Rafa Irizarry is a biostatistics professor and one of the three people behind SimplyStatistics.org (the others are Jeff Leek, Roger Peng). They post ideas that they find interesting and their blog contributes greatly to discussion of science/popular writing.
Rafa is the creator of many data visualization GIFs that have recently trended on the web, and in a recent post he provides all the source code behind the beautiful imagery. I sincerely recommend you check out the orginal blog if you want to find out more, but here are the GIFS:
Simpson’s paradox is a statistical phenomenon where an observed relationship within a population reverses within all subgroups that make up that population. Rafa visualized it wonderfully in a GIF that took only twenty-some lines of R code:
A different statistical phenomenon is discussed at the end of the original blog: namely the ecological fallacy. It occurs when correlations that occur on the group-level are erroneously extrapolated to the individual-level. Rafa used the gapminder data included in the dslabs package to illustrate the fallacy: there is a very high correlation at the region level and a lower correlation at the individual country level:
A next visualization demonstrates how the UN voting data (of Erik Voeten and Anton Strezhnev) can be used to examine different voting behaviors. It seems to reduce the voting data to a two-dimensional factor structure, and seemingly there are three distinct groups of voters these days, with particularly the USA and Israel far removed from other members:
The next GIFs are more statistical. The one below demonstrates how the local regression (LOESS) works. Simply speaking, LOESS determines the relationship for a local subset of the population and when you iteratively repeat this for all local subsets in a population you get a nicely fitting LOESS curve, the red line in Rafa’s GIF:
Not quite sure how to interpret the next one, but Rafa explains it visualized a random forest’s predictions using only one predictor variable. I think that different trees would then provide different predictions because they leverage different training samples, and an ensemble of those trees would then improve predictive accuracy?
The next one is my favorite I think. This animation illustrates how a highly accurate test would function in a population with low prevalence of true values (e.g., disease, applicant success). More details are in the original blog or here.
The blog ends with a rather funny animation of the only good use of pie charts, according to Rafa:
Multilevel models (also known as hierarchical linear models, nested data models, mixed models, random coefficient, random-effects models, random parameter models, or split-plot designs) are statistical models of parameters that vary at more than one level (Wikipedia). They are very useful in Social Sciences, where we are often interested in individuals that reside in nations, organizations, teams, or other higher-level units. Next to their individuals characteristics, the characteristics of these units they belong to may also have effects. To take into account effects from variables residing at multiple levels, we can use multilevel or hierarchical models.
If you want to practice hierarchical modeling in R, I recommend the lesson by Page Paccini (first video) or the more elaborate video series by Statistics of DOOM (second):
Simpson (1951) demonstrated that a statistical relationship observed within a population—i.e., a group of individuals—could be reversed within all subgroups that make up that population. This phenomenon, where X seems to relate to Y in a certain way, but flips direction when the population is split for W, has since been referred to as Simpson’s paradox. Others names, according to Wikipedia, include the Simpson-Yule effect, reversal paradox or amalgamation paradox.
The most famous example has to be the seemingly gender-biased Berkeley admission rates:
“Examination of aggregate data on graduate admissions to the University of California, Berkeley, for fall 1973 shows a clear but misleading pattern of bias against female applicants. Examination of the disaggregated data reveals few decision-making units that show statistically significant departures from expected frequencies of female admissions, and about as many units appear to favor women as to favor men. If the data are properly pooled, taking into account the autonomy of departmental decision making, thus correcting for the tendency of women to apply to graduate departments that are more difficult for applicants of either sex to enter, there is a small but statistically significant bias in favor of women. […] The bias in the aggregated data stems not from any pattern of discrimination on the part of admissions committees, which seem quite fair on the whole, but apparently from prior screening at earlier levels of the educational system.” – part of abstract of Bickel, Hammel, & O’Connel (1975)
In a table, the effect becomes clear. While it seems as if women are rejected more often overall, women are actually less often rejected on a departmental level. Women simply applied to more selective departments more often (E & C below), resulting in the overall lower admission rate for women (35% as opposed to 44% for men).
Simpsons Paradox can easily occur in organizational or human resources settings as well. Let me run you through two illustrated examples, I simulated:
Assume you run a company of 1000 employees and you have asked all of them to fill out a Big Five personality survey. Per individual, you therefore have a score depicting his/her personality characteristic Neuroticism, which can run from 0 (not at all neurotic) to 7 (very neurotic). Now you are interested in the extent to which this Neuroticism of employees relates to their Job Performance (measured 0 – 100) and their Salary (measured in Euro’s per Year). In order to get a sense of the effects, you may decide to visualize both these relations in scatter plots:
From these visualizations it would look like Neuroticism relates significantly and positively to both employees’ performance and their yearly salary. Should you select more neurotic people to improve your overall company performance? Or are you discriminating emotionally-stable (non-neurotic) employees when it comes to salary?
Taking a closer look at the subgroups in your data, you might however find very different relationships. For instance, the positive relationship between neuroticism and performance may only apply to technical positions, but not to those employees’ in service-oriented jobs.
Similarly, splitting the employees by education level, it becomes clear that there is a relationship between neuroticism and education level that may explain the earlier association with salary. More educated employees receive higher salaries and within these groups, neuroticism is actually related to lower yearly income.
If you’d like to see the code used to simulate these data and generate the examples, you can find the R markdown file here on Rpubs.
Solving the paradox
Kievit and colleagues (2013) argue that Simpsons paradox may occur in a wide variety of research designs, methods, and questions, particularly within the social and medical sciences. As such, they propose several means to “control” or minimize the risk of it occurring. The paradox may be prevented from occurring altogether by more rigorous research design: testing mechanisms in longitudinal or intervention studies. However, this is not always feasible. Alternatively, the researchers pose that data visualization may help recognize the patterns and subgroups and thereby diagnose paradoxes. This may be easy if your data looks like this:
But rather hard, or even impossible, when your data looks more like the below:
Clustering may nevertheless help to detect Simpson’s paradox when it is not directly observable in the data. To this end, Kievit and Epskamp (2012) have developed a tool to facilitate the detection of hitherto undetected patterns of association in existing datasets. It is written in R, a language specifically tailored for a wide variety of statistical analyses which makes it very suitable for integration into the regular analysis workflow. As an R package, the tool is is freely available and specializes in the detection of cases of Simpson’s paradox for bivariate continuous data with categorical grouping variables (also known as Robinson’s paradox), a very common inference type for psychologists. Finally, its code is open source and can be extended and improved upon depending on the nature of the data being studied.
One example of application is provided in the paper, for a dataset on coffee and neuroticism. A regression analysis would suggest a significant positive association between coffee and neuroticism overall. However, when the detection algorithm of the R package is applied, a different picture appears: the analysis shows that there are three latent clusters present and that the purported positive relationship only holds for one cluster whereas it is negative in the others.
Update 24-10-2017: minutephysics – one of my favorite YouTube channels – uploaded a video explaining Simpson’s paradox very intuitively in a medical context:
Update 01-11-2017: minutephysics uploaded a follow-up video:
The paradox is that we remain reluctant to fight our bias, even when they are put in plain sight.
Massachusetts Institute of Technology (MIT) hosts their entire 2010 course on artificial intelligence / machine learning by Professor Patrick Winston on YouTube. Although some parts seem already kind of dated seven years later, the videos on several evolving topics (e.g., Neural Networks) have been updated in the fall of 2015. The tutorial assignments you can find at the course website. Requirements for the course include experience with Python programming and an understanding of search algorithms (depth-first, breadth-first, uniform-cost, A*), basic probability, state estimation, the chain rule, partial derivatives, and dot products.
Below is the first, introductory lecture, which provides a short introduction to the history and concept of artificial intelligence: AI is about algorithms enabled by constraints exposed by representations that support models targeted at loops that tie together thinking, perception and action.
Welcome to my repository of data science, machine learning, and statistics resources. Software-specific material has to a large extent been listed under their respective overviews: R Resources & Python Resources. I also host a list of SQL Resources and datasets to practice programming. If you have any additions, please comment or contact me!
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