Prison rate among Black and White children by family income
17 February, 2023
This figure may seem mysterious, but it is not. Well, less so!
My attempt at explaining:
https://twitter.com/KirkegaardEmil/status/1626448996246888449
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.
Init
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## +load_packages(
ggeffects
)
theme_set(theme_bw())
options(
digits = 3
)Data
n = 10000
threshold = -1
family_transfer = .5
random_component_sd = .5
#assume 1 SD gap in human capital, simulate parents' data
set.seed(1)
d = tibble(
family = (1:(2*n)) %>% as.factor(),
race = c(
rep("black", n), rep("white", n)
),
hc_parent = rnorm(n = n*2, mean = 0, sd = 1),
hc_children = c(
hc_parent * family_transfer + rnorm(n = n*2, mean = 0, sd = sqrt(1 - family_transfer^2))
)
)
#adjust group means, add threshold binary
d %<>% mutate(
hc_parent = if_else(race=="black", true = hc_parent-1, false = hc_parent),
hc_children = if_else(race=="black", true = hc_children-1, false = hc_children),
below_threshold = ((rnorm(n, mean = 0, sd = random_component_sd) + hc_children) < threshold) %>% as.factor(),
delta_parent_child = hc_children - hc_parent
)
#stats
describe2(d)describe2(d, group = d$race)cor_matrix(d)## hc_parent hc_children delta_parent_child
## hc_parent 1.000 0.595 -0.454
## hc_children 0.595 1.000 0.446
## delta_parent_child -0.454 0.446 1.000cor_matrix(d, by = d$race)## $black
## hc_parent hc_children delta_parent_child
## hc_parent 1.000 0.501 -0.502
## hc_children 0.501 1.000 0.497
## delta_parent_child -0.502 0.497 1.000
##
## $white
## hc_parent hc_children delta_parent_child
## hc_parent 1.000 0.488 -0.506
## hc_children 0.488 1.000 0.506
## delta_parent_child -0.506 0.506 1.000#wide format for ratio
d_wide = pivot_wider(
d,
names_from = "race",
values_from = "hc_children"
)
#plot regression lines
d %>%
ggplot(aes(hc_parent, hc_children, color = race)) +
geom_smooth() +
scale_x_continuous(breaks = seq(-5, 5)) +
scale_y_continuous(breaks = seq(-5, 5))## `geom_smooth()` using method = 'gam' and formula = 'y ~ s(x, bs = "cs")'
#distance to children as function as parent
d %>%
ggplot(aes(hc_parent, delta_parent_child, color = race)) +
geom_smooth() +
scale_x_continuous(breaks = seq(-5, 5)) +
scale_y_continuous(breaks = seq(-5, 5, by = .5))## `geom_smooth()` using method = 'gam' and formula = 'y ~ s(x, bs = "cs")'
#ratio threshold as function of parent hc
#logistic function
logis_fit = glm(formula = below_threshold ~ hc_parent + race, data = d, family = "binomial")
logis_fit %>% summary()##
## Call:
## glm(formula = below_threshold ~ hc_parent + race, family = "binomial",
## data = d)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.121 -0.830 -0.509 0.967 2.726
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -0.8198 0.0286 -28.7 <2e-16 ***
## hc_parent -0.8096 0.0188 -43.1 <2e-16 ***
## racewhite -0.8161 0.0360 -22.7 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 25756 on 19999 degrees of freedom
## Residual deviance: 21377 on 19997 degrees of freedom
## AIC: 21383
##
## Number of Fisher Scoring iterations: 4#fancy package to ensure we did it correctly
ggpredict(logis_fit, terms = c("hc_parent [all]", "race")) %>%
plot()
#compute ratios from logistic prediction lines
d %<>% mutate(
pred_below = predict(logis_fit, newdata = d) %>% plogis()
)
#manual plot
d %>%
ggplot(aes(hc_parent, pred_below, color = race)) +
geom_line()
#new smooth data for ratios
d_pred = tibble(
hc_parent = c(
seq(-3, 3, length.out = n/2),
seq(-3, 3, length.out = n/2)
),
race = rep(c("black", "white"), each = n/2)
) %>% mutate(
pred_below = predict(logis_fit, newdata = .) %>% plogis()
)
#wide format so we can get the ratio
d_pred_wide = pivot_wider(
d_pred,
id_cols = c("hc_parent"),
names_from = "race",
values_from = "pred_below"
)
#add ratio of below
d_pred_wide %<>% mutate(
ratio_below = black / white
)
#final plot
d_pred_wide %>%
ggplot(aes(hc_parent, ratio_below)) +
geom_line()
In the real data, the ratio of Black/White is about 3x at the lowest parental income, and about 6-7x at the highest. These values correspond to ours if you multiply by 3. This scaling factor would reflect an extra Black proneness to crime, controlling for income.
Meta
#versions
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