Representing and learning stress: Grammatical constraints and neural networks

This new NSF grant is currently being processed so this information is here temporarily.

Joe Pater (PI), Gaja Jarosz (co-PI), total costs $386,226

Public summary: Languages are systems of remarkable complexity, and linguists and computer scientists have devoted considerable effort to the development of methods for representing those complex systems, as well as computational methods for learning the system of a given language. This effort is driven by the desires to better understand human cognition, and to build better language technologies. This project draws on the theories and methods of both linguistics and computer science to study the learning of word stress, the pattern of relative prominence of the syllables in a word. The stress systems of the world’s languages are relatively well described, and there are competing linguistic theories of how they are represented. This project applies learning methods from computer science to find new evidence to distinguish the competing linguistic theories. It also examines systems of language representation that have been developed in computer science and have received relatively little attention by linguists (neural networks). The research will engage undergraduate and graduate linguistics students at a public university. Linguistics has a much higher proportion of female students than computer science, and this project aims to address gender imbalance in STEM. 

From a linguistic perspective, learning stress involves learning hidden structure, parts of the representation that are not present in the observed data and that must be inferred by the learner. A given pattern of prominence over syllables is often consistent with multiple prosodic representations. The approach to hidden structure learning used in this project applies the general technique of Expectation Maximization, which in pilot work achieved good results on a standard test set. Intriguingly, many of the languages that this learner failed on in the test set are ones that are in fact cross-linguistically unattested. This project expands the set of tested languages to include more of the range of systems found cross-linguistically, and further explores the possibility that typological gaps have learning explanations. It compares hypotheses about the constraints responsible for stress placement by comparing how well they support the learning of attested systems, and whether they can help explain typological gaps. Pilot work also found indications that a neural network could learn generalizable representations of the data; the project is further testing this method. All of the software developed in this project is being made freely available, as is a database of the stress systems of the world’s languages. 


Linguistics as cognitive science

Presentation to the College of Humanities and Fine Arts’ 5 at 4 series
March 9, 2022

Pater, Joe. 2019. Generative linguistics and neural networks at 60: foundation, friction, and fusion. Language 95/1, pp. e41-e74. 

Leading questions

How is knowledge of language represented?

How is language learned?

The broader questions of how human knowledge is learned and represented have been given two general kinds of answer in cognitive science since the field emerged in the 1950s.

Birth of cognitive science

Noam Chomsky 1957
From Pater (2019)
Frank Rosenblatt with the Perceptron image sensor late 1950s
From Pater (2019)

1980s: Cognitive science as a field

Cognitive science became a recognized interdisciplinary field in the early 1980s, thanks partly to funding from the Sloan Foundation. Barbara Partee of Linguistics collaborated with Michael Arbib of Computer Science to secure Sloan funding to establish interdisciplinary CogSci at UMass Amherst.

Barbara Partee circa 1977. University Photograph Collection (RG 120_2). Special Collections and University Archives, University of Massachusetts Amherst Libraries

The fight over the English past tense

David Rumelhart from
James McClelland from

Rumelhart and McClelland (1986) present a Perceptron-based approach to learning and representing knowledge of the English past tense (e.g. love, loved; take, took):

Scholars of language and psycholinguistics have been among the first to stress the importance of rules in describing human behavior…

We suggest that lawful behavior and judgments may be produced by a mechanism in which there is no explicit representation of the rule. Instead, we suggest that the mechanisms that process language and make judgments of grammaticality are constructed in such a way that their performance is characterizable by rules, but that the rules themselves are not written in explicit form anywhere in the mechanism.

Rumlhart and McLellan (1986) On Learning the Past Tenses of English Verbs, pp. 216-217
Steven Pinker 1994
Alan Prince 2013 MIT

Twenty years ago, I began a collaboration with Alan Prince that has dominated the course of my research ever since. Alan sent me a list of comments on a paper by James McClelland and David Rumelhart. Not only had Alan identified some important flaws in their model, but pinpointed the rationale for the mechanisms that linguists and cognitive scientists had always taken for granted and that McClelland and Rumelhart were challenging — the armamentarium of lexical entries, structured representations, grammatical categories, symbol-manipulating rules, and modular organization that defined the symbol-manipulation approach to language and cognition. By pointing out the work that each of these assumptions did in explaining aspects of a single construction of language — the English past tense — Alan outlined a research program that could test the foundational assumptions of the dominant paradigm in cognitive science. 

Steven Pinker (2006) Whatever Happened to the Past Tense Debate

Fusion: Optimality Theory

Paul Smolensky from
John McCarthy from

Now: Neural networks’ third wave

Modern computers are getting remarkably good at producing and understanding human language. But do they accomplish this in the same way that humans do? To address these questions, the investigators will derive measures of the difficulty of sentence comprehension by computer systems that are based on deep-learning technology, a technology that increasingly powers applications such as automatic translation and speech recognition systems. They will then use eye-tracking technology to compare the difficulty that people experience when reading sentences that are temporarily misleading, such as “the horse raced past the barn fell,” with the difficulty encountered by the deep-learning systems. 

From Brian Dillon’s 2020 NSF award abstract

This project draws on the theories and methods of both linguistics and computer science to study the learning of word stress, the pattern of relative prominence of the syllables in a word. The stress systems of the world’s languages are relatively well described, and there are competing linguistic theories of how they are represented. This project applies learning methods from computer science to find new evidence to distinguish the competing linguistic theories. It also examines systems of language representation that have been developed in computer science and have received relatively little attention by linguists (neural networks).

From NSF project summary of “Representing and learning stress: Grammatical constraints and neural networks”, Joe Pater PI, Gaja Jarosz co-PI

CDC Covid transmission levels (2021 vs. 2022)

The Shoestring Covid tracker uses the CDC community transmission levels released as part of their Feb. 12 2021 guidance for school reopening. They are based on a combination of a per capita weekly new case rate and the test positivity rate, as shown in this table:

The Shoestring Covid tracker uses just the new case rate, since the test positivity rate is often not available, and because it usually wouldn’t matter (e.g. it would be very unusual to have a “low” new case rate and the greater than 5% positivity that would change the classification to “moderate”).

These transmission rates can be used for decision making, for communities, businesses and institutions, or individuals. For example, from July 27th 2021 until Feb. 25th 2022, the CDC recommended that masks be worn indoors in communities with substantial transmission or greater, that is, with 50 or more new cases per 100K in a week. On Feb. 19th 2022, Bob Wachter, Chair of the UCSF Department of medicine, published a Twitter thread explaining his reasoning for maintaining a similar level (10 per 100K per day = 70 per week) as a threshold for indoor mask wearing, even given the changed circumstances from mid-2021. (Update 3/24: this piece from Inside Medicine supports the 50 per 100K threshold for universal indoor masking).

On Feb. 25th 2022, the CDC released new community levels and masking guidance. The new metric uses a single new case rate threshold of 200 per 100K per week, combined with the per capita rate of new Covid-19 hospital admissions, and the percentage of staffed hospital beds occupied by Covid-19 patients, to classify communities as having Low, Medium or High levels. The indoor mask recommendation applies for communities with a High level, which is reached when new cases exceed the 200 per 100K per week rate, and there are either 10 or more new admissions per 100K, or 10% or more hospital beds occupied by Covid-19 patients. This is four times the previous new case rate threshold, plus an added hospitalization rate requirement.

The CDC 2022 guidance is controversial. On the day it was released, the president of the AMA issued a statement that included the following:

But even as some jurisdictions lift masking requirements, we must grapple with the fact that millions of people in the U.S. are immunocompromised, more susceptible to severe COVID outcomes, or still too young to be eligible for the vaccine. In light of those facts, I personally will continue to wear a mask in most indoor public settings, and I urge all Americans to consider doing the same, especially in places like pharmacies, grocery stores, on public transportation…

Gerald E. Harmon MD, President American Medical Association, Feb. 25 2022,

The new levels are of limited value for individual decision making. For example, at the Medium level, the guidance states that “[I]f you are immunocompromised or high risk for severe disease [t]alk to your healthcare provider about whether you need to wear a mask and take other precautions (e.g., testing)”. The Medium level could occur with any new case rate. Presumably, a healthcare provider’s advice on masking should take the community transmission level into account, but the new CDC guidance provides no basis on which that could be done. In addition, individuals may want to take precautions as new case rates rise, before hospitalizations begin to increase and trigger a change in the CDC 2022 community level. Furthermore, the new CDC guidance gives no help in determining the circumstances under which individuals might want to wear masks to protect community health, as the president of the AMA urges us to do in the above statement.


School mask mandates still make good sense in Hampshire County

Published in the Hampshire Gazette Feb. 23, 2022. This open letter was co-authored with Seth Cable, Summer Cable, Michael Stein and Susan Voss, and was also signed by 216 other people that live and work in Hampshire County, listed as the end of this letter. For further information on Covid safety in schools, please see the Urgency of Equity toolkit.

An opinion column published in the Hampshire Gazette on Feb. 17 2022 claims that “even if at one point in the pandemic it was possible to make a reasonable argument for the masking of children in school, that is no longer the case”. We disagree, since the following provides what we take to be a clearly reasonable basis for deciding to continue the school mask mandates until the levels of community transmission subside to a much lower level. We offer this as a statement of views that we believe are widespread, but are usually not made as vocally as those of the opponents of mask mandates and other public health measures.

1. It is reasonable to minimize spread in our community by using school masking. Our children interact with other members of the community, some of whom are relatively vulnerable to the effects of Covid-19 infection. By slowing spread in our schools, we are also slowing spread in our communities. The authors of the opinion column claim that “[t]here are no credible scientific data indicating that masking of children in schools has limited the spread of COVID-19”. They do not say why they do not consider the data presented by the CDC or other data to be credible. It is possible that they consider only data from randomized controlled trials (RCTs) to be credible, since they say in the next sentence that “[n]o randomized controlled trials of mandatory school masking have been carried out”. The CDC and other experts clearly consider sources of evidence other than RCTs to be useful, and it is not difficult to imagine why no RCTs have been run on school masking. For example, Institutional Review Boards may well balk at approving a study with a control group of unmasked students in a community with high transmission. 

2. It is reasonable to characterize the current local level of community transmission as high, and the risk to community health of that transmission as high as well. According to Mass-DPH data, there were 680 new cases in Hampshire County the week ending Feb. 17, which translates to a per capita rate that is over 4 times the CDC’s bar for “high transmission”, and over 8 times the bar for indoor mask wearing. Many of those cases are likely from an outbreak at UMass Amherst, which reported 456 cases in the week ending Feb. 15th. The future impact to the broader community of that outbreak is unknown. The Mass-DPH reports 37 Covid-19 deaths in the last 28 days in Hampshire County, which can reasonably be taken to indicate a high community health risk.

3. It is reasonable to minimize Covid-19 infections in our children. While most children recover quickly from Covid-19 and have mild symptoms, some wind up in hospital, and some die. That the proportion of deaths is lower than in adults, or that the number is lower than child deaths from some other cause, does not make it any less desirable to avoid those deaths. In addition, the long-term effects of these infections is unknown. There are clearly long-term effects of Covid-19 infections in general. We can only hope that childhood infections with Omicron, especially in vaccinated children, will have fewer long-term effects.

4. Mandates maximize the protection of each individual. If everyone in a room wears a mask, the amount of airborne virus is minimized, maximizing the protection for everyone. It is a less effective protection for an individual if others are maskless. This is especially true if that individual does not have the mask perfectly fitted, or occasionally takes it off to eat or drink, circumstances that seem common in a school. Wearing a mask is not just about protecting oneself, it is also for the protection of the community, including its most vulnerable members. For example, universal masking allows children who are immune compromised or otherwise at high risk for severe disease and children who have family members who are immune compromised to attend school when it would otherwise be unsafe to do so. 

5. It is reasonable to decide that real or potential negative effects of masking are outweighed by their positive benefits in minimizing Covid-19 infections. There seems to be no good evidence of negative effects of masking on child development. It is quite possible that speakers of non-mainstream varieties of English (e.g. second language speakers) may be more impacted than others by mask wearing. Real and potential negative effects should be taken into account in any decision about a mask mandate, and attempts should be made to address them when masking is in effect. But it may well be that the benefits of masks outweigh any risks.  

Signed by:

Joe Pater, Northampton resident and Professor of Linguistics, UMass Amherst

Summer Cable, Northampton resident

Seth Cable, Florence resident and UMass Faculty

Susan Voss, Northampton resident and Professor of Engineering, Smith College

Michael Stein, Northampton Resident, Ward 4 School Committee Member

Jennifer Ritz Sullivan, COVIDJustice Leader for Massachusetts with Marked By COVID  Goshen

Suzanne Theberge MPH, Northampton 

Tom Roeper, Amherst

Naomi Gerstel, Professor emerita UMass, resident Northampton

Rene Theberge, Retired Public Health Worker, Florence

Neil Kudler MD, Physician

Kirsten Leng, Resident of Northampton, Associate Professor, Women, Gender, Sexuality Studies, University of Massachusetts Amherst

Jean Potter, Doula, Northampton 

Frazer Ward, Northampton

Erica Kates, Florence, MA

Thomas Wartenberg, Professor of Philosophy, Emeritus, Mount Holyoke College

Jen Davis, Northampton

Lou Davis, Financial Planner and Advisor, Northampton

Wenona Rymond-Richmond, Northampton

Eric Poehler, Northampton

Karen Foster, Ward 2 City Councilor

Erin Kates, Resident of Florence 

Sarah Metcalf, writer, Northampton resident

Christopher Pye, teacher, Northampton resident

Andrew Kennard, Postdoctoral Fellow, UMass Amherst. Amherst resident

Tom Riddell, Northampton

Beth Adel, Teacher and resident of Northampton 

Elliot Fratkin, Professor Emeritus Smith College. Northampton

Sally Popper, Retired, Northampton

Robert Buscher, Northampton

Laura Briggs, Professor, University of Massachusetts and Northampton resident

Maureen Flannery, Northampton

Steven Goode, Northampton

Christopher Golden, parent and NOAA software engineer, Northampton

Hedy Rose, retired educator, Northampton resident

Norma Akamatsu, Social Worker, Psychotherapist, Northampton

Ian Goodman, MD, Pediatrician and Northampton Resident

Angela Silvia, CT technologist, Northampton, MA

Meg Robbins, Resident,  Northampton, MA

Traci Olsen, Northampton

Jennifer L. Nye, Northampton resident and UMass Amherst faculty member (History)

Anisa Schardl, Northampton Public Schools teacher and parent

Janet Gross, Retired

Nicolas Gross, Retired

Matthew Hine, Service Engineer (Aerospace), Northampton

John Selfridge, public school teacher, Northampton

Sara Lennox, Northampton

Jill de Villiers, Professor, Smith College, Northampton resident

Daniel Cannity, Northampton Resident

Rachel Merrell, Teacher

Cora Fernandez Anderson, Assistant Professor at Mount Holyoke College, Amherst resident

Melinda Buckwalter, Williamsburg

Emily Hamilton, Professor of history of science/medicine

Taylor Flynn, Parent & retired professor, Northampton MA

Deborah Keisch, Florence

Adele Franks, Public health physician, retired

Young Min Moon, Professor, UMass Amherst

Jude Almeida, School-Based Social Worker, Northampton resident

Karin Baker, Teacher, Northampton

Meghan  Armstrong-Abrami, Associate Professor of Hispanic Linguistics, Northampton resident

Lynn Posner Rice, Northampton

Justin Pizzoferrato, Father/self employed

Greg Lewis, Public Health Emergency Planner, Northampton

Alyssa Lovell, school-based OTR/L 

Kim Gerould, Northampton

Omar Dahi, College professor 

Kai Simon, Northampton 

Andrea Ayvazian, Pastor, Northampton resident

Jennifer Fronc, UMass Faculty; Northampton resident

Graciela Monteagudo, Senior Lecturer, UMass Amherst, Amherst Resident

Roberta Issler, Retired teacher

Cathy McNally consultant, Northampton

Rachel Wysoker, Northampton

T. Stephen Jones, MD, MPH retired public health physician 

Alison Morse, Educator

Cory Ellen Gatrall, Registered Nurse

John McNally, Attorney and grandparent, Northampton

Jeff Napolitano, Northampton, MA

Rebecca Busansky, Northampton

Rachel Yox, Amherst

Judd Gledhill, Director IT

Meg Bogdan, Parent of Northampton Public School Students

Roz Chapman, Northampton 

Lisa Weremeichik,  Northampton

Charles Dumont, MD MS Pulmonary and Critical Care physician

Tara Dumont, MD Physician

Rebecca Burwell, Professor 

Karen Sullivan, College staff, Northampton

Victoria  Dixon, Disabilities Advocate, Amherst 

Leah Greenberger, veterinarian, Belchertown MA

Annie Salsich, Self-employed 

Gabriel Phipps, Adjunct Professor

Karen Hodges, Florence

Katherine Fabel, DUA and Lecturer, UMass Amherst, Florence MA

Nykole Roche, Northampton resident w/3 kids in NPS

Garrett Warren, Amherst

Annabelle Link, Northampton

Capella Sherwood, Music teacher/ Northampton

Bertha Thorman, Northampton

Neha Kennard, Amherst

Kelly Link, Writer

Lesley Yalen, Florence, MA

David Arnold, UMass Professor of Psychological and Brain Sciences

Kristen Elde, Leeds

Lisa Harvey, Professor of Psychological & Brain Sciences at UMass, Resident of Amherst

Michael Becker, Hadley resident and UMass Faculty

Henry Rosenberg, Northampton

Andrew Gorry, Staff, UMass Amherst

Eddie (Erin) Gorry, UMass Staff, Resident of Florence, MA

Leeba Morse, Grant Writer

Jonathan Knapp, Northampton Public Schools educator

Alexis Callender, Works as faculty in Northampton, Lives in Easthampton

Megan Paik, Northampton

Mary Hoyer, Amherst resident and retired Hartford Public Schools teacher

Terianne Falcone, Writer / teacher

David Ball, Northampton

Renee Spring, Amherst Psychotherapist

Therese Kim, Social Worker

Anand Soorneedi, Amherst

Dorcas Grigg-Saito, Northampton, retired Community Health Center CEO

Erica Deighton, Retired educator, Amherst resident.

Steve Waksman, Elsie Irwin Sweeney Professor of Music, Smith College

Cornelia Daniel, Retired in Amherst

Christine Clark, Dental Hygienist 

Wendy Sutter, Amherst resident

Lijah Joyce, Amherst 

Patricia Maynard, Retired teacher. Northampton resident 

Heather Brown, Educator, Northampton 

Marissa Elkins, Attorney/City Councilor 

Mary Savarese, Retired Teacher 

Peggy Matthews-Nilsen, Amherst (Psychotherapist, Retired)

Julia Frisby, Hatfield MA

Karen Osborn, Anherst

Lisa Moos, Physical Therapist Assistant 

Patricia Duffy, Leverett

Barbara Palangi, Retired

Elizabeth Jimenez, Northampton

Sandy Oldershaw, South Hadley 

Tania Menz, Hatfield Resident, Hadley Family Physician

Kimberly Schlichting, resident of Hadley, teacher in Northampton

Elizabeth Hallstrom, resident of Amherst

Kasey Mimitz, Youth services coordinator

Scarlett Mimitz, Student

Nora Mimitz, Student

Emily Kawano, Non-profit Co-director

Jalen Michals Levy, EMT-B

Andrea Gaus, Farmer, Hatfield 

Melanie Miller, Northampton

Daniel and Angela Dee Amherst

Sandra Torrence, Teacher

Michelle Trim, Faculty at UMass Amherst/ South Hadley

Roberta Pato, Retired teacher, Northampton

Barbara Partee, Amherst resident and Professor Emerita, UMass Amherst

Norma Brunelle, Retired

Raymond R. Brunelle, Retired

Mark Brunelle, Laborer

Joanne Brunelle, Dental Assistant

Barbara Cooper, Retired teacher/librarian

Toni Brown, Hatfield

Faruk Akkus, Faculty at UMass Amherst

Felice Swados, South Hadley

Victoria Rosen, Northampton 

Felice Swados, South Hadley

Victoria Rosen, Northampton 

Jon Wynn, UMass Amherst, Associate Professor, Northampton Resident

Zelia Almeida, RN Pediatric ICU/ Belchertown 

Marci Linker, Occupational Therapist and Northampton resident

Lindsay Whittier-Liu, Northampton 

Sarah Wolfe, Northampton paralegal, resident of Belchertown

Jean Fay, Amherst educator

Lance Hodes, Pelham

Alex Robinson, Amherst

Barry Seth, Student in Amherst

Oliver Dubon, Amherst

Basil  Perkins, College Student

Ivonne Vidal, Belchertown, Attorney 

Tina Cornell, Florence 

Judith Trickey retired

Lisa  Packard, Amherst 

Bennett Lyons, Amherst, MA

Kate Matt, Shutesbury

Anne Hazzard, Amherst

Isolda Ortega-Bustamante, fundraiser; Amherst

Maureen Vezina, Belchertown 

Evelyn Trier, Mount Holyoke College Admission/ Amherst resident & parent

Katherine Kraft, retired, Amherst 

Marshall Cohen, retired, Amherst

Monroe Rabin, retired

George Collison, retired prof

Emilie Hamilton, Amherst

Stefan Gonick, Belchertown

John Hondrogen, retired and still masking in Pelham

David Gross, Pelham

Lili Kim, Amherst

Susan Watkins, Shutesbury

Amelia Vetter, Student, Amherst

Dan Levine, Business Owner

Theresa Ryan, Realtor

Jenny Miller Sechler, Psychotherapist, Northampton

Matthew Levin, retired pre-school/k teacher (Northampton)Hatfield (residence)

Robert Jackson, Amherst

Amy Dopp, Easthampton

Keri Heitner, Amherst

Anita Sarro, Retired Nurse-Attorney

Amy Hirsch, Psychologist

Emily Case, Amherst parent, Hatfield educator

Michelle McBride, UMASS Employee in Linguistics Department

Kimberly Stillwell, Speech Language Pathologist, Northampton

Delia Martinez, Retired teacher-keep masks in schools

Amy Martyn, Florence

Rebecca Leopold,Northampton, retired Amherst-Pelham HS teacher

Alicia Lopez, Teacher, Amherst

Sharon Moulton, Northampton

Louis Faassen, Architect

Scott Billups, Shutesbury

Jacqueline A. Faison, Pelham

Seth Lepore, Arts and Small Business Consultant, Easthampton, MA

Rachel Brod, Northampton

Stephanie and David Kraft, Retired

Jack Howe-Janssen, Florence

Annette Gates Teacher, Crocker Farm Elementary


Historical UMass Covid-19 data

This graph shows one week totals for new cases reported on the UMass Amherst COVID-19 Dashboard. The gap in summer 2021 is due to a lack of reporting. Similarly, the gaps in the post-summer data at Thanksgiving and Christmas are due to weeks with no reporting.

The data pre-summer 2021 were downloaded May 14th 2021 from the dashboard. They do not seem to be available any longer there, so I have put the .csv file here. To create the weeks, I summed seven days of reporting, with the last seven day period ending 2021-05-10. Because some days had no reporting, these periods are sometimes longer than a week. The post-summer 2021 data were copied from the dashboard’s weekly reports. A .csv file with the data used to make the graph is available here.


Materials for Summer Language and Music Research Group

Background reading on recent work by members of our group:

Moreton, Elliott, Brandon Prickett, Katya Pertsova, Josh Fennell, Joe Pater, and Lisa Sanders. Learning Repetition, but not Syllable Reversal. In Ryan Bennett, Richard Bibbs, Mykel Loren Brinkerhoff, Max J. Kaplan, Stephanie Rich, Nicholas Van Handel & Maya Wax Cavallaro (eds.), Supplemental Proceedings of the 2020 Annual Meeting on Phonology. Washington, DC: Linguistic Society of America. DOI:

Music versions of above experiment: work best in Chrome

No rule given:
Rule given:

White, Christopher, Joe Pater and Mara Breen. 2021/in submission. A Comparative Analysis of Melodic Rhythm in Two Corpora of American Popular Music. Ms., University of Massachusetts Amherst and Mount Holyoke University.

Judge Russell’s Praat scripts from his REU work last year

Readings on musical grouping (relevant to class music experiment)

Patel 2008 on grouping

Deutsch 2013 grouping overview – see esp. pp. 209-213

De Liège 1987

Hutchison et al. 2015 “Minding the Gap: An Experimental Assessment of Musical Segmentation Models”.


Three County Covid Tracker

This was a draft for the Shoestring Covid tracker, which is now live and updated every Friday by noon.

County and State Data

New case totals for week ending 3/11, per capita total using 2019 US census populations, CDC transmission level classification.

Hampshire – 223 new cases, 138.7 cases per 100K, CDC Red/High (prior week 271, 168.5 per 100K)

Hampden – 834 new cases, 178.8 cases per 100K, CDC Red/High (prior week 908, 194.7 per 100K)

Franklin – 48 new cases, 68.4 cases per 100K, CDC Orange/Substantial (prior week 30, 42.7 per 100K)

Massachusetts – 9353 new cases, 143.2 cases per 100K, CDC Red/High(prior week 9869, 143.2 per 100K)

Percentage of residents vaccinated

Hampshire – at least one dose 20%, fully vaccinated 12%

Hampden – at least one dose 18%, fully vaccinated 10%

Franklin – at least one dose 22%, fully vaccinated 11%

Current vaccination eligibility

Mass vaccination site sign-up (for all MA residents); check with your health care provider and local health board for other options.

Diagnosed cases of new variants of concern

From the CDC; the MassDPH does not maintain a public database. (Note that new variant testing rates remain very low in Massachusetts as well as the US as a whole in comparison to Canada and Europe).

Town and City Data

These tables provide totals of new Covid-19 cases for the weeks ending at the dates shown in the column headings. The per capita rates are color coded according to the CDC transmission level metric with Red indicating a High level, Orange Substantial, Yellow Moderate and Blue Low. Follow this link for an explanation of the metric, and a comparison with the one the MassDPH uses. Further details about the data and the tables are provided below.

These weekly totals are differences between totals from MassDPH public health reports from one week to the next (the reports came out two days after the dates shown above). Some inaccuracies in weekly totals calculated in this way may occur because of updates in earlier data (see e.g. –1 for Hatfield). Weekly totals per 100K based on populations supplied in the downloadable dashboard data. Asterisks indicate instances where one week’s total was given as <5, so the difference could not be calculated. “Change” is the current week over the last one; values in red are increases (values greater than 1). Blank values in Change indicate that one week had 0 cases, so a ratio could not be calculated. The county totals and populations in these tables are the sums over the cities and towns. “UMass” in the Hampshire table shows weekly totals from data downloaded from the UMass Amherst dashboard. The UMass weeks end a day earlier than the dates shown; these were the weeks that aligned best with Amherst in the state data.


The CDC Covid-19 metric

The CDC released a new color-coded Covid-19 transmission level metric as part of their Feb. 12 2021 guidance for school reopening. This post explains how it works, how it can be interpreted, and why it is better than the Massachusetts Covid-19 metric for community classification.

Community transmission levels from the CDC school reopening guidance.

The new case levels in the first row are given as a per capita weekly total (the number of cases times 100,000 divided by the population). To convert to a weekly total from a daily average as provided by the MassDPH, multiply by seven, and from a two-week total as you might find elsewhere, divide by two. An advantage of a weekly total over a daily average is that it’s more transparently related to the number that we really care about: how many cases of Covid-19 are in a community. (I’ve seen active cases estimated as anywhere from 10 to 21 days of new cases, and the number of infections has recently been estimated as about 4 times the new case number). All of these numbers have the advantage over raw daily counts that they smooth over irrelevant factors, like the differences between weekend and weekday test rates.

“New cases” is the number of people diagnosed with a positive molecular test. The second row in the above table “Percentage of NAATs…” shows positivity rates that are based on the number of tests that are positive, over the number of tests. Because it’s based on number of tests rather than people, the positivity rate is not a good measure of the incidence of the disease. Rather, it is a usually used as a measure of whether enough testing is being done; high positivity rates indicate a high proportion of (highly) symptomatic people being tested. In the CDC metric, the positivity rate measure is a safeguard against having a low new case count because you aren’t doing enough testing. The metric takes the higher of the classifications, for example placing a community in Yellow if the positivity rate is higher than 5%, even if the case rate is beneath ten per week. In Massachusetts there is now sufficient testing that we can pretty safely ignore this number, and just use new case rates for classification (e.g. in the April 1 report, all the 61 communities that had 5% or greater positivity also had 50 or more new cases per week, which would have placed them at least in the CDC Orange category already).

The CDC provided this metric as a part of its guidance on best practices for school reopening, as shown in the further tables appended at the end of this post. This kind of community transmission level metric is also useful for providing a rubric for quickly comparing across communities (on a color-coded map, for instance), and can also be used to guide other sorts of decision making, by officials, businesses, and even individuals. The CDC has not yet released guidance on how to use its metric in these ways (update: their late July 2021 revised mask guidance uses it), but the New York Times provides guidance for individuals relative to a current risk level “developed with public health experts at Johns Hopkins Bloomberg School of Public Health and Resolve to Save Lives, an initiative of Vital Strategies.” The NYTimes metric uses new case rates and percent test positivity in a similar way to the CDC metric, but its levels are defined differently, so we can’t map it directly. We can get a sense of how we can apply the CDC levels by comparing the NYTimes guidance on indoor activities for “Very High Risk” (> 80 new cases per 100K per week) and “Medium Risk” (5 – 20).

The NYTimes/Hopkins guidance for communities with “Very High Risk” of Covid-19 transmission (> 80 new cases per 100K per week).
The NYTimes/Hopkins guidance for communities with “Medium Risk” of Covid-19 transmission (5-20 new cases per 100K per week).

See this Atlantic article on discrepancies between state restrictions and advice for individuals (it also cites an epidemiologist as providing 70 per week per 100K as an upper bound on a personal decision to eat indoors at a restaurant).

The Massachusetts color-coded metric uses positivity rates in a different way than the CDC and the NYTimes/Hopkins metrics, in a way that doesn’t seem to make much sense. In the MA metric, for communities over 10K population, the difference between the Yellow and the Red classification is based on positivity rate alone (under 10K is done on raw counts of new cases). A community is classified as Yellow if it has a new case rate of 70 per week per 100K or more (note that this is a much higher bar than the CDC or NYTimes metrics). To be classified as Red, it must have in addition a positivity rate of greater than 4% (5% in communities smaller than 50K). Since the positivity rate is more a measure of testing than incidence, this would seem to mean: “As long as there is enough testing, there is only a moderate risk of transmission, no matter how high the new case rate is.”

The result of the MA metric’s unusual application of positivity rate is that apparently very different rates of incidence are all classified as yellow. This is shown in the following tables, which are based on bar graphs showing the average daily new case rates over the previous 2-weeks from the MassDPH public health reports, provided by The positivity rates are the dashed lines. I have indicated the classifications that the CDC metric would make on these rates.

Northampton as classified by the Massachusetts and CDC metrics.
Amherst as classified by the Massachusetts and CDC metrics.
Cambridge as classified by the Massachusetts and CDC metrics.

Northampton, Amherst and Cambridge were not chosen at random. These are all municipalities with a high concentration of individuals being tested at higher ed institutions, which artificially depresses the overall positivity rate. They therefore provide a particularly striking illustration of this general problem.

CDC transmission levels applied in school reopening guidance (from the March 19 update).


Hampshire County weekly new cases by town/city

I am no longer updating this page. An expanded version is being published every Friday by noon as the Shoestring Covid Tracker.

Total new cases for weeks ending on the dates shown as column headings. These are differences between totals from MassDPH public health reports from one week to the next (the reports came out two days after the dates shown above). Some inaccuracies in weekly totals calculated in this way may occur because of updates in earlier data (see e.g. –1 for Hatfield). Weekly totals per 100K based on populations supplied in the report. Color coding with the CDC categories from the Feb. 12 2021 guidance shown below. “Change” is the current week over the last one; values in red are increases (values greater than 1). “Hampshire” is the total over all the towns and cities except Greenfield and Holyoke, which are included as adjacent large municipalities (in Franklin and Hampden counties respectively). “UMass” shows weekly totals from data downloaded from the UMass Amherst dashboard. These weeks end a day earlier than the dates shown; these were the weeks that aligned best with Amherst in the state data.
The CDC transmission categories from the Feb. 12 2021 school reopening guidance. Positivity rates are not taken into account in the Hampshire County table above, and if they were they could lead to a higher risk categorization, but for at least Amherst, Easthampton and Northampton they would have no effect.

Some of the CDC guidance on school reopening is below. It should be noted that the guidance is controversial; see this New Republic article for an excellent discussion. It is clear, though, that the CDC metric does a better job of separating different degrees of disease spread than the Massachusetts one does, especially in Hampshire County – see this discussion and the Northampton and Amherst graphs here.

Some of the CDC school reopening guidance based on these categories (Table 2 in this document).


CDC guidance and local data

Letter to the Northampton Public School Committee, sent by e-mail March 3 2021.

Dear School Committeee members,

I am writing to share a couple graphs I made that illustrate how the CDC metric from the Feb. 12th school reopening guidance applies locally. The first graph shows the Northampton daily new case rates from the MassDPH public health reports. The dates are the ends of the two-week period over which the daily rate is averaged (the period ending Feb. 20 is from the Feb. 25 report). The CDC Red “High Transmission” category is greater or equal to 100 cases per week, so 14.3 cases per day. Orange “Substantial Transmission” is 50 per week or greater, so just over 7 per day. Yellow “Moderate Transmission” is 10-49, so 1.4 – 7 per day.


The colors of the bars correspond to the MassDPH classifications. This graph illustrates a flaw in that system, which is also illustrated by the fact that Amherst was categorized as yellow at the peak of the UMass outbreak, when it had 113 cases per 100K. The flaw has to do with how the MassDPH metric uses the positivity rate – I have a discussion here if you are interested. The most current data has us in the orange category, with almost exactly the same rate as we had in November. The graph on which I added the CDC categories comes from this very useful site created by a UMass Amherst computer science alum – you can find weekly updated interactive graphs for all Mass municipalities there.

The next graph shows weekly totals of new cases for Hampshire County from the MassDPH data, along with weekly totals with the data from UMass Amherst subtracted. There is more on this method here. The lines show how these totals correspond to the categories. With a population of 160K, 160 cases corresponds to 100 cases per 100K, the bottom end of the CDC Red category, and 80 cases corresponds to the bottom end of the CDC Orange category.


It is not totally obvious to me which of the three numbers are best to use is local decision making. Using Northampton alone, or Hampshire minus UMass ignores the added risk of a nearby outbreak, while using the Hampshire number perhaps exaggerates the risk that the UMass outbreak ads to our situation here in Northampton.

I’d like to encourage you to have a look at the guidance with these numbers in mind, if you aren’t familiar with it already. This NPR piece gives a good overview, calling it a “measured, data-driven effort”. It also links to the full guidance. Table 2 in the CDC document provides some guidance using the transmission levels – I’ve pasted it beneath my signature.

All the best,

Joe Pater

Northampton resident, father of a Jackson Street School student

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Update (not in e-mail): here are graphs for Cambridge and Amherst