text:
Coronavirus Makes Work from Home the New Normal
Employers and workers adapt to new ways of getting work done.
#Rita Zeidner
By Rita Zeidner March 21, 2020
When two new hires showed up for their first day on the job at New York City commercial real estate company SquareFoot in early March, their orientation was anything but normal.
At any other time, the open-space office would have been bustling with activity and the company's 70-some employees would have given the newbies a hearty welcome. But with employees urged to work at home as a result of the coronavirus pandemic, only a skeleton staff was on hand to greet their newest co-workers, says Eugenie Fanning, the company's vice president of people. After completing some routine paperwork, the workers were handed laptops and sent home to learn the ropes of their new jobs and meet their co-workers online via videoconferencing.
A new crop of SquareFoot workers is slated to come on board in April, and no one knows when they will meet any of their co-workers face to face.
"We're just going to send the laptops directly to the new employees' homes," says Fanning, whose company decided in mid-March to shutter the office completely until the virus threat passes.
It's a scenario that has been playing out across the country as the number of coronavirus cases swells and more employers commit to the social-distancing practices that health officials say will help slow the spread of COVID-19, the respiratory illness caused by the virus. Large tech employers such as Apple, Google, Facebook and Microsoft—with significant populations based in Seattle and other cities hit hard by the pandemic—were among the first to ramp up remote work plans for many or all of their employees around the globe.
"Here at Microsoft in the Puget Sound, we're encouraging our teams to work from home as much as possible, as are many organizations in this region," wrote Jared Spataro, a Microsoft vice president in a March 5 blog. "At Microsoft, our top priority is the health and safety of employees, customers, partners and communities."
Technology Travails
For Microsoft, the choice to send tens of thousands home to work may have been simplified by employees' familiarity with the company's own Microsoft Teams, a widely used suite of products that ties together videoconferencing tools and other software to make it easier for workers to share files and work together, even when they are thousands of miles apart.
But for companies outside the tech sector and those not yet ordered by health officials to enact social-distancing protocols, mandating work at home has been more complicated. In 2017-18, only about 25 percent of U.S. workers worked at home at least occasionally and only 15 percent had days on which they only worked at home, according to the U.S. Bureau of Labor Statistics.
"For these large tech companies, working at home is in their DNA," says Dara Conroy, chief human resources officer at Tax Analysts (TA), a small, nonprofit publisher in Falls Church, Va. When it comes to fully embracing remote work, "most companies don't have the budget or resources," she says.
TA has had a flexible workplace policy in place for years, but it's mostly designed to address short-term situations such as a snowstorm or a sick child. While the company routinely evaluates which jobs are best suited to regular work-at-home arrangements—a valued perk in the congested Washington, D.C., metro region—the majority of the company's 170 employees are accustomed to working out of the company's headquarters, Conroy says.
"We have a very team-oriented culture," she says. "We want to make sure that we can keep our culture."
Like many private-sector employers, TA leaders had been following the dire news about the spread of the virus locally in early March and notified employees in advance of the possibility that the physical office, would be closed temporarily. A final decision was made in the middle of the month. But before that happened, employees were urged to practice social distancing by using e-mail, even when it may have seemed easier to carry a piece of paper to an employee down the hall. Laptops were handed out to workers who didn't already have them.
Reporters were given a refresher course on how to use videoconferencing and chats to conduct interviews, and people managers were coached on using the technology as a replacement for in-office meetings. But the company decided to delay the annual performance review process that had been scheduled for the spring rather than have those highly sensitive meetings take place via videoconference, Conroy says.
George Washington Law School in Washington, D.C., was among the many colleges and universities around the world that opted to cancel traditional classes in favor of online learning.
Rather than attempting to distribute videoconferencing equipment to its widely scattered faculty, the law school suggested instructors buy the equipment they needed and promised to reimburse them. Equipment recommendations were posted on the law school's intranet, along with detailed instructions. Faculty needing additional help recording a class or conducting one live in front of a camera can still come to campus and meet one-on-one with a media specialist.
"The idea is to try to prevent the virus from spreading and keep everyone safe, while holding as many classes as possible," says Hank Molinengo, GW's senior associate dean for administrative affairs.
Having a detailed work-at-home policy and knowledge of its computer capacity made it easier for F5 Networks Inc. to respond to the health crisis, says Ana White, the company's chief human resource officer. The technical services provider urged thousands of its workers in the Seattle and San Francisco regions to work from home beginning in early March, after a large number of coronavirus cases were reported in those areas. The company has since shuttered all of its physical offices worldwide.
Technology Eases Transition for New Remote Workers
Companies are sending employees home to work in large numbers to curb transmission of COVID-19. Many staffers will be working remotely for the first time. Experts share technology tips that can help ease the transition and ensure that workers remain productive and engaged.
Long before the outbreak, F5 had embraced flexible workplace arrangements as a way to increase employee motivation and promote work/life balance. But it has become increasingly clear that the flexibility is also a key component in keeping workers safe, White says.
"I'm so glad we rolled out [flexible work arrangements], as it helped us a lot with the changes due to COVID-19," she says. "We are leveraging the learnings from our [flexible workplace] program and continuing to learn as the majority of our employees are working from home in cities with COVID-19 outbreaks around the world."
To help spread the word on best practices, F5 has posted articles and videos about remote work on its employee website and provides guidance to supervisors on managing remote teams.
Amid the chaos surrounding the health scare, HR professionals say it's important for business leaders to take steps to lighten things up and help reduce feelings of isolation that some workers might experience working alone at home.
SquareFoot and F5 have created Slack channels where employees can post pictures of their at-home setups, and some workers are hosting virtual lunches and coffee chats using apps such as Microsoft Teams and Zoom.
"We're doing our best to keep people engaged," SquareFoot's Fanning says.
Practical Advice
Thinking about sending your workers home to work during the coronavirus pandemic?Here are some tips for getting started.
Establish an information command center and chief communicator. Now, more than ever, employees need information. "You need someone who's focused on working with disbursed groups," says Mari Anne Snow, chief executive officer and founder of Boston-based companies Sophaya, a remote work consultancy, and Remote Nation, which offers remote work products. Pick someone "who can talk about and help reduce employee fears, as well as deal with logistical concerns," she advises.
Change your productivity mindset. "It's not realistic to think we can maintain pre-coronavirus levels of productivity right now," says Cali Williams Yost, founder of the Flex Strategy Group, a flexible workplace consultancy. Instead, "focus on keeping everyone safe and healthy while maintaining as much productivity as possible."
Work with what you have. Organizations with a long history of teleworking likely spent years developing policies and procedures. "Now is not the time to start setting up the remote access infrastructure to test new processes," says Simona Rollinson, chief technology officer at ISACA, an information technology and cybersecurity association. Companies that must make decisions quickly about how to handle e-mail and other data remotely should "leverage the cloud," she says.
Be secure. It's never too late to update virus protections and remind employees about safe remote work processes, especially if they're using their personal laptops for official business. If possible, conduct a crash course on cybersecurity, Rollinson says. "You don't want people to take their laptop and connect at Starbucks," she adds. Keep an IT staffer on hand to help workers with their security and other technical issues.
Apply policies consistently and fairly. Be mindful that telework policies that have a negative or disparate impact on legally protected groups such as women or racial minorities can increase your exposure to a discrimination claim. Many states are asking employers to allow telework as much as possible and, in some instances, are requiring businesses to reduce capacity and the number of employees in the workplace.
It's not illegal for an employer to decide that some jobs lend themselves to remote work while others don't, says lawyer Linda Carlozzi of Jackson Lewis in New York City. But any employer that makes a call about who gets to work from home and who goes without a paycheck must be able to show a legitimate business reason, she says.
Rita Zeidner is a freelance writer based in Falls Church, Va.
SHRM provides advice and resources to help employers implement a remote work program or expand an existing one.
SHRM Resource Page: Coronavirus and COVID-19
These resources will help employers prepare their workforce for business closures, working from home, quarantines and other outcomes of a disease outbreak.
SHRM Resource Page: Remote Work
A collection of resources that can help employers facilitate flexible work arrangements while navigating a sophisticated array of technologies and remote work policies.
Quiz: Test Your Knowledge of Remote Workers
Working with a distributed workforce makes hiring and managing more complex and requires different sets of leadership skills and worker attributes. Take this quiz to test your knowledge on managing remote workers.
Coronavirus and Teleworking: Tips for Preparing Your Workforce
The coronavirus is prompting a wide range of employers to embrace telecommuting, but many employees are not experienced in working remotely. Here are some tips that can help.
Technology to Support Remote Workers Evolves
Many new technologies have emerged to meet employers' and remote employees' needs when it comes to getting work done
Job Interviews Go Virtual in Response to COVID-19
As employers begin asking employees to work from home to stop the spread of the coronavirus outbreak, virtual interactions are taking the place of in-person job interviews.
Business Acumen Risk Management


text:
A Year of Extremes: 2022 Weather Overview for Saskatoon
2022 will be remembered for many reasons. We experienced the world beginning to reopen, but as per usual for Saskatoon, we also experienced record-setting weather. Follow along as we recap the weather Saskatoon experienced throughout 2022.
Small But Mighty Weather Station at Black Fox Farm and Distillery
SRC's Climate Services team installed a weather station at Black Fox Farm and Distillery, which provides Black Fox's owners accurate weather data for their business. Learn more about the weather station and SRC's nearly sixty-year history tracking Saskatchewan's changing climate.
A Decade of Data: Climate Station at the Conservation Learning Centre Turns Ten
SRC's Climate Reference Station at the Conservation Learning Centre became fully operational in 2012. Find out how the station was started and why the Centre has been the ideal home for over a decade.
Sun, Wind and Storms: 2021 Weather Highlights for Saskatoon
2021 was another year for the record books in more ways than one—just like 2020. Here's a look back at 2021 and the weather that came with it in Saskatoon.
Make a World of Difference on Earth Day and Every Day
We're celebrating Saskatchewan's landscape with a list of great activities and interesting facts. The diverse landscapes across our province create unique ecosystems, filled with colourful flora and fauna—read on to learn how you can help and enjoy the natural wonder of the place we call home.
How about this weather? A look at Saskatoon in 2020
2020 was a year for the record books, and one none of us will soon forget. We spent our days starting in March at home, in the yard or awfully close to home. Let's look back at the weather we experienced in a year we were so glad to see the end of.
Mitigating GHG Emissions: New tech could meet mounting concerns about climate change
Given both global and national pressure to decarbonize operations, Canadian industries are looking for new technologies and implementing more energy-efficient processes from proven, applicable mitigation options. Learn how SRC's CeDER platform helps companies sift through the available technologies.
Climate change vulnerability assessment sheds light on future of boreal forest
Climate change is affecting Canada's forests to a greater extent than other parts of the globe. A climate change vulnerability assessment is currently underway in Saskatchewan and Manitoba with the goal of identifying what climate change will look like in the boreal forest and how to reduce its impacts.
Saskatoon weather report for 2019 shows a fairly average year in the City of Bridges
2019 won't be remembered in Saskatoon's history books for its weather. When averaged over the year, the weather was very typical for Saskatoon and surrounding area. While the year may have been fairly unremarkable, there were still some notable weather statistics and events. Learn more.
A Dry Saskatoon in 2018: Yearly Weather Summary
Take a look back at Saskatoon's top weather statistics and events of 2018 using data from SRC's Climate Reference Station. Some might be obvious – who could forget the thick smoke blanketing our city in August or the beautiful frost (freezing fog or rime frost) that set in on Saskatoon in late November – but some might also surprise you.
How SRC is Using Metal Smelting for Rare Earths
Top 5 Stories of 2022
Meet Our Experts: Philip Rees, Chief Security Officer and Facilities Manager

redpajama_set_name: RedPajamaCommonCrawl


text:
Buhari, Osinbajo get asset declaration forms 24hrs to deadline
Local, Politics
President Muhammadu Buhari and Vice-President Yemi Osinbajo received their asset declaration forms from the Code of Conduct Bureau on Monday, about 24 hours to the deadline on the submission of their declared assets.
The Chairman of the CCB, Prof. Muhammed Isah, presented the forms to Buhari and Osinbajo at a short ceremony at the Presidential Villa, Abuja.
Isah was at the Villa in company with some board members, including Mr Murtala Kankia and Dr Emmanuel Atta.
Buhari, Osinbajo and all political office holders are required by law to declare their assets prior to assuming office.
The CCB cited the legal requirements on Monday as they presented the forms to them.
The chairman said, "Mr President, as part of the constitutional requirements, there is need for every public officer – President, Vice-President, minister, members of the National Assembly – to swear in his assets declaration and liabilities in compliance with Paragraph 11, sub 1 of Part 1 of Fifth Schedule to the constitution.
"Mr President, it is in view of this that we found it pertinent to present ourselves and also present forms to Mr President and the Vice-President for end of tenure and beginning of new tenure in office.
"The forms are readily here with us for presentation."
Buhari, in a response, said he would speedily fill his form and submit same to the CCB in record time.
However, he asked the bureau to keep his form safely as he was sure that many people planning to come after him in 2023, would go for the form to use information against him.
Buhari noted that among such persons were those who had been caught by his anti-corruption war.
The President said, "I assure you I will quickly fill this form and dispatch it back to you so that at the end of 2023, I believe there are a lot of people that will like to get back at me.
"So, please, make sure you keep it safely because there are people who believe they shouldn't be questioned, which they are being questioned and some of them are already in trouble.
"I expect them to fight back and this is one of the instruments. So, I hope you will keep it when I finish."
He also had kind words for the CCB chairman, whom he said he was meeting for the first time after his appointment.
The Attorney-General of the Federation and Minister of Justice, Mr Abubakar Malami, and some senior government officials witnessed the event.
The PUNCH recalls that Buhari in 2015, through a statement by his Senior Special Assistant on Media and Publicity, Mr Garba Shehu, mentioned the assets he had declared at the CCB before assuming office.
On the list released in September, 2015, Buhari stated that he had N30m in his bank account before he assumed office in May of that year.
In addition to owning 270 cows, 25 sheep, five horses, birds and economic trees, there were five houses in Kaduna, Daura, Kano and Abuja. He said the Daura houses were made of "mud."
He also owned a plot of land each in Port Harcourt and Kano; farms, an orchard and some cars.
For Osinbajo, the Presidency stated that he had, "A bank balance of about N94m and $900,000 in his bank accounts."
The houses owned by Osinbajo back then were listed as "4-bedroomed residence at Victoria Garden City, Lagos and a 3-bedroomed flat at 2 Mosley Road, Ikoyi; 2-bedroomed flat at Redemption Camp along Lagos-Ibadan Expressway and a 2-bedroomed mortgaged property in Bedford, England."
The Presidency stated further, "Apart from his law firm, known as SimmonsCooper, the Vice-President also declared shareholding in six private companies based in Lagos, including Octogenerium Ltd., Windsor Grant Ltd., Tarapolsa, Vistorion Ltd., Aviva Ltd. and MTN Nigeria."
Tags: 24hrs, asset, Buhari, Deadline, Declaration, forms, get, Osinbajo, To
Previous Post Stop enrolling unqualified graduates into NYSC, DG warns universities
Next Post We'll create 20 million new jobs in next four years –FG


text:
Occupational Licensing
New York Might License Elevator Mechanics to Solve a Public Safety Problem That Doesn't Exist
Those claiming that elevators are a public safety risk likely have ulterior motives.
Alex Muresianu | 6.6.2019 11:30 AM
(Marcel Kusch/dpa/picture-alliance/Newscom)
Residents of New York City will take 35 million elevator rides today. That works out to an astronomical 12.77 billion elevator rides per year. An average of 2.75 of those rides end in a fatal accident. That's 404 times less likely than an ocean swimmer getting attacked by a shark.
Is that a problem that demands government action? Some lawmakers seem to think so. The State of New York is considering a bill to require licenses for elevator mechanics. Some recent elevator accidents, the legislation's authors argue, were caused by insufficiently trained mechanics.
In fact, there's good reason to doubt that New York has less safe elevators than the rest of the country or that requiring licenses would make them safer. The chief effect of those licenses would be to make it harder for New Yorkers to find jobs in that field.
Thirty-five states require licenses for workers in elevator repair or installation. There's not much empirical research comparing elevator safety on a state-by-state basis, but it's illuminating to compare elevator safety in license-free New York City to elevator safety across the country.
Nationally, there are 325 million elevator rides per day, or 118.625 billion rides per year—and there are roughly 27 elevator-related deaths a year, according to ConsumerWatch.com. That comes to .00000000023 deaths per elevator ride across the United States.
In New York City, there are 35 million elevator rides per day, or 12.775 billion rides per year. According to The Real Deal, which covers New York real estate, the city has seen 22 confirmed deaths in elevator-related accidents since 2010, or an average of 2.75 elevator deaths per year. That comes to .000000000215 deaths per elevator ride in the Big Apple.
In other words, the death rate per elevator ride is actually a little lower in New York City than in the rest of the country.
Meanwhile, stairs cause 1,600 deaths a year in the United States. By LiveScience.com's calculations, that makes them significantly more dangerous than taking the elevator.
Can government licensing make New York's elevators marginally safer? It's hard to tell for sure when you're working with numbers this tiny.
But this certainly seems like another instance where occupational licensing laws are being justified on flimsy public safety or consumer welfare grounds, even though many analyses of specific licenses have shown little connection between the existence or stringency of the license and public health, safety, or service quality.
Instead, licensing laws tend to enrich license holders, by protecting them from competition, at the expense of consumers, who pay higher prices, and at the expense of potential new workers, who face higher barriers to entry.
The most likely result of this bill, if it is passed, is to create new barriers to entry for workers looking to find a new job. They would have to pay a fee (licensing fees average $279 in New York), and they would have to either complete a certification program from an industry group, hold an out-of-state elevator mechanic license, or have four full years of experience in elevator maintenance, construction, or repair. Elevator mechanics make well above the average wage, and are a growing field; new fees and certification requirements will reduce the availability of a good middle-class job.
Occupational licensing laws currently on the books already place a significant burden on New York's economy. According to an analysis from the University of Minnesota economist Morris Kleiner, excessive licensing laws cost the state over 100,000 full-time jobs and almost $13.1 billion in economic growth. Instead of stacking on another licensing requirement, New York lawmakers should consider rolling back some of those existing regulations.
NEXT: Foot-Voting Nation
Alex Muresianu is a journalism intern at Reason.
Occupational Licensing Regulation New York New York City
June.6.2019 at 11:45 am
New York is so up and down on its freedom boot-stomping.
If we could just get someone to lift up licensing schemes enough to see the cronyism underneath.
June.6.2019 at 12:00 pm
Way to elevate the conversation.
June.6.2019 at 9:12 pm
But that'd give the rest of us the shaft.
That's just weird. It's not the mechanic that requires licensing, it's the elevator itself. That's where the safety is verified.
Do those NYC elevator death stats include the deaths in Die Hard With A Vengeance?
https://www.youtube.com/watch?v=_dPHjsyQL54
There is a bit of an elevator problem. It's more about NYCHA not having them properly inspected.
You seem to be missing half the story – it's not just shitty mechanics, it's shitty inspectors. And it's the NYC Department of Buildings that's failing to properly inspect the elevators. The real story is government proposing a new fix to a problem that government is responsible for creating.
"That comes to .00000000023 deaths per elevator ride across the United States."
BUT WHAT IF THAT .00000000023 IS A CHILD!!!!!!???????
Someone's monocle will go unpolished. That's the real tragedy.
Licensing elevator mechanics will just mean fewer of them are available. This will cause backups in elevator maintenance, which will cause additional accidents, bringing NYC's elevator fatality rate up to the national average.
NYC will rightfully identify this as a problem, and mandate additional licensing to alleviate the problem.
So they have an industry group that already has a cert program.
This should be resolved by insurance companies giving lower rates to buildings that use mechanics with existing certifications and higher rates to justify the risk for those without certs.
You deal with risk thorough insurance, not through licensing.
North49
Most of those deaths are not from the elevator breakdown itself, but people trying to 'die-hard' their way out of stuck elevators, and learning the hard way that they in fact die-all-too-easy.
rsteinmetz
Across the country there is a shortage of elevator mechanics and installers as the older generation retires and there have been insufficient numbers of younger people entering that trade. It is a significant problem.
Most of those are employed by a few very large companies Otis, Schindler, Kone and ThyssenKrupp. Most of them are also unionized, I'd expect in NYC pretty much 100% unionized. There are some smaller companies, mostly Asian subsidiaries of large companies from Japan and Korea but they are not as well known in the US. There are also some independent companies who hire mostly people trained by the bigger companies. The Independents do a lot of maintenance work but also some installations. In most places all elevator companies are licensed like plumbers and electricians. I'd be surprised it the companies providing maintenance services aren't already licensed.
In most places elevators are subject to periodic (usually annual) inspections. The Owners are usually required to maintain maintenance logs and perform periodic testing based on the national codes which the Inspector should verify.
Consensus Codes for Elevator Installation and Maintenance are in place under ANSI/ASME A17{1-8). Those codes are effectively developed by the major manufacturers and the union, all you have to do is look at who is on the committees.
Given the way the industry is organized it is difficult to see how an additional level of licensing is going to have much effect on safety given the level of safety that already exists.
Finally while anyone can make a mistake that could lead to a fatal accident, poor maintenance usually shows up first in other problems.
Phil Kelley
I've worked on elevators in Los Angeles since 1982 and have been a mechanic since '85. The State of California started requiring licensing about 15 years ago or so. The City of LA since before I got into the trade. So I, as well as all Los Angeles mechanics, carry two certifications.
I personally am not a fan of government involvement in our lives, unless absolutely necessary. I fought getting my state CCCM (Certified Competent Conveyance Mechanic), until people started loosing their jobs for not being licensed. I now believe California's licensing program has had a positive effect on safety. I'll explain my thoughts.
Prior to mechanic licensing, a contractor holding a C-11 license (California Elevator Contractor's License), could plausibly hire anyone off the street and call them a mechanic. Just from the safety aspect alone, this should not be allowed. If people had, even the smallest, perception of how dangerous elevators can potentially be, they should be demanding that all people working on the vertical transportation equipment they use be certified.
For the skeptics who say it will widen the barrier for people who wish to enter a truly great trade, it depends on how New York plans to implement the program. The State of California as well as the City of Los Angeles only require mechanics to be licensed. To be a mechanic, you must serve as an apprentice for several years to be properly trained. To be an apprentice, you need nothing more than a high school diploma or GED and a little good luck.
The purpose of government is to provide safety and infrastructure. On this one occasion, I believe the safety aspect may outweigh the government overreach.
Anyone interested in joining the elevator industry apprenticeship program should visit this site: https://www.NEIEP.org It may change your life
TxLibertarian
"If people had, even the smallest, perception of how dangerous elevators can potentially be…"
How dangerous are they to passengers? I'm not trying to be argumentative but I would be interested to know how many deaths or injuries are directly attributable to elevator malfunctions where those malfunctions were caused by improper maintenance by unlicensed elevator mechanics . I'm not talking about deaths caused by people trying to "rescue" themselves from a stuck elevator or some unexplained control system malfunction (lack of explanation = no root cause = could not have been prevented).
New York Might License Elevator Mechanics to Solve a Public Safety Problem That Doesn't Exist
[…] New York Might License Elevator Mechanics to Solve a Public Safety Problem That Doesn't Exist […]


, Aug 7, 2012 / 04:08 am ().- Archbishop William E. Lori of Baltimore believes that Pope Benedict's praise for the Knights of Columbus' efforts to defend religious freedom shows that the issue is broader than the confines of American politics.
The Pope's recent letter affirming the Knights, Archbishop Lori added, puts to rest suggestions that the group's religious liberty efforts have right-wing affiliations.
"This letter from the Holy Father is exceedingly important," Archbishop Lori told CNA in an exclusive Aug. 6 interview.
The Pope made his remarks before the Knights' annual assembly, held this year in Anaheim, Calif., from Aug. 6-9.
"Religious liberty touches man at his root, in his relation to his Creator. And so, the right to life and religious liberty are two sides to one coin," Archbishop Lori said.
With more than 1.8 million members, the Knights of Columbus is the world's largest fraternal group and among the nation's most active charitable organizations. Last year, the Knights donated more than $158 million and 70 million volunteer hours to various humanitarian causes.
In recent months, the group has been an outspoken opponent of measures that threaten religious freedom – most notably a federal mandate that requires employers to offer health insurance that includes contraception, sterilization and early abortion-inducing drugs, even if doing so violates their religious beliefs.
Each of the Knights' state councils passed a resolution supporting religious freedom, and Supreme Knight Carl Anderson has publicly opposed the mandate as well as other threats to religious liberty.
However, at a June 13 press conference at the U.S. bishops' spring meeting in Atlanta, Archbishop Lori was questioned by Jerry Filteau of the National Catholic Reporter about funding for the bishops' campaign to defend religious liberty.
Filteau said that he had heard "rumors" that much of the funding for the bishops' effort is coming from the Knights, whose head, Anderson, is a former Reagan administration official.
"It is not in any way partisan, either in its spirit or in its funding," he said.
"Because, if religious liberty is compromised or violated, then a very unjust society emerges and that is not in anyone's interest of any party or of any political stripe," the archbishop said.


text:
\section{Introduction}\label{intro}
The National Health and Nutrition Examination Survey (NHANES) asks a critical question:  ``For how many days during the past 30 days was your mental health not good?" The responses (\texttt{DaysMentHlthNotGood}) provide critical insights on self-reporting of mental health issues and can be linked to demographic, socioeconomic, behavioral, and health-related covariates.
Mental health and mental disorders are key factors in quality of life, depression, and risk of self-harm, and are focal points of many research studies   \citep{Scheid2017}. Previous research has sought to associate mental health, mental disorders, or depression with
gender \citep{Seedat2009},
race \citep{Williams2010race},
socioeconomic status \citep{Ortega1990},
marital status \citep{Williams2010},
age \citep{Mirowsky1999},
smoking \citep{Klungsoyr2006},
and
blood pressure \citep{Tzourio1999}, among many other factors. Notably, NHANES data include relevant covariates for \emph{all} of these factors---and several others (see Table~\ref{tab:data})---on a large sample of individuals, and thus offers a unique opportunity for a joint analysis of multiple factors.

From a statistical modeling perspective, \texttt{DaysMentHlthNotGood} is an overdispersed and zero-inflated count variable. While these features are common in practice, \texttt{DaysMentHlthNotGood} also presents new challenges. Figure~\ref{fig:pred} shows the empirical probability mass function (PMF) for \texttt{DaysMentHlthNotGood}. The PMF has spikes at both the lower bound (zero days) and the upper bound (30 days) and, most uniquely, heaping in five- and seven-day increments. This is an expected consequence of \emph{self-reported} behavior: individuals are more likely to report 14 or 15 days than they are 16 or 17 days regardless of the exact truth. Hence, \texttt{DaysMentHlthNotGood} is characterized by discreteness, overdispersion, zero-inflation, boundedness, and heaping.

\begin{figure}[h]
\begin{center}
\includegraphics[width=.49\textwidth]{Plots/pred_zip}
\includegraphics[width=.49\textwidth]{Plots/pred_star}
\end{center}
\caption{Empirical probability mass function (PMF) for \texttt{DaysMentHlthNotGood} with estimated PMFs for zero-inflated Poisson (ZIP, left) and \textsc{star}  (right).
While \textsc{star}  neatly captures zero-inflation, heaping (light gray), and boundedness, ZIP is inadequate for the nonzero counts.
\label{fig:pred}}
\end{figure}

Our goal is to construct an adequate count regression model for these data and characterize the effects of key covariates on self-reported mental health. More broadly, regression models for count data are useful in many fields, including ecology \citep{dorazio2005improving}, epidemiology \citep{kowal2019measles}, call center demand \citep{shen2008interday}, and emergency medical services \citep{matteson2011forecasting}. These and other applications require interpretable statistical models that provide reliable inference of regression effects and accurate count-valued predictions.

The majority of count regression models build upon the Poisson distribution. While Poisson regression can be suitable for point estimation or point prediction (see Section~\ref{sims}), it is frequently inadequate for modeling and inference due to the rigid equidispersion requirements.
Extensions for overdispersion or underdispersion typically introduce additional parameters or latent variables, such as quasi-Poisson \citep{mccullagh1989generalized}, Negative Binomial \citep{hilbe2011negative}, or Conway-Maxwell-Poisson  \citep{sellers2010flexible}. Further features such as zero-inflation can be appended to the Poisson model or its generalizations.
However, the added complexity of generalized Poisson models introduces computational challenges,
undermines the interpretability of the regression model, and most important, often fails to produce an adequate model. Figure~\ref{fig:pred} illustrates this point for zero-inflated Poisson regression: the model is incapable of capturing the key features of \texttt{DaysMentHlthNotGood} and fails crucial residual diagnostic checks (see Figure~\ref{fig:qqplots}); similar results are observed for Negative Binomial models.

An alternative strategy eschews count data models in favor of more familiar and flexible continuous data models \citep{st2018count}. Indeed, some authors recommend applying a transformation to the count data (such as logarithmic or square-root) and proceeding with Gaussian linear models \citep{ives2015testing}; others recommend against this practice \citep{o2010not} or provide strict guidelines under which it may be reasonable \citep{warton2016three}. This debate highlights two important yet conflicting points: continuous data models are often preferable---and occasionally  satisfactory---but are  fundamentally incoherent for count data and clearly cannot be expected to capture the key properties of \texttt{DaysMentHlthNotGood}.

Informed by these benefits and limitations, we develop a semiparametric regression model for count data. The count data regression model is constructed by \emph{simultaneously transforming and rounding} (\textsc{star}) a latent continuous data regression model. The latent regression model infuses the benefits of continuous data modeling while the rounding operator ensures a coherent integer-valued process. The transformation provides essential distributional flexibility and can be specified in advance (e.g., logarithmic or square-root), estimated nonparametrically via the marginal empirical cumulative distribution function (CDF), or estimated parametrically to match a given marginal distribution for the data (e.g, Poisson or Negative Binomial). In conjunction, the transformation and rounding operations effectively map a count variable with challenging distributional features to a more convenient latent space for continuous regression modeling. This strategy allows the modeler to deploy interpretable and familiar continuous data regression models, such as Gaussian linear regression, for settings with difficult count-valued distributions. The effectiveness of the proposed approach for  \texttt{DaysMentHlthNotGood} is advertised in Figure~\ref{fig:pred}: the \textsc{star}  model captures the key features of \texttt{DaysMentHlthNotGood} and offers significant improvements relative to existing count regression models (see Section~\ref{apps}).
Using likelihood-based inference, we provide a complete toolbox for count data regression: maximum likelihood estimators (MLEs) for estimation and prediction, hypothesis testing, confidence intervals, variable selection via information criteria, and customized diagnostics for \textsc{star}  regression.

\textsc{star}  models were initially developed by \cite{Kowal2020a} for Bayesian analysis. Similarly, \cite{canale2011bayesian} and \cite{canale2013nonparametric} used rounded latent data models for nonparametric Bayesian inference and count data regression, respectively, while \cite{Kowal2020target} applied \textsc{star}  for Bayesian functional data analysis. While these methods demonstrate the promise of this modeling strategy, these approaches are exclusively Bayesian and do not consider frequentist estimation, inference, or diagnostics. Our framework offers two additional contributions. First, the proposed EM algorithm is compatible with any (weighted) least squares estimator, which immediately adapts many existing (non-Bayesian) machine learning models for continuous data to be more suitable for count data. Second, the proposed strategy for learning the transformation via the marginal CDF---estimated nonparametrically or matched to a parametric count distribution---is distinct from these previous approaches and offers an efficient and simple strategy for improving model flexibility

In Section~\ref{star}, we introduce the \textsc{star}  regression model; Section~\ref{estimation} develops the estimation and inference for \textsc{star}  regression; Section~\ref{apps} applies the \textsc{star}  regression model to the NHANES data; Section~\ref{sims} evaluates the proposed methods on simulated data; Section~\ref{discuss} concludes.

\section{STAR models for integer-valued data}\label{star}
Consider the paired observations $\{\bm x_i, y_i\}_{i=1}^n$ of predictors $\bm x_i \in \mathcal{X} \subset \mathbb{R}^p$ and (non-negative) integer-valued response variables $y_i = y_i(\bm x_i) \in \mathcal{N}= \{0,1,\ldots,\infty\}$. Let $\mu_\theta \!:\mathcal{X}\to \mathbb{R}$ denote a real-valued regression function with parameters $\theta \in \Theta$, such as a linear model $\mu_\theta(\bm x) = \bm x'\bm\theta$. The \textsc{star}  framework seeks to link the regression function $\mu_\theta$ to the integer-valued data $y_i$ in a way that is interpretable, computationally convenient for estimation and inference, and well-defined for discrete data.  \textsc{star}  achieves these goals by introducing continuous latent random variables that are carefully connected to each other and the observable data via \emph{transformation} and \emph{rounding} operations.

\textsc{star}  begins with a regression model for latent Gaussian random variables $z^*\!:\mathcal{X}\to \mathbb{R}$:
\begin{equation}\label{spn}
z_i^* = \mu_\theta(\bm x_i) +  \epsilon_i, \quad \epsilon_i \stackrel{iid}{\sim}N(0,\sigma^2), \quad i=1,\ldots,n.
\end{equation}
The latent data model \eqref{spn} is ideal: the regression function $\mu_\theta(\bm x_i) = \mathbb{E}(z_i^* | \bm x_i)$ is the conditional expectation of $z^*$ at $\bm x_i$ and the errors $\epsilon_i$ are iid Gaussian. If the latent data $z_i^*$ were observable, then the model parameters $\theta$ in \eqref{spn} could be estimated by maximum likelihood

Instead, the latent data $z_i^*$ are linked to the observable data $y_i$ via transformation and rounding. The transformation step introduces a redundant latent variable $y_i^*\!: \mathcal{X}\to \mathcal{T}$ via a monotone transformation $g\!: \mathcal{T}\to \mathbb{R}$:
\begin{equation}\label{transform}
z_i^* = g(y_i^*).
\end{equation}
The transformation $g$  may be  known, such as logarithmic or square-root, or unknown and learned from the data (see Section~\ref{learn-g}). The purpose of $g$ is to provide modeling flexibility: the latent $y_i^*$ are transformed to $z_i^*$ and modeled as Gaussian. An appropriate choice of $g$ can improve the suitability of model \eqref{spn} for any regression function $\mu_\theta$.

Lastly, the latent data $y_i^*$ are linked to the observed data via a rounding operation $h\!: \mathcal{T}\to \mathbb{N}$:
\begin{equation}
\label{round}
y_i = h(y_i^*)
\end{equation}
where $h(y^*) = j \in \mathcal{N}$ when $y^* \in \mathcal{A}_j$ and $\{\mathcal{A}_j\}_{j=0}^\infty$ is a known partition of $\mathcal{T}$. We use rounding operators of the form $\mathcal{A}_j = [a_j, a_{j+1})$ for known $a_j \in \mathcal{T}$, which maps intervals to integers. The latent data $y_i^*$  provide a continuous proxy for the count data $y_i$, which are matched to the regression model \eqref{spn} through a transformation in \eqref{transform}.

The latent data $y_i^*$  are redundant: we may compress \eqref{transform}--\eqref{round} into
$y_i = h\{ g^{-1}(z_i^*)\}$
which, combined with \eqref{spn}, constitutes the \textsc{star}  regression model. From this perspective, the composition $h \circ g^{-1}$ is sufficient for mapping the Gaussian model \eqref{spn} to $\mathcal{N}$. We prefer to decouple $h$ and $g$ for several reasons. First, there exist many approaches for modeling a smooth and continuous function, which we can leverage to estimate the transformation $g$ (see Section~\ref{learn-g}). Second, $h$ can be specified to provide key distributional features, such as zero-inflation, boundedness, or censored data (see Lemma~\ref{star-prop}) without requiring any modifications to the estimation algorithm (see Section~\ref{em-alg}). Finally, the decoupled representation \eqref{spn}--\eqref{round} in insightful: \eqref{spn}--\eqref{transform} is comparable to the common practice of transforming the observed counts $y_i$ and modeling the transformed data as Gaussian, which is obtained by replacing $y_i^*$ with $y_i$ in \eqref{transform}. This approach implicitly acknowledges the utility of both the transformation and the continuous regression model. However, such models are not coherent for integer-valued data. The rounding operation in \eqref{round} corrects for this incoherence while preserving the core attributes of the transformed regression model.

The \textsc{star}  model not only incorporates a broad class of regression models in \eqref{spn}, but also offers vital distributional features for modeling count data:
\begin{lemma}[\citealp{Kowal2020a}]\label{star-prop}
The \textsc{star}  model \eqref{spn}--\eqref{round} has the following properties:
\begin{enumerate}
\item \emph{Over/underdispersion:} for fixed $g$, there exist values of $(\mu_\theta, \sigma)$ such that $\mathbb{E}(y) < \mbox{Var}(y)$ (overdispersion) and  values of $(\mu_\theta, \sigma)$ such that $\mathbb{E}(y) > \mbox{Var}(y)$ (underdispersion).

\item \emph{Zero-inflation:} the probability of a zero count is
$\mathbb{P}\{y(\bm x) = 0\} = \mathbb{P}\{z^*(\bm x) < g(a_1)\} = \Phi \left[ \{g(a_1) - \mu_\theta(\bm x_i)\}/\sigma\right]$.

\item \emph{Boundedness:} suppose there exists an upper bound $y_{max} \in \{1, \ldots, \infty\}$ for the counts $y$. By specifying the rounding operator such that $a_{y_{max} + 1} = \infty$ and requiring $\lim_{t \rightarrow\infty}g(t) = \infty$, we have  $\mathbb{P}(y \in \{0,1,\ldots, y_{max}\}) = 1$.

\item \emph{(Right) Censoring:} suppose only $y_i^c = \min\{y_i, C\}$ is observed and let  $\delta_i = \mathbb{I}\{y_i \ge C\}$ be a known censoring indicator. The likelihood for $\{y_i\}_{i=1}^n$ is
\begin{align*}
L_c(\bm y; \theta, \sigma) &= \prod_{i: \delta_i = 0} \mathbb{P}(y_i) \prod_{i: \delta_i = 1} \mathbb{P}(y_i \ge C) \\
&=  \prod_{i: \delta_i = 0} \mathbb{P}\{z_i^*\in [g(a_{y_i}), g(a_{y_i + 1}))\} \prod_{i: \delta_i = 1} \mathbb{P}\{z_i^* \ge g(a_C)\}
\end{align*}
which is \emph{identical} to the likelihood for $\{y_i^c\}_{i=1}^n$ subject to $y_{max} = C$ with $a_{y_{max} + 1} = \infty$ and $\lim_{t \rightarrow\infty}g(t) = \infty$.
\end{enumerate}
\end{lemma}

The distributional features in Lemma~\ref{star-prop} are nontrivial. First, the capability to model over- or underdispersion immediately offers an advantage relative to Poisson models, which require $\mathbb{E}(y) = \mbox{Var}(y)$. Common generalizations of  Poisson models, such as Negative Binomial models, can capture overdispersion but not underdispersion.
Next, zero-inflation simply corresponds to an abundance of small values of $z^*$. For example, if $g(t) = \log(t)$ and $a_j = j$, then $\mathbb{P}\{y(\bm x) = 0\}  = \mathbb{P}\{z^*(\bm x) < 0\}$, which is the probability of negative latent data in the regression model \eqref{spn}. \textsc{star}  is also compatible with a mixture-based approach for modeling zeros analogous to zero-inflated models; however, we have found that the baseline \textsc{star}  model \eqref{spn} is adequate for modeling an abundance of zeros in both real and simulated datasets.

Among regression models for count data, the boundedness and censoring properties in Lemma~\ref{star-prop} are unique to \textsc{star}. Boundedness and censoring are common in practice: for example, boundedness occurs for any count of daily events within a month ($y_{max} = 30$) and censoring occurs whenever data are recorded as ``$C$ or more" counts ($y_{max} = C$). Each property is easily incorporated into the model by setting $a_{y_{max} + 1} = \infty$ and requiring $\lim_{t \rightarrow\infty}g(t) = \infty$, which provides valid likelihood-based estimation and inference for bounded or censored data. No additional modifications to the estimation algorithm are required and the regression  specification in \eqref{spn} is not affected. By comparison, boundedness or censoring of Poisson and related models require a reformulation of the likelihood and customized estimation algorithms.

The latent representation for the \textsc{star}  model in \eqref{spn}--\eqref{round} produces a convenient (log-) likelihood with important properties for estimation. The PMF implied by \eqref{spn}--\eqref{round} is
\begin{equation}\label{pmf}
\mathbb{P}(y_i = j)= \mathbb{P}(y_i^* \in \mathcal{A}_j) = \mathbb{P}\{z_i^* \in g(\mathcal{A}_j)\} = \int_{g(\mathcal{A}_j)} \phi(z; \mu_\theta(\bm x_i), \sigma^2) \, dz
\end{equation}
for $j \in \mathcal{N}$, where $\phi(\cdot)$ is the Gaussian density function implied by \eqref{spn}. For rounding operators of the form $\mathcal{A}_j = [a_j, a_{j+1})$, it follows that $g(\mathcal{A}_j) = [g(a_j), g(a_{j+1}))$ due to the monotonicity of $g$. Letting $\bm y = (y_1,\ldots, y_n)'$ denote the observed integer-valued data and noting the independence assumptions in \eqref{spn}, it follows that the log-likelihood is
\begin{equation}\label{log-like}
\ell(\bm y; \theta, \sigma)
=
\sum_{i=1}^n \log \left[
\Phi\left\{\frac{g(a_{y_i + 1}) - \mu_\theta(\bm x_i)}{\sigma}\right\} -
\Phi\left\{\frac{g(a_{y_i }) - \mu_\theta(\bm x_i)}{\sigma}\right\}
 \right]
\end{equation}
where $\Phi$ denotes the CDF of a standard normal random variable. The log-likelihood does not include latent data: the \textsc{star}  model can be constructed directly from \eqref{log-like} without reference to $y_i^*$  or $z_i^*$. Evaluation of \eqref{log-like} is critical for subsequent hypothesis testing, confidence intervals, variable selection (via information criteria), and residual diagnostics (see Sections~\ref{estimation}~and~\ref{apps}).

Despite the apparent complexity of the log-likelihood in \eqref{log-like}, we obtain useful model properties and provide an efficient maximum likelihood estimation algorithm (Section~\ref{estimation}). For the \textsc{star}  linear regression model $\mu_\theta(\bm x) = \bm x ' \bm \theta$, we have the following important result:
\begin{lemma}\label{global-concave}
Suppose $\mu_\theta(\bm x_i) = \bm x_i'\bm\theta$ and $\bm X = (\bm x_1, \ldots, \bm x_n)'$ is full rank. The log-likelihood \eqref{log-like} is strictly concave in $(\bm \theta/\sigma, 1/\sigma)$.
\end{lemma}
The result follows immediately from \cite{pratt1981concavity}. The primary implication of Lemma~\ref{global-concave} is that MLEs for the \textsc{star}  linear model can be computed easily and efficiently using a variety of algorithms. In Section~\ref{estimation}, we introduce an Expectation-Maximization (EM) algorithm to compute MLEs for \textsc{star}  models.  Lemma~\ref{global-concave} implies that the EM algorithm converges to the global MLE for \textsc{star}  linear models---and therefore avoids the local optima that often plague EM algorithms. Global concavity of the log-likelihood has other important implications for the MLEs:
\begin{lemma}\label{unique}
Under the conditions of Lemma~\ref{global-concave}, the MLEs for $(\bm \theta, \sigma)$ exist and are unique.
\end{lemma}
Lemmas~\ref{global-concave}~and~\ref{unique} do not consider joint estimation of the transformation $g$ with the parameters $(\bm \theta, \sigma)$. These results hold under one of two conditions on $g$: either the transformation is known or the transformation is estimated prior to computing the MLEs (see Section~\ref{learn-g}).

The log-likelihood in \eqref{log-like} resembles that of ordinal regression, which has been used for modeling count data \citep{valle2019ordinal}. However, there are important limitations of ordinal regression that are not present for \textsc{star}. First, ordinal regression does not incorporate the numeric values of the data $y_i$ and instead only uses the ranks. That approach is reasonable for modeling ordered categories absent any numeric meaning, but ignores useful information available for count data. Second, ordinal regression introduces a latent cut point for every unique data value, each of which must be estimated. The analogous quantity in \textsc{star}  is the transformation $g$, which may be known or unknown, but readily incorporates smoothness to provide regularized estimation. Lastly, ordinal regression is only supported on the \emph{observed} values (or levels) of $y_i$ rather than $\mathcal{N}$, which is unnecessarily restrictive and suboptimal for predictive modeling.

\section{Estimation and inference for STAR models}\label{estimation}

\subsection{Data-driven determination of the transformation}\label{learn-g}
The transformation $g$ is an essential component of the \textsc{star}  model \eqref{spn}--\eqref{round}. Although the rounding operation \eqref{round} ensures a \emph{valid} count-valued process, it does not guarantee an \emph{adequate model} for count data. The transformation $g$ assists in this endeavor and provides the vital link between the continuous regression model  \eqref{spn} and the count-valued coherence of \eqref{round}.

We propose to learn the transformation $g$ by linking it to the marginal CDF of $y$:
\begin{equation}\label{cdf}
F_y(j) = \mathbb{P}(y \le j) = \mathbb{P}\{z < g(a_{j+1})\} = F_z\{g(a_{j+1})\}
\end{equation}
where $F_z$ is the CDF of $z^*$ and $\mathcal{A}_j = [a_j, a_{j+1})$ for $j  \in \mathcal{N}$. Gaussianity is assumed for the latent $z^*$ in \eqref{spn} and determines $F_z$. Let  $z^* \sim N(\mu_z, \sigma_z^2)$ denote a working marginal distribution. The transformation $g$ is identifiable only up to location ($\mu_z$) and scale ($\sigma_z$). This parametrization will allow us to specify a location and scale of the transformation to match the first two moments of $z^*$ with those of $y$, which lends interpretability to the parameters $\mu_\theta$ and $\sigma$ in \eqref{spn}. Since $F_z(t) = \Phi\{(t - \mu_z)/\sigma_z\}$ for $t \in \mathbb{R}$, it follows from \eqref{cdf} that
$g(a_{j+1}) = \mu_z + \sigma_z \Phi^{-1}\{F_y(j)\}.$
For the default case of $a_0 = -\infty$, $a_j = j$ for $j=1,\ldots,y_{max}$, and $a_{y_{max}+1} = \infty$, we define the transformation more broadly:
\begin{equation}\label{g-R}
g(t) = \mu_z + \sigma_z \Phi^{-1}\{F_y(t - 1)\}, \quad  t \in [1,   y_{max} + 1)
\end{equation}
with $g(t) = -\infty$ for $t<1$ and $g(t) = \infty$ for $ t \ge y_{max} + 1$.
Similarly, we define the inverse to be $g^{-1}(s) = F_y^{-1}[\Phi\{(s - \mu_z)/\sigma_z\}] + 1$ and set $\lim_{s \rightarrow -\infty} g(s) = 0$ and $\lim_{s \rightarrow \infty} g(s) = y_{max} + 1$.

The representation in \eqref{g-R} suggests a simple estimation procedure. First, we match the marginal moments of $z^*$ and $y$: $\mu_z = \bar y$ is the sample mean and $\sigma_z = \hat s_y$ is the sample standard deviation of $\{y_i\}_{i=1}^n$.   Next, we substitute a point estimate for the CDF of $y$ using the nonparametric estimator \begin{equation}\label{g-np0}
\hat g_0(t) = \bar y + \hat s_y \Phi^{-1}\{\tilde F_y(t-1)\}, \quad t \in [1, y_{max} + 1)
\end{equation}
where $\tilde F_j(j) = n(n+1)^{-1} \hat F_y(j)$ is a rescaling of the empirical CDF  $\hat F_y(j) = n^{-1}\sum_{i=1}^n \mathbb{I}\{y_i \le j\}$ to avoid boundary issues. As in \eqref{g-R}, we set $\hat g_0(t) = -\infty$ for $t<1$ and $\hat g_0(t) = \infty$ for $t \ge y_{max}+1$. A similar strategy of pre-estimating a transformation using the empirical CDF has been applied for Gaussian graphical models  \citep{Liu2009}.  Parametric alternatives, such as Poisson or Negative Binomial CDFs, offer the ability to conform \textsc{star}  with a pre-specified marginal distribution for the counts. In this case, the accompanying parameters of $F_y$ must be estimated. By default, we use method-of-moment estimators that recycle $\bar y$ and $\hat s_y$.

The transformation  \eqref{g-np0} resembles a classical Gaussian copula model. In fact, $\hat g_0$ eliminates the need for the rounding operator $h$: since $F_y(t) = F_y(\lfloor t \rfloor )$ for $y \in \mathcal{N}$,  $\hat g_0(t) = \hat g_0(\lfloor t \rfloor )$ is a step function with integer-valued inverse $\hat g_0^{-1}(z^*) \in \mathcal{N}$ for $z^* \in \mathbb{R}$. The critical drawback of \eqref{g-np0} is that the data-generating process defined by $\hat g_0^{-1}(z^*)$ for Gaussian $z^*$ can \emph{only} generate the observed values $\{y_i\}_{i=1}^n$. Therefore, any value in $\mathcal{N}$ that is not observed in the data is also not supported by the \textsc{star}  model with transformation \eqref{g-np0}: $\mathbb{P}(y = j) = 0$ whenever $g(a_j) = g(a_{j+1})$, which occurs whenever $j \not \in \{y_i\}_{i=1}^n$. This restriction is unnecessary and can inhibit prediction.

To remedy this issue, we again decouple the rounding and transformation functions. We compute a smooth, monotonic interpolation of $\hat g_0$ on the observed data values $\{y_i\}_{i=1}^n$:
\begin{equation}\label{interpo}
\hat g(y_i) = \hat g_0(y_i), \quad i=1,\ldots,n
\end{equation}
where $\hat g$ is a monotonic cubic spline  estimated using the \texttt{R} function \texttt{splinefun(..., method = `monoH.FC')}  \citep{Fritsch1980}. The limits $\hat g(t) = -\infty$ for $t<1$ and $\hat g(t) = \infty$ for $t \ge y_{max}+1$ are imposed as well. The estimate $\hat g$ interpolates $\hat g_0$ at each increment, but connects these points smoothly and monotonically to induce a data-generating process supported wholly on $\mathcal{N}$. As a result, $\hat g$ preserves the best of both worlds: the smoothness expands the data-generating process to support all of $\mathcal{N}$, while the interpolation \eqref{interpo} ensures that $\hat g$ recovers the same (empirical) CDF as $\hat g_0$ at the observed data values $\{y_i\}_{i=1}^n$. Lastly, since $\hat g$ is estimated in advance of $\theta$ and $\sigma$, the log-likelihood concavity results from Lemma~\ref{global-concave} are unaffected. We refer to the \textsc{star}  model with this nonparametric estimator of $g$ as \textsc{star}-np.

For Bayesian \textsc{star}  models, \cite{Kowal2020a} proposed Box-Cox transformations $g(t ; \lambda) = (t^\lambda - 1)/\lambda$ with $\lambda > 0$ and $g(t; \lambda =0) = \log(t)$.  Important special cases include the (shifted) identity transformation $\lambda = 1$, the (shifted and scaled) square-root transformation $\lambda = 1/2$, and the log-transformation $\lambda = 0$. Box-Cox functions are popular for transforming continuous data towards Gaussianity, which in the present setting corresponds to \eqref{spn}--\eqref{transform}. However, these transformations impose a restrictive parametric form, often lack interpretability, and require selection or estimation of $\lambda$. Nonetheless, Box-Cox transformations are compatible with the proposed approach.

\subsection{An EM algorithm for maximum likelihood estimation}\label{em-alg}

We introduce an EM algorithm \citep{dempster1977maximum} for computing MLEs under the \textsc{star}  model.
EM algorithms use iterative optimizations that exploit the computational convenience of a \emph{complete data} likelihood. For \textsc{star}, we define the complete data to be $(\bm y, \bm z^*)$ with latent data $\bm z^* = (z_1^*,\ldots, z_n^*)'$. For simplicity, let $\theta$ denote all unknown parameters, including $\sigma^2$. The complete data log-likelihood for \textsc{star}  is
\begin{align}
\label{complete-1}
\log p(\bm y, \bm z^* | \theta) &=  \log p(\bm y | \bm z^*, \theta) + \log p(\bm z^* | \theta)\\
\label{complete-3}
&= C(\bm y, \bm z^*) - \frac{1}{2} \sum_{i=1}^n \log(\sigma^2) - \frac{1}{2} \sum_{i=1}^n \Big[\{z_i^* - \mu_\theta(\bm x_i)\}^2/\sigma^{2}\Big]
\end{align}
where $C(\bm y, \bm z^*)$ is a constant that does not depend on $\theta$.
The EM algorithm iteratively updates the parameters $\theta$ by  alternating the following steps at each iteration $s$:
\begin{itemize}
\item[] {\bf E-Step:} Compute  $Q(\theta, \theta^s) = \mathbb{E}_{[z^* | y, \theta = \theta^s]}\log p(\bm y, \bm z^* | \theta)$, where the expectation is taken with respect to the distribution $[\bm z^* | \bm y, \theta]$ for $\theta = \theta^s$ the parameter value at iteration $s$;
\item[] {\bf M-Step:} Maximize the conditional expectation, $\theta^{s+1} = \arg\max_{\theta} Q(\theta, \theta^s)$
\end{itemize}
The algorithm proceeds until convergence, which is typically assessed by monitoring the stability of the log-likelihood \eqref{log-like} across iterations.
Note that although the EM algorithm always produces non-decreasing log-likelihoods across iterations $s=1,\ldots,S$, in general the algorithm is not guaranteed to converge to a global maximum. However, for  \textsc{star}   linear models (Lemma~\ref{global-concave}) and other special cases, the EM algorithm will converge to the unique global MLEs.

Under the \textsc{star}  model, computation of $Q(\theta, \theta^s)$ is straightforward:
\begin{eqnarray} \label{estep}
Q(\theta, \theta^s) &= & C(\bm y) - \frac{1}{2} \sum_{i=1}^n \log(\sigma^2)- \frac{1}{2} \sum_{i=1}^n \mathbb{E}_{[z^* | y, \theta = \theta^s]}\left[\{z_i^* - \mu_\theta(\bm x_i)\}^2/\sigma^{2}\right] \nonumber \\
&=& C(\bm y) - \frac{n}{2} \log(\sigma^2)
- \frac{1}{2\sigma^2}\left\{ \sum_{i=1}^n  \hat z_i^{(2)}
-2  \sum_{i=1}^n \mu_\theta(\bm x_i)\hat z_i^{(1)}
+ \sum_{i=1}^n \mu_\theta(\bm x_i)^2 \right\}
\end{eqnarray}
 where $C(\bm y)$ is a constant that does not depend on $\theta$ and $\hat z_i^{(k)}  =\mathbb{E}\left\{(z_i^*)^k | y, \theta = \theta^s\right\}$, $k=1,2$ is the $k$th moment of the distribution $\left[z_i^*| \bm y,  \theta = \theta^s\right]$.
 Under model \eqref{spn}, this distribution is  $\left[z_i^*| \bm y, \theta\right] \stackrel{indep}{\sim} N(\mu_\theta(\bm x_i), \sigma^2)$ truncated to $g(\mathcal{A}_{y_i})$ and using the parameter values $\theta = \theta^s$. For the common special case of $\mathcal{A}_j = [a_j, a_{j+1})$,   the required moments are computable as follows:
 \begin{align}
 \label{m1}

 \hat z_i^{(1)}  &= \mu_\theta(\bm x_i) + \sigma \frac{\phi(a_i) - \phi(b_i)}{\Phi(b_i) - \Phi(a_i)} \\
  \label{m2}

 \hat z_i^{(2)} &=  \mu_\theta(\bm x_i)\left\{\mu_\theta(\bm x_i) + 2 \sigma \frac{\phi(a_i) - \phi(b_i)}{\Phi(b_i) - \Phi(a_i)}\right\} + \sigma^2 \left\{
 1 + \frac{a_i\phi(a_i) - b_i\phi(b_i)}{\Phi(b_i) - \Phi(a_i)}
 \right\}
 \end{align}
 where $\phi$ is the standard normal density function, $a_i = \{g(a_{y_i}) - \mu_\theta(\bm x_i)\}/\sigma$, and $b_i = \{g(a_{y_i + 1}) - \mu_\theta(\bm x_i)\}/\sigma$. Note that the parameters in \eqref{m1}--\eqref{m2} will update at each iteration $s$ in the M-step.

The form of \eqref{estep} produces important simplifications for the M-step:
\begin{lemma}\label{EM-gen}
Consider a \textsc{star}  model \eqref{spn} generalized to include heteroskedastic errors, $\epsilon_i \stackrel{indep}{\sim}N(0,\sigma^2/w_i)$, where $\theta \in \Theta$ and $\sigma^2 > 0$ are the unknown parameters and $w_i > 0$ are known weights.

  The M-step solutions for $(\theta, \sigma^2)$ are $\hat \theta = \hat \theta( \{\bm x_i,  \hat z_i^{(1)}\}_{i=1}^n)$ and \begin{equation}
\label{mle-sig}
\hat \sigma^2 = \frac{1}{n}\Big\{\sum_{i=1}^n w_i \hat z_i^{(2)}  + \sum_{i=1}^n  w_i \mu_{\hat \theta}^2(\bm x_i) -2 \sum_{i=1}^n w_i \mu_{\hat \theta}(\bm x_i)\hat z_i^{(1)} \Big\},
\end{equation}
where $\hat \theta( \cdot)$ is the weighted least squares solution to
\begin{equation}\label{sse}
\hat \theta( \{\bm x_i, d_i\}_{i=1}^n) = \arg\min_{\theta \in\Theta} \sum_{i=1}^n w_i \{d_i - \mu_\theta(\bm x_i)\}^2
\end{equation}
for paired observations $(\bm x_i, d_i)$ with $d_i \in \mathbb{R}$ and $w_i  > 0$.
\end{lemma}
The proof is in the supplementary material, along with a complete  description of the EM algorithm.  For \emph{any} model that can be estimated using weighted least squares, Lemma~\ref{EM-gen} implies that the same model can be adapted for integer-valued data using the \textsc{star}  model \eqref{spn} and that the algorithm solving the least squares problem \eqref{sse} is sufficient for
estimation in \textsc{star}

To compute fitted values from \textsc{star}, we note that the expected value of the response is
\begin{equation}\label{exp-count}
\mathbb{E}\{y(\bm x)\} = \sum_{j=0}^\infty j \mathbb{P}\{y(\bm x) = j\} = \sum_{j=0}^\infty j \Big\{\Phi\Big( \frac{g(a_{j+1}) - \mu_\theta(\bm x)}{\sigma}\Big) -  \Phi\Big( \frac{g(a_{j}) - \mu_\theta(\bm x)}{\sigma}\Big)\Big\}
\end{equation}
and is a function of $\theta$ and $\sigma$. The invariance property of MLEs ensures an MLE for $\mathbb{E}\{y(\bm x)\}$:
\begin{lemma}\label{mle}
The maximum likelihood estimator of $\mathbb{E} \{y(\bm x)\}$ conditional on $\bm x \in \mathcal{X}$ is
\begin{equation}\label{mle-Ey}
\hat y(\bm x) \equiv \sum_{j=0}^\infty j\Big[\Phi\Big\{ \frac{g(a_{j+1}) -  \mu_{\hat \theta}(\bm x)}{\hat\sigma}\Big\}-  \Phi\Big\{ \frac{g(a_{j}) - \mu_{\hat \theta}(\bm x)}{\hat\sigma}\Big\}\Big]
\end{equation}
where $\hat \theta$ and $\hat \sigma$ are MLEs for $\theta$ and $\sigma$, respectively.
\end{lemma}
Extensions for heteroskedastic errors as in Lemma~\ref{EM-gen} are straightforward. In practice, we truncate the summation at $J(\bm x)$ pointwise for each $\bm x$, where $J(\bm x)$ is the 99.99th quantile of the distribution of $y(\bm x)$ and computed as $h[g^{-1}\{z_q^*(\bm x)\}]$ where $z_q^*(\bm x)$ is the $q$th quantile of the distribution of $z^*(\bm x)$.

\subsection{Inference for STAR models}\label{inference}
We develop likelihood-based inference for the \textsc{star}  model \eqref{spn}. Under classical regularity conditions, which are satisfied for the linear regression model $\mu_\theta(\bm x) = \bm x' \bm\theta$, the \textsc{star}  MLEs are consistent and admit generalized likelihood ratio tests for asymptotic inference.
For the composite hypothesis test $H_0: \theta \in \Theta_0$ against the alternative $H_1: \theta \in \Theta \setminus \Theta_0$ for $\theta \in \Theta$ and $\Theta_0 \subset \Theta$, the likelihood ratio test statistic is
$\Lambda(\bm y) = \sup_{\theta \in \Theta_0} p(\bm y | \theta)/\sup_{\theta \in \Theta} p(\bm y | \theta)$,
where $p(\bm y|\theta)$ denotes the likelihood. The test statistic $\Lambda(\bm y)$ requires the global MLE for $\theta \in \Theta$, the restricted MLE for $\theta \in \Theta_0$, and the likelihood evaluated at these MLEs. Under \eqref{spn}, the
 distribution of $y$ is given by \eqref{pmf} and the log-likelihood \eqref{log-like} is sufficient. The rejection region for the likelihood ratio test is defined as $\{\bm y: \Lambda(\bm y) \le c\}$ for some $c \in [0,1]$.

The asymptotic distribution of the likelihood ratio test statistic is known: under the null hypothesis, $-2 \log \Lambda(\bm y) \sim \chi_r^2$ as $n \rightarrow \infty$, where $r = \dim \Theta - \dim \Theta_0$ \citep{wilks1938large}. The rejection region for an $\alpha$-level test is therefore $-2 \log \Lambda(\bm y) >  c_{\alpha, r}$, where $c_{\alpha, r}$ is the upper $\alpha$-quantile of the $\chi^2$-distribution with $r$ degrees of freedom. Given MLEs $\hat \theta_0$ and $\hat \theta$ computed over the sets $\Theta_0$ and $\Theta$, respectively, and for sufficiently large sample size $n$, we may therefore test composite hypotheses  under the \textsc{star}  model.

The likelihood ratio test statistic also provides a mechanism for constructing a confidence set for $\theta$. In particular, a $(1-\alpha)$ confidence set for $\theta$ is
$C(\bm y) = \{ \theta_0: \bm y \in A(\theta_0)\}$,  where $A(\theta_0)$ is the acceptance region of an $\alpha$-level  test of $H_0: \theta = \theta_0$ (\citealp{casella2002statistical}, Theorem 9.2.2). For each $\theta_0$, the acceptance region for the (asymptotic) $\alpha$-level test is
$\log p(\bm y | \hat \theta) - \log p(\bm y | \theta_0) < c_{\alpha, 1}/2$
which implies that the (asymptotic) confidence set is equivalently
\begin{equation}\label{confidence-2}
C(\bm y) = \big\{ \theta_0: \log p(\bm y|\theta_0) >  \log p(\bm y | \hat \theta) - c_{\alpha, \dim(\theta) - 1}/2\big\}.
\end{equation}
The confidence set in \eqref{confidence-2} may be constructed without computing derivatives of the log-likelihood \eqref{log-like}, which, although straightforward in general, would require case-specific derivations depending on the relationship between the conditional mean $\mu_\theta$ and the unknown parameters $\theta$. Note that this approach for constructing confidence intervals is identical to that used for non-Gaussian GLMs in \texttt{R} via the \texttt{MASS} package

 \subsection{Model diagnostics}
Model diagnostics typically rely on inspection of the residuals from the fitted model. For non-Gaussian models, various options for residuals are available, such as Pearson or deviance residuals \citep{mccullagh1989generalized}. However, for discrete data, standard residuals diagnostics are often misleading due to the nonuniqueness of the observations $\{y_i\}_{i=1}^n$. \cite{dunn1996randomized} proposed a randomization-based procedure that produces continuous and Gaussian residuals, which improves interpretability and diagnostic assessment. Letting $U^{(0,1)} \sim \mbox{Uniform}(0,1)$, Dunn-Smyth residuals for \textsc{star}  are given by $r_i = \Phi^{-1}(u_i)$, where
\begin{align*}
u_i &= \mathbb{P}\{y(\bm x) = y_i\}U^{(0,1)} + \mathbb{P}\{y(\bm x) < y_i\} \\%= \mathbb{P}\{z^*(\bm x) \in \mathcal{A}_{y_i}\}U^{(0,1)} + \mathbb{P}\{z^*(\bm x) \le g(a_{y_i - 1})\} \\
&= \Big[\Phi \Big\{ \frac{g(a_{y_i+1}) - \mu_{\hat \theta}(\bm x_i)}{\hat \sigma}\Big\} -  \Phi\Big\{ \frac{g(a_{y_i}) - \mu_{\hat \theta}(\bm x_i)}{\hat \sigma}\Big\}\Big]U^{(0,1)} + \Phi\Big\{ \frac{g(a_{y_i - 1}) - \mu_{\hat \theta}(\bm x_i)}{\hat \sigma}\Big\}
\end{align*}
is readily computable based on the MLEs $\hat \theta$ and $\hat \sigma$. We recommend examining multiple instances of $r_i$ to verify that conclusions are not specific to a particular realization of $U^{(0,1)}$.

\section{NHANES data analysis}\label{apps}
Using data from the 2011--2012 National Health and Nutrition Examination Survey (NHANES), we study the demographic, socioeconomic, behavioral, and health-related factors that predict self-reported mental health (\texttt{DaysMentHlthNotGood}).
The covariates ($p=23$) are summarized in Table~\ref{tab:data}. Individuals with missing,  refused, or ``don't know" responses are excluded; additional pre-processing details are provided in the supplementary material.  The analysis dataset includes $n=2311$ individuals.
We fit the \textsc{star}  linear model to these data using the smoothed empirical CDF from Section~\ref{learn-g} (\textsc{star}-np) and the Box-Cox transformation with fixed $\lambda = 1/2$ (\textsc{star}-sqrt). In addition, we include Gaussian regression on $\log(y+1)$ (Gauss-log), Poisson and zero-inflated Poisson (ZIP) regression,  Negative Binomial (NegBin) and zero-inflated Negative Binomial (ZINB) regression, and a logistic regression model that incorporates the upper bound 30 as the number of ``trials" in the binomial distribution (Binom).

Since NHANES data are collected by a complex multistage sampling design, we include several mechanisms to adjust for or quantify the effects of the sampling design. Following \cite{Lumley2017}, we include the primary sampling unit as a covariate to account for possible geographic clustering effects. In addition, the variables that determine the survey weights---gender, age, and race---are all included, which can provide a model-based adjustment to the design \citep{Gelman2007,Niu2019}.
For comparison, we compute MLEs for the heteroskedastic \textsc{star}-np model from Lemma~\ref{EM-gen} using the individual survey weights for $w_i$, which provides a design-consistent estimator for the \textsc{star}  regression coefficients via maximization of a \textsc{star}  pseudo-likelihood \citep{Lumley2017}. In this case, we also include the survey weights for the constituent quantities of the estimated transformation $\hat g$: the sample mean, sample standard deviation, and empirical CDF of $y$.

To summarize the competing regression models, we compute the -2 log-likelihood for each model in Table~\ref{tab:loglike}. The \textsc{star}  models definitively outperform all competitors by a wide margin. Although the models differ in the number of parameters,  we note that \textsc{star}-sqrt contains the same number of parameters as Gauss-log and NegBin yet fewer parameters than the zero-inflated models. Since $y \in \{0,1,\ldots, 30\}$ only admits 31 unique values, the smoothed empirical CDF for \textsc{star}-np counts no more than 31 additional parameters compared to \textsc{star}-sqrt. As a result, it is straightforward to verify that AIC and BIC rank \textsc{star}-np first and \textsc{star}-sqrt second ahead of the competing models.

\begin{table}[h]
\centering
\begin{tabular}{rrrrrrrr}
  \hline
Gauss-log & Poisson & ZIP & NegBin & ZINB & Binom & \textsc{star}-np & \textsc{star}-sqrt \\
  \hline
  11025 & 26208 & 14391 & 9636 & 9367 & 34321 & {\bf 8146} & 8691 \\
   \hline
\end{tabular}
\caption{\small -2 log-likelihood for each regression model. The \textsc{star}  models are clear favorites by likelihood metrics, including AIC and BIC. Note that Gauss-log includes a change-of-variables to the original data scale.\label{tab:loglike}}
\end{table}

Visually, the preference for \textsc{star}  is even more striking. Figure~\ref{fig:qqplots} presents the Dunn-Smyth residual diagnostics for \textsc{star}-np, \textsc{star}-sqrt, and ZIP. Most notably, \textsc{star}-np provides an excellent fit to the data. More surprising, \textsc{star}-sqrt---which cannot account for heaping---also provides a reasonable fit. Despite the simplicity of the transformation in \textsc{star}-sqrt, the rounding operator crucially incorporates both zero-inflation and boundedness ($y_{max} = 30$). By comparison, the ZIP model accounts only for zero-inflation and is decidedly inadequate for these data.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=.32\textwidth]{Plots/qqplots_star}
\includegraphics[width=.32\textwidth]{Plots/qqplots_sqrt}
\includegraphics[width=.32\textwidth]{Plots/qqplots_zip}
\end{center}
\caption{\small Dunn-Smyth residual QQ-plots for \textsc{star}-np (left), \textsc{star}-sqrt (center), and ZIP (right). Points along the line indicate goodness-of-fit. Each plot includes 10 sets of the randomized residuals (color-coded).  Both \textsc{star}  models provide excellent fits, while the ZIP model is clearly inadequate.
\label{fig:qqplots}}
\end{figure}

Proceeding with \textsc{star}-np, we summarize the results of the fitted model in Figure~\ref{fig:coef}. The \textsc{star}-np MLEs and 90\% confidence intervals for each coefficient present the magnitude and direction of each variable's effect with uncertainty quantification. Using a backward-elimination algorithm, we select (and annotate) variables according to AIC and BIC, both of which use the \textsc{star}-np likelihood. The selected variables with (directionally) positive effects on self-reported poor mental health include demographic (female) and socioeconomic  (low income) variables, behavioral attributes (heavy drinking, recent nicotine use, any use of marijuana, any use of hard drugs), and health variables (diabetes and high blood pressure). The only selected variable with a (directionally) negative effect is age, but the estimated effects of uninsured, Black, Hispanic, and high income status are also negative. Moreover, the survey-weighted \textsc{star}-np MLEs help assess the effect of the sampling design on the point estimates. The estimates are similar with the exception of the race categories, each of which is oversampled in the NHANES design.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=1\textwidth]{Plots/linear_coef_lm}
\end{center}
\caption{\small Estimated coefficients and 90\% confidence intervals for \textsc{star}-np (blue) along with the survey-weighted point estimates (yellow). The BIC-selected variables (slate gray) and AIC-selected variables (slate gray and gray) are annotated. The survey-adjusted estimates are similar to the unweighted estimates with the exception of the race categories, each of which is oversampled in the NHANES design.
\label{fig:coef}}
\end{figure}

To target the effect of drug use, we compute a \textsc{star}  likelihood ratio test  comparing the full \textsc{star}-np model against the reduced \textsc{star}-np model that excludes nicotine, marijuana, or hard drug use. The likelihood ratio test obtained a $p$-value $1.45\times 10^{-6}$, providing strong evidence that drug use is an important factor in self-reported mental health. The estimated effect of each variable is positive  (Figure~\ref{fig:coef}), which suggests that drug use is associated with more self-reported days of poor mental health---even after adjusting for other demographic, socioeconomic,  behavioral, and health-related factors.

The estimated nonparametric transformation \eqref{interpo} is presented in Figure~\ref{fig:star-g} along with the survey-weighted alternative. The most prominent changes in slope occur around $t \in \{7, 10, 15, 20\}$, which notably correspond to the primary heaping points observed in the PMF of $y$ (Figure~\ref{fig:pred}). This effect is by design: the nonparametric transformation \eqref{interpo} converts the empirical CDF of $y$ into a suitable transformation for the \textsc{star}  regression model  \eqref{spn}--\eqref{round}. Informally,  the nonparametric transformation ``stretches out" these heaps in the latent $z^*$-space for the continuous regression model \eqref{spn}. There is substantial agreement between the weighted and unweighted estimators.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=.6\textwidth]{Plots/star-g}
\end{center}
\caption{\small Nonparametric estimate of the transformation $\hat g$ along with the survey-weighted version. The heaping points are annotated.
The most prominent changes in slope occur around $t \in \{7, 10, 15, 20\}$ and correspond to the primary heaping points observed in the PMF of $y$ (Figure~\ref{fig:pred}). The unweighted and weighted estimates are similar.
\label{fig:star-g}}
\end{figure}

For completeness, Table~\ref{tab:pvals} compares the marginal $p$-values among Gauss-log, ZIP, NegBin, ZINB, and the \textsc{star}  models. We note that this is not our recommended procedure for variable selection (see the AIC and BIC selections in Figure~\ref{fig:coef}) and that these $p$-values have not been corrected for survey sampling or multiple comparisons. However, these statistics provide useful summaries of the degree of inferential agreement among the competing models.  In general, \textsc{star}-np and \textsc{star}-sqrt agree with each other and with Gauss-log. The alternative count data models---which are empirically inadequate according to residual diagnostics---offer little to no consensus with each other or the Gauss-log and \textsc{star}  models.

\begin{table}[h]
\centering
\begin{tabular}{rrrrrrr}
  \hline
 & Gauss-log & ZIP & NegBin & ZINB & \textsc{star}-np & \textsc{star}-sqrt \\
  \hline
Black & 0.005 & 0.129 & 0.089 & 0.680 & 0.008 & 0.008 \\
  Hispanic & 0.006 & 0.129 & 0.088 & 0.719 & 0.010 & 0.011 \\
  RaceOther & 0.375 & 0.024 & 0.635 & 0.596 & 0.424 & 0.429 \\
  Female & 0.000 & 0.000 & 0.000 & 0.048 & 0.000 & 0.000 \\
  Age & 0.000 & 0.511 & 0.047 & 0.893 & 0.000 & 0.000 \\
  Widow/Divorce/Sep & 0.261 & 0.000 & 0.218 & 0.101 & 0.233 & 0.213 \\
  MaritalSingle & 0.487 & 0.000 & 0.261 & 0.127 & 0.420 & 0.381 \\
  High Income & 0.054 & 0.008 & 0.732 & 0.232 & 0.032 & 0.042 \\
  Low Income & 0.000 & 0.000 & 0.000 & 0.003 & 0.000 & 0.000 \\
  Education= HS & 0.820 & 0.043 & 0.813 & 0.413 & 0.995 & 0.953 \\
  Education$>$ HS & 0.544 & 0.011 & 0.408 & 0.412 & 0.587 & 0.562 \\
  HeavyDrinker & 0.000 & 0.000 & 0.006 & 0.035 & 0.000 & 0.000 \\
  HighBP & 0.000 & 0.000 & 0.007 & 0.110 & 0.000 & 0.000 \\
  HighChol & 0.089 & 0.976 & 0.160 & 0.950 & 0.084 & 0.085 \\
  Diabetes & 0.007 & 0.000 & 0.021 & 0.053 & 0.006 & 0.005 \\
  BMI & 0.493 & 0.005 & 0.555 & 0.311 & 0.533 & 0.519 \\
  Cholesterol & 0.872 & 0.497 & 0.673 & 0.730 & 0.768 & 0.790 \\
  UseNicotine & 0.001 & 0.000 & 0.025 & 0.169 & 0.004 & 0.004 \\
  UsedMarijuana & 0.019 & 0.072 & 0.159 & 0.703 & 0.012 & 0.015 \\
  UsedHardDrug & 0.059 & 0.002 & 0.271 & 0.384 & 0.055 & 0.048 \\
  Uninsured & 0.043 & 0.713 & 0.285 & 0.756 & 0.038 & 0.045 \\
  svypsu2 & 0.222 & 0.397 & 0.325 & 0.554 & 0.246 & 0.240 \\
  svypsu3 & 0.514 & 0.000 & 0.557 & 0.102 & 0.720 & 0.734 \\
   \hline
\end{tabular}
\caption{\small Marginal $p$-values for each regression coefficient ($H_{0j}: \beta_j = 0$ for  $j=1,\ldots,p$) under various models. There is strong agreement among \textsc{star}-np, \textsc{star}-sqrt, and Gauss-log.
\label{tab:pvals}}
\end{table}

\section{Simulation study}\label{sims}
The \textsc{star}  models are evaluated for point prediction,  distributional fitness, and hypothesis testing using simulated data. Among \textsc{star}  linear models, we compare the smoothed empirical CDF from Section~\ref{learn-g} (\textsc{star}-np), Box-Cox with estimated $\lambda$ (\textsc{star}-bc), and Box-Cox with fixed $\lambda = 1/2$ (\textsc{star}-sqrt). Competing methods are Negative Binomial regression (NegBin), Poisson regression (Poisson), and Gaussian regression on $\log(y+1)$  (Gauss-log).

We consider two count-valued data-generating processes: (i) \textsc{star}  data (Mixture-CDF) designed to match the empirical features of \texttt{DaysMentHlthNotGood} and (ii) Negative Binomial data with overdispersion.  For Mixture-CDF data, we specify the rounding operator to support $y \in \{0,\ldots, 30\}$. The transformation is constructed by smoothing the CDF of a random variable  that places $1/2$ mass on a $\mbox{Poisson}(10)$ distribution, $1/4$ mass uniformly on the set of heaps $\{5, 10, \ldots, 25\}$, and $1/4$ mass uniformly on the set of boundaries $\{0, 30\}$. The latent data-generating model \eqref{spn} is completed by fixing $\sigma = 0.7$. The Negative Binomial data are generated independently from  $ y_i \sim \mbox{NB}\{r^*,  \lambda_i^*/(r^* + \lambda_i^*) \}$ with expectation $\lambda_i^*$ and variance $\lambda_i^*(1 + \lambda_i^*/r^*)$. We specify $\log( \lambda_i^* )= \bm x_i'\bm \beta^*$ and set  $r^* = 3$ to include overdispersion.

In both cases, we use $n = 500$ (see the supplement for $n=100$)
and $p = 10$ with covariates generated from marginal standard normal distributions with $\mbox{Cor}(x_{i,j}, x_{i,j'}) = (0.75)^{|j - j'|}$. The $p$ columns are randomly permuted and augmented with an intercept. The true regression coefficients are defined by the intercept $\beta_0^* = \log(1.5)$, signals $\beta_j^* = \log(1.25)$ for  $j=1,\ldots, p/2 = 5$, and the remaining coefficients fixed at zero. The simulations are repeated for 100 iterations.

Point prediction and distributional fitness are summarized in Figure~\ref{fig:sims-500}. Point prediction is evaluated using root mean squared error (RMSE) on $\mathbb{E}(y_i | \bm x_i)$, while distributional fitness is evaluated by computing -2 log-likelihood on a test set of 1000 points. The log-likelihood evaluations use each model likelihood (e.g., Poisson) with the parameters fixed at the MLEs. As expected, the \textsc{star}  models demonstrate vastly superior point prediction and distributional accuracy for the data exhibiting zero-inflation, boundedness, and heaping (Mixture-CDF). Similarly, the Negative Binomial data is best modeled by NegBin; however, \textsc{star}  is surprisingly competitive for both point predictions and distributional accuracy. In both cases, the Poisson model is among the best for point prediction yet the clear worst for distributional fitness. Note that Gauss-log is included in the likelihood comparisons but is not a coherent model for count data.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=.49\textwidth]{Plots/rmse_Mixture-CDF_n=500_p=11}
\includegraphics[width=.49\textwidth]{Plots/rmse_NegBin_n=500_p=11}

\includegraphics[width=.49\textwidth]{Plots/n2loglike_Mixture-CDF_n=500_p=11}
\includegraphics[width=.49\textwidth]{Plots/n2loglike_NegBin_n=500_p=11}
\end{center}
\caption{\small Simulation results for Mixture-CDF data (left) and Negative Binomial data (right) evaluated for point prediction (RMSE,  top) and distributional fitness (-2 log-likelihood, bottom). \textsc{star}  and NegBin are best for their respective data-generating processes, but \textsc{star}  is surprisingly competitive for both point predictions and distributional accuracy with Negative Binomial data.
\label{fig:sims-500}}
\end{figure}

Table~\ref{table:tests-500} reports the Type I error and power at $\alpha = 0.10$ for testing the null hypothesis $H_{0j}: \beta_j = 0$ against the two-sided alternative. These values are averaged across $j=1,\ldots, p$ and all simulations. Most notably, the Poisson model is unable to control Type I error at the designated level. Despite the distinct data-generating processes, \textsc{star}  and NegBin perform similarly along with Gauss-log, with each method slightly favored under its own data-generating process.

\begin{table}[h] \small
\centering
\begin{tabular}{rrrrrrrr}
  \hline
 & & Gauss-log & Poisson & NegBin & \textsc{star}-sqrt & \textsc{star}-bc & \textsc{star}-np \\
  \hline
Mixture-CDF
& Type I Error & 0.086 & 0.412 & 0.086 & 0.102 & 0.114 & 0.110 \\
&   Power & 0.896 & 0.974 & 0.916 & 0.938 & 0.938 & 0.948 \\

NegBin
&        Type I Error & 0.078 & 0.252 & 0.086 & 0.084 & 0.082 & 0.080 \\
 & Power & 0.830 & 0.940 & 0.882 & 0.854 & 0.844 & 0.838 \\

     \hline
\end{tabular}
\caption{\small Type I error rates and power for hypothesis testing in the linear model at the $\alpha = 0.10$ level for Mixture-CDF and Negative Binomial data. \textsc{star}  and NegBin perform similarly, while Poisson cannot control Type I error at the designated level. \label{table:tests-500}}
\end{table}

\section{Discussion}\label{discuss}
We introduced semiparametric estimation and inference for count data regression. The proposed simultaneous transformation and rounding (\textsc{star}) framework  ``sanitizes" continuous data models for use with discrete data exhibiting a variety of distributional phenomena.  Empirically, \textsc{star}  regression has demonstrated the ability to model zero-inflation, over/underdispersion, boundedness (or censoring), and heaping. This modeling flexibility is enabled by the transformation and rounding operators: the transformation is estimated nonparametrically while the rounding operator ensures the correct support for the data (e.g., count-valued and if necessary, bounded). By design, \textsc{star}  leverages existing continuous data models and estimation algorithms: within our EM algorithm, a (weighted) least squares solution is sufficient to produce \textsc{star}  maximum likelihood estimators. Likelihood-based inference provides hypothesis testing, confidence intervals, variable selection via information criteria, and regression diagnostics. Simulation results confirmed the utility of the regression modeling approach under multiple designs.

\textsc{star}  regression models were applied to study self-reported mental health using a large NHANES dataset. While the response variable \texttt{DaysMentHlthNotGood} presented challenges for existing count regression models, \textsc{star}  was capable of providing an adequate model fit that demonstrated substantial improvements in both likelihood-based comparisons and visual diagnostics. Using a \textsc{star}  linear model with a nonparametric transformation, we estimated effects and confidence intervals, incorporated adjustments for survey weights, selected variables (via AIC and BIC backward-elimination), and tested hypotheses for a broad set of demographic, socioeconomic, behavioral, and health-related covariates.

We emphasize that our results are neither causal nor direct measurements of poor mental health, but rather estimated associations with \emph{self-reported} mental health. In particular, our estimated effects (Figure~\ref{fig:coef}) do not directly imply that certain groups (e.g., Black or Hispanic individuals) are less likely to  \emph{experience} poor mental health  (all else equal), but rather that they are less likely to \emph{report} poor mental health (all else equal). Nonetheless, self-reporting of poor mental health remains an independently interesting topic; future research to decouple experienced and self-reported poor mental health would be valuable and complementary.

Lastly, we have used survey-weighted \textsc{star}  (Section~\ref{apps}) to visualize the impact of the NHANES sampling design on point estimation. However, valid uncertainty quantification and inference for general survey data remain open challenges for \textsc{star}  and other regression models
\citep{Lumley2017}.

\section{Software}
\texttt{R} code to reproduce the NHANES data pre-processing, NHANES data modeling and analysis, and simulated data comparisons is available online. An \texttt{R} package for \textsc{star}  regression is available at \url{https://github.com/drkowal/rSTAR}.

\bibliographystyle{apalike}

\section{Proofs}\label{proofs}

Recall Lemma 3.1:
\emph{Consider a heteroskedastic \textsc{star}  model
\begin{equation}\label{spn-w}
z_i^* = \mu_\theta(\bm x_i) +  \epsilon_i, \quad \epsilon_i \stackrel{indep}{\sim}N(0,\sigma^2/w_i), \quad i=1,\ldots,n
\end{equation}
  with unknown parameters $\theta \in \Theta$ and $\sigma^2 > 0$ and known weights $w_i > 0$. The M-step solutions for $\theta$ and $\sigma$ are given by
\begin{align}
\label{mle-mu}
\hat \theta &= \hat \theta( \{\bm x_i,  \hat z_i^{(1)}\}_{i=1}^n) \\
\label{mle-sig}
\hat \sigma^2 &= \frac{1}{n}\left\{\sum_{i=1}^n w_i \hat z_i^{(2)}  + \sum_{i=1}^n  w_i \mu_{\hat \theta}^2(\bm x_i) -2 \sum_{i=1}^n w_i \mu_{\hat \theta}(\bm x_i)\hat z_i^{(1)} \right\}
\end{align}
where $\hat \theta( \cdot)$ is the weighted least squares solution to
\begin{equation}\label{sse}
\hat \theta( \{\bm x_i, d_i\}_{i=1}^n) = \arg\min_{\theta \in\Theta} \sum_{i=1}^n w_i \{d_i - \mu_\theta(\bm x_i)\}^2
\end{equation}
for paired observations $(\bm x_i, d_i)$ with $d_i \in \mathbb{R}$ and $w_i  > 0$.}

\begin{proof}
For heteroskedastic errors, the E-step is
$
Q(\theta, \theta^s)  =
C(\bm y) + \frac{1}{2}\sum_{i=1}^n \log (w_i) - \frac{n}{2} \log(\sigma^2) - \frac{1}{2\sigma^2} \big\{\sum_{i=1}^n w_i \hat z_i^{(2)} + 2\sum_{i=1}^n w_i \mu_\theta(\bm x_i)\hat z_i^{(1)} +  \sum_{i=1}^n w_i\mu_\theta^2(\bm x_i)\big\}$. Invariant of $\sigma$, $Q(\theta, \theta^s)$ is maximized by $\hat \theta = \arg\min_{\theta \in \Theta}\big\{2\sum_{i=1}^n w_i\mu_\theta(\bm x_i)\hat z_i^{(1)} +  \sum_{i=1}^n w_i\mu_\theta^2(\bm x_i)\big\} = \arg\min_{\theta \in \Theta}\sum_{i=1}^n w_i\big\{ \hat z_i^{(1)} - \mu_\theta(\bm x_i)\big\}^2$, which is equivalent to \eqref{sse} with $d_i =\hat z_i^{(1)}$. Given $\hat \theta$ and therefore $\mu_{\hat\theta}$, derivation of the estimator \eqref{mle-sig} to maximize $Q$ is straightforward.
\end{proof}

\section{Estimation algorithm}\label{sec-alg}
For completeness, we provide a cumulative description of the EM algorithm in Algorithm~\ref{em-alg}. We initialize by computing the Gaussian MLEs using $y_i^* = y_i$, which omits the rounding step, and adding a random perturbation to the estimated $\hat z_i$ values given the initialized parameters $\theta^0$ and $\sigma^0$.  For the case of the Box-Cox transformation with unknown $\lambda$, we include an additional step at each iteration
to optimize the (incomplete data) log-likelihood $\ell(\bm y; \theta = \hat \theta^s, \sigma = \hat \sigma^s)$ with respect to $\lambda$ using Brent's method (We thank Jeffrey Jiang for the \texttt{R} code implementation).

\begin{algorithm} \label{em-alg}
\SetAlgoLined
 initialize $\theta^0, \sigma^0, \hat z_i^{(1)}$\;
 set $s=1$, $\varepsilon > 0$\;
 \While{$\log p(\bm y | \theta^{s}, \sigma^{s}) - \log p(\bm y | \theta^{s-1}, \sigma^{s-1}) > \varepsilon$}{
  compute $ \theta^{s+1} = \hat \theta( \{\bm x_i,  \hat z_i^{(1)}\}_{i=1}^n)$ from \eqref{sse} and set $\mu_{\hat \theta} = \mu_{\theta^{s+1}}$\;
  compute $\sigma^{s+1}$ from \eqref{mle-sig} and set $\hat \sigma = \sigma^{s+1}$\;
  update $\hat z_i^{(1)}$ and $\hat z_i^{(2)}$ using
   \begin{align*}

 \hat z_i^{(1)}  &= \mu_{\hat \theta}(\bm x_i) + \hat\sigma \frac{\phi(a_i) - \phi(b_i)}{\Phi(b_i) - \Phi(a_i)} \\

 \hat z_i^{(2)} &=  \mu_{\hat \theta}(\bm x_i)\left\{\mu_{\hat \theta}(\bm x_i) + 2 \hat\sigma \frac{\phi(a_i) - \phi(b_i)}{\Phi(b_i) - \Phi(a_i)}\right\} + \hat\sigma^2 \left\{
 1 + \frac{a_i\phi(a_i) - b_i\phi(b_i)}{\Phi(b_i) - \Phi(a_i)}
 \right\}
 \end{align*}
 where $a_i = \{g(a_{y_i}) - \mu_{\hat \theta}(\bm x_i)\}/\hat\sigma$ and $b_i = \{g(a_{y_i + 1}) - \mu_{\hat \theta}(\bm x_i)\}/\hat\sigma$\;
  increment $s = s+1$;
 }
 \caption{EM Algorithm for \textsc{star} }
\end{algorithm}

A key feature of the EM algorithm is the modularity. First, any valid rounding operator or (estimated) transformation can be substituted and only impacts the terms $a_i$ and $b_i$. Hence, we may consider nonparametric or parametric estimators of $g$ as well as fixed transformations (such as logarithmic or square root) with minimal adjustment to the algorithm: only  $g(a_{y_i})$ and $g(a_{y_i + 1})$ must be updated. Second, the latent Gaussian model  \eqref{spn-w} can be modified for any continuous regression model estimable by least squares. In this case, only the least squares update \eqref{sse} is changed.

\section{Additional simulation results}\label{sec-sims}
The \textsc{star}  models are evaluated for point prediction,  distributional fitness, and hypothesis testing using simulated data with a smaller sample size, $n=100$. Among \textsc{star}  linear models, we compare the smoothed empirical CDF  (\textsc{star}-np), Box-Cox with estimated $\lambda$ (\textsc{star}-bc), and Box-Cox with fixed $\lambda = 1/2$ (\textsc{star}-sqrt). Competing methods are Negative Binomial regression (NegBin), Poisson regression (Poisson), and Gaussian regression on $\log(y+1)$  (Gauss-log).

We consider two count-valued data-generating processes: (i) \textsc{star}  data (Mixture-CDF) designed to match the empirical features of \texttt{DaysMentHlthNotGood} and (ii) Negative Binomial data with overdispersion.  For Mixture-CDF data, we specify the rounding operator to support $y \in \{0,\ldots, 30\}$. The transformation is constructed by smoothing the CDF of a random variable  that places $1/2$ mass on a $\mbox{Poisson}(10)$ distribution, $1/4$ mass uniformly on the set of heaps $\{5, 10, \ldots, 25\}$, and $1/4$ mass uniformly on the set of boundaries $\{0, 30\}$. The latent data-generating model is completed by fixing $\sigma = 0.5$. The Negative Binomial data are generated independently from  $ y_i \sim \mbox{NB}\{r^*,  \lambda_i^*/(r^* + \lambda_i^*) \}$ with expectation $\lambda_i^*$ and variance $\lambda_i^*(1 + \lambda_i^*/r^*)$. We specify $\log( \lambda_i^* )= \bm x_i'\bm \beta^*$ and set  $r^* = 3$ to include overdispersion.

In both cases, we use $n = 100$ and $p = 10$ with covariates generated from marginal standard normal distributions with $\mbox{Cor}(x_{i,j}, x_{i,j'}) = (0.75)^{|j - j'|}$. The $p$ columns are randomly permuted and augmented with an intercept. The true regression coefficients are defined by the intercept $\beta_0^* = \log(1.5)$, signals $\beta_j^* = \log(1.25)$ for  $j=1,\ldots, p/2 = 5$, and the remaining coefficients fixed at zero. The simulations are repeated for 100 iterations.

Point prediction and distributional fitness are summarized in Figure~\ref{fig:sims-100}. Point prediction is evaluated using root mean squared error (RMSE) on $\mathbb{E}(y_i | \bm x_i)$, while distributional fitness is evaluated by computing -2 log-likelihood on a test set of 1000 points. The log-likelihood evaluations use each model likelihood (e.g., Poisson) with the parameters fixed at the MLEs. As expected, the \textsc{star}  models demonstrate vastly superior point prediction and distributional accuracy for the data exhibiting zero-inflation, boundedness, and heaping (Mixture-CDF). For the Negative Binomial data, there is less separation among the methods for point or distributional accuracy.

Table~\ref{table:tests-100} reports the Type I error and power at $\alpha = 0.10$ for testing the null hypothesis $H_{0j}: \beta_j = 0$ against the two-sided alternative. These values are averaged across $j=1,\ldots, p$ and all simulations. As in the case of $n=500$ (see the main paper), the Poisson model is unable to control Type I error at the designated level, while \textsc{star}  and NegBin perform similarly. Naturally, the smaller sample size reduces the power of the tests, but the Type I error control is acceptable for all methods except the Poisson model.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=.49\textwidth]{Plots/rmse_Mixture-CDF_n=100_p=11}
\includegraphics[width=.49\textwidth]{Plots/rmse_NegBin_n=100_p=11}

\includegraphics[width=.49\textwidth]{Plots/n2loglike_Mixture-CDF_n=100_p=11}
\includegraphics[width=.49\textwidth]{Plots/n2loglike_NegBin_n=100_p=11}
\end{center}
\caption{Simulation results for Mixture-CDF data (left) and Negative Binomial data (right) evaluated for point prediction (RMSE,  top) and distributional fitness (-2 log-likelihood, bottom). \textsc{star}  performs best for Mixture-CDF, while there is less separation among competing methods for the Negative Binomial data.
\label{fig:sims-100}}
\end{figure}

\begin{table}[!p]
\tblcaption{Type I error rates and power for hypothesis testing in the linear model at the $\alpha = 0.10$ level for Mixture-CDF and Negative Binomial data. \textsc{star}  and NegBin perform similarly, while Poisson cannot control Type I error at the designated level.\label{table:tests-100}}
\centering
\begin{tabular}{rrrrrrrr}
  \hline
 & & Gauss-log & Poisson & NegBin & \textsc{star}-sqrt & \textsc{star}-bc & \textsc{star}-np \\
  \hline
Mixture-CDF
&	Type I Error & 0.076 & 0.294 & 0.128 & 0.106 & 0.106 & 0.128 \\
&	  Power & 0.598 & 0.826 & 0.726 & 0.750 & 0.742 & 0.778 \\

NegBin
& Type I Error & 0.074 & 0.240 & 0.120 & 0.100 & 0.096 & 0.096 \\
&  Power & 0.280 & 0.520 & 0.372 & 0.316 & 0.326 & 0.324 \\

     \hline
\end{tabular}
\end{table}

 \end{document}

redpajama_set_name: RedPajamaArXiv