Orphan drug

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Orphan_drug 

An orphan drug is a pharmaceutical agent developed to treat medical conditions which, because they are so rare, would not be profitable to produce without government assistance. The conditions are referred to as orphan diseases.

The assignment of orphan status to a disease and to drugs developed to treat it is a matter of public policy in many countries and has yielded medical breakthroughs that might not otherwise have been achieved, due to the economics of drug research and development.

In the U.S. and the EU, it is easier to gain marketing approval for an orphan drug. There may be other financial incentives, such as an extended period of exclusivity, during which the producer has sole rights to market the drug. All are intended to encourage development of drugs which would otherwise lack sufficient profit motive to attract corporate research budgets and personnel.

Contents

• 1 Definition
• 2 Global statistics
• 3 Effect on investment, sales and profit
• 4 Legislation
  • 4.1 United States
  • 4.2 European Union
  • 4.3 Other countries
  • 4.4 Numbers of new drugs
• 5 Examples for selected diseases
  • 5.1 Cystic fibrosis
  • 5.2 Familial hypercholesterolemia
  • 5.3 Wilson's disease
  • 5.4 Phospholipase 2G6-associated neurodegeneration
  • 5.5 Transthyretin-related hereditary amyloidosis
• 6 Activism, research centers
• 7 Cost
• 8 Public funding
  • 8.1 Evaluation criteria
  • 8.2 Abuse potential
  • 8.3 NICE
• 9 See also
• 10 References
• 11 External links

Definition

According to the US Food and Drug Administration (FDA), an orphan drug is defined as one "intended for the treatment, prevention or diagnosis of a rare disease or condition, which is one that affects less than 200,000 persons in the US" (which equates to approximately 6 cases per 10,000 population) "or meets cost recovery provisions of the act." In the European Union (EU), the European Medicines Agency (EMA) defines a drug as "orphan" if it is intended for the diagnosis, prevention or treatment of a life-threatening or chronically and seriously debilitating condition affecting not more than 5 in 10,000 EU people. EMA also qualifies a drug as orphan if – without incentives – it would be unlikely that marketing the drug in the EU would generate sufficient benefit for the affected people and for the drug manufacturer to justify the investment. As of 2017, there was no official integration of the orphan drug programs between the FDA and EMA.

Global statistics

As of 2014, there were 281 marketed orphan drugs and more than 400 orphan-designated drugs in clinical trials. More than 60% of orphan drugs were biologics. The U.S. dominated development of orphan drugs, with more than 300 in clinical trials, followed by Europe. Cancer treatment was the indication in more than 30% of orphan drug trials.

• Number of orphan drugs in clinical trials: 600
• Number of orphan drugs in phase 2 trial: 231
• Number of orphan drugs in U.S. clinical trials: 350 in the pipeline from research until registration

Effect on investment, sales and profit

According to Thomson Reuters in their 2012 publication "The Economic Power of Orphan Drugs", there has been increased investment in orphan drug research and development, partly due to the U.S. Orphan Drug Act of 1983 (ODA) and similar acts in other regions of the world driven by "high-profile philanthropic funding".

According to Drug Discovery Today, the years 2001 to 2011 were the "most productive period in the history of orphan drug development, in terms of average annual orphan drug designations and orphan drug approvals". For the same decade the compound annual growth rate (CAGR) of the orphan drugs was an "impressive 25.8%, compared to only 20.1% for a matched control group of non-orphan drugs". By 2012, the market for orphan drugs was worth US$637 million, compared with US$638 million for a control group of non-orphan drugs.

By 2012,

the revenue-generating potential of orphan drugs [was] as great as for non-orphan drugs, even though patient populations for rare diseases are significantly smaller. Moreover, we suggest that orphan drugs have greater profitability when considered in the full context of developmental drivers, including government financial incentives, smaller clinical trial sizes, shorter clinical trial times and higher rates of regulatory success.

— Gaze and Breen 2012

According to a 2014 report, the orphan drug market has become increasingly lucrative for a number of reasons. The cost of clinical trials for orphan drugs is substantially lower than for other diseases because trial sizes are naturally much smaller than for more diseases with larger numbers of patients. Small clinical trials and minimal competition place orphan agents at an advantage in regulatory review.

Tax incentives reduce the cost of development. On average the cost per patient for orphan drugs is "six times that of non-orphan drugs, a clear indication of their pricing power". The cost of per-person outlays are large and are expected to increase with wider use of public subsidies.

The 2014 Orphan Drug report stated that the percentage of orphan drug sales as part of all prescription drug sales had been increasing at rapid rate. The report projected a total of US$176 billion by 2020. Although orphan disease populations are the smallest, the cost of per-patient outlays among them are the largest and are expected to increase as more people with rare diseases become eligible for subsidies – in the U.S., for example, through the Affordable Care Act.

Legislation

Orphan drugs generally follow the same regulatory development path as any other pharmaceutical product, in which testing focuses on pharmacokinetics and pharmacodynamics, dosing, stability, safety and efficacy. However, some statistical burdens are lessened to maintain development momentum. For example, orphan drug regulations generally acknowledge the fact that it may not be possible to test 1,000 patients in a phase III clinical trial if fewer than that number are afflicted with the disease.

Government intervention on behalf of orphan drug development takes several forms:

• Tax incentives
• Exclusivity (enhanced patent protection and marketing rights)
• Research subsidies
• Creating a government-run enterprise to engage in research and development as in a Crown corporation

A 2015 study of "34 key Canadian stakeholders, including drug regulators, funders, scientists, policy experts, pharmaceutical industry representatives, and patient advocates" investigated factors behind the pharmaceutical industry growing interest in "niche markets" such as orphan drugs.

United States

The Orphan Drug Act (ODA) of January 1983, passed in the United States, with lobbying from the National Organization for Rare Disorders and many other organizations, is meant to encourage pharmaceutical companies to develop drugs for diseases that have a small market. Under the ODA drugs, vaccines, and diagnostic agents would qualify for orphan status if they were intended to treat a disease affecting fewer than 200,000 American citizens. Under the ODA orphan drug sponsors qualify for seven-year FDA-administered market Orphan Drug Exclusivity (ODE), "tax credits of up to 50% of R&D costs, R&D grants, waived FDA fees, protocol assistance and may get clinical trial tax incentives.

In the U.S., orphan drug designation means that the sponsor qualifies for certain benefits, but it does not mean the drug is safe, effective or legal.

In 2002, the Rare Diseases Act was signed into law. It amended the Public Health Service Act to establish the Office of Rare Diseases. It also increased funding for the development of treatments for people with rare diseases.

European Union

In 2000, the European Union (EU) enacted similar legislation, Regulation(EC) No 141/2000, which refers to drugs developed to treat rare diseases to as "orphan medicinal products". The EU's definition of an orphan condition is broader than that of the US, in that it also covers some tropical diseases that are primarily found in developing nations. Orphan drug status granted by the European Commission gives marketing exclusivity in the EU for 10 years after approval. The EU's legislation is administered by the Committee on Orphan Medicinal Products of the European Medicines Agency (EMA).

In late 2007 the FDA and EMA agreed to use a common application process for both agencies to make it easier for manufacturers to apply for orphan drug status but, while continuing two separate approval processes.

Other countries

Legislation has been implemented by Japan, Singapore, and Australia that offers subsidies and other incentives to encourage the development of drugs that treat orphan diseases.

Numbers of new drugs

Under the ODA and EU legislation, many orphan drugs have been developed, including drugs to treat glioma, multiple myeloma, cystic fibrosis, phenylketonuria, snake venom poisoning, and idiopathic thrombocytopenic purpura.

The Pharmaceutical Executive opines, that the "ODA is nearly universally acknowledged to be a success".

Before the US Congress enacted the ODA in 1983, only 38 drugs were approved in the US specifically to treat orphan diseases. In the US, from January 1983 to June 2004, 249 orphan drugs received marketing authorization and 1,129 received different orphan drug designations, compared to fewer than ten such products in the decade prior to 1983. From 1983 until May 2010, the FDA approved 353 orphan drugs and granted orphan designations to 2,116 compounds. As of 2010, 200 of the roughly 7,000 officially designated orphan diseases have become treatable.

Critics have questioned whether orphan drug legislation was the real cause of this increase, claiming that many of the new drugs were for disorders which were already being researched anyway, and would have had drugs developed regardless of the legislation, and whether the ODA has truly stimulated the production of non-profitable drugs; the act also has been criticised for allowing some pharmaceutical companies to make a large profit off drugs which have a small market but sell for a high price.

While the European Medicines Agency grants orphan drugs market access in all member states, in practice, they only reach the market when a member state decides that its national health system will reimburse for the drug. For example, in 2008, 44 orphan drugs reached the market in the Netherlands, 35 in Belgium, and 28 in Sweden, while in 2007, 35 such drugs reached the market in France and 23 in Italy.

Though not technically an orphan disease, research and development into the treatment for AIDS has been heavily linked to the Orphan Drug Act. In the beginning of the AIDS epidemic the lack of treatment for the disease was often accredited to a believed lack of commercial base for a medication linked to HIV infection. This encouraged the FDA to use the Orphan Drug Act to help bolster research in this field, and by 1995 13 of the 19 drugs approved by the FDA to treat AIDS had received orphan drug designation, with 10 receiving marketing rights. These are in addition to the 70 designated orphan drugs designed to treat other HIV related illnesses.

Examples for selected diseases

Cystic fibrosis

In the 1980s, people with cystic fibrosis rarely lived beyond their early teens. Drugs like Pulmozyme and tobramycin, both developed with aid from the ODA, revolutionized treatment for cystic fibrosis patients by significantly improving their quality of life and extending their life expectancies. Now, cystic fibrosis patients often survive into their thirties and some into their fifties.

Familial hypercholesterolemia

The 1985 Nobel Prize for medicine went to two researchers for their work related to familial hypercholesterolemia, which causes large and rapid increases in cholesterol levels. Their research led to the development of statin drugs which are now commonly used to treat high cholesterol.

Wilson's disease

Penicillamine was developed to treat Wilson's disease, a rare hereditary disease that can lead to a fatal accumulation of copper in the body. This drug was later found to be effective in treating arthritis. Bis-choline tetrathiomolybdate is currently under investigation as a therapy against Wilson's disease.

Phospholipase 2G6-associated neurodegeneration

In 2017, FDA granted RT001 orphan drug designation in the treatment of phospholipase 2G6-associated neurodegeneration (PLAN).

Transthyretin-related hereditary amyloidosis

The FDA granted Patisiran (Onpattro) orphan drug status and breakthrough therapy designation due to its novel mechanism involving RNA therapy to block the production of an abnormal form of transthyretin. Patisiran received full FDA approval in 2018 and its RNA lipid nanoparticle drug delivery system was later used in the Pfizer–BioNTech COVID-19 vaccine and Moderna RNA vaccines.

Activism, research centers

The Center for Orphan Drug Research at the University of Minnesota College of Pharmacy helps small companies with insufficient in-house expertise and resources in drug synthesis, formulation, pharmacometrics, and bio-analysis. The Keck Graduate Institute Center for Rare Disease Therapies (CRDT) in Claremont, California, supports projects to revive potential orphan drugs whose development has stalled by identifying barriers to commercialization, such as problems with formulation and bio-processing.

Numerous advocacy groups such as the National Organization for Rare Disorders, Global Genes Project, Children's Rare Disease Network, Abetalipoproteinemia Collaboration Foundation, Zellweger Baby Support Network, and the Friedreich's Ataxia Research Alliance have been founded in order to advocate on behalf of patients suffering from rare diseases with a particular emphasis on diseases that afflict children.

Cost

According to a 2015 report published by EvaluatePharma, the economics of orphan drugs mirrors the economics of the pharmaceutical market as a whole but has a few very large differences. The market for orphan drugs is by definition very small, but while the customer base is drastically smaller the cost of research and development is very much the same as for non orphan drugs. This, the producers have claimed, causes them to charge extremely high amounts for treatment, sometimes as high as $700,000 a year, as in the case of Spinraza (Biogen), FDA approved in December 2016 for spinal muscular atrophy, placing a large amount of stress on insurance companies and patients. An analysis of 12 orphan drugs that were approved in the US between 1990 and 2000 estimated a price reduction of on average 50% upon loss of marketing exclusivity, with a range of price reductions from 14% to 95%.

Governments have implemented steps to reduce high research and development cost with subsidies and other forms of financial assistance. The largest assistance are tax breaks which can be as high as 50% of research and development costs. Orphan drug manufacturers are also able to take advantage of the small customer base to cut cost on clinical trials due to the small number of cases to have smaller trials which reduces cost. These smaller clinical trials also allow orphan drugs to move to market faster as the average time to receive FDA approval for an orphan drug is 10 months compared to 13 months for non-orphan drugs. This is especially true in the market for cancer drugs, as a 2011 study found that between 2004 and 2010 orphan drug trials were more likely to be smaller and less randomized than their non-orphan counterparts, but still had a higher FDA approval rate, with 15 orphan cancer drugs being approved, while only 12 non-orphan drugs were approved. This allows manufactures to get cost to the point that it is economically feasible to produce these treatments. The subsidies can total up to $30 million per fiscal year in the United States alone.

By 2015, industry analysts and academic researchers agreed, that the sky-high price of orphan drugs, such as eculizumab, was not related to research, development and manufacturing costs. Their price is arbitrary and they have become more profitable than traditional medicines.

Public resources went into understanding the molecular basis of the disease, public resources went into the technology to make antibodies and finally, Alexion, to their credit, kind of picked up the pieces.

— Sachdev Sidhu 2015

Public funding

Evaluation criteria

By 2007 the use of economic evaluation methods regarding public-funding of orphan drugs, using estimates of the incremental cost-effectiveness, for example, became more established internationally. The QALY has often been used in cost-utility analysis to calculate the ratio of cost to QALYs saved for a particular health care intervention. By 2008 the National Institute for Health and Care Excellence (NICE) in England and Wales, for example, operated with a threshold range of £20,000–30,000 per quality-adjusted life year (QALY). By 2005 doubts were raised about the use of economic evaluations in orphan drugs. By 2008 most of the orphan drugs appraised had cost-effectiveness thresholds "well in excess of the 'accepted' level and would not be reimbursed according to conventional criteria". As early as 2005 McCabe et al. argued that rarity should not have a premium and orphan drugs should be treated like other pharmaceuticals in general. Drummond et al. argued that the social value of health technologies should also be included in the assessment along with the estimation of the incremental cost-effectiveness ratio.

Abuse potential

Rosuvastatin (brand name Crestor) is an example of a drug that received Orphan Drug funding but was later marketed to a large consumer base.

The very large incentives given to pharmaceutical companies to produce orphan drugs have led to the impression that the financial support afforded to make these drugs possible is akin to abuse. Because drugs can be used to treat multiple conditions, companies can take drugs that were filed with their government agency as orphan drugs to receive financial assistance, and then market it to a wide population to increase their profit margin. For example AstraZeneca's cholesterol drug Crestor was filed as a treatment for the rare disease pediatric familial hypercholesterolemia. After the drug was approved for orphan drug designation, and AstraZeneca had received tax breaks and other advantages, AstraZeneca later applied and received FDA approval for the drug to be used to treat cholesterol in all diabetics.

NICE

The UK's National Institute for Health and Care Excellence (NICE) can pay from £100,000 to £300,000 per QALY (Quality Adjusted Life Year) for treatments of "very rare conditions". This is compared to under £20,000 for non-orphan drugs.

In 2015, NICE held consultations with "patient groups, the Department of Health, companies, learned societies, charities and researchers" regarding the appraisal of medicines and other technologies. There was a call for more research into new processes, including: 

the model of pharmaceutical research and development, the expectations that companies and patient groups have about how risk and reward is shared between the industry and a publicly funded NHS, and in the arrangements for commissioning expensive new treatments.

— NICE 2014

 

Expanded access

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Expanded_access 

Expanded access or compassionate use is the use of an unapproved drug or medical device under special forms of investigational new drug applications (IND) or IDE application for devices, outside of a clinical trial, by people with serious or life-threatening conditions who do not meet the enrollment criteria for the clinical trial in progress.

These programs go under various names, including early access, special access, or managed access program, compassionate use, compassionate access, named-patient access, temporary authorization for use, cohort access, and pre-approval access.

In general the person and their doctor must apply for access to the investigational product, the company has to choose to cooperate, and the medicines regulatory agency needs to agree that the risks and possible benefits of the drug or device are understood well enough to determine if putting the person at risk has sufficient potential benefit. In some countries the government will pay for the drug or device, but in many countries the person must pay for the drug or device, as well as medical services necessary to receive it.

In the US, compassionate use started with the provision of investigational medicine to certain patients in the late 1970s, and a formal program was established in 1987 in response to HIV/AIDS patients requesting access to drugs in development. An important legal case was Abigail Alliance v. von Eschenbach, in which the Abigail Alliance, a group that advocates for access to investigational drugs for people who are terminally ill, tried to establish such access as a legal right. The Supreme Court declined to hear the case, effectively upholding previous cases that have maintained that there is not a constitutional right to unapproved medical products.

Programs

As of 2016, regulation of access to pharmaceuticals that were not approved for marketing was handled on a country by country basis, including in the European Union, where the European Medicines Agency issued guidelines for national regulatory agencies to follow. In the US, Europe, and the EU, no company could be compelled to provide a drug or device that it was developing.

Companies sometimes provide drugs under these programs to people who were in clinical trials and who responded to the drug, after the clinical trial ends.

United States

In the US as of 2018, people could try obtain unapproved drugs or medical devices that were in development under specific conditions.

These conditions were:

• The person wanting the drug or device and a licensed physician are both willing to participate.
• The person's physician determines that there is no comparable or satisfactory therapy available to diagnose, monitor, or treat the patient's disease or condition.
• That the probable risk to the person from the investigational product is not greater than the probable risk from the disease or condition.
• The FDA determines that there is sufficient evidence of the safety and effectiveness of the investigational product to support its use in the particular circumstance;
• The FDA determines that providing the investigational product will not interfere with the initiation, conduct, or completion of clinical investigations to support marketing approval;
• The sponsor (generally the company developing the investigational product for commercial use) or the clinical investigator (or the patient's physician in the case of a single patient expanded access request) submits a clinical protocol (a document that describes the treatment plan for the patient) that is consistent with FDA's statute and applicable regulations for INDs or investigational device exemption applications (IDEs), describing the use of the investigational product; and
• The person is unable to obtain the investigational drug or device under another IND application (for drugs), IDE application (for devices), or to participate in a clinical trial.

Drugs can be made available to individuals, small groups, or large groups.

In the US, actual provision of the drug depends on the manufacturer's willingness to provide it, as well as the person's ability to pay for it; it is the company's decision whether to require payment or to provide the drug or device for free. The manufacturer can only charge direct costs for individual INDs; it can add some but not all indirect costs for small group or larger expanded access programs. To the extent that a doctor or clinic is required for use of the drug or device, they too may require payment.

In some cases, it may be in the manufacturer's commercial interest to provide access under an EA program; this is a way, for example, for a company to make money before the drug or device is approved. Companies must provide data collected from people getting the drug or device under EA programs to the FDA annually; this data may be helpful with regard to getting the drug or device approved, or may be harmful, should unexpected adverse events occur. The manufacturer remains legally liable as well. If the manufacturer chooses to charge for the investigational product, that price influences later discussions about the price if the product is approved for marketing.

State law

As of February 2019, 41 states have passed right-to-try laws that permit manufacturers to provide experimental medicines to terminally ill people without US FDA authorization. Legal, medical, and bioethics scholars, including Jonathan Darrow and Arthur Caplan, have argued that these state laws have little practical significance because people can already obtain pre-approval access through the FDA's expanded access program, and because the FDA is generally not the limiting factor in obtaining pre-approval access.

Europe

In Europe, the European Medicines Agency issued guidelines that members may follow. Each country has its own regulations, and they vary. In the UK, for example, the program is called "early access to medicine scheme" or EAMS and was established in 2014. If a company that wants to provide a drug under EAMS, it must submit its Phase I data to the Medicines and Healthcare products Regulatory Agency and apply for what is called a "promising innovative medicine" (PIM) designation. If that designation is approved, the data is reviewed, if that review is positive, the National Health Service is obligated to pay for people who fit the criteria to have access to the drug. As of 2016, governments also paid for early access to drugs in Austria, Germany, Greece, and Spain.

Companies sometimes make use of expanded programs in Europe even after they receive EMA approval to market a drug, because drugs also must go through regulatory processes in each member state, and in some countries this process can take nearly a year; companies can start making sales earlier under these programs.

History

Medicinal cannabis farmed by the University of Mississippi for the government

In the US, one of the earliest expanded access programs was a compassionate use IND that was established in 1978, which allowed a limited number of people to use medical cannabis grown at the University of Mississippi. It is administered by the National Institute on Drug Abuse.

The program was started after Robert C. Randall brought a lawsuit (Randall v. U.S) against the FDA, the Drug Enforcement Administration, the National Institute on Drug Abuse, the Department of Justice, and the Department of Health, Education & Welfare. Randall, who had glaucoma, had successfully used the Common Law doctrine of necessity to argue against criminal charges of marijuana cultivation that had been brought against him, because his use of cannabis was deemed a medical necessity (U.S. v. Randall). On November 24, 1976, federal Judge James Washington ruled in his favor.

The settlement in Randall v. U.S. became the legal basis for the FDA's compassionate IND program. People were only allowed to use cannabis under the program who had certain conditions, like glaucoma, known to be alleviated with cannabis. The scope was later expanded to include people with AIDS in the mid-1980s. At its peak, fifteen people received the drug. 43 people were approved for the program, but 28 of the people whose doctors completed the necessary paperwork never received any cannabis. The program stopped accepting new people in 1992 after public health authorities concluded there was no scientific value to it, and due to President George H.W. Bush administration's policies. As of 2011, four people continued to receive cannabis from the government under the program.

The closure of the program during the height of the AIDS epidemic led to the formation of the medical cannabis movement in the United States, a movement which initially sought to provide cannabis for treating anorexia and wasting syndrome in people with AIDS.

In November 2001 the Abigail Alliance for Better Access to Developmental Drugs was established by Frank Burroughs in memory of his daughter, Abigail. The Alliance seeks broader availability of investigational drugs on behalf of people with terminal illnesses. It is best known for a legal case, which it lost, Abigail Alliance v. von Eschenbach, in which it was represented by the Washington Legal Foundation. On August 7, 2007, in an 8–2 ruling, the U.S. Court of Appeals for the District of Columbia Circuit reversed an earlier ruling in favor of the Alliance. In 2008, the Supreme Court of the United States declined to hear their appeal. This decision left standing the appellate court decision that people who are terminal ill patients have no legal right to demand "a potentially toxic drug with no proven therapeutic benefit".

In March 2014, Josh Hardy, a 7-year-old boy from Virginia, made national headlines that sparked a conversation on pediatric access to investigational drugs when his family's request for brincidofovir was declined by the drug manufacturer, Chimerix. The company reversed its decision after pressure from cancer advocacy organizations, and Josh received the drug that saved his life. In 2016 Kids v Cancer, a pediatric cancer advocacy organization, launched the Compassionate Use Navigator to assist physicians and guide families about the application process. Since then, FDA simplified the application process, but stressed that it cannot require a manufacturer to provide a product. FDA receives about 1,500 expanded access requests per year and authorizes 99% of it.

Odds algorithm

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Odds_algorithm#See_also 

The odds-algorithm is a mathematical method for computing optimal strategies for a class of problems that belong to the domain of optimal stopping problems. Their solution follows from the odds-strategy, and the importance of the odds-strategy lies in its optimality, as explained below.

The odds-algorithm applies to a class of problems called last-success-problems. Formally, the objective in these problems is to maximize the probability of identifying in a sequence of sequentially observed independent events the last event satisfying a specific criterion (a "specific event"). This identification must be done at the time of observation. No revisiting of preceding observations is permitted. Usually, a specific event is defined by the decision maker as an event that is of true interest in the view of "stopping" to take a well-defined action. Such problems are encountered in several situations.

Examples

Two different situations exemplify the interest in maximizing the probability to stop on a last specific event.

1. Suppose a car is advertised for sale to the highest bidder (best "offer"). Let n potential buyers respond and ask to see the car. Each insists upon an immediate decision from the seller to accept the bid, or not. Define a bid as interesting, and coded 1 if it is better than all preceding bids, and coded 0 otherwise. The bids will form a random sequence of 0s and 1s. Only 1s interest the seller, who may fear that each successive 1 might be the last. It follows from the definition that the very last 1 is the highest bid. Maximizing the probability of selling on the last 1 therefore means maximizing the probability of selling best.
2. A physician, using a special treatment, may use the code 1 for a successful treatment, 0 otherwise. The physician treats a sequence of n patients the same way, and wants to minimize any suffering, and to treat every responsive patient in the sequence. Stopping on the last 1 in such a random sequence of 0s and 1s would achieve this objective. Since the physician is no prophet, the objective is to maximize the probability of stopping on the last 1. (See Compassionate use.)

Definitions

Consider a sequence of ${\displaystyle n}$ independent events. Associate with this sequence another sequence ${\displaystyle I_{1},\,I_{2},\,\dots ,\,I_{n}}$ with values 1 or 0. Here ${\displaystyle \,I_{k}=1}$, called a success, stands for the event that the kth observation is interesting (as defined by the decision maker), and ${\displaystyle \,I_{k}=0}$ for non-interesting. We observe independent random variables ${\displaystyle I_{1},\,I_{2},\,\dots ,\,I_{n}}$ sequentially and want to select the last success.

Let ${\displaystyle \,p_{k}=P(\,I_{k}\,=1)}$ be the probability that the kth event is interesting. Further let ${\displaystyle \,q_{k}=\,1-p_{k}}$ and ${\displaystyle \,r_{k}=p_{k}/q_{k}}$.Note that ${\displaystyle \,r_{k}}$ represents the odds of the kth event turning out to be interesting, explaining the name of the odds-algorithm.

Algorithmic procedure

The odds-algorithm sums up the odds in reverse order

${\displaystyle r_{n}+r_{n-1}+r_{n-2}\,+\cdots ,\,}$

until this sum reaches or exceeds the value 1 for the first time. If this happens at index s, it saves s and the corresponding sum

${\displaystyle R_{s}=\,r_{n}+r_{n-1}+r_{n-2}+\cdots +r_{s}.\,}$

If the sum of the odds does not reach 1, it sets s = 1. At the same time it computes

${\displaystyle Q_{s}=q_{n}q_{n-1}\cdots q_{s}.\,}$

The output is

1. ${\displaystyle \,s}$, the stopping threshold
2. ${\displaystyle \,w=Q_{s}R_{s}}$, the win probability.

Odds-strategy

The odds-strategy is the rule to observe the events one after the other and to stop on the first interesting event from index s onwards (if any), where s is the stopping threshold of output a.

The importance of the odds-strategy, and hence of the odds-algorithm, lies in the following odds-theorem.

Odds-theorem

The odds-theorem states that

1. The odds-strategy is optimal, that is, it maximizes the probability of stopping on the last 1.
2. The win probability of the odds-strategy equals ${\displaystyle \,w=Q_{s}R_{s}}$
3. If ${\displaystyle \,R_{s}\geq \,1}$, the win probability ${\displaystyle \,w}$ is always at least ${\displaystyle \,1/e=0.368\dots }$, and this lower bound is best possible.

Features

The odds-algorithm computes the optimal strategy and the optimal win probability at the same time. Also, the number of operations of the odds-algorithm is (sub)linear in n. Hence no quicker algorithm can possibly exist for all sequences, so that the odds-algorithm is, at the same time, optimal as an algorithm.

Sources

Bruss 2000 devised the odd-algorithm, and coined its name. It is also known as Bruss-algorithm (strategy). Free implementations can be found on the web.

Applications

Applications reach from medical questions in clinical trials over sales problems, secretary problems, portfolio selection, (one-way) search strategies, trajectory problems and the parking problem to problems in on-line maintenance and others.

There exists, in the same spirit, an Odds-Theorem for continuous-time arrival processes with independent increments such as the Poisson process (Bruss 2000). In some cases, the odds are not necessarily known in advance (as in Example 2 above) so that the application of the odds-algorithm is not directly possible. In this case each step can use sequential estimates of the odds. This is meaningful, if the number of unknown parameters is not large compared with the number n of observations. The question of optimality is then more complicated, however, and requires additional studies. Generalizations of the odds-algorithm allow for different rewards for failing to stop and wrong stops as well as replacing independence assumptions by weaker ones (Ferguson (2008)).

Variations

Bruss & Paindaveine 2000 discussed a problem of selecting the last ${\displaystyle k}$ successes.

Tamaki 2010 proved a multiplicative odds theorem which deals with a problem of stopping at any of the last ${\displaystyle \ell }$ successes. A tight lower bound of win probability is obtained by Matsui & Ano 2014.

Matsui & Ano 2017 discussed a problem of selecting ${\displaystyle k}$ out of the last ${\displaystyle \ell }$ successes and obtained a tight lower bound of win probability. When ${\displaystyle \ell =k=1,}$ the problem is equivalent to Bruss' odds problem. If ${\displaystyle \ell =k\geq 1,}$ the problem is equivalent to that in Bruss & Paindaveine 2000. A problem discussed by Tamaki 2010 is obtained by setting ${\displaystyle \ell \geq k=1.}$

multiple choice problem: A player is allowed ${\displaystyle r}$ choices, and he wins if any choice is the last success. For classical secretary problem, Gilbert & Mosteller 1966 discussed the cases ${\displaystyle r=2,3,4}$. The odds problem with ${\displaystyle r=2,3}$ is discussed by Ano, Kakinuma & Miyoshi 2010. For further cases of odds problem, see Matsui & Ano 2016.

An optimal strategy belongs to the class of strategies defined by a set of threshold numbers ${\displaystyle (a_{1},a_{2},...,a_{r})}$, where ${\displaystyle a_{1}<a_{2}<\cdots <a_{r}}$. The first choice is to be used on the first candidates starting with ${\displaystyle a_{1}}$th applicant, and once the first choice is used, second choice is to be used on the first candidate starting with ${\displaystyle a_{2}}$th applicant, and so on.

When ${\displaystyle r=2}$, Ano, Kakinuma & Miyoshi 2010 showed that the tight lower bound of win probability is equal to ${\displaystyle e^{-1}+e^{-{\frac {3}{2}}}.}$ For general positive integer ${\displaystyle r}$, Matsui & Ano 2016 discussed the tight lower bound of win probability. When ${\displaystyle r=3,4,5}$, tight lower bounds of win probabilities are equal to ${\displaystyle e^{-1}+e^{-{\frac {3}{2}}}+e^{-{\frac {47}{24}}}}$, ${\displaystyle e^{-1}+e^{-{\frac {3}{2}}}+e^{-{\frac {47}{24}}}+e^{-{\frac {2761}{1152}}}}$ and

 ${\displaystyle e^{-1}+e^{-{\frac {3}{2}}}+e^{-{\frac {47}{24}}}+e^{-{\frac {2761}{1152}}}+e^{-{\frac {4162637}{1474560}}},}$ 

 respectively. For further cases that ${\displaystyle r=6,...,10}$, see Matsui & Ano 2016.