{"id":1618,"date":"2025-10-16T19:57:01","date_gmt":"2025-10-16T19:57:01","guid":{"rendered":"https:\/\/www.allendowney.com\/blog\/?p=1618"},"modified":"2025-10-16T20:23:01","modified_gmt":"2025-10-16T20:23:01","slug":"simpsons-what","status":"publish","type":"post","link":"https:\/\/www.allendowney.com\/blog\/2025\/10\/16\/simpsons-what\/","title":{"rendered":"Simpson’s What?"},"content":{"rendered":"\n

I like Simpson\u2019s paradox so much I wrote three chapters about it in Probably Overthinking It<\/em><\/a>. In fact, I like it so much I have a Google alert that notifies me when someone publishes a new example (or when the horse named Simpson\u2019s Paradox<\/a> wins a race).<\/p>\n\n\n\n

So I was initially excited about this paper that appeared recently in Nature: \u201cThe geographic association of multiple sclerosis and amyotrophic lateral sclerosis\u201d<\/a>. But sadly, I\u2019m pretty sure it\u2019s bogus.<\/p>\n\n\n\n

The paper compares death rates due to multiple sclerosis (MS) and amyotrophic lateral sclerosis (ALS) across 50 states and the District of Columbia, and reports a strong correlation.<\/p>\n\n\n\n

This result is contrary to all previous work on these diseases \u2013 which might be a warning sign. But the author explains that this correlation has not been detected in previous work because it is masked when the analysis combines male and female death rates.<\/p>\n\n\n\n

This could<\/strong> make sense, because death rates due to MS are higher for women, and death rates due to ALS are higher for men. So if we compare different groups with different proportions of males and females<\/strong>, it\u2019s possible we could see something like Simpson\u2019s paradox.<\/p>\n\n\n\n

But as far as I know, the proportions of men and women are the same in all 50 states, plus the District Columbia \u2013 or maybe a little higher in Alaska. So an essential element of Simpson\u2019s paradox \u2013 different composition of the subgroups \u2013 is missing.<\/p>\n\n\n\n

Annoyingly, the \u201cData Availability\u201d section of the paper only identifies the public sources of the data \u2013 it does not provide the processed data. But we can use synthesized data to figure out what\u2019s going on.<\/p>\n\n\n\n

Specifically, let\u2019s try to replicate this key figure from the paper:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The x-axis is age adjusted death rates from MS; the y-axis is age-adjusted death rates from ALS. Each dot corresponds to one gender group in one state. The blue line fits the male data, with correlation 0.7. The pink line fits the female data, with correlation 0.75.<\/p>\n\n\n\n

The black line is supposed to be a fit to all the data, showing the non-correlation we supposedly get if we combine the two groups. But I\u2019m pretty sure that line is a mistake.<\/p>\n\n\n\n

Click here to read this article with the Python code<\/a>, or if you want to replicate my analysis, you can click here to run the notebook on Colab.<\/a><\/p>\n\n\n\n

Synthetic Data<\/h2>\n\n\n\n

I used a random number generator to synthesize correlated data with the approximate distribution of the date in the figure. The following figure shows a linear regression for the male and female data separately, and a third line that is my attempt to replicate the black line in the original figure.<\/p>\n\n\n\n

I thought the author might have combined the dots from the male and female groups into a collection of 102 points, and fit a line to that. That is a nonsensical thing to do, but it does yield a Simpson-like reversal in the slope of the line — and the sign of the correlation.<\/p>\n\n\n\n

\"_images\/3e7af43cbca8d3b134f479a3aa1cdd26131edd1562308344099d9f17782e7d57.png\"<\/figure>\n\n\n\n

The line for the combined data has a non-negligible negative slope, and the correlation is about -0.4 \u2013 so this is not the line that appears in the original figure, which has a very small correlation. So, I don\u2019t know where that line came from.<\/p>\n\n\n\n

In any case, the correct way to combine the data is not to plot a line through 102 points in the scatter plot, but to fit a line to the combined death rates in the 51 states. Assuming that the gender ratios in the states are close to 50\/50, the combined rates are just the means of the male and female rates. The following figure shows what we get if we combine the rates correctly.<\/p>\n\n\n\n

\"_images\/21fd70a6c01747e4906e2c0e87286110ce9a96b1e6d1e145b5e69c2e987bf9f4.png\"<\/figure>\n\n\n\n

So there\u2019s no Simpson\u2019s paradox here \u2013 there\u2019s a positive correlation among the subgroups, and there\u2019s a positive correlation when we combine them. I love a good Simpson\u2019s paradox, but this isn\u2019t one of them.<\/p>\n\n\n\n

On a quick skim, I think the rest of the paper is also likely to be nonsensical, but I\u2019ll leave that for other people to debunk. Also, peer review is dead.<\/p>\n\n\n\n

It gets worse<\/h2>\n\n\n\n

UPDATE: After I published the first draft of this article, I noticed that there are an unknown number of data points at (0, 0) in the original figure. They are probably states with missing data, but if they were included in the analysis as zeros — which they absolutely should not be — that would explain the flat line.<\/p>\n\n\n\n

If we assume there are two states with missing data, that strengthens the effect in the subgroups, and weakens the effect in the combined groups. The result is a line with a small negative slope, as in the original paper.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

I like Simpson\u2019s paradox so much I wrote three chapters about it in Probably Overthinking It. In fact, I like it so much I have a Google alert that notifies me when someone publishes a new example (or when the horse named Simpson\u2019s Paradox wins a race). So I was initially excited about this paper that appeared recently in Nature: \u201cThe geographic association of multiple sclerosis and amyotrophic lateral sclerosis\u201d. But sadly, I\u2019m pretty sure it\u2019s bogus. The paper compares…<\/p>\n

Read More Read More<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[1],"tags":[102,82],"class_list":["post-1618","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-epidemiology","tag-simpsons-paradox"],"yoast_head":"\nSimpson's What? - Probably Overthinking It<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.allendowney.com\/blog\/2025\/10\/16\/simpsons-what\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Simpson's What? - Probably Overthinking It\" \/>\n<meta property=\"og:description\" content=\"I like Simpson\u2019s paradox so much I wrote three chapters about it in Probably Overthinking It. In fact, I like it so much I have a Google alert that notifies me when someone publishes a new example (or when the horse named Simpson\u2019s Paradox wins a race). So I was initially excited about this paper that appeared recently in Nature: \u201cThe geographic association of multiple sclerosis and amyotrophic lateral sclerosis\u201d. But sadly, I\u2019m pretty sure it\u2019s bogus. The paper compares... 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And if it happens, what does it mean? To answer these questions, I've been searching for natural examples in data from the General Social Survey (GSS). 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I said I didn't spend much time on it because, I opined, it is a mathematical curiosity unlikely to come up in practice. My colleague was shocked and dismayed\u2026","rel":"","context":"In \"general social survey\"","block_context":{"text":"general social survey","link":"https:\/\/www.allendowney.com\/blog\/tag\/general-social-survey\/"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2021\/04\/grass_vs_year_by_cohort10.jpg?resize=350%2C200&ssl=1","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2021\/04\/grass_vs_year_by_cohort10.jpg?resize=350%2C200&ssl=1 1x, https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2021\/04\/grass_vs_year_by_cohort10.jpg?resize=525%2C300&ssl=1 1.5x"},"classes":[]},{"id":590,"url":"https:\/\/www.allendowney.com\/blog\/2021\/05\/01\/simpsons-paradox-and-education\/","url_meta":{"origin":1618,"position":2},"title":"Simpson’s Paradox and Education","author":"AllenDowney","date":"May 1, 2021","format":false,"excerpt":"Is Simpson's paradox a mathematical curiosity or something that matters in practice? To answer this question, I'm searching the General Social Survey (GSS) for examples. Last week I published the first batch, examples where we group people by decade of birth and plot their opinions over time. In this article\u2026","rel":"","context":"In \"general social survey\"","block_context":{"text":"general social survey","link":"https:\/\/www.allendowney.com\/blog\/tag\/general-social-survey\/"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2021\/05\/libath_vs_year_by_degree.jpg?resize=350%2C200&ssl=1","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2021\/05\/libath_vs_year_by_degree.jpg?resize=350%2C200&ssl=1 1x, https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2021\/05\/libath_vs_year_by_degree.jpg?resize=525%2C300&ssl=1 1.5x"},"classes":[]},{"id":596,"url":"https:\/\/www.allendowney.com\/blog\/2021\/05\/03\/simpsons-paradox-and-age-effects\/","url_meta":{"origin":1618,"position":3},"title":"Simpson’s Paradox and Age Effects","author":"AllenDowney","date":"May 3, 2021","format":false,"excerpt":"As people get older, do they become more racist, sexist, and homophobic? To find out, you could use data from the General Social Survey (GSS), which asks questions like: Do you think there should be laws against marriages between Blacks\/African-Americans and whites?Should a man who admits[mfn]If you find the wording\u2026","rel":"","context":"In \"general social survey\"","block_context":{"text":"general social survey","link":"https:\/\/www.allendowney.com\/blog\/tag\/general-social-survey\/"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2021\/05\/fepol_vs_age_by_cohort10.jpg?resize=350%2C200&ssl=1","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2021\/05\/fepol_vs_age_by_cohort10.jpg?resize=350%2C200&ssl=1 1x, https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2021\/05\/fepol_vs_age_by_cohort10.jpg?resize=525%2C300&ssl=1 1.5x"},"classes":[]},{"id":555,"url":"https:\/\/www.allendowney.com\/blog\/2021\/04\/15\/simpsons-paradox-and-real-wages\/","url_meta":{"origin":1618,"position":4},"title":"Simpson’s paradox and real wages","author":"AllenDowney","date":"April 15, 2021","format":false,"excerpt":"I have good news and bad news. First the good news: after a decade of stagnation, real wages have been rising since 2010. The following figure shows weekly wages for full-time employees (source), which I adjusted for inflation and indexed so the series starts at 100. Real wages in 2019\u2026","rel":"","context":"In \"Simpson's paradox\"","block_context":{"text":"Simpson's paradox","link":"https:\/\/www.allendowney.com\/blog\/tag\/simpsons-paradox\/"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2021\/04\/simpson_wages6.png?resize=350%2C200&ssl=1","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2021\/04\/simpson_wages6.png?resize=350%2C200&ssl=1 1x, https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2021\/04\/simpson_wages6.png?resize=525%2C300&ssl=1 1.5x"},"classes":[]},{"id":748,"url":"https:\/\/www.allendowney.com\/blog\/2022\/07\/23\/almost-done\/","url_meta":{"origin":1618,"position":5},"title":"Almost done?","author":"AllenDowney","date":"July 23, 2022","format":false,"excerpt":"I thought I would be done this week, but it looks like there will be one more chapter. If you don't know, I am working on a book that includes updated articles from this blog, plus new examples, and pulls the whole thing together. So far, it's going well. I\u2026","rel":"","context":"In \"book\"","block_context":{"text":"book","link":"https:\/\/www.allendowney.com\/blog\/tag\/book\/"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2022\/07\/simpson_penguin.png?resize=350%2C200&ssl=1","width":350,"height":200},"classes":[]}],"_links":{"self":[{"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/posts\/1618","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/comments?post=1618"}],"version-history":[{"count":3,"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/posts\/1618\/revisions"}],"predecessor-version":[{"id":1626,"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/posts\/1618\/revisions\/1626"}],"wp:attachment":[{"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/media?parent=1618"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/categories?post=1618"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/tags?post=1618"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}