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Welcome to VSNi Blogs
We regularly release blogs written by statisticians, researchers, scientists and industry leaders. From how-to guides to the application of data analytics and opinion pieces, you will find a wealth of content below
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The VSNi Team
5 days agoAndrew Day joins the board of leading statistical software business, VSN International Limited
VSNi is delighted to announce that Andrew Day has joined the VSNi board as non-executive director. Andy has extensive experience driving transformational change through the exploitation of big data across diverse industries, including telecommunications, media, retail and financial services.
Andy’s full-time role is Group Chief Data Officer with Pepper Global, the innovative lending-solutions business. Prior to that having already held senior roles with Telefonica O2 and Sky plc, Andy pioneered the application of data as Business Intelligence Director with UK media group News UK before moving to take up the position of Chief Data Officer at Sainsbury’s in the UK. He is also a non-executive director with GlobalData plc, a leading provider of data, analytics, and insights on the world's largest industries.
Commenting, Andy said, “I’m delighted to be joining the board of VSNi. It’s a company with a rich heritage in the application of data and science in biosciences and, most excitingly, an innovation agenda to put information and analytics at the heart of food production globally.”
VSNi CEO Stewart Andrews said, “We consider ourselves extremely privileged to welcome Andy to the VSNi board, and feel sure that his experience and approach will help us drive towards our vision – to see the challenges of global food production eradicated through the effective application of data, analytics and technology”.
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Prof. Stephen Senn
8 days agoThe Rule of Three
The importance of deciding whether it is necessary to use nitrogen in manures needs no further comment. It was to settle definitely questions like this that John Bennet Lawes began his experiments at Rothamsted in Hertfordshire on the manuring of crops.
T B Wood (1913), The Story of a Loaf of Bread, p4.
The towering trio
Three great heads of statistics at Rothamsted made important contributions to the design and analysis of experiments. Ronald Aylmer Fisher (1890-1962) was at Rothamsted Experimental Station from 1919 to 1933, initially as the sole statistician and then as the head of statistics. When he left to become Galton Professor of Eugenics at University College London he was succeeded as head of statistics by Frank Yates (1902-1994), who had only arrived to work at Rothamsted two years earlier. Yates was to remain the head of statistics for 35 years. When he retired in 1968, his successor was John Nelder (1924-2010), who had previously worked briefly at Rothamsted but who was, at the time of his appointment, head of statistics at The National Vegetable Research Station in Wellesbourne. John Nelder remained head until his retirement in 1984.
All three made considerable contributions to many fields of statistics and Fisher also to genetics and evolutionary biology. Yates worked on sampling theory and computing, Nelder on computing and modelling and Fisher on just about everything. A common interest of all three, however, was the design and analysis of experiments. Together they created what I like to think of as The Rothamsted School of design and analysis of experiments. Of course, they were not the only statisticians who did this. Many others who worked at Rothamsted made important contributions, as did others elsewhere. Nevertheless, the work of these three was crucial and the theory they created has been extremely influential, although not, as I shall explain in due course, as influential as it deserves to be.
Field work
An important context for the development of this theory was that agriculture was the field (a word that always causes one to pause, given the subject) of application. Agricultural scientists were ingenious and ambitious in constructing complex experiments. Typically a field would be subdivided into plots, to which treatments would be applied, but it could be that other treatments were applied at a lower subplot level. Due to spatial autocorrelation in fertility, variation between plots would generally be higher than variation between subplots. Thus, care had to be taken in judging whether the effects of the treatments applied differed from each other by more than could be expected by chance. Discovering exactly how this should be done is something that took half a century, and all the three heads made important contributions.
In 1910 Thomas Barlow Wood (1869-1929), an agricultural scientist, and Frederick John Marrian Stratton (1881-1960), an astronomer, collaborated to write a paper that described how the accuracy of results from an agricultural experiment could be estimated [1]. They showed how a technique that astronomers had long been using to assess the reliability of a mean of a number of differing observations could be applied to agricultural yields also. (They took an example of the percentage of dry matter in 160 roots of a variety of Golden Globe mangold.) They also showed how two treatments could be compared.
The theory of errors was well established amongst astronomers. George Bidell Airy (1801-1892) had written a monograph on the subject [2] in 1862 that had become a standard work of reference. From a modern perspective, it is slightly surprising that Wood and Stratton felt it necessary to explain a common technique amongst astronomers to agronomists but as they put it:
It might seem at first that no two branches of study could be more widely separated than Agriculture and Astronomy. A moment's consideration, however, will show that they have one point in common: both are at the mercy of the weather. (p425)
Furthermore, only two years earlier, Student (William Sealy Gosset, 1976-1937), whose work involved regular contact with problems of agricultural experiments, had published his later-to-become-famous paper The Probable Error of A Mean [3]. This was in many ways in advance of that of Wood and Stratton. Presumably, they were unaware of Student’s work but we now know that Student himself had been anticipated in 1876 by Jakob Luroth(1844-1910) [4], a German mathematician who was originally an astronomer, so the story of astronomers and agronomists advancing the theory of errors by stumbling past each other has some history.
An interesting connection (it has perhaps a causal explanation and cannot be marked down definitively as a coincidence) is that both Wood and Stratton had connections to Caius College Cambridge, as did Fisher.
Starting in the middle
I am going to pick up the story with the second of these statisticians. Frank Yates studied mathematics at Cambridge, graduating in 1924 and after a brief period teaching at Malvern college worked from 1927-1931 as a surveyor in what is now Ghana [5]. This either honed or provided an outlet for a talent for efficient computation and developing effective algorithms. Surveying required a lot of calculation using least squares and as David Finney put it [6]:
Gaussian least squares was not a topic then taught to undergraduate mathematicians; the need for regularly using this technique undoubtedly developed in him the concern for efficient, well-organised, and accurate computation that characterised his later career. (p2)
Interestingly, Yates never saw the need for matrix algebra and generations of statisticians working at Rothamsted subsequently had to hide their interest in matrices from the head of statistics!
On his arrival at Rothamsted, Yates started collaborating with Fisher, developing, in particular, the work on the design and analysis of experiments; he achieved much rapidly. A good example is given by his Royal Statistical Society(RSS) read paper of 1935, ‘_Complex Experiments_’ [7]. This presents a dazzling array of ideas with much of what has become standard theory to support them, but is also grounded in application. Many of the ideas come directly from Fisher, some indirectly, but there are also many felicitous and ingenious touches that are clearly due to Yates. In it he covers complex treatment structures, in particular for factorial designs, but also how to deal with different sources of variation in the experimental material, including their influence on efficient estimation and appropriate error estimation, for example for incomplete block designs, a topic he was to develop more fully the following year. [8]
As was usual for a read paper, a number of commentaries were also published. Neyman pointed out that interactive effects in factorial experiments would be estimated with low precision. Yates changed his definition in the published version of the paper from the version read to the RSS and in reply to Neyman added the remark:
Since the meeting, I have altered my definition of an interaction by the inclusion of a factor 1/2, for reasons stated in the text. (p247)
This had the effect of reducing the standard error. However, this response was not quite fair. I once discussed this with Michael Healy, a statistician who also worked at Rothamsted, and he agreed with me that however useful this modification might be algorithmically, it was not an answer to Neyman’s criticism.
Back to the beginning
In his published comment on Yates’s read paper, Fisher drew attention to two aspects of any experiment (in Genstat we now call these the block structure and the treatment structure). He gave an example of a field with plots arranged in five rows and five columns, with each of the 25 plots subdivided into two, giving 50 units and thus 49 degrees of freedom in total. As an example of the second kind, he considered studying two factors: one with five levels and one with two with each combination studied with five replications, making 5 x 2 x 5 = 50 applications and again 49 degrees of freedom. He then stated:
The choice of the experimental design might be regarded as the choice of which items in the first analysis were to correspond to any chosen items in the second, and this could be represented by a two-way analysis of the 49 elements.
In other words, it was the way that the treatment structure mapped onto the block structure that guided the way that the experiment was to be analysed and, of course, the anticipated analysis would guide the way the experiment should be designed.
An example of a modern application of Fisher’s insight is shown in the following image, which gives the Genstat code I used to carry out analysis of variances for three possible treatment models, defined by TREATMENTSTRUCTURE commands, on a cross-over design for which the basic experimental units were defined by the BLOCKSTRUCTURE command.
Use of the ANOVA command without mentioning an outcome variable gives me a so-called dummy analysis, showing how the degrees of freedom should be apportioned but not, of course, giving me a full analysis since no outcome data are used. The example is described in a blog of mine: https://www.linkedin.com/pulse/designed-inferences-stephen-senn/.
Well before Yates’s arrival at Rothamsted, Fisher had realised that these distinctions between block and treatment structure were crucial and that in particular careful attention had to be paid to the former when calculating errors. He had, however, learned by making mistakes. Two years after Fisher’s death, in reviewing his contributions to experimental design, in commenting on an early example dating from 1923 of Fisher analysing a complex experiment, Yates, having first criticised the design, wrote:
To obtain a reasonable estimate of error for these interactions, however, the fact that the varietal plots were split for the potash treatments should have been taken into account. This was not done in the original analysis, a single pooled estimate being used...
But adding:
The need for the partition of error into whole-plot and sub-plot components was recognised by 1925. Part of the data of the above experiment was re-analysed in Statistical Methods for Research Workers in the now conventional form. (P311-312)
Fisher had taught himself fast. [9]
In fact, by the appearance of his classic text Statistical Methods for Research Workers [10], Fisher had developed analysis of variance (indeed, the term variance is due to him), the principles of blocking and replication, and his most controversial innovation, randomisation. An important point about this is still regularly misunderstood. As Fisher put it:
In a well-planned experiment, certain restrictions may be imposed upon the random arrangement of the plots in such a way that the experimental error may still be accurately estimated, while the greater part of the influence of heterogeneity may be eliminated. [11] (p232)
Thus, randomisation was not an alternative to balancing known influences but an adjunct to it.
As Yates put it in summing up what Fisher had achieved:
Apart from factorial design, therefore, all the principles of sound experimental design and analysis were established by 1925. [9] (p312)
On to the end
One day John Nelder was analysing a complex experiment. He was doing so in the tradition of Fisher and Yates. This is what he subsequently had to say about it:
During my first employment at Rothamsted, I was given the job of analyzing some relatively complex structured experiments on trace elements. There were crossed and nested classifications with confounding and all the rest of it, and I could produce analyses of variance for these designs. I then began to wonder how I knew what the proper analyses were and I thought that there must be some general principles that would allow one to deduce the form of the analysis from the structure of the design. The idea went underground for about 10 years. I finally resurrected it and constructed the theory of generally balanced designs, which took in virtually all the work of Fisher and Yates and Finney and put them into a single framework so that any design could be described in terms of two formulas. The first was for the block structure, which was the structure of the experimental units before you inserted the treatments. The second was the treatment structure—the treatments that were put on these units. The specification was completed by the data matrix showing which treatments went on to which unit. [12] (P125)
I have quoted this at length because it leaves me little else to say. John was able to unify the developments of Fisher and Yates and others, (David Finney is mentioned) so that a wide range of experimental designs could be analysed using a single general approach. The results were published in two papers in the Proceedings of the Royal Society [13], [14] in 1965, one of which did, indeed cover block structure and the other treatment structure.
Was that it?
No. Not at all. What Nelder established was that a general algorithm could be used and that hence a computer package could be written to implement it. After his arrival as head of statistics at Rothamsted, he was able to direct the development of Genstat, the software that was designed to implement his theory. However, many others worked on this [15], particularly notable being the contributions of Roger Payne, who continues to develop it to this day. An irony is that whereas one of John Nelder’s other seminal contributions to statistics, Generalised Linear Models, has been taken up by every major statistical package, (as far as I am aware) Genstat is the only one to have implemented the Rothamsted School approach to analysing designed experiments. Thus, when the Genstat user proceeds to analyse such an experiment by first declaring a BLOCKSTRUCTURE and then a TREATMENTSTRUCTURE before proceeding to request an ANOVA they are using software that is still ahead of its time but based on a theory with a century of tradition.
About the author
Professor Stephen Senn has worked as a statistician but also as an academic in various positions in Switzerland, Scotland, England and Luxembourg. From 2011-2018 he was head of the Competence Center for Methodology and Statistics at the Luxembourg Institute of Health. He is the author of Cross-over Trials in Clinical Research (1993, 2002), Statistical Issues in Drug Development (1997, 2007,2021), and Dicing with Death (2003). In 2009 he was awarded the Bradford Hill Medal of the Royal Statistical Society. In 2017 he gave the Fisher Memorial Lecture. He is an honorary life member of PSI and ISCB.
Stephen Senn: Blogs and Web Papers http://www.senns.uk/Blogs.html
References
1. Wood TB, Stratton F. The interpretation of experimental results. The Journal of Agricultural Science. 1910;3(4):417-440.
2. Airy GB. On the Algebraical and Numerical Theory of Errors of Observations and the Combination of Observations. MacMillan and Co; 1862.
3. Student. The probable error of a mean. Biometrika. 1908;6:1-25.
4. Pfanzagl J, Sheynin O. Studies in the history of probability and statistics .44. A forerunner of the t-distribution. Biometrika. Dec 1996;83(4):891-898.
5. Dyke G. Obituary: Frank Yates. Journal of the Royal Statistical Society Series A (Statistics in Society. 1995;158(2):333-338.
6. Finney DJ. Remember a pioneer: Frank Yates (1902‐1994). Teaching Statistics. 1998;20(1):2-5.
7. Yates F. Complex Experiments (with discussion). Supplement to the Journal of the Royal Statistical Society. 1935;2(2):181-247.
8. Yates F. Incomplete randomized blocks. Annals of Eugenics. Sep 1936;7:121-140.
9. Yates F. Sir Ronald Fisher and the design of experiments. Biometrics. 1964;20(2):307-321.
10. Fisher RA. Statistical Methods for Research Workers. Oliver and Boyd; 1925.
11. Fisher RA. Statistical Methods for Research Workers. In: Bennett JH, ed. Statistical Methods, Experimental Design and Scientific Inference. Oxford University; 1925.
12. Senn SJ. A conversation with John Nelder. Research Paper. Statistical Science. 2003;18(1):118-131.
13. Nelder JA. The analysis of randomised experiments with orthogonal block structure I. Block structure and the null analysis of variance. Proceedings of the Royal Society of London Series A. 1965;283:147-162.
14. Nelder JA. The analysis of randomised experiments with orthogonal block structure II. Treatment structure and the general analysis of variance. Proceedings of the Royal Society of London Series A. 1965;283:163-178.
15. Senn S. John Ashworth Nelder. 8 October 1924—7 August 2010. The Royal Society Publishing; 2019.
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Dr. Vanessa Cave
a month agoANOVA, LM, LMM, GLM, GLMM, HGLM? Which statistical method should I use?
Unsure which statistical method is appropriate for your data set? Want to know how the different methods relate to each one another?
The simple diagram below may help you.
| Treatment factor | Categorical explanatory variable defining the treatment groups. In an experiment, the experimental units are randomly assigned to the different treatment groups (i.e., the levels of the treatment factor). |
| Blocking variable | Factor created during the design of the experiment whereby the experimental units are arranged in groups (i.e., blocks) that are similar to one another. You can learn more about blocking in the blog Using blocking to improve precision and avoid bias. |
| Continuous predictor | A numeric explanatory variable (x) used to predict changes in a response variable (y). Check out the blog Pearson correlation vs simple linear regression to learn more. |
| Unbalanced design | An experimental design is unbalanced if there are unequal sample sizes for the different treatments. Genstat provides users with a tool to automatically determine whether ANOVA, LM (i.e., regression) or LMM (i.e., a REML analysis) is most appropriate for a given data set. Watch this YouTube video to learn more. |
| Temporal correlation | Occurs when repeated measurements have been taken on the same experimental unit over time, and thus measurements closer in time are more similar to one another than those further apart. To learn more, check out our blog A brief introduction to modelling the correlation structure of repeated measures data. |
| Spatial correlation | Occurs when experimental units are laid out in a grid, for example in a field trial or greenhouse, and experimental units that are closer together experience more similar environmental conditions than those which are further apart. For more information, read our blog A brief look at spatial modelling. |
| Random effects | Represents the effect of a sample of conditions observed from some wider population, and it is the variability of the population that is of interest. The blog FAQ: Is it a fixed or random effect? can help you understand the difference between fixed and random effects. |
About the author
Dr Vanessa Cave is an applied statistician, interested in the application of statistics to the biosciences, in particular agriculture and ecology. She is a team leader of Data Science at AgResearch Ltd, New Zealand's government-funded agricultural institute, and is a developer of the Genstat statistical software package. Vanessa is currently President of the Australasian Region of the International Biometric Society, past-President of the New Zealand Statistical Association, an Associate Editor for the Agronomy Journal, on the Editorial Board of The New Zealand Veterinary Journal and a member of the Data Science Industry Advisory Group for the University of Auckland. She has a PhD in statistics from the University of St Andrew.
Vanessa has over a decade of experience collaborating with scientists, using statistics to solve real-world problems. She provides expertise on experiment and survey design, data collection and management, statistical analysis, and the interpretation of statistical findings. Her interests include statistical consultancy, mixed models, multivariate methods, statistical ecology, statistical graphics and data visualisation, and the statistical challenges related to digital agriculture.
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Dr. Vanessa Cave
2 months agoA lunchtime conversation: technical, biological and pseudo-replication
What do you talk about over a workday lockdown lunch with a partner who’s also a statistician? Well, today in my home it was the three flavours of replication: technical, biological (or “true”) and pseudo replication. Why was this a topic of interest? Because knowing what type of replication you have not only impacts how your data should be analysed but also the conclusions that can be drawn. Unfortunately, this is not always well understood.
So, let’s examine the different types of replication in turn:
Technical replication
What? Technical replicates are repeated measurements taken on the same sample.
Why? Technical replicates are used to understand the noise (i.e., variability) associated with a protocol, procedure or piece of equipment.
Example? A blood diagnostic company running repeated measurements on a patient’s sample to study the reproducibility of their testing procedure.
Therefore? If the technical replicates are highly variable the observed effects will be difficult to distinguish from the background noise.
Biological replication
What? Biological replicates are independent measurements taken on distinct biological samples (ideally a random sample from the studied population).
Why? Biological replicates are used to understand the biological variation in the population under study.
Example? In a clinical trial, blood measurements are collected on many patients to understand the effect of a new drug treatment in the studied population.
Therefore? Biological replication allows us to generalize our results to the wider population of interest.
Pseudo-replication
What? Pseudo-replication occurs when data are treated as independent when they are not.
Why? Pseudo-replication arises due to errors in the planning or execution stage of an experiment, or during the statistical analyses.
Example? A clinical trial in which patients have been recruited from several medical centres and the treatments (either the control or new drug) are applied at the medical centre level but this structure hasn’t been accounted for in the analysis.
Therefore? If not correctly accounted for in the analysis, pseudo-replication will lead to invalid inferences.
You can learn more about pseudo-replication and how to identify and deal with it in Salvador Gezan’s excellent blog, Dealing with pseudo-replication in linear mixed models.
About the author
Dr Vanessa Cave is an applied statistician, interested in the application of statistics to the biosciences, in particular agriculture and ecology. She is a team leader of Data Science at AgResearch Ltd, New Zealand's government-funded agricultural institute, and is a developer of the Genstat statistical software package. Vanessa is currently President of the Australasian Region of the International Biometric Society, past-President of the New Zealand Statistical Association, an Associate Editor for the Agronomy Journal, on the Editorial Board of The New Zealand Veterinary Journal and a member of the Data Science Industry Advisory Group for the University of Auckland. She has a PhD in statistics from the University of St Andrew.
Vanessa has over a decade of experience collaborating with scientists, using statistics to solve real-world problems. She provides expertise on experiment and survey design, data collection and management, statistical analysis, and the interpretation of statistical findings. Her interests include statistical consultancy, mixed models, multivariate methods, statistical ecology, statistical graphics and data visualisation, and the statistical challenges related to digital agriculture.
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Dr. Vanessa Cave
2 months agoA plethora of podcasts to delight statisticians
I’m a fan of podcasts, especially those that make me think, intrigue me, teach me something new, and/or challenge my world view. And when it comes to entertaining and engaging podcasts, we are spoilt for choice! So, if you’re on the hunt for a new podcast to become engrossed in, here are a few gems that offer a statistical flavour.
More or Less
This renowned BBC Radio 4 podcast is “devoted to the powerful, sometimes beautiful, often abused but ever ubiquitous world of numbers". An investigative podcast, More or Less takes you on a journey to explain and explore the accuracy of statistics encountered in our daily lives in an entertaining and accessible way. As the podcast typically tackles statistical issues related to news topics in the UK, many of the recent episodes have focussed on COVID-19. Fascinating as these episodes are (and they are extremely interesting and informative!), I’d encourage you to checkout the back catalogue. There’s a wealth of intriguing episodes, such as:
- Are married women flipping miserable? Measuring happiness, university access in Scotland, plus will one in two people get cancer?
- Hay Festival Special What does it mean to say that the UK is the fifth largest economy in the world? Plus, how to answer multiple choice questions.
- Sex, coal, missing people and mice Are we having less sex? And what happened to coal?
- Are natural disasters on the rise? Investigating the numbers in the news; has the number of natural disasters really quadrupled in the last forty years?
Stats + Stories
A collaboration between the American Statistical Association and Miami University, Stats + Stories explores the “statistics behind the stories and the stories behind the statistics”. In each episode an expert guest (or guests) shares a story on a topical statistical idea, issue or myth. In a nutshell, they use statistics to study a story and use stories to bring statistics to life. Recent episodes include:
- Conspiracy dissemination dilemma The ways in which misinformation and conspiracies move through social media networks (guest: Sander van der Linden)
- A chance for good Use of data, statistics and data science for public good (guests: Caitlin Augustin and Matt Brems)
- No one is poisoning your kids' candy, trust the numbers Along with Halloween comes the worry, the concern, the fear that in fact, someone will be poisoning our kids’ candy.
- The right to be left alone With the ubiquity of technology in our lives, and Big Data, have come with concerns over privacy, security, and surveillance.
The Random Sample
Funded by the Australian Research Council Centre of Excellence for Mathematical and Statistical frontiers, this podcast shares stories about mathematics, statistics and the people involved. It provides a fascinating glimpse into the world of statistics, and how statistics often play a crucial role in areas we don’t expect. Some of my favourite episodes include:
- Pioneering women - Helen Newton Turner A look back at the life of Helen Newton Turner, an Australian statistician and statistical geneticist who made some amazing contributions to science starting in the 1930s and spanning over a 40-year career.
- Weighing up evidence in criminal courts and at a king's burial site Few things have changed the legal system over the past few decades as much as the use of forensic evidence – most notably, DNA.
- Hacking your lower brain to Improve predictions Investigates the disturbing possibility that companies get us to do what they were predicting - or at least push us closer to our predicted behaviour.
- Success, luck and second chancesWhen things go your way – or don’t go your way – how much of that is luck? Do we overvalue or undervalue a process or a person's performance because of how things eventually turn out? Do we need to take into account things out of our control?
Royal Statistical Society
The Royal Statistical Society is one of most renowned and prestigious statistical societies in the world. Its events often feature distinguished statisticians talking about interesting things. Luckily for us, many of these talks have been recorded and been made available as a podcast, including:
- Podcast with Richard Saldanha on statistical machine learning and quantitative finance.
- Use of data and research at the World Health Organisation with Dr Maria Van Kerkhove.
- Real-time epidemiology, statistics and modelling with Professor Christl Donnelly.
- A Bayesian analysis of animal movement data with Dr Colin Gillespie.
Planet Money
This popular NPR produced podcast uses playful storytelling to explore important and complex topics related to the economy – generally that of the USA’s. Although focused on economics, statistics permeates throughout this podcast. Thus, listening to this podcast not only teaches you about popular economic issues but it also provides you with insights into how individuals, businesses and governments use statistics in their decision making. Some episodes from 2021 are:
- The wheat whisperer Southeast Asia is one of the biggest growth markets for American wheat. Where did this taste for wheat come from and who is responsible?
- Two indicators: will remote work kill the office? Armed with studies, historical examples, theories on efficiency and happiness and from their closet studios, the presenters bring their indicators for the future of the office.
- How uncle Jamie broke jeopardyJames's Jeopardy! strategy wasn't a reckless gamble. It was informed by probability.
- Emission impossible Carbon offsets have become a popular tool to combat climate change. But how effective are they?
Freakonomics Radio
Ok, so like Planet Money, another USA podcast that discusses economics with statistics and statistical thinking featuring strongly. A great podcast for those of us interested in socio-economic issues and statistics, Freakonomics offers new perspectives that may change the way you think. One of the first episodes I listened to, provocatively titled How much do we really care about children? discusses car seats, the unconvincing safety data and the unintended consequence of the policies we create. To help whet your appetite, here are a few other episodes:
- Is there really a “Loneliness epidemic”? That’s what some health officials are saying, but the data aren’t so clear. What’s known (and not known) about the prevalence and effects of loneliness — including the possible upsides.
- Does Hollywood still have a princess problem? For decades, there’s been a huge gender disparity both on-screen and behind the scenes. But it seems like cold, hard data — with an assist from the actor Geena Davis — may finally be moving the needle.
- Speak softly and carry Big Data Do economic sanctions work? Are big democracies any good at spreading democracy? What is the root cause of terrorism? It turns out that data analysis can help answer all these questions — and make better foreign-policy decisions.
- The data-driven guide to sane parenting Humans have been having kids forever, so why are modern parents so bewildered? The economist Emily Oster marshals the evidence on the most contentious topics — breastfeeding and sleep training, vaccines and screen time — and tells her fellow parents to calm the heck down.
- Hacking your lower brain to improve predictions Do you ever stop and think about how much time you spend on your mobile phone, or binge-watching your favourite TV show? Could you predict how much time that will be tomorrow, or the next day, or the next week?
About the author
Dr Vanessa Cave is an applied statistician, interested in the application of statistics to the biosciences, in particular agriculture and ecology. She is a team leader of Data Science at AgResearch Ltd, New Zealand's government-funded agricultural institute, and is a developer of the Genstat statistical software package. Vanessa is currently President of the Australasian Region of the International Biometric Society, past-President of the New Zealand Statistical Association, an Associate Editor for the Agronomy Journal, on the Editorial Board of The New Zealand Veterinary Journal and a member of the Data Science Industry Advisory Group for the University of Auckland. She has a PhD in statistics from the University of St Andrew.
Vanessa has over a decade of experience collaborating with scientists, using statistics to solve real-world problems. She provides expertise on experiment and survey design, data collection and management, statistical analysis, and the interpretation of statistical findings. Her interests include statistical consultancy, mixed models, multivariate methods, statistical ecology, statistical graphics and data visualisation, and the statistical challenges related to digital agriculture.
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The VSNi Team
2 months agoRandom coefficient regression models for repeated measures data
A random coefficient regression is a special type of linear mixed model. They can be used when we want to explore the relationship between a response variable (y) and a continuous explanatory variable (x) and we have repeated measurements of x and y on individual subjects. Whereas in ordinary regression there is a single fixed value for each parameter (e.g., the intercept and slope), random coefficient regression allows these parameters to be unique for each subject. This is done by modelling all the coefficients of the regression model for each subject simultaneously using random effects and, importantly, allowing for correlation among these random effects.
 |  |
The conceptual difference between ordinary regression (left) and random coefficient regression (right) | |
To illustrate, let’s consider some repeated measures data on the orthodontic growth rate of children (Potthoff and Roy, 1964). [1] In this study, researchers at the University of North Carolina Dental School tracked the orthodontic growth of 27 children by measuring the distance between the pituitary and the pterygomaxillary fissure every 2 years from the ages of 8 to 14 years.
The data set contains two variates:
distance, the response variableage, age of the child in years
And two factors:
Subject, identifying the individual children on whom repeated measures were takenSex, the sex of the child
Of interest is comparing the orthodontic growth profiles between female and male children. We can do this by using random coefficient regression. Here, we aim to model the orthodontic growth profiles of female and male children over age, allowing for random variation about the regression parameters for the individual children.
 |
Graph of the orthodontic growth profiles of the individual children coloured by sex. |
To do this we fit a fixed effect for Sex and allow the fixed age covariate effect to differ between the two sexes. That is: Sex + age + Sex.age, where Sex.age represents an interaction term.
We also need to fit correlated random intercept and slope deviations for the individual children. That is, we specify our random model such it generates
- a set of child intercepts with a common variance (
) - a set of child slopes with a common, but different, variance (
)
such that
- the intercepts and/or slope from any two different children are independent
BUT
- the intercept and slope deviations from any given child are correlated (
).
Variance-covariance matrix = I ⊗ C where C = |
Our tutorial videos will teach you more about how to analyse your data using a random coefficient regression in Genstat or ASReml-R.
Random Coefficient Regression in Genstat
Random coefficient regression in ASReml-R 4
Defining complex variance structures in MMA: Part 2-Random Coefficient Regression
Citation
[1] Potthoff, R. F. and Roy, S. N. (1964), A generalized multivariate analysis of variance model useful especially for growth curve problems, Biometrika, 51, 313–326.
