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sponsored links
India, on 2 August, invited Belgium to invest in the upcoming national manufacturing and investments zones (NMIZs) with a focused approach in energy, water disposal as well as treatment, and research and training. This was conveyed to the visiting Belgian Deputy Prime Minister and Minister of Foreign Affairs, Foreign Trade and European Affairs, Didier Reynders, by Commerce and Industry Minister, Anand Sharma, during his meeting in New Delhi. An official statement issued after the meeting said the visiting Belgian Minister offered expertise in water treatment, waste disposal and energy sector which would be useful in NMIZs .NMIZs will be mega industrial zones with world-class supporting infrastructure.The government is offering incentives such as exemption from capital gains tax and a liberalised labour and environment norms to promote these zones.
India and Israel on 2 August, entered into a memorandum of understanding (MoU) for launching a programme that would promote collaborative research across a wide range of disciplines from medical and information technology to social sciences, humanities and arts. The programme, which will run for five years, will provide support up to $1,00,000 a research project for three years. The MoU was signed by Professor Geiger Chairman of the Israel Science Foundation and Chairman of University Grants Commission Ved Prakash. After the signing ceremony, Israeli Ambassador Alon Ushpiz noted that both nations hosted some of the best academic institutions. "The linking of our researchers together comes only naturally," he said. The programme was initiated during a recent visit of Minister for Human Resource Development Kapil Sibal to Israel, when he met with Israel's Finance Minister Yuval Steinitz, Education Minister Gideon Sa'ar and Chairman of Planning and Budgeting Committee Manuel Trajtenberg.
Giving a big boost to trade and bilateral relations with Pakistan, the Government of India on 1 August , notified law permitting investments by citizens and companies, under the government route, in sectors/activities other than defence, space and atomic energy, of the neighbouring country. The move is likely to give new direction to South Asian economic integration. Pakistan was the only country in the negative list under the Foreign Exchange Management Act (FEMA), which prohibits investments in India. Sri Lanka was removed from the list in 2006 and Bangladesh in 2007. The initiative is part of a road map drawn by Trade Ministers and leading chambers of commerce of the two countries to give a fillip to trade and economic engagement. Officials in the Industry Ministry said that to address security concerns over investments from Pakistan, FDI proposals from Islamabad would be routed through the Foreign Investment Promotion Board (FIPB), which is headed by the Economic Affairs Secretary in the Finance Ministry.
The first meeting of the India-CELAC (The Community of Latin American and Caribbean States) Troika foreign ministers was held in New Delhi on 7 August 2012. The Minister of Foreign Affairs of Chile Alfredo Moreno Charme, the pro-tempore President of CELAC, led the CELAC Troika. The Foreign Minister of Venezuela Nicolas Maduro and the Vice Foreign Minister of Cuba, Rogelio Sierra participated in the meeting as members of the Troika. The External Affairs Minister of India, SM Krishna, led the Indian delegation. The meeting was held with the objective of strengthening India-CELAC relationship India and the Community of Latin American and Caribbean countries (CELAC) agreed to work for raising their relationship to the level of Strategic Partnership and having regular Summit Level Meetings in the near future. Pending an agreement on full-fledged Strategic Partnership, both sides agreed to cooperate in diverse sectors like trade and investment, energy, minerals, agriculture, science and technology, culture, education and people-to-people contacts. They agreed to work together on common global challenges in the interest of developing countries by maintaining coordinated approach on UN reforms, financial crisis, climate change and international terrorism. They urged the international community to adopt a Comprehensive Convention on International Terrorism at the earliest. Both sides acknowledged the efforts of the UN Conference on Sustainable Development (Rio+20) for a renewed political commitment towards Sustainable Development to achieve poverty eradication through a balanced integration of economic, social and environmental aspects.
On the sidelines of the First India-CELAC Foreign Ministers Troika Meeting in New Delhi on August 7, 2012, Vice Foreign Minister of Cuba ,Mr. Rogelio Sierra Diaz, met with External Affairs of India, S.M. Krishna. During the meeting, both sides reviewed the entire gamut of bilateral relations and expressed satisfaction. Both sides reiterated their desire to sustain and consolidate these relations. Both sides noted that the total bilateral trade of US$ 40 million between India and Cuba did not correspond to the excellent bilateral ties. They emphasized that the trade turnover should be enhanced through concerted efforts and underlined that new opportunities for cooperation existed in the area of trade and investment. The Cuban side appreciated the importance of the role played by the Indian Company OVL (ONGC Videsh Ltd.) in the field of petroleum exploration in Cuba and sought OVL's continued engagement in Cuba in this area. The Indian side reiterated its offer of Lines of Credit announced during the visit of its Minister of External Affairs to Cuba in June 2012. India and Cuba agreed on the need for the reform of the United Nations Organisation. The Indian side appreciated Cuba's expression of support to India's candidature for permanent membership of the United Nations Security Council. Both sides emphasised the continued relevance of the Non-Aligned Movement in the current context of global relations and agreed to work towards the success of the forthcoming NAM Summit in Tehran. India and Cuba expressed appreciation over the importance and significance of the First India-CELAC Troika Ministerial Dialogue and recalled the considerable opportunities available for mutually beneficial cooperation between India and CELAC both in a multilateral manner and bilaterally.
India and Belgium on 8 August decided to cooperate in the railway sector. A MoU was signed for this purpose in the presence of visiting Deputy Prime Minister Didier Reynders, who also holds the portfolio of Foreign Affairs, Foreign Trade and European Affairs. Speaking on the occasion, Minister of External Affairs SM Krishna said India wanted to collaborate with Belgium in newer areas, particularly in infrastructure and energy sector, including port development, renewable energy, both solar and wind power. He said his discussions with Mr Reyders touched the entire gamut of their bilateral relations and touched on their cooperation in further strengthening the strategic partnership with the European Union. They took note of the fact that despite global economic slowdown, there was substantial increase in their bilateral trade, which has now crossed the 13 billion Euro mark in 2011. Belgium has now emerged as the second largest trading partner in goods sector within the European Union.


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Tags Rashid
Tag: Rashid
Artivists in Action and Solidarity: Rattle the KAGE Dec. 7, 4-7pm
Artivists in Action and Solidarity: Rattle the KAGE Dec. 7, 4-7pm! Busting down walls and opening doors through art, culture and education at the first annual Ratcliff Award celebration benefiting SF Bay View and Prison Focus. Y'all come!
On Pan Afrikanism: Interview with Comrade Rashid by JR Valrey of...
We need to concentrate and blend the various strains of the Afrikan experience and our adaptations to the Diaspora and cross-cultural and economic exchange into a Pan Afrikan culture and consciousness and productive relations that are rooted in proletarian intercommunalism, internationalism and humanism.
Aramark: Multibillion dollar food vendor starves and exploits prisoners
"In Ohio prisons, maggots were also found in Aramark's foods." | Corporate profit and Aramark's starvation diets
The Black Panther Party of 1966 and the New United Panther...
Meet Shaka Zulu and George Katsiaficas, learn more, get involved: Thursday, Oct. 17, 7 p.m., at Freedom Archives, 522 Valencia St., San Francisco.
A Black Panther barbecue: Serving the people on Juneteenth
For Juneteenth 2019 the New Afrikan Black Panther Party and its mass organization, the United Panther Movement, organized a People's Barbecue at Westside Park in Newark, N.J., where the NABPP is headquartered.
A modest proposal for the abolition of slavery in America in...
We declare all elections not based upon full universal suffrage to be invalid and powers not derived from the consent of the governed to be usurpations.
No Prison Fridays: Panthers organize shutdown of prison construction project
The persistent turnouts of people's power community outrage each week compelled New Jersey Gov. Phil Murphy and his attorney general to announce in early May that they had scrapped plans to build the youth prison at the planned site.
Fight Toxic Prisons Convergence report back
Prisons are both an economic project and a counterinsurgency program. Their neverending goal is to continue "locking people up who are trying to be free." And a reoccurring theme and recognition of the discussion was that "we can't destroy prisons without destroying capitalism."
Thoughts on 'Democracy Denied'
During outside recreation, as we all were locked in individual dog cages that are side by side, I initiated a conversation into what we read in "Democracy Denied."
Independent oversight of Texas prisons: Fantasy or reality?
Lorie Davis has created a culture within TDCJ by which jailhouse lawyers, also known as "writ-writers," are subjected to manifold reprisals for their peaceful and legal activities. Activities which are supposed to be protected under the U.S. Constitution's First and 14th Amendments!
Hunger strike logo artist transferred again – to Texas – for...
I was flown here to Texas on June 14. This is a bit more dire than the February ordeal, so it calls for at least as much sustained and broad help. (Rashid was abruptly transferred from Virginia, where he'd been held, mostly in solitary confinement, since 1990, to Oregon in February 2012, where he was placed in Oregon's Orwellian Snake River Correctional Unit, an unvarnished behavior modification program.)

subset: commoncrawl


text: Fresh Food Markets – Laos Published:June 18, 2014Published In:Asia, LaosAuthor:The Wandering CapitalistDiscussion:2 CommentsTime to read:1 minute WARNING – Some of the following images may not be suitable for young or queasy viewers (in the Blog Business, this is known as a TEASE…) Whole Foods Bad camera work, but hopefully you get the gist…. Snakes Tastes like chicken? Tastes like iguana? in all of Asia people do not say "takeout"….this is "take away"… Luang Prabang morning market Frogs DO taste like chicken, don't they? Meat section. Post Script – Please congratulate Alison Svoboda as the "winner, winner, chicken dinner" of the Find Waldo – 2nd Edition. Tee shirt on the way, Alison! Share this:TwitterFacebookEmailPrintLike this:Like Loading... Previous ArticleThe Finding Waldo Game Wandering Capitalist Style – 2nd EditionNext ArticleKathmandu Field Trip – Saying Hello To Everest


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Born in Thun (Switzerland) in 1981. Julian Sartorius was taught by musicians such as Fabian Kuratli, Pierre Favre and Norbert Pfammatter at the Jazz Schools in Bern and Lucerne. He has collaborated with Matthew Herbert, Shahzad Ismaily, Sylvie Courvoisier, Dimlite, Merz, Fred Frith, Sophie Hunger, Rhys Chatham and many others, and has toured throughout Europe, South America, Canada and the US. With his solo show he has toured widely throughout Europe, playing on bills with the likes of Deerhoof, Faust, Marc Ribot, Jaki Liebezeit, Arto Lindsay, and has released several solo albums, including the 12-LP-Box set «Beat Diary», consisting of 365 beats and a companion photo book. His video installation «Schläft ein Lied in allen Dingen» was shown in several art galleries. The web-project «Morph», is another audiovisual work: Sartorius, on a daily basis, adds an element to his collage and modifies an 8-second-audioloop. He has won several prestigious awards and in 2014 was nominated for the inaugural Swiss Music Prize along with a selection of the most influential Swiss musicians.
Ouch. It seems we can't find what you're looking for.


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Tradition meets modernity.
Plumbing / Roofing Tools
Anyone who manufactures tools for traditional occupations knows: Changes on tools are performed only when the user explicitly asks and if they provide a real added value. We understand this balancing act: we preserve tradition and at the same time, provide the craft with a highly professional tool for daily use.


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I swear! Almost all the pickup truck drivers I have encountered on the road are genuine assholes. Just a few hours ago on my way home from work it was pouring rain due to scattered thunderstorm and its almost 25% visibility on the highway and most cars are keeping their distance and had their headlights on when this dick just cut in between 2 cars and splashed the car behind him.
The idiot doesn't have any respect for the other drivers nor had his headlights on.
And come winter when snow and ice dominates the road and all vehicles starts to drive slow since nobody wants to slip and slide to each other the mighty pickup trucks comes barreling down apparently showing off his wide ass back and tailgates everyone on its path! Like move over bitch! You all driving too slow!
Well, then you see them at the bridge spinning like skaters! LOL!Google+


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HIV testing during the COVID-19 pandemic
Editor – We support the recommendations made in the recent report by Kelly et al for increased awareness of importance of HIV testing during the COVID-19 pandemic.1
During the UK's first wave, our hospital trust in north west London experienced a high burden of COVID-19 patients, with clinical staff quickly becoming familiar with the clinical and radiological features of COVID-19. Studies that have compared pooled nasopharyngeal aspirate results with cross-sectional imaging indicate that the sensitivity of reverse transcriptase polymerase chain reaction (RT-PCR) for SARS-CoV-2 may be very poor.2 As a result, a significant proportion of COVID-19 patients are diagnosed on the basis of typical clinical and radiological features, despite negative tests.3
Between March and June 2020, three patients were admitted with signs and symptoms consistent with COVID-19 (respiratory failure and bilateral infiltrates on chest imaging) but were COVID-19 RT-PCR negative. These patients were subsequently diagnosed with advanced HIV (mean CD4 count 62 × 106/L (range 12–108 × 106/L)) and Pneumocystis jirovecii pneumonia (PCP). Due to the initial presumed clinical diagnosis of COVID-19 there was a delay in testing for HIV and diagnosing and treating PCP.
PCP and COVID-19 are difficult to distinguish on clinical grounds, with shared symptoms including fever, dyspnoea, dry cough and fatigue; although, in COVID-19 the onset is more acute.4,5 The similar radiological findings include bilateral infiltrates with COVID-19 typically involving the lung peripheries whereas PCP is often peripherally sparing.5
Our experience highlights the importance of maintaining diagnostic vigilance when pursuing a diagnosis of COVID-19.1 We strongly advocate HIV testing all individuals presenting with acute respiratory illness, including those with 'typical' features of COVID-19 as per UK national guidelines.6
- © Royal College of Physicians 2021. All rights reserved.
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- British HIV Association, British Association for Sexual Health and HIV, British Infection Association.
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Samuel P. Holocaust testimony (HVT-2131) interviewed by Elliot Perry and Gillian Green Douek,
Videotape testimony of Samuel P., who was born in Będzin, Poland in 1926, the third child of seven. He recalls his family's orthodoxy; especially enjoying Passover and Sukkot; cordial relations with non-Jews; participating in Gordonyah; his brother's bar mitzvah (German invasion precluded his); increasing anti-Jewish restrictions; confiscation of the family business; forced labor with his older brother; his brother's deportation; ghettoization; receiving extra food from his German supervisor; hiding in a bunker with his family during deportations; having to leave the bunker during the ghetto's liquidation; deportation to Birkenau; separation from his family (he never saw them again); assistance from a prisoner when he had typhus; transfer after four months to Myslowice (Fürstengrube); public hangings; a death march to Gleiwitz; transfer to Nordhausen in open train cars; Czechs throwing bread to them; cannibalism; transfer to Magdeburg, then Ahrensburg; forced labor on a former commandant's farm; being loaded on ships; landing near Neustadt; liberation by British troops; assistance from UNRRA; reunion with his brother; and the two of them joining relatives in London in 1946. Mr. P. discusses his appreciation for freedom; finding it too difficult to share all his suffering; many SS who were never punished; and filing complaints with German prosecutors which were never acted upon.
- Published
- London, England : British Video Archive for Holocaust Testimonies, 1992
- Interview Date
- April 29, 1992.
- Locale
- Będzin (Poland)
Ahrensburg (Germany)
Neustadt an der Aisch (Germany)
- Language
- Copies
- 2 copies: 3/4 in. dub; and 1/2 in. VHS with time coding.
- Cite As
- Samuel P. Holocaust Testimony (HVT-2131). Fortunoff Video Archive for Holocaust Testimonies, Yale University Library.
View in Yale University Library Catalog: http://hdl.handle.net/10079/bibid/4296568
Record last modified: 2018-06-04 13:28:00
This page: https://collections.ushmm.org/search/catalog/hvt4296568


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Characterization of cellular prion protein (PrPC) in Caenorhabditis elegans
vts_7796_11247.pdf (1.756Mb)
Stahl, Marcel Werner
FacultiesMedizinische Fakultät
Cellular prion protein (PrPC) is a conserved glycoprotein predominantly expressed in neurons, glial and lymphatic cells. It is associated with the broad group of prion diseases in which PrPC becomes misfolded. However, the purpose of PrPC in living cells is still enigmatic. To elucidate its debated function in oxidative stress resistance, human PrPC was extrachromosomally expressed in the nematode Caenorhabditis elegans (C. elegans). It was co-expressed with GFP and the thermoselective worm gene pha-1 for verification of correct expression. C. elegans strains expressing PrPC were compared with controls in a variety of lifespan experiments. Under non stressful conditions, nematodes expressing PrPC had a positive effect after population size shrank to about 50%. For further investigation, strains were exposed to Paraquat causing the creation of superoxide anions. PrPC expressing nematodes showed a significant advantage compared to controls. Interaction with DAF-16/FoxO and SKN1 were excluded by knock-out or iRNA inhibited strains. Expression of PrPC showed no effect in handling other stressors like hydrogen peroxide, heat and copper ions intoxication. Further testing showed Paraquat resistance being SOD-1 dependent. However, advantageous resistance of PrPC strains is only achieved via SOD-4 and SOD-5 expression. It can be inferred that additional resistance of prion protein strains is caused by SOD-1 overexpression regulated by SOD-4 and SOD-5 which is coherent to the SOD Enzyme Assay results. False positive results by changes in chemosensory neurons were excluded by dye fill procedure. Hence, it can be concluded that C. elegans was established as a new model system in prion research. It was further proven that human cellular PrPC is capable of improving resistance of C. elegans against superoxide anions, while it is shown that PrPC modifies the expression or activity level of SOD-1 by interacting with SOD-4 and SOD-5, which, in turn, act as regulator enzymes.
Date created
Subject headings[GND]: Caenorhabditis elegans | Paraquat | Prion | Superoxiddismutasen
[MeSH]: Oxidative stress
[Free subject headings]: DAF-16 | PrPC | SKN-1 | SOD
[DDC subject group]: DDC 610 / Medicine & health
LicenseStandard (Fassung vom 01.10.2008)
MetadataShow full item record
DOI & citation
Please use this identifier to cite or link to this item: http://dx.doi.org/10.18725/OPARU-2276
Stahl, Marcel Werner (2011): Characterization of cellular prion protein (PrPC) in Caenorhabditis elegans. Open Access Repositorium der Universität Ulm und Technischen Hochschule Ulm. Dissertation. http://dx.doi.org/10.18725/OPARU-2276
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Monday - Friday
Time: 8:00 AM to 4:30:00 PM
(970) 565-3751
Economic Development
Economic Development responsibilities include economic development initiatives from within the Reservation and from national and international enterprises interested in developing sustainable business relationships with the Ute Mountain Ute Tribe. To this end the Division may serve as the lead funding development agency for Economic Development projects.
The Division oversees Tribal Enterprises which include: The Ute Mountain Ute Casino and Resort Weeminuche Construction Authority The Ute Mountain Ute Farm & Ranch The Ute Mountain Ute Pottery /Gallery The Ute Mountain Ute Travel Plaza.
There is an old legend that a Ute Chief once stood in the San Juan Mountains and said, "From where I stand, all the land this water touches belongs to the Ute people."
Weeminuche Construction Authority
WCA was formed in 1985 by the Tribal Council of the Ute Mountain Ute Indian Tribe. WCA is a commercial construction enterprise owned and operated by the Ute Mountain Ute Indian Tribe. It is bonded to $100 Million. WCA provides comprehensive construction services to a wide range of clientele, and a wide range of projects.
Contact Info: (970) 565-7430
Read More
Farm and Ranch Enterprise
The Ute Mountain Ute Tribe's Farm and Ranch Enterprise is an irrigated agricultural project designed for 7,634 acres of Ute Mountain Reservation land in Southwest Colorado. A tribal owned farm consisting of 7700 irrigated acres producing alfalfa, corn, wheat; and manages a 700 head cow/calf operation.
Contact Info: (970) 565-6412Read More
Retail/Travel Center
The Travel Center offers a variety of services to you. We are open 24-hours a day with two full fuel islands,convenience/gift/grocery store, showers and laundry facilities, vacuum and, car/truck air service. We also offer easy diesel fueling stations for trucks and RVs,fast food service outlets, and Free WI-FI.
Contact Info:Toll Free: 1-800-889-5072
Phone: (970) 565-6544 Read More
Ute Mountain Ute Casino/Hotel
The Resort has spacious and luxurious rooms featuring Native American Anasazi decor. We also have a Banquet & Convention Center for your convience. For those who like a little excitement in their life! We have a Casino and bingo. Additionally the resort offers Kuchu's restaurant, Ute Mountain cafe snack bar, gift shop and free shuttle services, and an RV park.
Contact Info:Toll Free: 1-800-258-8007 Tel: 1-970-565-8800Read More
If you have questions or concers contact them directly.


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Scott Golueke
Position title: Assistant University Registrar for Credit Evaluation Services
Pronouns: he/him/his
Email: email@example.com
B.S. Biology, UW-Madison
Scott has worked for the Division of Enrollment Management since 2007 and has held previous roles as a Transfer Admissions Counselor and Course Specialist. He also had a student position in the Office of Student Financial Aid during my undergraduate career. Scott is from Green Bay, Wisconsin, and is a diehard Packers fan. He enjoys being outdoors, no matter the season – camping, hiking, biking, water skiing, cross country skiing, and snowshoeing.


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Rationale for Aspirin
Risk prediction can also be used to establish whether low-dose aspirin is needed in adults aged 40–70 years. However, the 2019 guidelines do not stipulate a risk cut-off beyond which aspirin is routinely recommended for primary prevention of CVD. Rather, the risk of bleeding should also be considered, and aspirin should be avoided for primary prevention of CVD in adults aged >70 years. These recommendations are the result of trials in patients without CVD published in the past 2 years, which reported minimal or no benefit of prophylactic aspirin at the expense of increased risk of bleeding


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We thrive in a competitive, candidate-driven job market. Organizations need to find the right candidates for their job positions fast. A recruitment agency assists businesses to fulfill their human resource requirements and at the same time aids job-seekers in finding a suitable job for them. These agencies act as a bridge between companies looking for new candidates to be hired and people searching for job opportunities.
How can a Recruitment Agency help you?
In today's highly competitive world, it becomes difficult to search for the right candidate from the pool of applications. In order to save time, improve the quality of hire, and additional security, a recruitment agency's role becomes significant. If you are planning to employ the services of a recruitment agency, you might be concerned about how to get the most out of your money.
A recruitment agency will be able to provide you with access to the best available talent. This will include candidates actively seeking work, as well as currently employed candidates. They may come along with a host of benefits and add-on services. At the same time, there may be risks involved in hiring their services.
Here are the things to keep in mind while selecting a recruitment agency
1. Network and Connections
The most important advantage a recruitment agency offers is the broad base of clients and network of connections they have established with various colleges and universities. The wider the network, the better are the chances of hiring the perfect fit for the job. So, checking the level of networking an agency has become essential.
Their database should be vast, with proper sectioning, highlighting the expertise of every college and university. Based on your requirement, the agency should be able to provide the appropriate applicant for a job position with the right skillset.
2. Personal Guide with market expertise
The recruitment agency should act as a personal guide for the company or the business looking to hire candidates. Their knowledge and experience must act as a guidance system for locating the right candidate. It should know how to narrow down the search area and look at the right places to find candidates with the precise set of skills required.
Your recruitment agency's market knowledge and industry expertise will come in handy when you have specific candidate requirements like Project Managers or members of the C-suite. For example, if you are hiring a Senior Manager, then you may need an expert opinion and knowledge of current recruitment legislation. In such cases, the agency should know all the legalities applicable and be aware of the relevant requirements.
3. Valuable Industry Insight
The experiences of the recruiters come in handy in providing valuable industry insight, which the amateur job-seekers lack. They should have enough experience to plan for your strategic growth in the company and the kind of salary packages offered currently.
Based on the company and their working environment, the recruiter must know if it matches your profile and expectations. Your Recruitment agency should be an industry specialist in the market. They should be fully aware of where and how to reach out to the best available talent. At the same time, they should be knowledgeable about various salary rates, career expectations, available skillsets and shortages, and any hiring complexities.
4. Interview Skills
Understand how the recruitment agency evaluates and assesses the candidates present in their database. The company/ business should conduct an interview with the recruitment agency just as they would interview a candidate for employment. You can ask questions like how they remain up to date about the candidate's profile and the duration of time required to fill the position.
An efficient recruitment agency should be able to fill your company's job openings and not just book interviews. The candidates hired by the company should be your organization's right fit and not just a rushed hire. Make sure they don't rush to fill in the spots to find candidates who contribute to the employee turnovers.
5. Evaluate their terms
The recruitment agency must have flexible terms and should not be rigid with their talent procurement contracts. The recruitment agency must possess qualities like being nimble, think creatively, and assist in hiring and retaining talent for your company.
The company would not want to get stuck with a recruitment agency that refuses to bend its terms and bogs down the aim of hiring a candidate with the correct profile for the job. Also, you should clearly evaluate the new hire retention terms. If a new hire for your company leaves before the expiry of a guarantee period or gets terminated, they should provide you with a full refund or find a replacement candidate for no extra charge.
6. Analyze the recruiters thoroughly
A background check is essential for the recruitment agency, too, same as the one done while hiring a candidate. Cross-check their experience and make sure they have the kind of knowledge which they claim to possess. Look for an indication of their experience in the field of your interest. Their expertise must match your requirements for fruitful results.
Usually, a recruitment agency is hired by a company when they are looking for candidates with a unique or hard-to-fill skill set. Thus, it is essential the recruitment agency has a vast network of great candidates in such a case. A fine quality of candidates who are hard to find on recruiting sites must be included in the agency's database to provide the company with the best. Hence a recruitment agency with experience and expertise will be beneficial for the company.
Recruitment agencies improve your hire quality and have a talent pool of highly skilled candidates at their disposal. They can assist you in the hiring process professionally and efficiently with cost savings. You should be aware of the scope of their services before giving them the responsibility of searching for the perfect candidate for your business. Organizations should study in detail and research about the recruitment agency in depth before hiring their services.


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Open Journal of Statistics
Vol.3 No.2(2013), Article ID:30157,24 pages DOI:10.4236/ojs.2013.32011
Cross-Validation, Shrinkage and Variable Selection in Linear Regression Revisited
1Department of Medical Statistics and Bioinformatics, Leiden University Medical Center, Leiden, The Netherlands
2Institut fuer Medizinische Biometrie und Medizinische Informatik, Universitaetsklinikum Freiburg, Freiburg, Germany
Email: firstname.lastname@example.com
Copyright © 2013 Hans C. van Houwelingen, Willi Sauerbrei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received December 8, 2012; revised January 10, 2013; accepted January 26, 2013
Keywords: Cross-Validation; LASSO; Shrinkage; Simulation Study; Variable Selection
In deriving a regression model analysts often have to use variable selection, despite of problems introduced by datadependent model building. Resampling approaches are proposed to handle some of the critical issues. In order to assess and compare several strategies, we will conduct a simulation study with 15 predictors and a complex correlation structure in the linear regression model. Using sample sizes of 100 and 400 and estimates of the residual variance corresponding to R2 of 0.50 and 0.71, we consider 4 scenarios with varying amount of information. We also consider two examples with 24 and 13 predictors, respectively. We will discuss the value of cross-validation, shrinkage and backward elimination (BE) with varying significance level. We will assess whether 2-step approaches using global or parameterwise shrinkage (PWSF) can improve selected models and will compare results to models derived with the LASSO procedure. Beside of MSE we will use model sparsity and further criteria for model assessment. The amount of information in the data has an influence on the selected models and the comparison of the procedures. None of the approaches was best in all scenarios. The performance of backward elimination with a suitably chosen significance level was not worse compared to the LASSO and BE models selected were much sparser, an important advantage for interpretation and transportability. Compared to global shrinkage, PWSF had better performance. Provided that the amount of information is not too small, we conclude that BE followed by PWSF is a suitable approach when variable selection is a key part of data analysis.
1. Introduction
In deriving a suitable regression model analysts are often faced with many predictors which may have an influence on the outcome. We will consider the low-dimensional situation with about 10 to 30 variables, the much more difficult task of analyzing 'omics' data with thousands of measured variables will be ignored. Even for 10+ variables selection of a more relevant subset of these variables may have advantages as it results in simpler models which are easier to interpret and which are often more useful in practice. However, variable selection can introduce severe problems such as biases in estimates of regression parameters and corresponding standard errors, instability of selected variables or an overoptimistic estimate of the predictive value [1-4].
To overcome some of theses difficulties several proposals were made during the last decades. To assess the predictive value of regression model cross-validation is often recommended [2]. For models with a main interest in a good predictor the LASSO by [5] has gained some popularity. By minimizing residuals under a constraint it combines variable selection with shrinkage. It can be regarded, in a wider sense, as a generalization of an approach by [2], who propose to improve predictors with respect to the average prediction error by multiplying the estimated effect of each covariate with a constant, an estimated shrinkage factor. As the bias caused by variable selection is usually different for individual covariates, [4] extends their idea by proposing parameterwise shrinkage factors. The latter approach is intended as a post-estimation shrinkage procedure after selection of variables. To estimate shrinkage factors the latter two approaches use cross-validation calibration and can also be used for GLMs and regression models for survival data.
When building regression models it has to be distinguished whether the only interest is a model for prediction or whether an explanatory model, in which it is also important to assess the effect of each individual covariate on the outcome, is required. Whereas the mean square error of prediction (MSE) is the main criterion for the earlier situation, it is important to consider further quality criteria for a selected model in the latter case. At least interpretability, model complexity and practical usefulness are relevant [6]. For the low-dimensional situation we consider backward elimination (BE) as the most suitable variable selection procedure. Advantages compared to other stepwise procedure were given by [7]. For a more general discussion of issue in variable selection and arguments to favor BE to other stepwise procedures and to subset selection procedures using various penalties (e.g. AIC and BIC) see [4] and [8]. To handle the important issue of model complexity we will use different nominal significance levels of BE. The two post-estimation shrinkage approaches mentioned above will be used to correct parameter estimates of models selected by BE. There are many other approaches for model building. Despite of its enormous practical importance hardly any properties are known and the number of informative simulation studies is limited. As a result many issues are hardly understood, guidance to built multivariable regression models is limited and a large variety of approaches is used in practice.
We will focus on a simple regression model
with X a p-dimensional covariate. Let there be n observations used to obtain estimates
and of the regression parameters.
The standard approach without variable selection is classic ordinary least squares (OLS). In a simulation study we will investigate how much model building can be improved by variable selection and cross-validated based shrinkage. The paper reviews and extends early work by the authors [2,4,9]. Elements added are a thorough reflection on the value of cross-validation and a comparison with Tibshirani's LASSO [5]. With an interest in deriving explanatory models we will not only use the MSE as criteria, but will also consider model complexity and the effects of individual variables. Two larger studies analyzed several times in the literature will also be used to illustrate some issues and to compare results of the procedures considered.
The paper is structured in the following way. Section 2 describes the design of the simulation study. Section 3 reviews the role of cross-validation in assessing the prediction error of a regression model and studies its behavior in the simulation study. Section 4 reviews global and parameterwise shrinkage and assesses the performance of cross-validation based shrinkage in the simulation data. The next Sections 5 and 6 discuss the effect of model selection by BE and the usefulness of crossvalidation and shrinkage after selection. Section 7 compares the performance of post-selection shrinkage with the LASSO. Two real-life examples are given in Section 8. Finally, the findings of the paper are summarized and discussed in Section 9.
2. Simulation Design
The properties of the different procedures are investigated by simulation using the same design as in [10]. In that design the number of covariates, the covariates have a multivariate normal distribution with mean, standard deviation for all covariates. Most correlations are zero, except R1,5 = 0.7, R1,10 = 0.5, R2,6 = 0.5, R4,8 = −0.7, and. The covariates and are uncorrelated with all other ones. The regression coefficients are taken to be (intercept), β1 = β2 = β3 = 0, β4 = −0.5, β5 = β6 = β7 = 0.5, β8 = β9 = 1 and,.
The variance of the linear predictor
in the model equals
, where CX is the covariance matrix of the X's. The residual variances are taken to be or. The corresponding values of the multiple correlation coefficient
are and
, respectively. Sample sizes are n = 100 or. For each of the four combinations, called scenarios, samples are generated and analyzed. The scenarios are ordered on the amount of information they carry on the regression coeffients. Scenario 1 is the combination, scenario 2 is, scenario 3 is
and scenario 4 is
Since the covariates are not independent, the contribution of Xj to the variance of the linear predictor
is not simply equal to. Moreover, the regression coefficients have no absolute meaning, but depend on which other covariates are in the model. To demonstrate this, it is studied how dropping one of the covariates influences the optimal regression coefficients of the other covariates, the variance of the linear predictor and the increase of the residual variance, which is equal to the decrease of. This is only done for which have non-zero coefficients in the full model. The results are shown in Table 1.
Table 1. Effects of dropping one covariate with non-zero β's. The other 14 covariates remain in the model. The main body of the table gives the regression coefficients. The last rows show the resulting values of, the increase in the residual variance and the multiple correlation R2. The latter is computed for the case of. Covariates X1, X2, X3, X11, ..., X15 have β = 0. Dropping them will not affect the β's of the model under "none".
The table also shows the resulting for the case that. Apparently, the effect of each covariate is partly "inherited" by some of the other covariates. A simple pattern of inheritance is seen for X6. It only correlates with X2 and the pair is independent of the rest. If X6 is dropped, gets the regression coefficient. This saves a little bit of the variance of the linear predictor. It drops from 6.250 to 6.063, while it would have dropped to 6.000 if X6 were independent of the other predictors. A more complicated pattern is seen for X7. If that one is dropped, and inherite the effects. The covariates X14 and X8 show up because they are directly correlated with X7. Covariate X4 shows up because it is correlated with X8. The variance of the linear predictor drops from 6.250 to 6.107.
Since are independent of the other covariates, they cannot inherit effects. However, can partly substitute, although they have coefficients in the full model.
3. The Value of Cross-Validation
Cross-validation is often recommended as a robust way of assessing the predictive value of a statistical model. The simplest approach is leave-one-out cross-validation in which each observation is predicted from a model using all the other observations. The generalization is -fold cross-validation in which the observations are randomly divided into "folds" of approximately equal size and observation in one fold are predicted using the observations in the other folds. In the paper leave-one-out crossvalidation will be used, but the formulas presented apply more generally. Let be obtained in the cross-validation subset, in which observation is not included. The cross-validation based estimate of the prediction error is defined as
The true prediction error of the model with estimates b0 and b1 from the "original" model using all n observations is defined as
In the simulation study it is given by
The results in the simulation study using all covariates without any selection are given in Table 2.
The results show that does a good job in estimating the mean value of over all simulations. However, since the correlation between and over all simulation runs is virtually equal to zero, it must be concluded that it does a very poor job in estimating the prediction error of the individual models.
Notice that the standard deviation of is much larger than that of. The explanation is that a lot of the variation in is due to the estimation of the unknown. Cross-validation might be do a better job in picking up the systematic prediction part of the prediction error caused by the error in the estimated's. That can be checked by studying the behavior of
which is an estimate of the systematic part
. Here is the usual unbiased estimator of. The results are shown in Table 3. It nicely shows that the systematic error decreases monotonically from scenario 1 to scenario 4.
Means are very similar but standard deviations from the CV estimates are much smaller. CV somehow shrinks the estimate of the systematic error towards the mean. The table shows that the correlations between the estimate and the true value are still very low. The warning issued in Section 4 of [2] still holds. It is nearly impossible to estimate the prediction error of a particular regression model. Cross-validation is of very little help in estimating the actual error. It can only estimate the mean error, averaged over all potential
"training sets". However, it might be helpful in selecting procedures that reduce the prediction error.
Finally, it should be pointed out that the cross-validation results are in close agreement with the model based estimates of the prediction error as discussed in the same section of [2].
4. Cross-Validation Based Shrinkage without Selection
4.1. Global Shrinkage
As argued by [2,11], the predictive performance of the resulting model can be improved by shrinkage of the model towards the mean. This gives the predictor
with shrinkage factor c,. In the following c will be called global shrinkage factor. Under the assumption of homo-skedasticity, the optimal value for c can be estimated as
with the explained sum of squares, the estimate of the residual variance and p the number of predictors.
A model free estimator can be obtained by means of cross-validation. Let be obtained in the cross-validation subset, in which observation is not included, then can be estimated by minimizing
Table 2. Simulation results for and and their correlation (corr.) in models without selection.
Table 3. Simulation results for and and their correlation (corr.) in models without selection.
resulting in
This estimate can be obtained by regressing
on in a model without an intercept. It differs slightly from the one obtained by regressing on as proposed in [2]. The definition allows k-fold cross-validation and is not restricted to leave-one-out cross-validation. The results of application of global shrinkage in the simulation data, ignoring the restriction, are shown in Table 4. Actually, was never observed and only occasionally.
The table shows that global shrinkage can help to reduce the prediction error if the amount of information in the data is low. For scenario 1 the mean of the shrinkage factor is 0.84 and the mean reduction of prediction error is 0.14. Corresponding values for scenario 4 are 0.98 and 0.001. For the latter all shrinkage factors are close to one and predictors with and without shrinkage are nearly identical. However, the positive correlation between the shrinkage factor c and the reduction in prediction error is counter-intuitive. To get more insight the data for scenario 1 with a small amount of information is shown in Figure 1.
The relation between reduction in prediction error due to shrinkage and the prediction error of the OLS models are shown for three categories of the shrinkage factor c, namely and. The frequencies of these categories among the 10,000 simulations are 1754, 7740 and 506, respectively. The upper panel shows the apparent (estimated) prediction errors based on cross-validation and the apparent reduction achieved by global shrinkage. The differences between the three categories are small, but they are in line with the intuition that the largest reduction is achieved when the shrinkage factor is small. The quartiles (25%, 50%, 75%) of the apparent reduction are 0.09, 0.15, 0.27 for for and −0.07−0.02, 010 for. The middle panel shows the actual (true) prediction error based on our knowledge of the true model. Here, the picture is completely different. Reduction of the prediction error only occurs when the shrinkage factor is close to one and the OLS prediction error is large. Substantial shrinkage with tends to increase the prediction error. The quartiles of the true reduction are −0.29, −0.13, 0.04 for c < 0.8, 0.05, 0.18, 0.31 for and 0.19, 0.28, 0.38 for. The lower panel shows the relation between the apparent and the actual reduction. At first sight the results our counter-intuitive. This phenomenon is extensively discussed in [9]. What happens could be understood from the heuristic shrinkage factor. If is "large" by random fluctuation, the observed explained sum of squares is large and stays close to 1 and does not "push" in the direction of the true. If is "small" by random fluctuation, is small and will be closer to 0 and might "push" in the wrong direction. This explains the overall negative correlation between apparent and actual reduction of the prediction error. It must be concluded that it is impossible to predict from the data whether shrinkage will be helpful for a particular data set or not. The chances are given under "frac. pos." in Table 4. They are quite high in noisy data, but that gives no guarantee for a particular data set.
4.2. Parameterwise Shrinkage
[4] suggested a covariate specific shrinkage factor, coined parameterwise shrinkage factor (PWSF), to be defined as
Here, is a vector of shrinkage factors with for and "" stands for coordinate-wise multiplication:. This way of regulation is in the spirit of Breiman's Garrote [12]. See also [9,13]. Sauerbrei suggests to use the parameterwise shrinkage after model selection and to estimate the vector by cross-validation. As for the global shrinkage
Table 4. Simulation results for the reduction in prediction error (compared with OLS) achieved by global shrinkage in models without selection; "frac. pos." stands for the fraction with positive reduction, "corr." stand for the correlation between the shrinkage factor and the reduction.
Figure 1. Reduction of prediction error achieved by global shrinkage for different categories of the shrinkage factor c; data from scenario 1. The upper panel shows the apparent prediction errors obtained through crossvalidation, the middle panel shows the actual (true) prediction errors and the lower panel shows the relation between the apparent and the actual reduction.
this could be obtained by regression without intercept of on.
Although this is against the advice of [4] parameterwise shrinkage was applied in the simulation data in models without selection, ignoring the restrictions for. A summary of the results is given in Table 5.
Using PWSF the average prediction error increases when compared with the OLS predictor. The increase is large (about 10%) in scenario 1. In scenario 4 the increase is moderate, but still present. Moreover, the estimated prediction error obtained from the cross-validation fit is far too optimistic. (Data not shown). The explanation is that parameterwise shrinkage is not able to handle the redundant covariates with no effect at all. This can be seen from the box plots in Figure 2.
For the redundant covariates the shrinkage factors are all over the place. Even variables with a weak effect have sometimes negative PWSF values. For the strong covariates they are quiet well-behaved despite the erratic behavior for the other ones. The conclusion must be that
Table 5. Comparison of the prediction errors of OLS, global shrinkage and parameterwise shrinkage in models without selection.
Figure 2. Box plot of parameterwise shrinkage factors in models without selection. Results from scenario 4.
in models with many predictors without selection on stronger predictors parameterwise shrinkage cannot be recommended. The behavior is a bit better if negative shrinkage values will be set to zero, but altogether such a constraint is not sufficient. This is completely in line with Sauerbrei's original suggestion.
5. Model Selection
Following [10] models are selected by backward elimination at significance levels of and. An impression of which covariates are selected is, shown in Figure 3.
5.1. Number of Variables Selected, Type I and Type II Error
The "softest" definition of model selection is the selection of covariates to be used in further research. The "optimal" model contains only the important covariates, the ones that have an influence on the outcome in the full model. In the simulation those are the covariates. If these are selected the other ones are redundant. However if one of the important covariates is not selected, other non-important covariates can come to the rescue if they are correlated with the non-selected important covariate(s). In the simulation data non-important covariates that can play such a role are covariates as can be seen from Table 1. The effect of omitting important covariates is the loss of explained variation in the optimal model after selection or, equivalently, the introduction of additional random error. If no selection takes place there is no loss of explained variation, but there is a large number of non-important covariates, leading to larger estimation errors. Even more important is a severe loss in general usability of such predictors [4,6]. Consider, for example, a prognostic model comprising many variables. All constituent variables would have to be measured in an identical or at least in a
Figure 3. Frequency of selection per covariate.
similar way, even when their effects were very small. Such a model is impractical, therefore "not clinically useful" and likely to be "quickly forgotten" [14].
From Figure 3 we learn that in the easy situation with a lot of information (scenario 4) all important variables are selected in nearly all replications. For the 8 variables without an influence selection frequencies agree closely to the nominal significance level. For selection frequencies are between 15.7% and 19.0%. The corresponding relative frequencies are between 5.1% and 6.55% for and between 0.9% and 3.3% for. Results are much worse for situations with less information. For the most extreme scenario 1 it is obvious that all selected models deviate substantially from the true model. For selection frequencies are only between 23.4% and 25.1% for the 4 relevant variables with a weak effect. Even for these frequencies are only between 46.1% and 57.3%. Of these has the lowest frequency, which is probably caused by the strong correlation with. With 28.9% has the largest frequency of selection among the non-important covariates. That is explained by its strong correlation with X5. In 19.4% of the simulations X1 is selected while the important variable X5 is not selected.
The effect of selection at the three levels is shown in Figures 4-6.
Figure 4 summarizes the number of included variables for the different scenarios. In the simple scenario 4 all seven relevant variables were nearly always selected. In addition irrelevant variables were selected in close agreement to the significance level. In the correct model was selected for whereas redundant variables without influence were added when using For scenarios with less information the number of selected variables is usually smaller and the significance level has a stronger influence on the inclusion frequencies. Strong predictors are still selected in most of the replications, but the rest of the selected model does hardly ever agree to the true model. For each of the variables with a weaker effect the power for variable inclusion is low.
The comparison of Figures 5 and 6 nicely shows the balance between allowing redundant covariates (covariates without effect in the selected model) and allowing loss of explained variation. The number of redundant covariates shown in Figure 5, corresponds directly to the type I error. It hardly depends on the residual variance and the sample size. Despite of some stronger correlations in our design the distribution is very close to binomial. The type II error is reflected by the loss of the variance of the optimal linear predictor , or ,equivalently, the increase in residual variance, caused by not selecting all important covariates. It could be translated into a loss of by dividing it by the marginal variance
, which is equal to 12.5 in scenarios 1 and 3 (σ2 = 6.25) and 8.75 in scenarios 2 and 4 (σ2 = 2.5). This is shown in Figure 6.
Generally speaking, loss of depends on both the significance level used in the selection process and the
Figure 4. Number of included covariates.
Figure 5. Number of redundant covariates.
amount of information in the data reflected by residual variance and the sample size. In scenario 4 with a high amount of information, the loss of is negligible and the number of redundant covariates can be controlled by taking. In scenario 1 there is a substantial loss of if the selection is too strict and might be more appropriate.
As mentioned above and seen in Table 1 the variable can partly take over the role of (correlation coefficient is 0.7) if the latter is not selected. That is shown in Table 6 for scenario 2 and. It has to be kept in mind that other variables are also deleted making a direct comparison difficult. Concerning these two variables the correct model includes X5 and excludes X1, which is the case in 69% of the replications. Here the mean loss is smallest. X5 is erroneously excluded in about 28% of the models. In about half of them the correlated variable X1 is included, reducing the R2 loss
Figure 6. Loss of R2 in comparison to the full model. The bars show mean ± 1 st.dev. in the population of replications.
Table 6. Inclusion frequencies and average loss of for combinations of and for scenario 2 and.
caused by exclusion of X5 substantially.
5.2. Assessment of Prediction Error
The prediction error of a particular model depends on the number of redundant covariates and the loss of explained variation. The average reduction of prediction error when compared with no selection (significance level) is shown in Figure 7. Those numbers could be best compared with the systematic component of the prediction error, , as reported in Table 3.
It nicely shows that the optimal level depends on the amount of information in the data. It also shows that moderate selection at, in a univariate situation equivalent with AIC or Mallows' CP, can do very little harm. Even gives always better results than no selection. For a small sample size the relative reduction in prediction error is small. For the large sample size elimination of several variables reduces the relative prediction error substantially for. The average number of selected variables is 7.02 for scenario 3 and 7.40 for scenario 4.
Figure 8 shows the prediction error ranking of the different levels in the individual simulation data sets. Mean ranks were used in replications where the same model was selected. As observed before, the variation is very big and there is no outspoken winner, but there are some outspoken losers. In all scenarios it is bad not to select at all. However, for a small sample size it is even worse to use a very small selection level.
The prediction errors of the selected models can be estimated by cross-validation in such a way that in each cross-validation data set the whole selection procedure is carried out. As in Section 3 this will yield a correct estimate of the average prediction. Thus, it could be used to select the "optimal" significance level in general, but it will not necessarily yield the best procedure for the actual data set at hand.
6. Post-Selection Cross-Validation
A common error in model selection is to use crossvalidation after selection to estimate the prediction error and to select the best procedure. As pointed out by [15], this is a bad thing to do. That is exemplified by Figures 9 and 10.
Comparing Figures 7 and 9 shows that cross-validation after selection is far too optimistic about the reduction of the prediction error and is not able to notice the poor performance of selection at for the scenarios 1 and 2. Moreover, as can be seen from Figure 10, post-selection cross-validation tends to favor selection at for all scenarios. This is no surprise because in univariate selection is equivalent with using AIC, which is very close to using cross-validated prediction error if the normal model holds true.
6.1. Post-Selection Shrinkage
While cross-validation after selection is not able to select
Figure 7. Mean reduction of the true prediction error as reported in Tables 2 and 3 for different α-levels. Values of Err − s2 are 1.21, 0.48, 0.26 and 0.104 in scenarios 1-4, respectively (see Table 3).
Figure 8. Ranks of prediction error for different levels; rank = 1 is best, rank = 4 is worst.
Figure 9. Reduction of estimated prediction error, obtained through post-selection cross-validation, after backward elimination with three selection levels.
Figure 10. Ranks of estimated prediction error obtained from post-selection cross-validation for different levels; rank = 1 is best, rank = 4 is worst. The ranks of the true prediction error are shown in Figure 8.
the best model, it might be of interest to see whether cross-validation based shrinkage after selection can help to improve the model. The results are shown in Figure 11.
On the average, parameterwise shrinkage gives better predictions than global shrinkage, when applied after selection. An intuitive explanation is that small effects that just survive the selection, are more prone to selection bias and, therefore, can profit from shrinkage. In contrast, selection bias plays no role for large effects and shrinkage is not needed, see [16] and chapter 2 of [8]. Whereas global shrinkage "averages" over all effects, parameterwise shrinkage aims to shrink according to these individual needs.
This can also be investigated by looking into the mean squared estimation errors of the regression coefficients conditional on the selection of covariates in the model. The coefficients are the optimal regression coefficients in the selected model. As discussed in Section 2 the coefficients can differ from the coefficients if there is correlation between the covariates. Figure 12 shows the mean squared errors of the shrinkage based estimators relative to the mean squared errors of the OLS estimators for sample size, scenarios 1 and 2 . Sample size is not shown, because post-selection shrinkage has hardly any effect.
It is clear that the parameterwise shrinkage helps to reduce the effect of redundant covariates that only enter by chance, while the global shrinkage is not able to make that distinction. The precise mechanism is not quite clear yet. To get some better feeling what is going on, the observed scatterplot of parameterwise shrinkage factors versus the OLS estimator are shown in Figure 13 for covariate X3 (no effect), X6 (weak effect) and X9 (strong effect). These covariates are selected because the optimal parameter value when selected does not depend on which other covariates are selected as well. X3 is independent of all other variables and the other two variables are only correlated with one variable without influence. Therefore parameter estimates are theoretically equal to the true value in the full model. If the optimal value varies with the selection the graphs are a bit harder to interpret.
For X3, the covariate without effect and no correlation, the inclusion frequency is close to the type I error and in about half of these cases parameter estimates are positive and negative. The variable is selected in replications in which the estimated regression coefficient is by chance most heavily overestimated (in absolute terms) compared to the true (null) effect. One would hope that PWFS would correct these chance inclusions by a rule like "the larger the absolute effect, the smaller the shrinkage factor". Although most shrinkage factors are much lower than one, Figure 13 shows a different cloud: "the larger, the larger".
A similar observation transfers to the plot for X9, which is selected in all replications. Therefore, selection bias is no issue for this covariate. The hope is that PWSF would move the estimate close to the true value β = 1.0.
Figure 11. Reduction in prediction error obtained through shrinkage after selection. Error bars show mean ± 1 standard deviation.
Figure 12. Relative mean squared estimation errors of the partial regression coefficient per covariate (compared to OLS in the selected model) for the parameter estimates obtained by global (black) or parameterwise post-selection shrinkage (grey). Covariates 1-3 and 11-15 have no effect in the full model.
Generally speaking that does not happen: c increases slowly with b. Most values are close to 1, indicating that shrinkage is not required. The only "hoped for" observation can be made for the cases where the correlated
Figure 13. Parameterwise shrinkage factors versus OLS estimates from selected models for covariates 3, 6 and 9; α = 0:05 and scenario 2; reference lines refer to the true value of the parameter.
variable X13 is included (plot not shown). X13 has no effect and the selection frequency (5.9%) agrees well to the type I error. If X13 is included, the shrinkage factor for X9 show a decreasing trend with -values clearly below 1 if the estimate overestimates the true value, and values around if. One might say that parameter shrinkages helps to correct for chance inclusions of X13 but not for estimation errors.
X6 is a covariate with a weak effect. It is not included in 19% of the replications, certainly cases in which the true effect was underestimated by chance. The overall picture for the case where it is included shows a stronger increasing trend (compared with X9) tending to a value of about, if b is large. Here, X2 plays the role of a confounding redundant covariate. In cases where it is included (plot not shown), the shrinkage factor for X6 is rather stable with median value about.
Some understanding can be obtained by the observation in [2] that in univariate models the optimal shrinkage factor is given by. If is smallthis quantity is very hard to estimate. If is large it could be estimated by. The parameterwise shrinkage factor behaves very similarly. This could be seen from plotting PWSF against (Graphs not shown). Such a plot clearly shows that (for) is the cut-point for an inclusion and that PWSF tends to increase with for included variables. For large absolute t-values, PWSF's are close to one. whereas they drop to about 0.8 for close to 2. The relation between PWSF and is similar for all three covariates. The difference between the covariates is the size of the effect and correspondingly the range of after selection.
The conclusion so far is that coordinate-wise shrinkage is helpful after selection. However it is not clear how to select the significance level. In a real analysis, the level should be determined subjectively by the aim of the study [4]. In the following we will compare backwards elimination with a procedure like the LASSO, that combines selection, shrinkage and fine-tuning.
7. Comparison with LASSO
The simulations discussed above were compared with the results of the LASSO with cross-validation based selection of the penalty parameter. Because LASSO is quite time-consuming it was only applied on the first 2000 data sets for each combination of and. Figure 14 shows the distribution of the cross-validation based for each combination. The variation in the penalty parameter, even in the simple situation of scenario 4 is surprisingly large. There is some correlation with the estimated variance in the full model, but that does not explain the huge variation.
The next Figure 15 shows the inclusion frequencies
Figure 14. LASSO: histogram of the cross-validation based λ's for the different scenarios.
Figure 15. LASSO: inclusion frequencies of the covariates.
for the different covariates. Relevant variables are nearly always included, but LASSO is not able to exclude redundant covariates if there is much signal in the other ones. For example, in scenario 4 inclusion frequencies are 52% for and 54% for, the two uncorrelated variables without influence. The probable reason is that selection and shrinkage are controlled by the same penalty term. The phenomenon is also nicely illustrated by Figure 16.
Finally, the question how the prediction error of LASSO compares with the models based on selection and shrinkage is answered by Figure 17.
The conclusion must be that LASSO is no panacea. Concerning prediction error, it seems to be OK for noisy data (scenarios 1 and 2), but it is beaten by variable selection followed by some form of shrinkage if the data are less noisy (scenario 4). Most likely, that is caused by the inclusion of two many variables without effect. Variable selection combined with parameterwise shrinkage performs quit well. The choice of a suitable significance level seems to depend on the amount of information in the data. Whereas has the best performance in scenario 4, this level seems to be too low in the other scenarios. In these cases selections with or have better prediction performance. Using post selection shrinkage slightly reduces the prediction errors with an advantage for parameterwise shrinkage.
8. Examples
8.1. Ozone Data
For illustration, we consider one specific aspect of a study on ozone effects on school children's lung growth. The study was carried out from February 1996 to October 1999 in Germany on 1101 school children in first and second primary school classes (6 - 8 years). For more details see [17]. As in [18] we use a subpopulation of 496 children and 24 variables. None of the continuous variables exhibited a strong non-linear effect, allowing to assume a linear effect for continuous variables in our analyses.
First, the whole data set is analyzed using backward elimination in combination with global and parameterwise shrinkage and LASSO. Selected variables with corresponding parameter estimates are given in Table 7 and mean squared prediction errors are shown in Table 8. The t-values are only given for the full model to illustrate the relation between the t-value and the parameterwise shrinkage factor. For variables with very large -values, PWSF are close to 1. In contrast, PWSFs are all over the place if is small, a good indication that variables should be eliminated.
Mean squared prediction errors for the full model and the BE models were obtained through double cross-validation in the sense that for each cross-validated prediction, the shrinkage factors were determined by crossvalidation within the cross-validation training set. Prediction error for the LASSO is based on single crossvalidation because double cross-validation turned out to be too time-consuming. Therefore, the LASSO prediction error might be too optimistic.
MSE is very similar for all models, irrespective of applying shrinkage or not (range 0.449 - 0.475; the full model with PWSF is the only exception), but the number
Figure 16. LASSO: distribution of the number of redundant covariates. There are eight redundant covariates in the design.
Figure 17. Average prediction errors for different strategies.
Table 7. Analysis of full ozone data set using standardized covariates. The unadjusted R2 equals 0.67 in the full model and drops only slightly to 0.64 for the BE(0.01) model. For the LASSO model it is 0.66.
Table 8. Mean squared prediction errors.
of variables in the model is very different. BE(0.01) selects a model with 4 variables, corresponding PWSF are all close to 1. Three variables are added if 0.05 is used as significance level. Using 0.157 selects a model with 12 variables. Two of them have a very low (below 0.3) PWSF, indicating that these variables may better be excluded. LASSO selects a complex model with 17 variables.
Although they carry relevant information the double cross-validation results for the full data set lack the intuitive appeal of the split-sample approach. To get closer to that intuition the following "dynamic" analysis scheme is applied. First the data are sorted randomly, next the first observations are used to derive a prediction model which is used to predict the remaining observations. This is done for ntrain = 100, 150, 200, 250, 300, 350 and repeated 100 times. In that way an impression is obtained how the different approaches behave with growing information. The results are shown in graphs. Figure 18 shows the mean number of covariates included. More variables are included with increasing sample size (larger power) and differences between the procedures are substantial. For ntrain = 350 LASSO selects on average 14.4 variables, whereas BE(0.01) selects only 4.0 variables. Figure 19 shows the evolution of the global shrinkage factor. For models selected with BE(0.01) it is always around 0.98 and for BE(0.157) it varies between 0.96 and 0.98, but for the full model the global shrinkage factor is much lower, starting from 0.83 with an increase to 0.95. Figure 20 shows the mean squared prediction errors for the different strategies. Without selection PWSF has a very bad performance, whereas the global shrinkage factor can slightly reduce MSE of prediction. PWSF improves the predictor if variable selection is performed and has smaller MSE than using a global shrinkage factor. The most important conclusion is that "BE(0.01) followed by PWSF" and LASSO have very similar prediction MSE's, but the LASSO has to include many more covariates to achieve that.
8.2. Body Fat Data
In a second example we will illustrate some issues in a study with one dominating variable. The data were first analysed in [19] and later used in many papers. 13 continuous covariates (age, weight, height and 10 body circumference measurements) are available as predictors for percentage of body fat. As in the book of [8] we excluded case 39 from the analysis. The data are available on the website of the book. In Table 9 we give mean squared prediction errors for several models and shrinkage approaches. Furthermore we give these estimates for variables excluding, the dominating predictor. This analysis assumes that would not have been measured or that it would not have been publicly available. A related analysis is presented in chapter 2.7 of [8] with the aim to illustrate the influence of a dominating predictor and to raise the issue about the term "full model" and whether a full model approach has advantageous properties compared with variable selection procedures. Using all variable MSEs of the models are very similar with a range from 18.76 - 20.80. Excluding leads to a severe increase, but differences between models are still negligible (25.87 - 27.47); with the full model followed by PWSF as an exception. This agrees well with the results of the ozone data. For BE(0.157), BE(0.01) and LASSO we give parameter estimates in Table 10.
Excluding results in the inclusion of other variables for all approaches. As in the ozone data LASSO hardly eliminates any variable, but the MSE is not better than from BE(0.01) followed by PWSF. PWSF of all variables selected by BE(0.01) are close to 1, whereas variables selected additionally by BE(0.157) all have PWSF values below 0.9 and sometimes around 0.6. This example confirms that BE(0.01) followed by PWSF gives similar prediction MSE's, but includes a much smaller number of variables.
9. Discussion and Conclusions
Building a suitable regression model is a challenging task if a larger number of candidate predictors is available. Having a situation with about 10 to 30 variables in mind the full model is often unsuitable and some type of variable selection is required. Obviously subject matter knowledge has to play a key role in model building, but often it is limited [20] and data-driven model-building is required. Despite of some critique in the literature [3,20,
Figure 18. Mean number of covariates included under different strategies.
Figure 19. Evolution of the global shrinkage factor for different selection levels. From top to bottom BE(0.010), BE(0.050), BE(0.157) and BE(1.000) = "No selection".
21] we consider backward elimination as a suitable approach, provided the sample size is not too small and the significance level is sensibly chosen according to the aim of a study. For a more detailed discussion see chapter 2 of [8]. Underand overfitting, model instability and bias of parameter estimates are well known issues of selected regression models.
In a simulation study and two examples we discuss the value of cross-validation, assess a global and a parameterwise cross-validation shrinkage approach, both without and with variable selection, and compare results with the LASSO procedure which combines variable
Figure 20. Mean squared prediction error for the different strategies.
Table 9. Mean squared prediction errors.
selection with shrinkage [2,4,5]. As discussed in the introduction it is often necessary to derive a suitable explanatory model which means that the effects of individual variables are important. In this respect a sparse model has advantages, both from a statistical point of view and from a clinical point of view. In a related context [22] refer to parameter sparsity and practical sparsity.
9.1. Design of Simulation Study
Our simulation design was used before for investigations of different issues [10]. We consider 15 covariates with seven of them having an effect on the outcome. In addition, multicollinearity between variables introduces indirect effects of irrelevant variables if a relevant variable is not included. It seems consensus that stepwise and other variable selection procedures have limited value as tools for model building in small studies ([8,21] Section 2.3) and 10 observations per variable is often considered as a lower boundary to derive an explanatory model [23]. Based on this knowledge it is obvious that a sample size of (6.7 per variable) is low. As a more realistic scenario we also consider. Concerning the residual variance we have chosen two scenarios resulting in and. The 4 scenarios reflect the situation with a low to a large(r) amount of information in the data. As expected from related studies the amount of information in the data has an influence on the results
Table 10. Parameter estimates for the standardized variables.
and therefore on the comparison between different procedures.
9.2. Cross-Validation and Shrinkage without Selection
The findings of Section 3 confirm that cross-validation does not estimate the performance of the model at hand but the average performance over all possible "training sets". The results of Section 4 confirm that global shrinkage can help to improve prediction performance in data with little information [2,11] like in the first scenario with and. However, the results show that the actual value of the global shrinkage factor is hard to interpret [9]. Shrinkage is a bit counterintuitive. Considerable shrinkage is a sign that something is wrong and application of shrinkage might even increase the prediction error. That is evident from the negative correlation between apparent and actual reduction in prediction error in the simulations from scenario 1. For the more informative scenarios 2-4 all shrinkage factors are close to one and predictors with and without shrinkage are nearly identical. It must be concluded that it is impossible to predict from the data whether shrinkage will be helpful for a particular data set or not.
Our results confirm that it does not make any sense to use parameterwise shrinkage in the full model [4]. Estimated shrinkage factors are not able to handle redundant covariates with no effect at all. Therefore they can have positive and negative signs and some of them have values far away from the intended range between 0 and 1.
9.3. Variable Selection and Post Selection Shrinkage
Most often prediction error is the main criteria to compare results from variable selection procedures. This implies that a suitable predictor with favorable statistical criteria are the main (or only) interest of an analysis. In contrast to such a statistically guided analysis researchers have often the aim to derive a suitable explanatory model and are willing to accept a model with slightly inferior prediction performance [6]. Excluding relevant variables results in a loss of and the inclusion of variables without effect complicates the model unnecessarily and usually increases the variance of a predictor. We used several criteria to compare the full model and models derived by using backward elimination with several values of the nominal significance level, the key criterion to determine complexity of a selected model, and the LASSO procedure. The number of variables is a key criterion for the interpretability and the practical usefulness of a predictor. BE(0.01) always selects the sparsest model, but such a low significance level may be dangerous in studies with a low amount of information. Our results confirm that BE(0.01) is very well suited if a lot of information is available. All stronger predictors are always included and only a small number of irrelevant variables is selected. Altogether BE with significance levels of 0.05 or 0.157 selected reasonable models. For studies with a sample size of 400 the loss in is acceptable and more than compensated when using the prediction error as criterion. Obviously, considering several statistical criteria the full model does not have advantages and the simulation study illustrates that some selection is always sensible. The results of Section 5 show that parameterwise shrinkage after selection can help to improve predictive performance by only correcting the regression coefficients that are borderline significant.
9.4. Comparison with LASSO and Similar Procedures
With the hype for high-dimensional data the LASSO approach [5] became popular. However, results in our simulation study and in the two examples are disappointing. As originally proposed we used cross-validation to select the penalty parameter lambda. However, even in the simplest situation of scenario 4 the variation is surprisingly large. In all scenarios a large number of redundant variables were selected which means that this approach is less suitable for variable selection. This is confirmed in the examples. From 24 candidate variables 17 were included in the model derived for the ozone data. In contrast, BE(0.01) selected a small model with only 4 variables, but the MSE was similar. The simultaneous combination of variable selection with shrinkage is often considered as an important advantage of LASSO and some of its followers, such as SCAD [24] or the elastic net [25]. We compare the LASSO results with post selection shrinkage procedures, in principle two-step procedures combining variable selection and shrinkage. In contrast to LASSO and related procedures post selection shrinkage is not based on optimizing any criteria under a given constraint and the approaches are somehow adhoc. Using cross-validation a global shrinkage approach was proposed [2] and later extended to parameterwise shrinkage [4]. These post selection shrinkage approaches can be easily used in all types of GLMs or regression models for survival data after the selection of a model. Whereas global shrinkage "averages" over all effects, parameterwise shrinkage aims to shrink according to individual needs caused by the selection bias. Our results confirm that parameterwise shrinkage helps to reduce the effect of redundant covariates that only enter by chance, while the global shrinkage is not able to make that distinction. The better performance concerning the individual factors results also in better predictions for parameterwise shrinkage compared to global shrinkage. The PWSF results confirm observations from another type of study on the use of cross-validation to reduce bias caused by model building [16]. Small effects that just survived selection are prone to selection bias and can profit from shrinkage. In contrast, large effects are always selected and do not need any shrinkage.
From the large number of newer proposals combining variable selection and shrinkage simultaneously, we considered only the LASSO in this work. In a comparison of approaches in low-dimensional survival settings [26] also compared results from the elastic net, SCAD and some boosting approaches. Using stability, sparseness, bias and prediction performance as criteria they conclude "overall results did not differ much. Especially in prediction performance, ..., the variation of them was not too large". They did not consider backward elimination followed by post-selection shrinkage, the approach which gave better results than the LASSO in our investigation, both in simulation and the examples. With a suitably chosen significance level for BE prediction performance was not worse compared to the LASSO and models selected were much sparser, an important advantage for interpretation and transportability of models. The investigation also shows that the PWSF approach has advantages compared to the global shrinkage factor. As our two step approach, BE followed by PWSF, can generally be used in regression models we consider it as a suitable approach when variable selection is a key part of data analysis.
9.5. Directions for Future Research
Like the choice of in LASSO, the choice of the significance level in the variable selection is crucial. Double cross-validation might be helpful in selecting, but reflection is needed about the criterion to be used. Prediction error is the obvious choice but does not reflect the need for a sparse model [27]. In order to improve research on selection procedures for high-dimensional data, several approaches to determine a more suitable or use two penalty parameters were proposed during the last years. It would be important to investigate whether they can improve model building in the easier low-dimensional situations As mentioned above, the approach of this paper can be easily implied for generalized linear models like model logistic regression and survival analysis. It would be interesting to see such applications.
10. Acknowledgements
We thank Gabi Ihorst for providing the ozone study data.
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