits derived by those with “real” disease.

The other important bias of screening programmes is lead-time bias, which occurs when a disease is diagnosed earlier than it would have been without screening. Even if the natural history of the case is not improved, the patient appears to survive longer than otherwise they would have but dies at the same date. Because of this it is vital that for proper evaluation of a screening programme the endpoint taken should be mortality difference in comparison with a control group.

## 2. Lung cancer screening trials

Several large studies were conducted in the 1960s and 1970s using plain chest radiography, with or without sputum cytology [410]. These studies all had methodological weaknesses [11], including possessing limited power and, rather remarkably, even the control group in several having three-year radiographs. No study provided evidence that mortality was reduced. Recently, the PLCO study has reported [12] and has finally and convincingly shown an absence of any mortality benefit of plain chest radiography screening, compared to no screening.

The advent of low-dose CT (LDCT) scanning provided a new modality applicable to lung cancer screening. Initial studies indicated that LDCT was able to identify early lung cancers with a high rate of resectability [13]. The landmark National Lung Screening Trial (NLST) was a large and adequately powered, randomised, controlled trial of screening with LDCT against plain chest radiography, undertaken in the USA. The headline result was a reduction in lung cancer mortality, by the apparently impressive figure of 20% [14]. Does this mean that CT screening immediately to be implemented for lung cancer or that lung cancer mortality could be reduced by anything like this amount? I believe that the answer is no to both.

### 2.1. The National Lung Screening Trial (NLST)

Of the 26,722 volunteers screened with LDCT, 1060 participants were found to have lung cancer in comparison to 941 of the group screened with chest radiography. Overall, when comparing the two groups, the detection rate of lung cancer was 13% greater in the group screened with LDCT than with chest radiography. More early stage (IA and IB) cancers were diagnosed with CT. Most importantly in the evaluation of a screening study, there was evidence of reduced mortality: a total of 356 lung cancer deaths occurred in the LDCT vs. 443 deaths in the plain radiography group, over a median of 6.5 years of follow-up (P = 0.004). Whilst the relative reduction in mortality rate from lung cancer was 20% in individuals screened using LDCT, the absolute risk reduction in mortality was only 0.33% less over the study period in the LDCT group (87 avoided deaths over 26,722 screened participants), meaning 310 individuals needed to participate in screening for typically three rounds to prevent one lung cancer death. Given the conclusion of the PLCO trial that plain radiography screening is ineffective, it can be assumed that this represents a similar benefit to eligible participants in comparison with no screening. However this is evaluated in terms of cost-effectiveness, this study does indicate that there exists a subgroup of patients with lung cancer that can be cured if it is identified earlier and, conversely, who will die from their disease if it is not. Despite the large number of participants who underwent further diagnostic testing, the authors noted that the testing resulted in only a small number of adverse events.

### 2.2. European studies

An important issue is to what extent the findings of the NSLT might hold for a non-US population. There are important differences in epidemiology in Europe, including a difference in distribution of histological subtypes and a much lower frequency of non-calcified pulmonary nodules arising from fungal pathogens. In the UK, squamous cancers represent about 40% of cancers and adenocarcinomas 18%, whilst in the USA, squamous cancers only represent 27% with adenocarcinoma being the most prevalent type at 31% [15]. As squamous cancers tend to arise in proximal airways, they are less amenable to identification as a lung nodule on CT, unlike adenocarcinomas, which more often present as intrapulmonary nodules.

The DANTE and DLCST studies each compared five annual rounds of LDCT screening to usual care and were both considerably smaller than the NLST. The DANTE study randomised 2811 and the DLCST randomised 4104 men and women who were healthy, heavy and current or former smokers to LDCT screening or no screening. After medians of 34 and 58 months of follow-up, respectively, not even a trend towards reduced mortality was found: (DANTE, relative risk [RR], 0.97; 95% CI, 0.71–1.32; P = 0.84) (DLCST, RR, 1.15; 95% CI, 0.83–1.61; P = 0.430) [16, 17]. The NELSON study is the largest European study so far performed, with 15,822 participants, and has employed a volumetry-based LDCT screening protocol with longer intervals between screening rounds [18, 19]. These investigators greatly reduced their reported false-positive rate when compared with the figure reported in NLST (23.3% in NLST vs. 3.6% in NELSON) by changing the definition of “false positive” to include only those nodules that had a baseline appearance or interval growth that supported malignancy. This is probably justified, given that whilst being recalled for a repeat CT may generate some anxiety and distress, this would be anticipated to be much less than the need for urgent evaluation and is likely to be short-lived.

A comparison of the larger European and LDCT screening trails with the NLST with the trials of the plain chest X-ray in screening [4, 12] is shown in Table 1 , particularly in respect of their effects on mortality. It should be emphasised that the only trial producing a statistically significant result was the NLST.

Trial Modality Recruits No. of lung cancers detected (%) No. at stage 1 (%) No. of deaths from lung cancer (%) Relative mortality reduction from screening Absolute mortality reduction from screening
Negative figures denote mortality increase
NLST (14) Chest X-ray 26,035 941 (3.61) 131 (0.5) 443 (1.70) +20% +0.33%
LDCT 26,309 1060 (4.03) 400 (0.52) 356 (1.35)
DANTE (17) Clinical 1186 73 (6.07) 16 (1.35) 55 (4.64) −0.65% −0.03%
LDCT 1264 104 (8.23) 47 (3.72) 59 (4.67)
DLCST (16) Usual 2052 24 (1.17) 5 (0.24) 11 (0.54) −35% −0.19%
LDCT 2052 70 (3.41) 47 (2.29) 15 (0.73)
PLCO (12) Usual 77,456 1620 (2.09) 462 (0.6) 1230 (1.59) +1.26% −0.02%
Chest X-ray 77,445 1696 (2.19) 374 (0.48) 1213 (1.57)
MLP (4) Usual 4593 160 (3.48) 31 (0.67) 115 (2.50) −6% −0.14%
Chest X-ray 4618 206 (4.46) 68 (1.47) 122 (2.64)

### Table 1.

Comparison of lung cancer screening trials, showing rates of identification of early stage disease and effects on mortality.

### 2.3. Downsides of LDCT screening

Any benefits of CT screening need to be weighed against the harms. Besides the relatively small direct risk of cancers that are caused directly by the radiation exposure from the CT scans, CT screening suffers from all of the general limitations of screening in general. These include:

• Overdiagnosis—see above

• The costs—both psychological and financial arising particularly through the high rate of false-positive diagnoses or indeterminate findings

• Uncertainty regarding how such a programme would work in practice including its potential to reach/capture a reasonable proportion of incident cases

In the NLST, the substantial excess of cancers diagnosed in the CT group (1060 vs. 941) implies that overdiagnosis did occur. Patz and colleagues’ [20] analysis of the NLST study suggested that up to 18% of the cancers identified in the NLST may have been indolent and likely to have been overdiagnosed and indicated that for each cancer death avoided, 1.38 cases may have been overdiagnosed. Moreover, the figures for overdiagnosis may have been even worse if the control arm had received no chest radiographs, which can also be assumed to have resulted in some overdiagnosis. The risk of overdiagnosis, as might be expected, depends on histological subtype and is most striking in patients with a diagnosis of bronchoalveolar cell carcinoma (now called minimally invasive adenocarcinoma), in whom the risk of overdiagnosis was estimated to be 85% after 7 years of follow-up or 49% with lifetime follow-up [20]. These data also raise the question as to the necessity and type of therapy required if a diagnosis of minimally invasive adenocarcinoma is established.

Because the major risk factor for lung cancer is the smoking of tobacco, in order to qualify for entrance into a screening programme, individuals need to have smoked significantly, and many will be current cigarette smokers. Consequently, the target population of a lung cancer screening programme may be expected to have a relative disregard for its own health and a tendency to accept of risk, potentially predisposing to poor acceptability of, and adherence to, screening. This is likely to be much more evident in real life in comparison to a highly motivated volunteer population. It has been shown that smokers in the USA are significantly more likely than never smokers to be male, non-white and less educated; to report poor health status; and to be less likely to be able to identify a usual source of healthcare [21]. This study also indicated that current smokers were less likely than never smokers to believe that early detection would result in a good chance of survival and expressed relative reluctance to consider computed tomography screening for lung cancer. Interestingly, only half of these smokers stated that they would opt for surgery for a cancer diagnosed as a result of screening, further calling into question the value of early diagnosis in this group.

Importantly, even if all subjects meeting the NLST criteria were to accept and adhere to screening, it has been estimated that only 27% of incident lung cancer patients would be included [22]. This implies that the potential to limit lung cancer mortality could only at the very best be 20 of 27% of cases, i.e. a 5.4% mortality reduction overall.

It can also be argued that a screening programme represents a collusion with self-harming behaviour, particularly in relation to current smokers, and throws into focus the interrelationship between smoking cessation and lung cancer screening, particularly as CT screening is only designed to mitigate one of the many potential harms of smoking. This is all the more relevant because recruitment into a lung cancer screening programme does not appear to increase the likelihood of smoking cessation [2325] and reduce it [19]. Offering smoking cessation, which is one of the most cost-effective of all heath interventions, within a screening programme has been shown to improve the cost-effectiveness of the screening—by 20–45% [26, 27]. Is this “cheating”? It is only when expensive LDCT screening is combined with highly cost-effective smoking cessation that cost-utility ratios become comparable with those of other accepted cancer screening programmes!

No long-term psychological harm was found in the NELSON trial. In those with negative results, anxiety and distress fell from baseline; following an abnormal result, anxiety and distress were transient and tended to have returned to baseline by the next screening round [28]. However, the harms—psychological, physical and financial—suffered by those with overdiagnosed tumours have not been quantified and are likely to be substantial.

In the NLST, a major complication occurred in almost five of every 10,000 persons screened due to investigation of a benign lung nodule [14]. Whilst this is a low proportion, these were “normal” subjects in whom any harm is seriously to be regretted.

Based on the NSLT’s own size cut-offs, the average nodule detection rate per round of screening was very high at 20%. In most LDCT screening studies, more than 90% of nodules prove to be benign. Whilst there is a tendency towards lower nodule detection rates in repeat screening rounds, this appears only to be due to the discounting of nodules that had been present in the prior round, so screening, if embarked upon, should probably continue. Furthermore, from the evidence of the fourth round of screening in the NELSON study, lengthening the duration of intervals between rounds beyond 2 years does not appear to be an effective strategy.

In most screening study protocols, a detected nodule triggers further imaging, but the approaches have been inconsistent between studies, and it has been suggested that follow-up imaging rates may have been underestimated [29]. The frequency of further CT imaging among screened individuals has ranged from 1% in the study by Veronesi et al. [30] to 44.6% in the study by Sobue et al. [31]. The frequency of further positron emission tomography (PET) imaging among screened individuals has exhibited less variation, ranging from 2.5% in the study by Bastarrika et al. [32] to 5.5% in the NLST.

The frequency of invasive evaluation of detected nodules, although generally low, has shown marked variation in reported studies. In the NLST, in the patients not found to have lung cancer, 1.2% underwent an invasive procedure such as needle biopsy or bronchoscopy, and 0.7% had a thoracoscopy, mediastinoscopy or thoracotomy. In the NELSON study, these numbers were very similar at 1.2% and 0.6%, respectively [18].

A workshop undertaken by the International Association for the Study of Lung Cancer (IASLC) [33] identified a number of areas where improvements are needed to be made in relation to future implementation of LDCT screening, indicating that, whilst LDCT screening may have potential value, the science around the process remains preliminary. The specific areas requiring clarification were identified as optimization of the identification of high-risk individuals, development of radiological guidelines, development of guidelines for the clinical workup of indeterminate nodules, development of guidelines for pathology reporting, definition of criteria for surgical and therapeutic interventions of suspicious nodules identified through lung cancer CT screening programmes and development of recommendations for the integration of smoking cessation practices into future national lung cancer CT screening programmes.

### 2.4. Cost-effectiveness

Incremental cost-effectiveness ratio (ICER) is defined as = (C1 − C2)/(E1 − E2) , where C1 and E1 are the cost and effect in the intervention and where C2 and E2 are the cost and effect in the control group. Costs are usually described in monetary units, whilst benefits/effect in health status is measured in terms of quality-adjusted life years (QALYs).

Using such methodology, lung cancer LDCT screening was found to be considerably more expensive than other US screening programmes with an ICER of between $126,000 and$169,000/LY [34]. In comparison, colorectal cancer screening has an ICER of $13,000 to$32,000/LY. When a basic smoking cessation intervention is included, which, as outlined above, subsidises an expensive intervention with a cheap one, but which also adds to the total costs, an ICER of $23,185 per QALY is gained, falling further to$16,198 to per QALY gained with a more intensive regimen [26]. Given the huge benefits of smoking cessation for a wide range of diseases, the case for offering smoking cessation to all anyway is strong. Setting up an LDCT screening programme first and adding on smoking cessation to that seems to be putting the cart before the horse: if employed at all for smokers, it should be an add-on to a smoking cessation intervention.

In the UKLST, the cost-effectiveness analyses used data from life tables and modelled data on quality-adjusted life year (QALY) from the NLST, the validity of which is unclear. Given that no effect on mortality could be shown, the reliability of cost-utility analysis is highly questionable, although was estimated at only £8466 (approximately \$11,000 at today’s exchange rate) per QALY gained. This figure is substantially less than that quoted in the NLST and is well within the threshold deemed acceptable by the National Institute for Health and Care Excellence.

### 2.5. Applicability of trial findings to routine clinical practice

Most of the NLST sites were designated National Cancer Institute centres, and more than 80% were large, multidisciplinary academic centres with more than 400 beds [29]. It seems unlikely that the results obtained by less specialised centres will be directly comparable. Furthermore, screening trials are likely to attract, if not healthy volunteers, at least a group who may be more likely than the average to adhere to the screening protocol. Indeed, adherence to the three screening rounds in the NLST was 90%, which is highly unlikely to be achievable in routine practice.

Great difficulty was experienced in recruiting participants to UKLS, given that a system comparable to that that may be used in a real-life screening programme was employed: of the 247,354 questionnaires sent out, response rates to the initial questionnaire were low, with an initial positive response rate of only 15% in current smokers and about 40% in never or former smokers. Even then, a high attrition rate occurred with potential participants being lost at every stage of the recruitment process. Finally, only 4061 subjects (46.5% of all high-risk positive responders) consented and were recruited into the trial.

Another positive from the CT screening studies is that they have provided evidence underpinning the rational approach to the investigation of solitary pulmonary nodules, including the very helpful algorithm developed by the BTS [35]. This includes appreciation that minimally invasive adenocarcinomas may be benign in behaviour and may allow a less aggressive approach to management in comorbid or frail patients.

## 3. Conclusions

The American NLST, in finding a 20% relative reduction in mortality from screening with LDCT, in comparison to plain chest radiography, which itself is ineffective, suggests that screening may be one strategy for improving lung cancer outcomes in a well-motivated American population. This benefit was achieved at a cost comparable to those of other established screening programmes but only when smoking cessation was included. However, the absolute reduction in mortality achieved is small: 87 avoided deaths in 26,722 screened participants, representing a 0.33% lower risk of dying from lung cancer for each individual participant. Besides the harms resulting from the unnecessary treatment of overdiagnosed cases, 24% of participants were found to have a nodule over three rounds, leading to further diagnostic workup. Even in these large academic institutions, a major complication occurred in five of every 10,000 cases with a benign nodule. Many of the cancers diagnosed are small, minimally invasive adenocarcinomas, and these contribute significantly to overdiagnosed cases. Overall, for every cancer death avoided, 1.38 cases may have been overdiagnosed.

Disappointingly, none of the European studies (DANTE and DLCST and NELSON) have found evidence of any significant, or indeed even a trend towards, reduction in mortality, possibly reflecting different epidemiology of cancer subtypes. The use of volumetric techniques employed in the NELSON study seems attractive, but the efficacy of this approach remains completely unproven.

Concentration on harm reduction through screening potentially deflects attention from the need to improve diagnosis and treatment for the majority of cases falling outwith current screening eligibility criteria for smoking history and age. It is known that patients often tolerate lung cancer symptoms for long periods before presenting with them [36]. General practitioners also find identifying lung cancer cases challenging, and patients will often attend several times before the diagnosis is considered and a chest radiograph performed [37]. An early study revealed that educating the public and primary healthcare teams on the importance of cough as a lung cancer symptom resulted in a large increase in chest radiographs being performed and suggested earlier diagnosis [38]. This led on to the national “Be Clear on Cancer” campaign in the UK, the results of which were positive, leading to state funding of repeat programmes. Further work is taking place to look at the effects of lowering thresholds for the obtaining of chest radiographs for chest symptoms in primary care [39]. Facilitating earlier diagnosis of symptomatic disease should also minimise overdiagnosis.

I believe that screening in lung cancer is potentially able to improve lung cancer mortality, but our understanding of how to apply this in real populations, including those outside the USA, is in its infancy. Cost-effectiveness is much improved when screening is combined with smoking cessation, but this is in effect subsidising an otherwise unaffordable screening programme by combining it with another highly cost-effective intervention that could be provided anyway.

Besides the expense of screening, small but definite harms resulting from radiation exposure, investigations of benign lesions and the more significant difficulty of finding of inconsequential disease (overdiagnosis) also reduce its attractiveness. Until we have better understanding of these issues, I believe we should be concentrating more effort on the earlier diagnosis of symptomatic disease, at least in Europe.

Whatever we think of the weaknesses of the current attempts to reduce mortality though screening, the NLST does point to the potential for improving lung cancer outcomes through expeditious diagnosis. For now, at least in Europe, this must be based on improved identification of symptomatic disease. Initiatives to improve detection of early stage disease will rely on improving public and primary care awareness of lung cancer symptoms and reducing the impediments to diagnosis following recognition that lung cancer is a possible diagnosis.

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39. 39. Hurt CN, Roberts K, Rogers TK, Griffiths GO, Hood K, Prout H, et al. A feasibility study examining the effect on lung cancer diagnosis of offering a chest X-ray to higher-risk patients with chest symptoms: protocol for a randomized controlled trial. Trials 2013;14:405. PMID: 24279296.

Written By

Trevor Keith Rogers

Submitted: May 24th, 2016 Reviewed: October 17th, 2016 Published: March 1st, 2017


text:
Problem Set - Trapezoidal and Simpson approximations

## Problem Set - Trapezoidal and Simpson approximations

Last updates: 7 December 2009

## Trapezoidal and Simpson approximations

 Determine the area of a trapezoid with left edge at $x=l$, right edge at $x=l+\Delta x$, left height $f\left(l\right)$, and right height $f\left(l+\Delta x\right)$. Determine the area of a parabola topped slice with left edge at $x=l$, right edge at $x=l+2\Delta x$, middle at $x=l+\Delta x$, left height $f\left(l\right)$, middle height $f\left(l+\Delta x\right)$, and right height $f\left(l+2\Delta x\right)$. Let $N$ be a positive integer. Show that adding up $N$ trapezoidal slices gives the approximation to ${\int }_{a}^{b}f\left(x\right)dx$ given by $\frac{\Delta x}{2}\left(f\left(a\right)+2f\left(a+\Delta x\right)+2f\left(a+2\Delta x\right)+\cdots +2f\left(b-\Delta x\right)+f\left(b\right)\right)$, where $\Delta x=\frac{b-a}{N}$. Let $N$ be an even positive integer. Show that adding up $N$ parabola topped slices gives the approximation to ${\int }_{a}^{b}f\left(x\right)dx$ given by $\frac{\Delta x}{2}\left(f\left(a\right)+4f\left(a+\Delta x\right)+2f\left(a+2\Delta x\right)+\cdots +4f\left(b-\Delta x\right)+f\left(b\right)\right)$, where $\Delta x=\frac{b-a}{N}$. Compute a trapezoidal approximation with $N=4$ slices for the integral ${\int }_{0}^{2}\left(1+{x}^{2}\right)\mathrm{dx}$ and obtain a bound for the error. Compute a trapezoidal approximation with $N=8$ slices for the integral ${\int }_{0}^{2}\left(1+{x}^{2}\right)\mathrm{dx}$ and obtain a bound for the error. Compute a trapezoidal approximation with $N=4$ slices for the integral ${\int }_{0}^{1}{e}^{-x}\mathrm{dx}$ and obtain a bound for the error. Compute a trapezoidal approximation with $N=8$ slices for the integral ${\int }_{0}^{1}{e}^{-x}\mathrm{dx}$ and obtain a bound for the error. Compute a trapezoidal approximation with $N=4$ slices for the integral ${\int }_{0}^{\pi /2}\mathrm{sin}x\mathrm{dx}$ and obtain a bound for the error. Compute a trapezoidal approximation with $N=8$ slices for the integral ${\int }_{0}^{\pi /2}\mathrm{sin}x\mathrm{dx}$ and obtain a bound for the error. Compute a trapezoidal approximation with $N=4$ slices for the integral ${\int }_{0}^{1}{\left(1+{x}^{2}\right)}^{-1}\mathrm{dx}$ and obtain a bound for the error. Compute a trapezoidal approximation with $N=8$ slices for the integral ${\int }_{0}^{1}{\left(1+{x}^{2}\right)}^{-1}\mathrm{dx}$ and obtain a bound for the error. Compute a Simpson approximation with $N=4$ slices for the integral ${\int }_{0}^{2}\left(1+{x}^{2}\right)\mathrm{dx}$ and obtain a bound for the error. Compute a Simpson approximation with $N=8$ slices for the integral ${\int }_{0}^{2}\left(1+{x}^{2}\right)\mathrm{dx}$ and obtain a bound for the error. Compute a Simpson approximation with $N=4$ slices for the integral ${\int }_{0}^{1}{e}^{-x}\mathrm{dx}$ and obtain a bound for the error. Compute a Simpson approximation with $N=8$ slices for the integral ${\int }_{0}^{1}{e}^{-x}\mathrm{dx}$ and obtain a bound for the error. Compute a Simpson approximation with $N=4$ slices for the integral ${\int }_{0}^{\pi /2}\mathrm{sin}x\mathrm{dx}$ and obtain a bound for the error. Compute a Simpson approximation with $N=8$ slices for the integral ${\int }_{0}^{\pi /2}\mathrm{sin}x\mathrm{dx}$ and obtain a bound for the error. Compute a Simpson approximation with $N=4$ slices for the integral ${\int }_{0}^{1}{\left(1+{x}^{2}\right)}^{-1}\mathrm{dx}$ and obtain a bound for the error. Compute a Simpson approximation with $N=8$ slices for the integral ${\int }_{0}^{1}{\left(1+{x}^{2}\right)}^{-1}\mathrm{dx}$ and obtain a bound for the error. Let ${T}_{4}$ be the trapezoidal approximation with $N=8$ slices for the integral ${\int }_{0}^{1}{\left(1+{x}^{2}\right)}^{-1}\mathrm{dx}$. Show that $|f\mathrm{\prime \prime }\left(x\right)|\le 2$ for $x\in \left[0,1\right]$ and that $|{T}_{4}-\frac{1}{4}\pi |\le 1/96<0.0105$. Use the trapezoidal approximation with $N=4$ slices to approximate the integral $\mathrm{log}2={\int }_{1}^{2}{x}^{-1}\mathrm{dx}$. Show that $0.6866\le \mathrm{log}2\le 0.6958$. Use Simpson's approximation with $N=4$ slices to approximate $\mathrm{log}2$. Show that $0.6927\le \mathrm{log}2\le 0.6933$. Compute a trapezoidal approximation with $N=8$ slices for the integral ${\int }_{0}^{1}{e}^{-{x}^{2}}\mathrm{dx}$. Compute a trapezoidal approximation with $N=16$ slices for the integral ${\int }_{0}^{1}{e}^{-{x}^{2}}\mathrm{dx}$. Compute a Simpson approximation with $N=8$ slices for the integral ${\int }_{0}^{1}{e}^{-{x}^{2}}\mathrm{dx}$. Compute a Simpson approximation with $N=16$ slices for the integral ${\int }_{0}^{1}{e}^{-{x}^{2}}\mathrm{dx}$. Compute a trapezoidal approximation with $N=8$ slices for the integral ${\int }_{0}^{\pi /2}\frac{\mathrm{sin}x}{x}\phantom{,}\mathrm{dx}$. Compute a trapezoidal approximation with $N=16$ slices for the integral ${\int }_{0}^{\pi /2}\frac{\mathrm{sin}x}{x}\phantom{,}\mathrm{dx}$. Compute a Simpson approximation with $N=8$ slices for the integral ${\int }_{0}^{\pi /2}\frac{\mathrm{sin}x}{x}\phantom{,}\mathrm{dx}$. Compute a Simpson approximation with $N=16$ slices for the integral ${\int }_{0}^{\pi /2}\frac{\mathrm{sin}x}{x}\phantom{,}\mathrm{dx}$. Derive the midpoint approximation for ${\int }_{a}^{b}f\left(x\right)\phantom{,}\mathrm{dx}$. With $N$ slices it is obtained by adding up the areas of rectangles with height equal to the value of the function at the midpoint of the interval. Show that the error estimate is given by $|{\int }_{a}^{b}f\left(x\right)\phantom{,}\mathrm{dx}-{M}_{N}|\le \frac{{\left(b-a\right)}^{3}}{24{n}^{2}}M$, where $M$ is an upper bound for $|f\mathrm{\prime \prime }\left(x\right)|$ on $\left[a,b\right]$.

format: plain


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# Abraham Busset (1704-1761)

220,560pages on
this wiki

Abraham Busset (1704--1761) was a Palatine emigrant from Europe, who later became a notable land-holder in early New Bern, North Carolina. Despite strong evidence of his European (probably Swiss [1] ) ancestry, the exact location of his birth is unknown. The ancestry of Abraham and his sister Margaret Busset Simmons is an important area of interest for genealogists seeking to extend their family tree beyond New Bern.

## European AncestryEdit

Records linking the Bussets to their ancestral home in Europe are scarce. Jones[2] suggests that Abraham and Margaret are children of a Daniel Buschart/Boset/Busset who came to the US as part of the 4th Rotterdam Departure of Palatines. As Jones states, this is only a Possible scenario. Another possibility is that they are the children of Johann and Maria Buchse/Busset mentioned in a letter by Margaret's husband Johannes Simon to his grandfather in Switzerland [3].

It has been suggested[4] that the Abraham Busset born in 1658 in the Ormont-Dessus village of Vaud canton in Switzerland is the father of Abraham and Margaret. It is intriguing to note that church records of Ormont-Dessus show at least four Abraham Bussets[5] from the period 1650-1750. Finding a link between the Bussets and Ormont-Dessus would certainly explain Abraham's name.

## Palatine ConnectionEdit

Knittle gives a flowery description of Switzerland's role in accepting religious refugees: (Switzerland) was the common refuge for persecuted Protestants in the Reformation period. This status as refuge apparently was not always acceptable to the Swiss. Luck notes that Georg Ritter and others formed a group to settle land in Pennsylvania and Virginia: Ritter and Co. were to be paid 45 thalers a head for every Taufer (Baptist) they succeeded in carrying off to America and 500 thalers more for another group of about 100 paupers who desired to [emigrate to] America.

Luck[6] gives information on Christoph von Graffenried, with whom it appears Abraham emigrated from Switzerland:

In 1710, Christoph von Graffenried (1661-1743) of Bern joined in the act. As a man of considerable influence, he managed to obtain 17,500 acres of land in North Carolina on which he settled some of Michel's emigres, with still more from the Palatinate. He founded the town of New Bern, apparently assisted by Michel and Georg Ritter and Co. Battles with the Indians made life in New Bern precarious.

Von Graffenried gathered Palatine refugees (apparently including some Swiss) in England, and sent them on to Virgina and North Carolina[7]. He followed later with a contingent from Bern. When he arrived after a relatively safe and comfortable passage, he found his Palatine settlers decimated in number, and in poor health. One of their ships had been plundered by a French privateer, another had a very rough voyage, and in North Carolina they had been settled by a government official on his unhealthy swamp land so they could clear it for him.

The rough times were not over yet for the settlers. Graffenried chronicles in his appologetic account yet more intrigue among the settlers. One governor died as von Graffenried arrived, and the other was not immediately recognized. Graffenried was pursuing silver which Michel had told him of, which surely was a distraction if nothing else. Finally, the Tuscarora Indian war decimated the population again. Abraham signed a petition in 1747 stating that he was affiliated with the Palatine emmigrants; it seems safe to assume that he emigrated with Graffenreid.

## Bussets and SwitzerlandEdit

It is not clear whether the Bussets came with the Palatine group to London, and were recruited by von Graffenried there, or if they came later with his group from Bern. There is a paucity of documentation of the trip. Apparently no official ship's passenger list survived[8], though there may be something in Switzerland.

Jones suggests that a Daniel Buschart/Boset is the father of Margaret and Abraham:

Daniel Buschart with wife and 8 children was in the 4th Party Roterdam Departure List; this party sailed for London 21 June 1709. He appears in the 4th Party London Arrivals List, taken at St. Catherine's and Deptford 27 June 1709, as {\em Daniel Boset} aged 59, linen and cloth weaver, Reformed, with wife, sons aged 14 and 5, and daughters 21, 19, 13, 11, 9 and 1 month. I think this is the New Bern settler. Hank Jones also shows Daniel Pasch/Bouche who went to New York in 1710. He is not shown on the 1711 Swiss map, so if he was indeed Swiss, then he was an artisan living in the town of New Bern. His family probably was too large for them to have been living with another family listed on the map. He is not mentioned in any of the 1711 Swiss letters home. Jones finds no record of Daniel Buschart/Boset in North Carolina; the connection of this man to Abraham and Margaret remains tenuous.

Jones above apparently refers to Various Letters from North Carolina included in one edition of von Graffenied's account. Johann Busche and wife Maria Magdalena are mentioned in a letter written by (or at least in the name of) Margaret Busset's future husband, Johannes Simon, but the connection to the Bussets seems incidental. If Abraham and Margaret's parents did come to North Carolina, it seems reasonable that they were killed in the Indian wars or of other causes not long after arriving in North Carolina. At least there is almost no documentation of the parents in N.C.

As for Abraham and Margaret, there are records to give a few details. Jones suggests that they are the "Minnet Orphans" and blames the spelling of the name on a poorly educated clerk. By the 1730s Abraham Busset had bought land as denoted by many court records [9]. Abraham signed a petition in 1747 stating that he was affiliated with the Palatine emmigrants; it seems safe to assume that he did indeed emigrate with Graffenreid. Though Abraham was married to "Susannah" at his death, he apparently did not have any surviving children, since "John" (Johannes) Simmons was listed as his next-of-kin. Margaret married Johannes Simmons (around 1716, as a guess); they named a son Abraham Busset Simmons.

## NotesEdit

1. ^ H.Z. Jones, pp 911-919
2. ^ H.Z. Jones
3. ^ See the letters in Graffenried
4. ^ Genealogical records of Frances Bushman Peel, Abraham Busset researcher.
5. ^ See LDS films 0128745, 0128746
6. ^ Luck, 1985, citing Haas, 1940
7. ^ Grafenried, 1975
8. ^ H. Jones, p 112
9. ^ H.Z. Jones, 2002, p 912.

## ReferencesEdit

• Henry Z. Jones Jr. and Lewis Bunker Rohrbach, Even More Palatine Families : 18th Century Immigrants to the American Colonies and their German, Swiss, and Austrian Origins, Piction Press, 2002.
• Christoph von Graffenried, Account of the founding of New Bern, Reprint Co, 1975 (originaly published by Edwards & Broughton Printing, 1920.), english Translation by Vincent H. Todd. See http://docsouth.unc.edu/nc/graffenried/graffenried.html.
• Births, baptisms, catechumenes, reg. paroissiaux, mariages, annonces 1605–1750, Mormon Genealogical Film, film number: 0128745.
• Births, baptisms, catechumenes, reg. paroissiaux, mariages, annonces 1750-1875, Mormon Genealogical Film, film number: 0128746.
• Walter Allen Knittle, Early Eighteenth Century Palatine Emigration, Genealogical Publishing Company, 1997.
• J. Murray Luck, History of Switzerland, Palo Alto, California: The Society for the Promotion of Science and Scholarship (SPOSS), 1985.
• Victor T. Jones, A list of known persons who left switzerland and germany to settle in new bern, north carolina in 1710, North Carolina Genealogical Society Journal, February 1997, see also http://newbern.cpclib.org/research/settlers.htm.
• J. L. Webb, Commoners and courtiers in the New World: Von Graffenried’s settlement at New Bern. ??, 1987.


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### Contact

Web Hosting and SEO

# Upgrading Joomla to PHP 7.3

PHP 7.3 was released in December of 2018 at about the same time that PHP 7.1 moved to support for security fixes only; PHP 7.2 is the primary release, but 7.3 offers some incremental improvements. PHP 7 was claimed to be about a 2X performance improvement over PHP 5.6, and each release of the 7 series claims performance of 5-20% better than the previous release. CPanel announced that EasyApache 4 added PHP 7.3 in December, so I decided that three months was enough time for many of the problems to be worked out and took the plunge on moving to 7.3; it went fairly easily.

If you are still on PHP 5.6, or on a Joomla release prior to 3.9, you should probably look at Upgrading to Joomla 3.8, 3.9, 3.10 and 4.0 before reading this article.

## Backup

As always, make sure that you have backups before beginning any system level work.

## Upgrade Joomla Extensions, and Joomla

As always for a PHP upgrade, make sure that both Joomla and all extensions are current and that you are on the current release of Joomla.

## Add PHP 7.3 in CPanel

If you have a VPS, you will need to log in to Web Hosting Manager and add PHP 7.3 in EasyApache. It will prompt you to include the PHP packages that are used in the other versions of PHP; say “yes” unless you know that you no longer need some of the packages.

If you do not have a VPS, PHP 7.3 may be available in the Multi-PHP settings within CPanel. If not, call your web hosting technical support to make sure that it is added. This will probably take a day or two, and may be much longer.

## In WHM, Update PHP Configuration

You will need to update the PHP configuration so that it matches the settings for your working PHP 7.1 or 7.2:

• Make sure that the maximum file upload and post size are increased, otherwise you will not be able to upload many Joomla extensions.
• Depending upon your site and the defaults, you may need to increase the default memory. Some sites of mine that were fine on 32M in PHP 7.2 required an increase in PHP 7.3.
• Update the php.ini setting for disable_functions to disable at least "show_source, system, shell_exec, passthru, exec, phpinfo, popen, proc_open, ini_set". The CPanel security audit will complain if you do not.

## Turn on Error Messages in Joomla

If everything goes perfectly, you will not need error messages, but it will be easier if you turn this on before switching over the first time. If you do not turn it on now, you will have to log in to Joomla each time you switch PHP versions (assuming you are using PHP sessions for performance).

## In WHM, Switch to PHP 7.3

Once you have done all the prep work, switch to PHP 7.3 and test everything on your site. If you have extensions for managing photo galleries, make sure to test these; some of them are dependent upon PHP modules that are not installed by default.

## Turn off Error Messages in Joomla

Once you are done, make sure to turn error messages off in Joomla. You will probably still get a number of warning messages, but no fatal messages.

## Summary

If you are current on Joomla maintenance and have already switched to PHP 7.1 or 7.2, moving to 7.3 should be a relatively painless. The most likely problem area will be forgetting to change a memory settings.

This web site uses cookies to provide user authentication and improve your user experience through the use of Google Analytics and Matomo Analytics. It also uses contact information for email and phone communcation. For details, see the Privacy Policy.


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## Phase Modulation

Phase modulation (PM) is a linear baseband modulation technique in which the message modulates the phase of a constant amplitude signal. Communications Toolbox™ software includes these functions, System objects, and blocks to modulate digital baseband signals with these modulation methods:

• Binary, quadrature, and general phase shift keying (PSK)

• Binary, quadrature, and general differential phase shift keying (DPSK)

• Offset quadrature phase shift keying (OQPSK)

### BPSK

In binary phase shift keying (BPSK), the phase of a constant amplitude signal switches between two values corresponding to binary 1 and binary 0. The passband waveform of a BPSK signal is

`${s}_{n}\left(t\right)=\sqrt{\frac{2{E}_{b}}{{T}_{b}}}\mathrm{cos}\left(2\pi {f}_{c}t+{\varphi }_{n}\right),$`

where:

• Eb is the energy per bit.

• Tb is the bit duration.

• fc is the carrier frequency.

In MATLAB®, the baseband representation of a BPSK signal is

`${s}_{n}\left(t\right)={e}^{-i{\varphi }_{n}}=\mathrm{cos}\left(\pi n\right).$`

The BPSK signal has two phases: 0 and π.

The probability of a bit error in an AWGN channel is

`${P}_{b}=Q\left(\sqrt{\frac{2{E}_{b}}{{N}_{0}}}\right),$`

where N0 is the noise power spectral density.

### QPSK

In quadrature phase shift keying, the message bits are grouped into 2-bit symbols, which are transmitted as one of four phases of a constant amplitude baseband signal. This grouping provides a bandwidth efficiency that is twice as great as the efficiency of BPSK. The general QPSK signal is expressed as

`${s}_{n}\left(t\right)=\sqrt{\frac{2{E}_{s}}{{T}_{s}}}\mathrm{cos}\left(2\pi {f}_{c}t+\left(2n+1\right)\frac{\pi }{4}\right);\text{ }n\in \left\{0,1,2,3\right\},$`

where Es is the energy per symbol and Ts is the symbol duration. The complex baseband representation of a QPSK signal is

`${s}_{n}\left(t\right)=\mathrm{exp}\left(j\pi \left(\frac{2n+1}{4}\right)\right);\text{ }n\in \left\{0,1,2,3\right\}.$`

In this QPSK constellation diagram, each 2-bit sequence is mapped to one of four possible states. The states correspond to phases of π/4, 3π/4, 5π/4, and 7π/4.

To improve bit error rate performance, the incoming bits can be mapped to a Gray-coded ordering.

Binary-to-Gray Mapping

Binary SequenceGray-Coded Sequence
0000
0101
1011
1110

The primary advantage of the Gray code is that only one of the two bits changes when moving between adjacent constellation points. Gray codes can be applied to higher-order modulations, as shown in this Gray-coded QPSK constellation.

The bit error probability for QPSK in AWGN with Gray coding is

`${P}_{b}=Q\left(\sqrt{\frac{2{E}_{b}}{{N}_{0}}}\right),$`

which is the same as the expression for BPSK. As a result, QPSK provides the same performance with twice the bandwidth efficiency.

### Higher-Order PSK

You can modulate and demodulate higher-order PSK constellations. The complex baseband form for an M-ary PSK signal using binary-ordered symbol mapping is

`${s}_{n}\left(t\right)=\mathrm{exp}\left(j\pi \left(\frac{2n+1}{M}\right)\right);\text{ }n\in \left\{0,1,\dots ,M-1\right\}.$`

This 8-PSK constellation uses Gray-coded symbol mapping.

For modulation orders beyond 4, the bit error rate performance of PSK in AWGN worsens. In the following figure, the QPSK and BPSK curves overlap one another.

### DPSK

DPSK is a noncoherent form of phase shift keying that does not require a coherent reference signal at the receiver. With DPSK, the difference between successive input symbols is mapped to a specific phase. As an example, for binary DPSK (DBPSK), the modulation scheme operates such that the difference between successive bits is mapped to a binary 0 or 1. When the input bit is 1, the differentially encoded symbol remains the same as the previous symbol, while an incoming 0 toggles the output symbol.

The disadvantage of DPSK is that it is approximately 3 dB less energy efficient than coherent PSK. The bit error probability for DBPSK in AWGN is Pb = 1/2 exp(Eb/N0).

### OQPSK

Offset QPSK is similar to QPSK except that the time alignment of the in-phase and quadrature bit streams differs. In QPSK, the in-phase and quadrature bit streams transition at the same time. In OQPSK, the transitions have an offset of a half-symbol period as shown.

The in-phase and quadrature signals transition only on boundaries between symbols. These transitions occur at 1-second intervals because the sample rate is 1 Hz. The following figure shows the in-phase and quadrature signals for an OQPSK signal.

For OQPSK, the quadrature signal has a 1/2 symbol period offset (0.5 s).

The BER for an OQPSK signal in AWGN is identical to that of a QPSK signal. The BER is

`${P}_{b}=Q\left(\sqrt{\frac{2{E}_{b}}{{N}_{0}}}\right),$`

where Eb is the energy per bit and N0 is the noise power spectral density.

## References

[1] Rappaport, Theodore S. Wireless Communications: Principles and Practice. Upper Saddle River, NJ: Prentice Hall, 1996, pp. 238–248.

[2] Viterbi, A.J. “An Intuitive Justification and a Simplified Implementation of the MAP Decoder for Convolutional Codes.” IEEE Journal on Selected Areas in Communications 16, no. 2 (February 1998): 260–64. https://doi.org/10.1109/49.661114.


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### About Absolute Value & the Distance Between Two Points on a Number Line:

When we think about absolute value, we are thinking about the distance between a number and zero on the number line. Since a distance is always non-negative, meaning it is either zero or some positive value, the absolute value of a number is always non-negative. The absolute value of a number is just the number if it is a non-negative number or the opposite of the number if it's a negative number.

Test Objectives
• Demonstrate an understanding of how to Simplify an Absolute Value Expression
• Demonstrate an understanding of how to Solve a Simple Absolute Value Inequality
• Demonstrate an understanding of how to find the Distance Between Two Points on a Number Line
Absolute Value & the Distance Between Two Points on the Number Line Practice Test:

#1:

Instructions: Simplify each.

$$a)\hspace{.2em}{-}|{-}3|$$

$$b)\hspace{.2em}|{-}5 \cdot 7 + 2^2|$$

#2:

Instructions: Solve each inequality for x.

$$a)\hspace{.2em}|x| > -12$$

$$b)\hspace{.2em}|x| < -8$$

#3:

Instructions: Solve each inequality for x.

$$a)\hspace{.2em}|x| < 7$$

$$b)\hspace{.2em}|x| > 9$$

#4:

Instructions: Find the distance between Point "Q" and Point "R" on the number line.

$$a)\hspace{.2em}Q=-5, R=7$$

$$b)\hspace{.2em}Q=-1, R=8$$

#5:

Instructions: Find the distance between Point "Q" and Point "R" on the number line..

$$a)\hspace{.2em}Q=-3, R=-1$$

$$b)\hspace{.2em}Q=12, R=-15$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}{-}3$$

$$b)\hspace{.2em}31$$

#2:

Solutions:

$$a)\hspace{.2em}\text{All Real Numbers}$$ $$(-\infty, \infty)$$

$$b)\hspace{.2em}\text{No Solution}$$ $$∅$$

#3:

Solutions:

$$a)\hspace{.2em}{-}7 < x < 7$$ $$(-7, 7)$$

$$b)\hspace{.2em}x < -9 \hspace{.25em}\text{or}\hspace{.25em}x > 9$$ $$(-\infty, -9) ∪ (9, \infty)$$

#4:

Solutions:

$$a)\hspace{.2em}12$$

$$b)\hspace{.2em}9$$

#5:

Solutions:

$$a)\hspace{.2em}2$$

$$b)\hspace{.2em}27$$


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## arXiv Analytics

### arXiv:nucl-ex/0202003AbstractReferencesReviewsResources

#### Survey of experimental data

Published 2002-02-06Version 1

>A review of meson emission in heavy ion collisions at incident energies from SIS up to collider energies is presented. A statistical model assuming chemical equilibrium and local strangeness conservation (i.e. strangeness conservation per collision) explains most of the observed features. Emphasis is put onto the study of $K^+$ and $K^-$ emission at low incident energies. In the framework of this statistical model it is shown that the experimentally observed equality of $K^+$ and $K^-$ rates at threshold-corrected'' energies $\sqrt{s} - \sqrt{s_{th}}$ is due to a crossing of two excitation functions. Furthermore, the independence of the $K^+/K^-$ ratio on the number of participating nucleons observed between SIS and RHIC is consistent with this model. It is demonstrated that the $K^-$ production at SIS energies occurs predominantly via strangeness exchange and that this channel is approaching chemical equilibrium. The observed maximum in the $K^+/\pi^+$ excitation function is also seen in the ratio of strange to non-strange particle production. The appearance of this maximum around 30 $A\cdot$GeV is due to the energy dependence of the chemical freeze-out parameters $T$ and $\mu_B$.

Comments: 13 pages, 14 figures, SQM2001 in Frankfurt, Sept. 2001, submitted to IOP
Journal: J.Phys.G28:1787-1800,2002
Categories: nucl-ex
Related articles: Most relevant | Search more
arXiv:nucl-ex/0112005 (Published 2001-12-17)
Strange Particle Production from SIS to LHC
arXiv:nucl-ex/0008010 (Published 2000-08-17)
An Excitation Function of K$^-$ and K$^+$ Production in Au+Au Reactions at the AGS
arXiv:nucl-ex/0408023 (Published 2004-08-30, updated 2004-09-14)
Towards the excitation function of the pp -> ppK+K- reaction at COSY-11
P. Winter et al.


text:
• Ringing non-Gaussianity from inflation with a step in the second derivative of the potential

• # Fulltext

https://www.ias.ac.in/article/fulltext/pram/096/0056

• # Keywords

Inflation; 3-point function; non-Gaussianity.

• # Abstract

Inflationary model driven by a scalar field whose potential has a step in the second derivative with respect to the field is considered. For the best-fit potential parameter values, the 3-point function and the non-Gaussianity associated with the featured model is calculated. We study the shape and scale dependence of the 3-point function. The distinctive feature of this model is its characteristic ringing behaviour of f$_{NL}$. We can see that the oscillationsin f$_{NL}$ in this model last for a much longer range of k values, than the previously studied models. In that sense, this model is potentially distinguishable from models with other features in the potential.

• # Author Affiliations

1. Department of Physics, NSS College, Pandalam 689 501, India
2. Department of Physics, Alphonsa College, Pala 686 574, India

• # Pramana – Journal of Physics

Volume 97, 2023
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019


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# How many joules of heat are needed to change 50.0 grams of ice at -15.0 °C to steam at 120.0 °C?

Dec 14, 2014

The answer is $153.7 k J$.

What you are asked to determine is the total energy required to go from ice to water, and then from water to vapor - the phase changes underwent by the water molecules.

In order to do this, you'll need to know:

Heat of fusion of water: $\Delta {H}_{f}$ = $334$ $J$/$g$;
Heat of fusion vaporization of water: $\Delta {H}_{v}$ = $2257$ $J$/$g$;
Specific heat of ice: $c$ = $2.09$ $J$/${g}^{\circ} C$;
Specific heat of water: $c$ = $4.18$ $J$/${g}^{\circ} C$;
Specific heat of steam: $c$ = $2.09$ $J$/${g}^{\circ} C$;

So, the following steps describe the overall process:

1. Determine the heat required to raise the temperature of the ice from $- {15.0}^{\circ} C$ to ${0}^{\circ} C$:

${q}_{1} = m \cdot {c}_{i c e} \cdot \Delta T = 50.0 g \cdot 2.09 \frac{J}{g {\cdot}^{\circ} C} \cdot \left({0}^{\circ} C - \left(- {15}^{\circ} C\right)\right) = 1567.5 J$

2. Determine the heat required to convert ${0}^{\circ} C$ ice to ${0}^{\circ} C$ water:

${q}_{2} = m \cdot \Delta {H}_{f} = 50.0 g \cdot 334 \frac{J}{g} = 16700 J$

3. Determine the heat required to go from water at ${0}^{\circ} C$ to water at ${100}^{\circ} C$:

${q}_{3} = m \cdot {c}_{w a t e r} \cdot \Delta T = 50.0 g \cdot 4.18 \frac{J}{g {\cdot}^{\circ} C} \cdot \left({100}^{\circ} C - {0}^{\circ} C\right) = 20900 J$

4. Determine the heat required to convert ${100}^{\circ} C$ water to ${100}^{\circ} C$ vapor:

${q}_{4} = m \cdot \Delta {H}_{v} = 50.0 g \cdot 2257 \frac{J}{g} = 112850 J$

5. Determine the heat required to go from ${100}^{\circ} C$ vapor to ${120}^{\circ} C$ vapor:

${q}_{5} = m \cdot {c}_{v a p o r} \cdot \Delta T = 50.0 g \cdot 2.09 \frac{J}{g {\cdot}^{\circ} C} \cdot \left({120}^{\circ} C - {100}^{\circ} C\right) = 2090 J$

Therefore, the total heat required is

${q}_{T O T A L} = {q}_{1} + {q}_{2} + {q}_{3} + {q}_{4} + {q}_{5} = 152696.5 J = 153.7 k J$


text:
Violation of newtons Third law

1. Aug 15, 2011

TheBlackNinja

2. Aug 15, 2011

D H

Staff Emeritus
It is violated, either that or F=ma is violated.

I beg to differ. What is true is that when that momentum is included (an advanced topic), the conservation laws are again obeyed.

One way to look at Newton's third law is that it is a special case of the more generic laws of conservation of momentum and angular momentum. The circumstances under which Newton's third law derives from the conservation laws are when forces are central in nature and depend on position only, and when the field that mediates the force does not itself store momentum or angular momentum. Various electromagnetic forces fail on all three accounts, so why would you expect Newton's third law to hold in such a case?

Last edited: Aug 15, 2011
3. Aug 15, 2011

TurtleMeister

If this is true then why don't we have reactionless drives?

4. Aug 15, 2011

D H

Staff Emeritus
Because momentum and angular momentum are conserved quantities.

5. Aug 15, 2011

TurtleMeister

Probably too advanced for me, so I'll just take your word for it. I've always thought of the third law and the conservation of momentum to be simply two different ways of viewing the same thing. Violate one then you violate the other.

6. Aug 15, 2011

atyy

Newton's third law is true for Newton's gravity - the sun attracts the earth and the earth attracts the sun at the same time - even though both are far from each other. How can each know how hard the other is pulling immediately? In Newtonian physics, there is no upper limit to the speed of things, so there's no problem there. But this is obviously doesn't fit in with special relativity - so Newton's third law doesn't hold in special relativity.

In special relativity, the local exchange of momentum is between fields and particles. A particle feels a force due to a field. But there isn't a simple way to say that a field feels a force due to a particle. So to keep the spirit of Newton's third law for fields and particles, we ascribe momentum to both, and say that momentum is conserved.

7. Aug 16, 2011

TurtleMeister

Oh, I see. Thanks for the simple explanation.


text:
# More lighting tests. . .

I have almost found the type of lightbulb that will work however, the fixtures are proving to be tricky, you will see why at the bottom. I think I need to utilize the track lights instead of trying to install lighting somehow because the room has a ceiling unlike in the rest of the gallery. This is actually a challenge, you will see why. . . I need to find out if we have any MR16 fixtures laying around here somewhere. . .  They come in all shapes and sizes, some with barn doors, adjustable zooms (to create more of a spot or more flood light). Let’s hope I can get lighting figured out this week!! It’s taken forever to get this far and I still don’t have the fixtures and I still need to try lower wattage!!

What I’ve tested:

All for now.

## One Reply to “More lighting tests. . .”

1. I had no idea how much was involved! Good luck as you continue to work it out.

Sent from my iPhone

>


text:
# Sum of squared Negative Binomial probability masses

Let $(p_k)_{k=0, \dots, \infty}$ denote the probability masses of a Negative Binomial distribution with parameters $r>0$ and $p\in]0,1[$. I'm looking for the sum of their squares, $$\sum_{k=0}^\infty p_k^2$$ as a function of $r$ and $p$ (other parameterizations are also fine). In other words I am interested in (the exponential of) the second-order Renyi entropy of a Negative Binomial distribution.

Background:

• As per @whuber's comment here, this question is related, but not a duplicate... I'd still be interested in an analytical solution. – Stephan Kolassa Jul 21 '15 at 11:44
• I have edited your post to add the name second-order Renyi entropy. I think this could be useful for web search engines. I hope this is fine for you. – Stéphane Laurent Jul 21 '15 at 16:13

Take two independent Poisson random variables $X$ and $X'$, with means $\lambda$ and $\lambda'$. The formula answering your question in the Poisson case is a particular case of the identity $$\Pr(X = X' \mid \lambda, \lambda') = \exp(-\lambda)\exp(-\lambda')I_0(2\sqrt{\lambda}{\sqrt{\lambda'}}).$$

> lambda <- 1; lambdaa <- 2
> sum(dpois(0:100,lambda)*dpois(0:100,lambdaa))
[1] 0.2117121
> gsl::bessel_I0(2*sqrt(lambda*lambdaa)) * exp(-lambda-lambdaa)
[1] 0.2117121

Your problem is the same as calculating $\int\int\Pr(X= X' \mid \lambda, \lambda') f_{a,b}(\lambda)f_{a,b}(\lambda')d\lambda d\lambda'$ where $f_{a,b}$ is the $\Gamma(a,b)$ pdf, setting $r=a$ and $p=\frac{b}{1+b}$ with your notations, because of the link between the negative binomial distribution and the Poisson-Gamma distribution.

Let's start by $$\int \exp(-\lambda)I_0(2\sqrt{\lambda}{\sqrt{\lambda'}})f_{a,b}(\lambda)d\lambda = \frac{b^a}{\Gamma(a)}\int \lambda^{a-1}\exp\bigl(-(b+1)\lambda\bigr)I_0(2\sqrt{\lambda}{\sqrt{\lambda'}})d\lambda.$$

According to Mathematica this is equal to $${\left(\frac{b}{1+b}\right)}^a {}_1\!F_1\left(a, 1, \frac{\lambda'}{b+1}\right)$$ where ${}_1\!F_1$ is the Kummer hypergeometric function.

Now we can even get something for $$\begin{multline} \int \exp(-\lambda'){}_1\!F_1\left(a, 1, \frac{\lambda'}{b+1}\right)f_{a',b'}(\lambda')d\lambda' \\ = \frac{{b'}^{a'}}{\Gamma(a')}\int {\lambda'}^{a'-1}\exp\bigl(-(b'+1)\lambda'\bigr) {{}_1\!F_1}\left(a, 1, \frac{\lambda'}{b+1}\right)d\lambda'. \end{multline}$$ Indeed, Mathematica gives $${\left(\frac{b'}{1+b'}\right)}^{a'} {}_2\!F_1\left(a, a', 1, \frac{1}{(b+1)(b'+1)}\right)$$ where ${}_2\!F_1$ is the Gauss hypergeometric function. The final result is beautiful: $${\left(\frac{b}{1+b}\right)}^{a}{\left(\frac{b'}{1+b'}\right)}^{a'}{}_2\!F_1\left(a, a', 1, \frac{1}{(b+1)(b'+1)}\right),$$ and even a bit more beautiful with your notations: $$p^{a}{p'}^{a'}{}_2\!F_1\left(a, a', 1, (1-p)(1-p')\right)$$

Check:

> a <- 2; A <- 3; b <- 5; B <- 8
> (b/(1+b))^a*(B/(1+B))^A*gsl::hyperg_2F1(a,A,1,1/(b+1)/(B+1))
[1] 0.5450618
> sum(dnbinom(0:100, a, b/(1+b))*dnbinom(0:100, A, B/(1+B)))
[1] 0.5450618

In the special case you are interested in, the sum of squares is $$p^{2a}{}_2\!F_1\left(a, a, 1, {(1-p)}^2\right),$$ and the second-order Renyi entropy is $$-2a \log p - \log {}_2\!F_1\left(a, a, 1, {(1-p)}^2\right).$$

• Thank you. This looks both intriguing and intimidating. I'll need a little time to digest it. In the meantime, could you please introduce me to your friend ${}_kF_\ell$? – Stephan Kolassa Jul 21 '15 at 15:28
• @StephanKolassa I have edited my answer to add their names. You will easily find some information about them. – Stéphane Laurent Jul 21 '15 at 15:54


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# Is this a valid heuristic for Dots and Boxes to reduce the branching factor of the search tree?

I am implementing an AI based on the MiniMax algorithm that plays the game Dots and Boxes. I would like to reduce the branching factor of the search tree, by introducing a heuristic rule that limits the number of successor states. However, I am not sure whether there exist any situations, where this rule would exclude the optimal move and make the AI play worse.

In the game 2 players take turns, until one of them captures a box, after which they have to make another move, possibly capturing another box etc. For this purpose, I define successor states of $$S$$ to be all states, where the opponent gets the move, or the terminal state. If a player made a capture and still has to make another move, then the current state is not yet a successor state of $$S$$ - the player has to continue making moves until they're unable to.

My rule is as follows:
Assume a player makes a series of captures from a starting state $$S$$ and ends it with a move $$m$$, after which the turn goes to the opponent. Then it's always better or at least not worse for the player to have made as many captures as possible before finishing the sequence with $$m$$.

This means that any state $$S$$ always has less successors, then there are legal moves. All successors where the player captures $$n$$ boxes and finishes with move $$m$$ are discarded, unless $$n$$ is the maximum number of boxes that can be captured such that $$m$$ can still be a finishing move. For example, in the following state:

immediately capturing the 4 boxes leads to a loss, while dividing them into 2 small chains, 2 boxes each, with the move (2,2)-(2,3) leads to a victory. The optimal strategy is consistent with the rule, as it would not be possible to finish with the move (2,2)-(2,3), if any of the boxes were captured. The rule excludes all successors, where the player captures 1 or 2 boxes, from being considered.

• This seems like a question of pure mathematics to me. (Is Sprague-Grundy theory helpful?)
– D.W.
Jan 23, 2020 at 21:57
• @D.W. unfortunately not, since in Dots and Boxes the player who makes the last legal move doesn't necessarily win - instead the player with the most captured boxes wins. Jan 23, 2020 at 22:07


text:
Article Text

Should we consider non-compliance a medical error?
Free
1. N Barber
1. Correspondence to:
 Professor N Barber, Centre for Practice and Policy, The School of Pharmacy, 29–39 Brunswick Square, London WC1N 1AX, UK;
 nick.barber{at}ulsop.ac.uk

## Abstract

Non-compliance is an extensive intractable problem. This paper argues that we can gain significant insight into non-compliance if we apply theories developed to explain human error in organisations. The resultant framework encompasses intentional and unintentional non-compliance, shifts blame from the patient, and recognises the influence of other factors, including organisational ones. There are also consequences for the measurement of compliance and new strategies to improve it. Terminology will need to be addressed, particularly whether intentional non-compliance by a patient should be considered an error. If empirical research supports the arguments in this paper then, with some further theory development, the application of human error theory will offer a useful new approach to understanding and reducing undesired non-compliance.

• non-compliance
• medical error
• human error theory
View Full Text

## Statistics from Altmetric.com

Around one third to one half of patients do not take their medicines as directed, yet this is not usually considered to be a medical error. Should this be the case? This article explores whether it should, and argues that there are benefits in applying the literature on errors to non-compliance.

Non-compliance is a substantial and, as yet, intractable issue; any new insights into it are to be welcomed. Estimates have remained constant over the years, that 30–50% of patients on chronic medication do not take their medicines as directed.1, 2 We do not know the full consequences of non-compliance on the population as a whole. The economic studies have been limited, but all of them suggest substantial consequences. In 1985 Smith estimated cardiovascular non-compliance alone resulted in 125 000 deaths in the USA annually and a further $1.5bn in lost earnings from hospitalisations.3 Another study suggested non-compliance to just 10 drugs cost the USA between$396 and $792 million each year,4 and the overall cost to the USA of all non-compliance is estimated at$100bn annually.5

The mutual exclusion of non-compliance and errors is fairly comprehensive. I failed to find any mention of non-compliance in a quick skim of my books on error, so I searched Medline over the last quarter century. Using the terms “patient compliance OR medication errors” there were 24 702 references; searching for “patient compliance AND medication errors” and restricting the search to “English” and “journal article” reduced the number to just 50. Most of these were studies in which the authors simply regarded non-compliance as an error but did not address it with reference to any literature on errors. An exception was the American Society of Hospital Pharmacists which did include non-compliance in its guidelines on preventing medication errors in hospitals.6 Considering the development of the literature on the causes of errors, the growing application of human error theory in medicine, and national initiatives to reduce errors in the UK and USA, it seems a good time to reassess whether the overlap should be so small.

## NON-COMPLIANCE

To understand the reasons for the separation of these two literatures, we need first to understand the definitions and causes of non-compliance. Compliance is usually defined as the extent to which patients follow medical advice.7 While often used with respect to medicine taking (as I will use it in this article), it need not be so, and could apply to the following of any advice on health. This commonly used definition is based on patient behaviour (an outcome based definition also exists8), and is rather like saying someone has a “stomach ache”—while it expresses a condition, it is not clear how serious it is and there may be many causes, each of which may require different solutions. Non-compliance to medication is often set at some level—for example, that less than 80% of doses are taken correctly. While this definition is precise, if somewhat arbitrary, it is difficult to assess as it is hard to find the true nature of patients' behaviour.

The term “compliance” is itself controversial, and this controversy is informative. In the 1970s it was felt that doctors told patients the right thing to do, and that patients should comply with these directions. However, Stimson,9 in a seminal paper, argued that it was inappropriate to see patients as “defaulters” and he put forward the patient's side of the argument. More recently the term “adherence” has been suggested as being more suitable than compliance, as it is seen to lack the implied one sided value implications of “compliance”. Adherence became more popular, particularly with researchers, but it was never that familiar to practitioners and so has been limited in its use. More recently, with the growth of interest in patients participating in decision making, the objections to “compliance” are weakened (as it is compliance with, ideally, an agreed regime), and the term seems to be becoming popular again. “Concordance” is another term which is sometimes used, inappropriately, as a synonym for compliance. In a document entitled “From compliance to concordance10 a specific form of doctor-patient interaction about medicines was suggested, and it was inferred that this would improve compliance. Perhaps because of the title of the article, some people incorrectly assume concordance is a modern word for compliance. These definitional arguments are not just academic semantics, but have at their core the role of the patient's free will, and whether the exercising of free will can be considered an error. This issue will be discussed later in this paper.

About 200 factors have been attributed to cause non-compliance.11 However, the causes—and hence the patients—are usually split into two groups: intentional and unintentional non-compliers. Those who intentionally do not comply have made the decision consciously; the others wish to comply but cannot do so for some reason. In reality there is some overlap between these groups. For example, some patients who forget to take their medicine (which would be classed as unintentional non-compliance) will do so because their view of the medicine, its importance and risks, may have made taking the medicine a low priority, and hence more easily driven out of mind by other matters.

What do we know of patients' intentions? The trend in recent times has been for psychologists to focus on the causes of intentional, rather than unintentional, non-compliance. They have applied several theoretical models to explain non-compliance12—for example, social cognition models, stage models, and Leventhal's self-regulatory theory. These do have several limitations and are not particularly good at predicting actual behaviour. When models allow regression modelling, in which the extent of compliance is the outcome variable, they generally account for a relatively small proportion of the observed variability. One reason for their limited success may be that they apply a theory to all non-compliers, whether intentional or unintentional. We would expect these models to predict intentional non-compliance much more than unintentional. Unless a substantial proportion were intentionally non-compliant, we would expect the noise from the others to weaken the explanatory power of the model. Although the division of non-compliance into intentional and unintentional is of long standing, I have been unable to find any papers that report the proportion of non-compliance in each group, so we still do not know how well we would expect the models to work. This is obviously of importance if we wish to reduce non-compliance. Different causes will require different solutions. While these theoretical approaches have helped our understanding of compliance, as Horne and Weinman note,12 the capacities of the models to generate effective interventions have yet to be fully evaluated.

The intractability of non-compliance is another clue that a new approach may be welcome. A Cochrane review13 found that there had been few well conducted studies, and that many interventions failed. Only one study of short term treatment was effective and, of the 17 randomised controlled trials of long term treatment that were of sufficient quality, only nine led to improvements in treatment outcome. Nearly all the successful studies used complex mixtures of interventions.

The current explanatory models of non-compliance are limited and there is a lack of effective solutions; there is room for a new approach. Having conducted research into non-compliance and, separately, into human factors in medical error, I find the stories of non-compliant patients remind me of the stories of those who have made errors. So far I have been unable to find any application of human error theory to non-compliance. In the next sections I describe current thinking on the causes of errors, illustrate how these explanatory models and techniques could explain non-compliance, suggest how the approaches could inform research, and outline some difficulties with the approach.

## MEDICAL ERROR THEORY

This analysis is based on Reason's descriptions of the causes of human error14, 15 which are adapted to non-compliance in fig 1. In Reason's terms, unsafe human acts follow intended or unintended actions (equivalent to intentional or unintentional non-compliance). Unintentional actions are made up of slips and lapses. These occur when people intend to do an action but do not do it properly; they occur largely in the automatic carrying out of routine tasks. Slips are a result of a lack of attention—for example, inadvertently taking the wrong tablet. Lapses result from failure of memory such as forgetting to take a dose.

Figure 1

Non-compliance illustrated using Reason's framework.

Mistakes are intended actions that are wrong, but are carried out with the intention of doing the right thing, so the perpetrator is unaware that what they are doing is wrong; these would be classified as intentional non-compliance. They occur once a problem has been recognised in the current plan. There are two types of mistake—rule based and knowledge based. Rule based mistakes, by far the more common, occur when the person is operating to a set of rules, but the rule they use is wrong; knowledge based mistakes are rarer, and occur in novel situations when the person is faced with a problem and has to work out what to do from their existing knowledge, rather than knowing a rule that fits the situation. Rule based mistakes include misapplication of a good rule and application of a bad rule. An example from non-compliance would be that some patients on chronic medication become concerned about addiction16 and therefore deliberately stop taking a drug for a period each year (sometimes called a “drug holiday”) even though the drug is not addictive. Application of a bad rule can occur in the case of inhaled steroids; I have met patients who, having heard about the effects of steroid abuse in sport, fear all steroids so choose not to take their medicine as frequently as directed. Knowledge based failures would include a patient who runs out of an important medicine (for some reason that is not their fault) and is faced with a decision of what to do; they may err by choosing to wait several days before seeing their doctor rather than getting an emergency supply from a pharmacy.

The final category of intentional acts/non-compliance is that of violations; these are deliberate deviations from safe practice. The boundary between violations and mistakes is rather blurred, but a guide is obtained by considering whether the error occurs solely from “internal processing”, in which the person is trying to do the right thing but does not (a mistake), or whether the issue is more motivational and caused by their operating in a wider social context (a violation). A deliberate violation would be choosing not to get a prescription dispensed because the patient felt the doctor hadn't listened and so could not have understood the problem. Other violations include taking “short cuts” that bend the rules to make life easier, such as taking several medicines together which should be taken at different times, or taking a non-steroidal anti-inflammatory drug on an empty stomach. Violations need not be bad and, indeed, can avoid harm. Deliberate non-compliance can be the right thing for a patient to do, such as not taking a diuretic before a long bus journey.

Following the categorisation of errors and violations, Reason and others have gone on to consider the aetiology of accidents in an organisation. Organisational processes and management decisions create an environment that may encourage errors; these failures are known as latent errors as they create the background against which errors are more likely to occur. Partly as a result of the latent failures, there are also local conditions at the time of the accident which may increase the likelihood of a worker making an error. While much of this theory has been derived from studies of accidents in areas such as nuclear power, aviation and shipping, it also applies to health care. The patient, even though at home, has engaged with and is part of—indeed, the end point of—a large organisation with the goal of improving health. Human error theory is not the only body of learning to address these areas. Theories in psychology include attitude and “norms”; sociology includes organisational and social issues; however, error theories seem best to pull them all together.

The organisational model presents the people who make the error as “the inheritors rather than the instigators” of an accident.15 Consequently, in health care there is a widening from studying the person who made the error to also seeking organisational and local factors that caused them to make the error. There is the need for a similar shift in non-compliance—we should shift the focus from the patient to their local environment and the organisation of care supporting them. There are differences, of course. Contributing local factors in a hospital, such as the shift system, being short staffed that day, poor lighting etc, may have equivalents in the patient's home, work or life in general, but may well not be the responsibility of the healthcare system. However, this approach can direct our attention to the causes of non-compliance and to suggestions of ways in which the patient or others could resolve them.

Examples of latent failures could be the doctor's style of communication with the patient17 or the remuneration system for pharmacists, which leads to many of them dispensing large numbers of prescriptions at the cost of helping smaller numbers of patients become compliant (although there are suggestions which would change this in the current NHS plans for pharmacy18). Non-compliance resulting from the system of information transfer on discharge of patients from secondary to primary care is common.19 Local factors could be related to where the patient kept their medicines or how they decided to ask for repeat prescriptions (for example, the length of time before their medicines ran out that they asked for them), or the use of a carer to ensure the medicines were given.

## IMPLICATIONS FOR THEORY AND PRACTICE

One advantage of the recognition that organisational factors can produce errors has been a shift away from blaming the individual who made the error. This is not only fairer (in the majority of cases), but also has the pragmatic benefit that it encourages people to be open about errors and, if linked to a monitoring mechanism, can lead to safer systems. A similar benefit could accrue if we adopted the same mechanism for non-compliant patients. I suspect little has changed since Stimson observed that non-compliant patients were seen as “deviants” by doctors.9 We need a mechanism to remove this stigma from patients and to encourage them to speak openly about how they use their medicines. We need to capture the information and learn from it.

This is reinforced by another area in which non-compliance research can benefit from the study of errors—that of deciding what we should measure and be concerned about. In studies of non-compliance the term is usually defined as a patient taking less than 80% of their doses as intended. This proportion presumably derives from the knowledge that most drugs follow a log dose response curve, so the reduction in clinical effect would only be likely to be evident when the total dose fell below about 80%. This suggests the term “non-compliance” has been used to conflate two terms, one reflecting incidence and the other severity. As with all such measures, it provides a simple dichotomy but it raises the issue of whether this dichotomy is helpful. I would suggest that it is limited as it imposes a threshold before studying errors, and its link to outcome is tenuous. In contrast, it has been recognised in organisations that, while major errors are rare and therefore hard to study, they are often unusual concatenations of commoner minor errors. There is a direct relationship (often drawn as a pyramid) that, for every case of an error causing major harm (shown at the tip of the pyramid), there are proportionately larger and larger numbers of incidents producing less and less harm (shown in layers beneath the tip). One study showed that, for every case of major harm, there were seven injuries and 189 non-injury accidents.20 The number of harmful incidents can therefore be reduced be reducing the number of minor errors.

What can this teach us about non-compliance? Firstly, that all cases of non-compliance are of interest, whether one dose or many, so the use of the 80% cut off is unhelpful. Minor errors are easier to study and hence to understand their causes and devise remedial solutions; by reducing their incidence it is possible to reduce the incidence of major errors. This approach would seem likely to bear fruit in studying non-compliance. If all cases of missed and inappropriate doses are studied, we will be likely to identify common causes including intentional, unintentional, and latent organisational ones. No theoretical explanation of non-compliance has been able to do this so far. We can then devise better solutions that, by reducing the incidence of minor “errors”, will also reduce the harm caused by non-compliance.

Removal of the 80% threshold has another advantage. It forces us to make explicit the relationship between non-compliance and harm. Instead of just counting the incidence, we should also be focusing on the effect it has on patient health. Why do some cases lead to hospitalisation or death and others do not? More empirical work is needed in this area.

One final implication of the influence of the organisational model on non-compliance is that it affects the design and interpretation of research. A study of several hundred patients of one doctor is less likely to be useful than a study of fewer patients from a greater number of doctors' practices.

While I have been emphasising similarities between the application of human error theory to errors in complex organisations and a patient taking their medicines, there are some significant differences. An organisation is trying to produce some end point in a safe manner, which it does by controlling the systems and activities of its staff. If someone delivering health care commits an error, it is usually somebody else who suffers. A patient is different; they are both the actor in a process and the end point of that process, and if they make an error, it is usually they who suffer most. What is more, the end point of that process is defined by the patient, or at least influenced by them; it is the patient who first seeks treatment, initiating the process which leads to them being directed to take a medicine. If they choose to violate the dose directions, then in the majority of cases it is their right—they chose to start the process, so they can exercise their free will and withdraw from it. The concept of error works best in a technical rational system in which “right” is clearly defined on technical rather than moral grounds. Error is a much harder idea to apply to patients taking medication where several conflicting concepts of “good” might be at work. It would not be acceptable to label a patient's actions as an error if they were exercising an informed judgement such as stopping treatment. If we are to apply human error theory to non-compliance, we will need to amend its terminology.

### Key messages

• Non-compliance is an extensive, intractable problem.

• Methods of studying human error in organisations may be applicable to non-compliance.

• The consequences may be new insights into the causes of and solutions to non-compliance.

• This paper is based on an argument; theory development and empirical support are required next.

It is interesting that the dominant explanatory theory of human error comes from psychology, and that psychology is probably the dominant explanatory discipline in non-compliance research. It seems likely that there could be some fruitful merging within the discipline. For example, a patient's condition changes and their response to a medicine emerges, particularly at the start of treatment when all is new. This results in the patient going through a reflexive process of reassessing their beliefs, and may result in them altering their medicine taking behaviour. While this could be categorised as a knowledge based mistake, it hardly seems to do justice to such a complex evolving process. An alternative approach is offered by some psychological models used in studying non-compliance—for example, Leventhal's self-regulatory theory.16 Work is needed to show whether these groups of theories can, or should, be effectively spliced together.

## CONCLUSIONS

Non-compliance is a substantial, expensive, intractable problem in health care which can lead to misery and harm. I have argued that the theories of the causes of human error in organisations could provide a powerful explanatory model of non-compliance. If my argument is supported by empirical research, then it would lead to different approaches to reducing non-compliance in practice, to conducting research into its causes, and to the development of theory. Under this new paradigm, all deviations from directions would be taken as informative causes of non-compliance, and there would be the need to understand the relationship between minor errors and harm. Errors and non-compliance are supported by substantial literatures in the field of psychology—it seems likely that there could be some joint development of theory in this discipline. There is also a need to develop the terminology. The combined issues of the patient being the instigator of the process, the actor in the process, the recipient of the consequences of the process, and an autonomous individual (arguably, given their lack of information) would all lead to the need to reconsider the terms we use and their implicit values.

Human error theory has made significant advances over the last decade. If we can re-configure it to non-compliance and adapt it to remove terminology that may be a barrier to its adoption, then this promises to be a significant advance in our understanding and solution of non-compliance. To answer the question posed in the title of this paper, we should not consider non-compliance as a medical error as the terminology is inappropriate. However, non-compliance has a substantial amount to learn from medical error theory, and a merging of the approaches could significantly benefit patients.

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text:
# Gibbs sampling where I can only find the mode of conditionals?

I'm trying to solve a problem with Gibbs Sampling, so I'm trying to do:

$$x_1^1 \sim p(x_1 | x_2^0, x_3^0)\\ x_2^1 \sim p(x_2 | x_1^1, x_3^0)\\ x_3^1 \sim p(x_3 | x_1^1, x_2^1)\\ x_1^2 \sim p(x_1 | x_2^1, x_3^1)\\ ...$$

But I can't compute the full distribution of each conditional, eg. $$p(x_1 | x_2, x_3)$$. However, I can find the MAP estimate of each conditional. So I'm thinking about doing this:

$$x_1^1 = \arg\max_{x_1} p(x_1 | x_2^0, x_3^0)\\ x_2^1 \sim p(x_2 | x_1^1, x_3^0)\\ x_3^1 \sim p(x_3 | x_1^1, x_2^1)\\ x_1^2 \sim p(x_1 | x_2^1, x_3^1)\\ ...$$

I couldn't find any theoretical results for this (or maybe I'm not searching the right terms). If I "sample" like this, would I end up with a mode for joint $$p(x_1, x_2, x_3)$$?

• There is no theoretical result to validate this method if the target is not unimodal . If it is unimodal, then each maximisation increases the value of the target and the sequence of $x$'s can only converge to the mode. – Xi'an Feb 20 at 18:51