{"id":11686,"date":"2023-02-17T11:56:42","date_gmt":"2023-02-17T10:56:42","guid":{"rendered":"https:\/\/emilkirkegaard.dk\/en\/?p=11686"},"modified":"2023-02-17T11:56:42","modified_gmt":"2023-02-17T10:56:42","slug":"prison-rates-as-function-of-family-income-by-race","status":"publish","type":"post","link":"https:\/\/emilkirkegaard.dk\/en\/2023\/02\/prison-rates-as-function-of-family-income-by-race\/","title":{"rendered":"Prison rates as function of family income by race"},"content":{"rendered":"<p>Steve Sailer tells the story of this figure: <a href=\"https:\/\/www.takimag.com\/article\/americas-black-male-problem\/\"><em>Why Is the Ratio of Black Incarceration to White Incarceration Worse in Higher Social Classes?<\/em><\/a><\/p>\n<p><a href=\"https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/Sailer-plot.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-11687\" src=\"https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/Sailer-plot.jpg\" alt=\"\" width=\"1200\" height=\"700\" srcset=\"https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/Sailer-plot.jpg 1200w, https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/Sailer-plot-300x175.jpg 300w, https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/Sailer-plot-1024x597.jpg 1024w, https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/Sailer-plot-768x448.jpg 768w\" sizes=\"auto, (max-width: 1200px) 100vw, 1200px\" \/><\/a><\/p>\n<p>He muses:<\/p>\n<blockquote><p>Why does the black-to-white ratio get steadily worse with higher income?<\/p>\n<p>I don\u2019t know. Before seeing Chetty\u2019s data, I might have guessed it shrank.<\/p>\n<p>Is the cause racism?<\/p><\/blockquote>\n<p>I think the cause of this increasing ratio is regression towards the mean. <a href=\"https:\/\/twitter.com\/KirkegaardEmil\/status\/1626448996246888449\">That&#8217;s what I proposed on Twitter at least<\/a>:<\/p>\n<blockquote>\n<div class=\"css-1dbjc4n\">\n<div class=\"css-1dbjc4n r-1s2bzr4\">\n<div id=\"id__w8r5ydgnwxe\" class=\"css-901oao r-1nao33i r-37j5jr r-1inkyih r-16dba41 r-rjixqe r-bcqeeo r-bnwqim r-qvutc0\" dir=\"auto\" lang=\"en\" data-testid=\"tweetText\"><span class=\"css-901oao css-16my406 r-poiln3 r-bcqeeo r-qvutc0\">Blacks from 98th centile income homes are far above their racial mean, so they regress a lot. White kids less so. The relative effect size of the regression becomes larger the stronger the selection, so will the difference between the children. Thus, the ratio goes up.<\/span><\/div>\n<\/div>\n<\/div>\n<\/blockquote>\n<p dir=\"auto\" lang=\"en\" data-testid=\"tweetText\">But one thing is statistical intuition and the other thing is doing the math. <a href=\"https:\/\/rpubs.com\/EmilOWK\/parent_income_sons_in_prison\">So let&#8217;s do the math.<\/a> I will assume these assumptions:<\/p>\n<ul>\n<li dir=\"auto\" lang=\"en\" data-testid=\"tweetText\">Black and Whites differ by 1 SD on a broad human capital phenotype. Whites have a mean of 0, Blacks of -1.<\/li>\n<li dir=\"auto\" lang=\"en\" data-testid=\"tweetText\">This runs in families with a parent-child correlation of .50<\/li>\n<li dir=\"auto\" lang=\"en\" data-testid=\"tweetText\">There is a threshold below which a person has a chance of being a criminal. I set this arbitrarily at -1. I say chance because it has a random noise component which I have arbitrarily set to SD 0.50.<\/li>\n<\/ul>\n<p>Given these assumptions, we can look at the regression towards the mean plot:<\/p>\n<p><a href=\"https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/regression-towards-the-mean-by-race.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-11688\" src=\"https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/regression-towards-the-mean-by-race.png\" alt=\"\" width=\"1344\" height=\"960\" srcset=\"https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/regression-towards-the-mean-by-race.png 1344w, https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/regression-towards-the-mean-by-race-300x214.png 300w, https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/regression-towards-the-mean-by-race-1024x731.png 1024w, https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/regression-towards-the-mean-by-race-768x549.png 768w\" sizes=\"auto, (max-width: 1344px) 100vw, 1344px\" \/><\/a><\/p>\n<p>So we can visually verify the results by looking up some point. E.g., look at the White parent human capital of 0 on X, which shouldn&#8217;t regress anywhere in the children, and indeed the Y is also 0. Or look up White X = 1, which should have children with Y = 0.5, which is what we see. Similarly, look up Black parents X = 1, and their children should regress halfway to -1, so to 0, and that&#8217;s what we see too.<\/p>\n<p>Equivalently, we can look at the distance to the children as a function of parental human capital:<\/p>\n<p><a href=\"https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/race-delta-to-child.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-11689\" src=\"https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/race-delta-to-child.png\" alt=\"\" width=\"1344\" height=\"960\" srcset=\"https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/race-delta-to-child.png 1344w, https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/race-delta-to-child-300x214.png 300w, https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/race-delta-to-child-1024x731.png 1024w, https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/race-delta-to-child-768x549.png 768w\" sizes=\"auto, (max-width: 1344px) 100vw, 1344px\" \/><\/a><\/p>\n<p>This shows the same thing as before, but in another form. So if we look up Black X = 2, that&#8217;s 3 d above their mean, so the value should be always back down, so -1.5 and that&#8217;s what we see. Looking up 2 for Whites though shows it is only -1, as that&#8217;s halfway to their mean.<\/p>\n<p>Then, we randomly assign some people to the below threshold group with a chance element. With these data binary data, we can fit a logistic regression model:<\/p>\n<p><a href=\"https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/logistic-plot-threshold.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-11690\" src=\"https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/logistic-plot-threshold.png\" alt=\"\" width=\"1344\" height=\"960\" srcset=\"https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/logistic-plot-threshold.png 1344w, https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/logistic-plot-threshold-300x214.png 300w, https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/logistic-plot-threshold-1024x731.png 1024w, https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/logistic-plot-threshold-768x549.png 768w\" sizes=\"auto, (max-width: 1344px) 100vw, 1344px\" \/><\/a><\/p>\n<p>So this is what the probability of being in the undesirable category is as a function of parental human capital and race. Finally, we can look at the ratios of these two lines:<\/p>\n<p><a href=\"https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/ratios-of-below-threshold-race.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-11691\" src=\"https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/ratios-of-below-threshold-race.png\" alt=\"\" width=\"1344\" height=\"960\" srcset=\"https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/ratios-of-below-threshold-race.png 1344w, https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/ratios-of-below-threshold-race-300x214.png 300w, https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/ratios-of-below-threshold-race-1024x731.png 1024w, https:\/\/emilkirkegaard.dk\/en\/wp-content\/uploads\/ratios-of-below-threshold-race-768x549.png 768w\" sizes=\"auto, (max-width: 1344px) 100vw, 1344px\" \/><\/a><\/p>\n<p>So indeed, as a function of parental human capital, the ratio grows with a logistic function. But the values don&#8217;t fit the Chetty plot! They are only 1.X and 2.X at the extremes. Why? First, note that the numbers do roughly fit if you use a scaling factor of 3. In Chetty&#8217;s plot, the ratio for 1st centile parental income is about 3, and for the 99th centile, it is about 6, the outlier flukes notwithstanding. I think the reason for the need for the scaling factor is that in my simulation, I used only one phenotype. In reality, criminality and income ability are not perfectly genetically correlated, so the control will be only partial. I think this would produce larger ratios across the board. So, we don&#8217;t need to be <em>too<\/em> mystified by the growing ratio. Regression towards the mean provides the main explanation.<\/p>\n<p>For those more curious about race and regression towards the mean, see <a href=\"https:\/\/humanvarieties.org\/2017\/07\/01\/measurement-error-regression-to-the-mean-and-group-differences\/\">Dailliard&#8217;s masterful review<\/a> of regression towards the mean, and <a href=\"https:\/\/www.mdpi.com\/2624-8611\/1\/1\/17\">our 2019 study<\/a> investigating and replicating Arthur Jensen&#8217;s old findings. You can also <a href=\"https:\/\/emilkirkegaard.dk\/en\/2023\/01\/interactive-statistics-breedings-equation-aka-regression-towards-the-mean\/\">play around with this simulator I made<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Steve Sailer tells the story of this figure: Why Is the Ratio of Black Incarceration to White Incarceration Worse in Higher Social Classes? He muses: Why does the black-to-white ratio get steadily worse with higher income? I don\u2019t know. Before seeing Chetty\u2019s data, I might have guessed it shrank. Is the cause racism? I think [&hellip;]<\/p>\n","protected":false},"author":17,"featured_media":11687,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-11686","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-uncategorized","entry","has-media"],"_links":{"self":[{"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/posts\/11686","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/users\/17"}],"replies":[{"embeddable":true,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/comments?post=11686"}],"version-history":[{"count":1,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/posts\/11686\/revisions"}],"predecessor-version":[{"id":11692,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/posts\/11686\/revisions\/11692"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/media\/11687"}],"wp:attachment":[{"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/media?parent=11686"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/categories?post=11686"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/tags?post=11686"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}