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The springtime emergence of vast swarms of cicadas can be explained by a mathematical model of collective decision-making that has similarities to models describing stock market crashes.

Pick almost any location in the eastern United States—say, Columbus Ohio. Every 13 or 17 years, as the soil warms in springtime, vast swarms of cicadas emerge from their underground burrows singing their deafening song, take flight and mate, producing offspring for the next cycle.

This noisy phenomenon repeats all over the eastern and southeastern U.S. as 17 distinct broods emerge in staggered years. In spring 2024, billions of cicadas are expected as two different broods—one that appears every 13 years and another that appears every 17 years—emerge simultaneously.

Previous research has suggested that cicadas emerge once the soil temperature reaches 18°C, but even within a small geographical area, differences in sun exposure, foliage cover or humidity can lead to variations in temperature.

Now, in a paper published in the journal Physical Review E, researchers from the University of Cambridge have discovered how such synchronous cicada swarms can emerge despite these temperature differences.

The researchers developed a mathematical model for decision-making in an environment with variations in temperature and found that communication between cicada nymphs allows the group to come to a consensus about the local average temperature that then leads to large-scale swarms. The model is closely related to one that has been used to describe “avalanches” in decision-making like those among stock market traders, leading to crashes.

Mathematicians have been captivated by the appearance of 17- and 13-year cycles in various species of cicadas, and have previously developed mathematical models that showed how the appearance of such large prime numbers is a consequence of evolutionary pressures to avoid predation. However, the mechanism by which swarms emerge coherently in a given year has not been understood.

In developing their model, the Cambridge team was inspired by previous research on decision-making that represents each member of a group by a “spin” like that in a magnet, but instead of pointing up or down, the two states represent the decision to “remain” or “emerge.”

The local temperature experienced by the cicadas is then like a magnetic field that tends to align the spins and varies slowly from place to place on the scale of hundreds of meters, from sunny hilltops to shaded valleys in a forest. Communication between nearby nymphs is represented by an interaction between the spins that leads to local agreement of neighbours.

The researchers showed that in the presence of such interactions the swarms are large and space-filling, involving every member of the population in a range of local temperature environments, unlike the case without communication in which every nymph is on its own, responding to every subtle variation in microclimate.

The research was carried out Professor Raymond E Goldstein, the Alan Turing Professor of Complex Physical Systems in the Department of Applied Mathematics and Theoretical Physics (DAMTP), Professor Robert L Jack of DAMTP and the Yusuf Hamied Department of Chemistry, and Dr. Adriana I Pesci, a Senior Research Associate in DAMTP.

“As an applied mathematician, there is nothing more interesting than finding a model capable of explaining the behaviour of living beings, even in the simplest of cases,” said Pesci.

The researchers say that while their model does not require any particular means of communication between underground nymphs, acoustical signaling is a likely candidate, given the ear-splitting sounds that the swarms make once they emerge from underground.

The researchers hope that their conjecture regarding the role of communication will stimulate field research to test the hypothesis.

“If our conjecture that communication between nymphs plays a role in swarm emergence is confirmed, it would provide a striking example of how Darwinian evolution can act for the benefit of the group, not just the individual,” said Goldstein.

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Credit of the article given to Sarah Collins, University of Cambridge

 

Have you ever wondered about the best way to pack a finite number of identical spheres into a shape-shifting flexible container, like a convex hull?

Researchers from the University of Twente, Active Soft Matter Lab led by Dr. Hanumantha Rao Vutukuri in the TNW Faculty, along with Utrecht University, have investigated this fascinating mathematical sphere-packing problem by combining experiments and computer simulations. Their research has been published in Nature Communications.

An intuitively simple problem concerning the best way to pack a set of spheres has a long history of studies dating back to the 17th century. The British sailor Raleigh, for instance, contemplated this issue while trying to find an efficient method for stacking cannonballs on his ship.

Later, Kepler conjectured that the densest packing for an infinite number of spheres would be the face-centered cubic (FCC) crystal structure, akin to the hexagonal arrangement of oranges and apples seen in supermarkets. Remarkably, this hypothesis was only proven in the 21st century.

The ‘sausage catastrophe’

When you have a finite number of spheres, everything gets more complicated; surprisingly, packing the “finite” spheres in a compact cluster does not always yield the densest packing. Mathematicians already conjectured decades ago that a linear, sausage-like arrangement provides the best packing, however, not for all numbers of spheres.

There’s a peculiar phenomenon at play: The sausage-shaped arrangement is the most efficient packing, but only with up to 55 spheres. Beyond that number, a clustered arrangement becomes the best packing. This abrupt transition is known as the “sausage catastrophe.”

In three-dimensional space, packing up to 55 spheres linearly forms a “sausage” that is denser than any cluster arrangement. However, in four dimensions, this scenario changes dramatically. About 300,000 spheres are needed for the “sausage” to transform into a spherelike cluster.

Rao was curious about whether this fundamentally intriguing problem could be observed and resolved in the lab using a model system. This system includes micron-sized spherical particles (colloids) and giant unilamellar vesicles (GUVs), which serve as flexible containers that are the main ingredients in Vutukuri’s lab.

“This curiosity led us to explore the finite sphere packing problem through experiments in 3D real space, specifically using colloids in GUVs. By varying the number of particles and the volume of the vesicles, we were able to examine the different particle arrangements inside these vesicles using a confocal microscope.

“We identified stable arrangements for specific combinations of vesicle volume and particle number: 1D (sausage), 2D (plate, with particles in one plane), and 3D (cluster). Notably, we also observed bistability; the configurations alternated between 1D and 2D arrangements or between 2D and 3D structures. However, our experiments were limited to observing a maximum of nine particles, as packing a larger number of particles resulted in the rupture of the vesicles.”

Vutukuri says that they then contacted Dijkstra’s lab at Utrecht University to delve deeper into this problem using simulations. Strikingly, the simulations predicted that packing spheres in a sausage configuration is most efficient for up to 55 spheres. However, when they attempted to pack 56 spheres into a vesicle, they discovered that a compact three-dimensional cluster was the more efficient arrangement.

Remarkably, for 57 spheres, the packing reverted back to a sausage configuration. While mathematicians have previously determined that a sausage configuration is the most efficient for 58 and 64 spheres, their study contradicts this, demonstrating that compact clusters are more effective.

The findings show that the “sausage catastrophe,” a phenomenon previously described by mathematicians, is not just a theoretical scenario but can also be observed experimentally.

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Credit of the article given to University of Twente

 

 

Research by Social Statistics has highlighted new methods to improve the accuracy of cross-national surveys. Cross-national surveys run the risk of differential survey errors, where data collected vary in quality from country to country. The study is published in the Journal of Survey Statistics and Methodology.

Responsive and adaptive survey designs (RASDs) have been proposed as a way to reduce survey errors, by leveraging auxiliary variables to inform fieldwork efforts, but have rarely been considered in the context of cross-national surveys.

Using data from the European Social Survey, Dr. Alex Cernat, Dr. Hafsteinn Einarsson and Professor Natalie Shlomo from Social Statistics simulate fieldwork in a repeated cross-national survey using RASD where fieldwork efforts are ended early for selected units in the final stage of data collection.

Demographic variables, paradata (interviewer observations), and contact data are used to inform fieldwork efforts.

Eight combinations of response propensity models and selection mechanisms are evaluated in terms of sample composition (as measured by the coefficient of variation of response propensities), response rates, number of contact attempts saved, and effects on estimates of target variables in the survey.

The researchers find that sample balance can be improved in many country-round combinations. Response rates can be increased marginally and targeting high propensity respondents could lead to significant cost savings associated with making fewer contact attempts.

Estimates of target variables are not changed by the case prioritizations used in the simulations, indicating that they do not impact non-response bias.

They conclude that RASDs should be considered in cross-national surveys, but that more work is needed to identify suitable covariates to inform fieldwork efforts.

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Credit of the article given to University of Manchester

Although a typical mathematical task has a single correct answer, in some cases, the assumptions of a proposition determine its truth. Such relativity of truth plays a major role in the development of mathematics.

Furthermore, in our daily lives, we must identify assumptions that underlie each other’s discussions and clarify such assumptions for better communication. Hence, students’ recognition of the relativity of truth involving assumptions must be developed; however, how to encourage such development in primary and secondary education remains unclear.

To address this issue, the researchers have developed principles that support the design of mathematics tasks. Contrary to typical mathematical tasks, the researchers introduced an innovation in which the conditions of tasks are intentionally made ambiguous, directing students’ attention to the task assumptions.

The study is published in Cognition and Instruction.

The researchers collaborated with primary and secondary school teachers to implement research cycles, each of which composed of designing a mathematical task, implementing it in one or more classrooms, and evaluating such implementation. Based on these research cycles, they developed task design principles, which involved creating a task open to different legitimate assumptions and conclusions by intentionally leaving some of the task’s assumptions implicit or unspecified, and demonstrated the effectiveness of these principles.

The task design principles developed in this study allow teachers to design relevant tasks independently and implement them in their classrooms, which would foster their students’ recognition of the relativity of truth in different circumstances.

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Credit of the article given to University of Tsukuba

 

After discovering that a Ralph Rowlett was in charge of the Royal Mint in 1540, Peter Rowlett runs the genealogy calculations to find out if he could be related.

In 1540, Henry VIII’s coins were made in the Tower of London. One of the Masters of the Mint was Ralph Rowlett, a goldsmith from St Albans with six children. I wondered: am I descended from Ralph? My Rowlett ancestors were Sheffield steelworkers, ever since my three-times great grandfather moved north in search of work. The trail goes cold in a line of Bedfordshire farm labourers in the 18th century, offering no evidence of a direct relationship.

My instincts as a mathematician led me to investigate this in a more mathematical way. I have two parents. They each have two parents, so I have four grandparents. So, I have eight great-grandparents, 16 great-great-grandparents and 2ⁿ ancestors n generations ago. This exponential growth doubles each generation and takes 20 generations to reach a million ancestors.

Ralph lived 20 to 25 generations before me in an England of about 2 million people. The exponential growth argument says I have several million ancestors in his generation, so, because we run out of people otherwise, he is one of them.

But this model is based on the assumption that everyone is equally likely to reproduce with anyone else. In reality, especially at certain points in history, people were likely to reproduce with someone from the same geographic area and demographic group as themselves.

But I am not sure this makes a huge difference here because we are dealing with something called a small-world network: most people are in highly clustered groups, tending to pair up with nearby people, but a small number are connected over greater distances. An illegitimate child of a nobleman would have a different social class to their father. A migrant seeking work could reproduce in a different geographic area.

We don’t need many of these more remote connections to allow a great amount of spread around the network. This is the origin of the six degrees of separation concept – that you can link two people through a surprisingly short chain of friend-of-a-friend relationships.

I ran a simulation with 15 towns of a thousand people, where everyone has only a 5 per cent chance of moving to another town to reproduce. It took about 20 generations for everyone to be descended from a specific person in the first generation. I ran the same simulation with 15,000 people living in one town, and the spread took about 18 generations. So the 15-town structure slowed the spread, but only slightly.

What does this mean for Ralph and me? There is a very good chance we are related, whether through Rowletts or another route. And if you have recent ancestors from England, there is a good chance you are too.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Peter Rowlett*

Our lives are becoming increasingly data driven. Our phones monitor our time and internet usage and online surveys discern our opinions and likes. These data harvests are used for telling us how well we’ve slept or what we might like to buy.

Numbers are becoming more important for everyday life, yet people’s numerical skills are falling behind. For example, the percentage of Year 12 schoolchildren in Australia taking higher and intermediate mathematics has been declining for decades.

To help the average person understand big data and numbers, we often use visual summaries, such as pie charts. But while non-numerate folk will avoid numbers, most numerate folk will avoid pie charts. Here’s why.

What is a pie chart?

A pie chart is a circular diagram that represents numerical percentages. The circle is divided into slices, with the size of each slice proportional to the category it represents. It is named because it resembles a sliced pie and can be “served” in many different ways.

An example pie chart below shows Australia’s two-party preferred vote before the last election, with Labor on 55% and the the Coalition on 45%. The two near semi-circles show the relatively tight race—this is a useful example of a pie chart.

What’s wrong with pie charts?

Once we have more than two categories, pie charts can easily misrepresent percentages and become hard to read.

The three charts below are a good example—it is very hard to work out which of the five areas is the largest. The pie chart’s circularity means the areas lack a common reference point.

Pie charts also do badly when there are lots of categories. For example, this chart from a study on data sources used for COVID data visualization shows hundreds of categories in one pie.

The tiny slices, lack of clear labeling and the kaleidoscope of colors make interpretation difficult for anyone.

It’s even harder for a color blind person. For example, this is a simulation of what the above chart would look like to a person with deuteranomaly or reduced sensitivity to green light. This is the most common type of color blindness, affecting roughly 4.6% of the population.

It can get even worse if we take pie charts and make them three-dimensional. This can lead to egregious misrepresentations of data.

Below, the yellow, red and green areas are all the same size (one-third), but appear to be different based on the angle and which slice is placed at the bottom of the pie.

So why are pie charts everywhere?

Despite the well known problems with pie charts, they are everywhere. They are in journal articles, Ph.D. theses, political polling, books, newspapers and government reports. They’ve even been used by the Australian Bureau of Statistics.

While statisticians have criticized them for decades, it’s hard to argue with this logic: “If pie charts are so bad, why are there so many of them?”

Possibly they are popular because they are popular, which is a circular argument that suits a pie chart.

What’s a good alternative to pie charts?

There’s a simple fix that can effectively summarize big data in a small space and still allow creative color schemes.

It’s the humble bar chart. Remember the brain-aching pie chart example above with the five categories? Here’s the same example using bars—we can now instantly see which category is the largest.

Linear bars are easier on the eye than the non-linear segments of a pie chart. But beware the temptation to make a humble bar chart look more interesting by adding a 3D effect. As you already saw, 3D charts distort perception and make it harder to find a reference point.

Below is a standard bar chart and a 3D alternative of the number of voters in the 1992 US presidential election split by family income (from under US$15K to over $75k). Using the 3D version, can you tell the number of voters for each candidate in the highest income category? Not easily.

Is it ever okay to use a pie chart?

We’ve shown some of the worst examples of pie charts to make a point. Pie charts can be okay when there are just a few categories and the percentages are dissimilar, for example with one large and one small category.

Overall, it is best to use pie charts sparingly, especially when there is a more “digestible” alternative—the bar chart.

Whenever we see pie charts, we think one of two things: their creators don’t know what they’re doing, or they know what they are doing and are deliberately trying to mislead.

A graphical summary aims to easily and quickly communicate the data. If you feel the need to spruce it up, you’re likely reducing understanding without meaning to do so.

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Credit of the article given to Adrian Barnett and Victor Oguoma, The Conversation

 

 

In fields such as physics and engineering, partial differential equations (PDEs) are used to model complex physical processes to generate insight into how some of the most complicated physical and natural systems in the world function.

To solve these difficult equations, researchers use high-fidelity numerical solvers, which can be very time consuming and computationally expensive to run. The current simplified alternative, data-driven surrogate models, compute the goal property of a solution to PDEs rather than the whole solution. Those are trained on a set of data that has been generated by the high-fidelity solver, to predict the output of the PDEs for new inputs. This is data-intensive and expensive because complex physical systems require a large number of simulations to generate enough data.

In a new paper, “Physics-enhanced deep surrogates for partial differential equations,” published in December in Nature Machine Intelligence, a new method is proposed for developing data-driven surrogate models for complex physical systems in such fields as mechanics, optics, thermal transport, fluid dynamics, physical chemistry, and climate models.

The paper was authored by MIT’s professor of applied mathematics Steven G. Johnson along with Payel Das and Youssef Mroueh of the MIT-IBM Watson AI Lab and IBM Research; Chris Rackauckas of Julia Lab; and Raphaël Pestourie, a former MIT postdoc who is now at Georgia Tech. The authors call their method “physics-enhanced deep surrogate” (PEDS), which combines a low-fidelity, explainable physics simulator with a neural network generator. The neural network generator is trained end-to-end to match the output of the high-fidelity numerical solver.

“My aspiration is to replace the inefficient process of trial and error with systematic, computer-aided simulation and optimization,” says Pestourie. “Recent breakthroughs in AI like the large language model of ChatGPT rely on hundreds of billions of parameters and require vast amounts of resources to train and evaluate. In contrast, PEDS is affordable to all because it is incredibly efficient in computing resources and has a very low barrier in terms of infrastructure needed to use it.”

In the article, they show that PEDS surrogates can be up to three times more accurate than an ensemble of feedforward neural networks with limited data (approximately 1,000 training points), and reduce the training data needed by at least a factor of 100 to achieve a target error of 5%. Developed using the MIT-designed Julia programming language, this scientific machine-learning method is thus efficient in both computing and data.

The authors also report that PEDS provides a general, data-driven strategy to bridge the gap between a vast array of simplified physical models with corresponding brute-force numerical solvers modeling complex systems. This technique offers accuracy, speed, data efficiency, and physical insights into the process.

Says Pestourie, “Since the 2000s, as computing capabilities improved, the trend of scientific models has been to increase the number of parameters to fit the data better, sometimes at the cost of a lower predictive accuracy. PEDS does the opposite by choosing its parameters smartly. It leverages the technology of automatic differentiation to train a neural network that makes a model with few parameters accurate.”

“The main challenge that prevents surrogate models from being used more widely in engineering is the curse of dimensionality—the fact that the needed data to train a model increases exponentially with the number of model variables,” says Pestourie. “PEDS reduces this curse by incorporating information from the data and from the field knowledge in the form of a low-fidelity model solver.”

The researchers say that PEDS has the potential to revive a whole body of the pre-2000 literature dedicated to minimal models—intuitive models that PEDS could make more accurate while also being predictive for surrogate model applications.

“The application of the PEDS framework is beyond what we showed in this study,” says Das. “Complex physical systems governed by PDEs are ubiquitous, from climate modeling to seismic modeling and beyond. Our physics-inspired fast and explainable surrogate models will be of great use in those applications, and play a complementary role to other emerging techniques, like foundation models.”

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Credit of the article given to Sandi Miller, Massachusetts Institute of Technology

 

When children start school, they learn how to recite their numbers (“one, two, three…”) and how to write them (1, 2, 3…). Learning about what those numbers mean is even more challenging, and this becomes trickier yet when numbers have more than one digit — such as 42 and 608.

It turns out that the meaning of such “multidigit” numbers cannot be gleaned from simply looking at them or by performing calculations with them. Our number system has many hidden meanings that are not transparent, making it difficult for children to comprehend it.

In collaboration with elementary teachers, the Mathematics Teaching and Learning Lab at Concordia University explores tools that can support young children’s understanding of multidigit numbers.

We investigate the impact of using concrete objects (like bundling straws into groups of 10). We also investigate the use of visual tools, such as number lines and charts, or words to represent numbers (the word for 40 is “forty”) and written notation (for example, 42).

Our recent research examined whether the “hundreds chart” — 10 by 10 grids containing numbers from one to 100, with each row in the chart containing numbers in groups of 10 — could be useful for teaching children about counting by 10, something foundational for understanding how numbers work.

When children start learning about numbers, they do not naturally see tens and ones in a number like 42. (Shutterstock)

What’s in a number?

Most adults know that the placement of the “4” and “2” in 42 means four tens and two ones, respectively.

But when young children start learning about numbers, they do not naturally see 10s and ones in a number like 42. They think the number represents 42 things counted from one to 42 without distinguishing between the meaning of the digits “4” and “2.” Over time, through counting and other activities, children see the four as a collection of 40 ones.

This realization is not sufficient, however, for learning more advanced topics in math.

An important next step is to see that 42 is made up of four distinct groups of 10 and two ones, and that the four 10s can be counted as if they were ones (for example, 42 is one, two, three, four 10s and one, two, “ones”).

Ultimately, one of the most challenging aspects of understanding numbers is that groups of ten and ones are different kinds of units.

Numbers can be arranged in different ways

The numbers in hundreds charts can be arranged in different ways. A top-down hundreds chart has the digit “1” in the top-left corner and 100 in the bottom-right corner.

A top-down hundreds chart. (Vera Wagner), Author provided (no reuse)

The numbers increase by 10 moving downward one row at a time, like going from 24 to 34 using one hop down, for instance. A second type of chart is the “bottom-up” chart, which has the numbers increasing in the opposite direction.

A bottom-up hundreds chart. (Vera Wagner), Author provided (no reuse)

Counting by 10s

Children can move from one number to another in the chart to solve problems. Considering 24 + 20, for example, children could start on 24 and move 20 spaces to land on 44.

Another way would be to move up (or down, depending on the chart) two rows (for example, counting “one,” “two”) until they land on 44. This second method shows a developing understanding of multidigit numbers being composed of distinct groups of 10, which is critical for an advanced knowledge of the number system.

For her master’s research at Concordia University, Vera Wagner, one of the authors of this story, thought children might find it more intuitive to solve problems with the bottom-up chart, where the numbers get larger with upward movement.

After all, plants grow taller and liquid rises in a glass as it is filled. Because of such familiar experiences, she thought children would move by tens more frequently in the bottom-up chart than in the top-down chart.

 

Study with kindergarteners, Grade 1 students

To examine this hypothesis, we worked with 47 kindergarten and first grade students in Canada and the United States. All the children but one spoke English at home. In addition to English, 14 also spoke French, four spoke Spanish, one spoke Russian, one spoke Arabic, one spoke Mandarin and one communicated to some extent in ASL at home.

We assigned all child participants in the study an online version of either a top-down or bottom-up hundreds chart, programmed by research assistant André Loiselle, to solve arithmetic word problems.

What we found surprised us: children counted by tens more often with the top-down chart than the bottom-up one. This was the exact opposite of what we thought they might do!

This finding suggests that the top-down chart fosters children’s counting by tens as if they were ones (that is, up or down one row at a time), an important step in their mathematical development. Children using the bottom-up chart were more likely to confuse the digits and move in the wrong direction.

Tools can impact learning

Tools used in the math classroom can impact children’s learning. (Shutterstock)

Our research suggests that the types of tools used in the math classroom can impact children’s learning in different ways.

One advantage of the top-down chart could be the corresponding

Our research suggests that the types of tools used in the math classroom can impact children’s learning in different ways.

One advantage of the top-down chart could be the corresponding left-to-right and downward movement that matches the direction in which children learn to read in English and French, the official languages of instruction in the schools in our study. Children who learn to read in a different direction (for example, from right to left, as in Arabic) may interact with some math tools differently from children whose first language is English or French.

The role of cultural experiences in math learning opens up questions about the design of teaching tools for the classroom, and the relevance of culturally responsive mathematics teaching. Future research could seek to directly examine the relation between reading direction and the use of the hundreds chart.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to The Conversation

 

Researchers from Northwestern University, University of Pennsylvania, and University of Colorado published a new Journal of Marketing study that proposes abandoning null hypothesis significance testing (NHST) as the default approach to statistical analysis and reporting.

The study is titled “‘Statistical Significance’ and Statistical Reporting: Moving Beyond Binary” and is authored by Blakeley B. McShane, Eric T. Bradlow, John G. Lynch, Jr., and Robert J. Meyer.

Null hypothesis significance testing (NHST) is the default approach to statistical analysis and reporting in marketing and, more broadly, in the biomedical and social sciences. As practiced, NHST involves

1. assuming that the intervention under investigation has no effect along with other assumptions,
2. computing a statistical measure known as a P-value based on these assumptions, and
3. comparing the computed P-value to the arbitrary threshold value of 0.05.

If the P-value is less than 0.05, the effect is declared “statistically significant,” the assumption of no effect is rejected, and it is concluded that the intervention has an effect in the real world. If the P-value is above 0.05, the effect is declared “statistically nonsignificant,” the assumption of no effect is not rejected, and it is concluded that the intervention has no effect in the real world.

Criticisms of NHST

Despite its default role, NHST has long been criticized by both statisticians and applied researchers, including those within marketing. The most prominent criticisms relate to the dichotomization of results into “statistically significant” and “statistically nonsignificant.”

For example, authors, editors, and reviewers use “statistical (non)significance” as a filter to select which results to publish. Meyer says that “this creates a distorted literature because the effects of published interventions are biased upward in magnitude. It also encourages harmful research practices that yield results that attain so-called statistical significance.”

Lynch adds that “NHST has no basis because no intervention has precisely zero effect in the real world and small P-values and ‘statistical significance’ are guaranteed with sufficient sample sizes. Put differently, there is no need to reject a hypothesis of zero effect when it is already known to be false.”

Perhaps the most widespread abuse of statistics is to ascertain where some statistical measure such as a P-value stands relative to 0.05 and take it as a basis to declare “statistical (non)significance” and to make general and certain conclusions from a single study.

“Single studies are never definitive and thus can never demonstrate an effect or no effect. The aim of studies should be to report results in an unfiltered manner so that they can later be used to make more general conclusions based on cumulative evidence from multiple studies. NHST leads researchers to wrongly make general and certain conclusions and to wrongly filter results,” says Bradlow.

“P-values naturally vary a great deal from study to study,” explains McShane. As an example, a “statistically significant” original study with an observed P-value of p = 0.005 (far below the 0.05 threshold) and a “statistically nonsignificant” replication study with an observed P-value of p = 0.194 (far above the 0.05 threshold) are highly compatible with one another in the sense that the observed P-value, assuming no difference between them, is p= 0.289.

He adds that “however when viewed through the lens of ‘statistical (non)significance,’ these two studies appear categorically different and are thus in contradiction because they are categorized differently.”

Recommended changes to statistical analysis

The authors propose a major transition in statistical analysis and reporting. Specifically, they propose abandoning NHST—and the P-value thresholds intrinsic to it—as the default approach to statistical analysis and reporting. Their recommendations are as follows:

• “Statistical (non)significance” should never be used as a basis to make general and certain conclusions.
• “Statistical (non)significance” should also never be used as a filter to select which results to publish.
• Instead, all studies should be published in some form or another.
• Reporting should focus on quantifying study results via point and interval estimates. All of the values inside conventional interval estimates are at least reasonably compatible with the data given all of the assumptions used to compute them; therefore, it makes no sense to single out a specific value, such as the null value.
• General conclusions should be made based on the cumulative evidence from multiple studies.
• Studies need to treat P-values continuously and as just one factor among many—including prior evidence, the plausibility of mechanism, study design, data quality, and others that vary by research domain—that require joint consideration and holistic integration.
• Researchers must also respect the fact that such conclusions are necessarily tentative and subject to revision as new studies are conducted.

Decisions are seldom necessary in scientific reporting and are best left to end-users such as managers and clinicians when necessary.

In such cases, they should be made using a decision analysis that integrates the costs, benefits, and probabilities of all possible consequences via a loss function (which typically varies dramatically across stakeholders)—not via arbitrary thresholds applied to statistical summaries such as P-values (“statistical (non)significance”) which, outside of certain specialized applications such as industrial quality control, are insufficient for this purpose.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to American Marketing Association

 

Mathematicians have long been fascinated by the most efficient way of packing spheres in a space, and now physicists have confirmed that the best place to put them is into a sausage shape, at least for small numbers of balls.

Simulations show microscopic plastic balls within a cell membrane

What is the most space-efficient way to pack tennis balls or oranges? Mathematicians have studied this “sphere-packing” problem for centuries, but surprisingly little attention has been paid to replicating the results in the real world. Now, physical experiments involving microscopic plastic balls have confirmed what mathematicians had long suspected – with a small number of balls, it is best to stick them in a sausage.

Kepler was the first person to tackle sphere packing, suggesting in 1611 that a pyramid would be the best way to pack cannonballs for long voyages, but this answer was only fully proven by mathematicians in 2014.

This proof only considers the best way of arranging an infinite number of spheres, however. For finite sphere packings, simply placing the balls in a line, or sausage, is more efficient until there are around 56 spheres. At this point, the balls experience what mathematicians call the “sausage catastrophe” and something closer to pyramid packing becomes more efficient.

But what about back in the real world? Sphere-packing theories assume that the balls are perfectly hard and don’t attract or repel each other, but this is rarely true in real life – think of the squish of a tennis ball or an orange.

One exception is microscopic polystyrene balls, which are very hard and basically inert. Hanumantha Rao Vutukuri at the University of Twente in the Netherlands and his team, who were unaware of mathematical sphere-packing theories, were experimenting with inserting these balls into empty cell membranes and were surprised to find them forming sausages.

“One of my students observed a linear packing, but it was quite puzzling,” says Vutukuri. “We thought that there was some fluke, so he repeated it a couple of times and every time he observed similar results,” says Vutukuri. “I was wondering, ‘why is this happening?’ It’s a bit counterintuitive.”

After reading up on sphere packing, Vutukuri and his team decided to investigate and carried out simulations for different numbers of polystyrene balls in a bag. They then compared their predictions with experiments using up to nine real polystyrene balls that had been squeezed into cell membranes immersed in a liquid solution. They could then shrink-wrap the balls by changing the concentration of the solution, causing the membranes to tighten, and see what formation the balls settled in using a microscope.

“For up to nine spheres, we showed, both experimentally and in simulations, that the sausage is the best packed,” says team member Marjolein Dijkstra at Utrecht University, the Netherlands. With more than nine balls, the membrane became deformed by the pressure of the balls. The team ran simulations for up to 150 balls and reproduced the sausage catastrophe, where it suddenly becomes more efficient to pack things in polyhedrons, with between 56  and 70 balls.

The sausage formation for a small number of balls is unintuitive, says Erich Müller at Imperial College London, but makes sense because of the large surface area of the membrane with respect to the balls at low numbers. “When dimensions become really, really small, then the wall effects become very important,” he says.

The findings could have applications in drug delivery, such as how to most efficiently fit hard antibiotic molecules, like gold, inside cell-like membranes, but the work doesn’t obviously translate at this point, says Müller.

 

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*Credit for article given to Alex Wilkins*