Causal salad of crime and immigration status

Although it wasn't published, I sent an email, for the first time, to The Guardian's letter section pointing out that getting the causal reasoning correct is of as much concern as is collecting accurate data. In particular, I gave two examples where the causal inferences are wrong and lead to opposite conclusions, even when data is uncontaminated. I won't regurgitate the letter here, but I will briefly summarise the examples below along with some example code.

I'm sharing this purely from an interest in statistical inference, not for any political motivation. The question of whether immigration status has any direct causal effect on crime is immeasurably more complex than what I wrote in the letter, and what I write here in this post. Any statement that a variable as broad as immigration status is linked to crime should cause alarm bells to ring. At the same time, simple examples can be highly instructive in checking our causal reasoning. As I mentioned in the letter, inferring any causal effect demands incredible caution and trained statisticians.

Erroneous inference from common causes

If sex and age both increase the chances of committing violent crime and of being an immigrant, any analysis that does not condition upon sex and age will find a correlation between violent crime and immigration status. The code below simulates some data under this scenario. Note, I am using a sample of 5000 here to reduce the noise in estimates later.

import numpy as np

rng = np.random.default_rng(1234)

def ilogit(x):
  return np.exp(x) / (1 + np.exp(x))

N = 5000
sex = rng.binomial(1, 0.5, size=N)
age = rng.lognormal(np.log(30), 0.2, N)
0
sex_z = sex - 0.5
age_z = (age - age.mean()) / age.std()

immigration = rng.binomial(1, ilogit(sex_z + -0.5 * age_z))
crime = rng.binomial(1, ilogit(-1 + sex_z + -0.5 * age_z))

I then fit a simple Bernoulli regression model with crime as the response variable and a single predictor of immigration.

import pymc as pm

with pm.Model() as model:
  alpha = pm.Normal("alpha", 0, 10)
  beta = pm.Normal("beta", 0, 2)
  eta = pm.math.invlogit(alpha + beta * (immigration - 0.5))
  likelihood = pm.Binomial("crime", n=1, p=eta, observed=crime)
  idata = pm.sample(1000, tune=1000, random_seed=rng)

This model returns a coefficient for $\beta$ with a mean of 0.41 and a 95% highest density interval (HDI) of [0.29, 0.53]. After exponentiating to the odds scale, we'd conclude that being an immigrant increases the odds of crime by an average of 51% (95% HDI: [33, 70]). We know this is wrong, however. The causal effects come from the common causes of sex and age, not any direct effect of immigration on crime.

Adding sex and age as predictor variables in this model returns a coefficient for immigration status as covering zero (mean: -0.01; 95% HDI [-0.14, 0.13]) because the common causes have been accounted for, and our conclusions would be correct.

These results are pertinent to any potential common causes. Sex and age are just two suggestions, but there are others I'm sure you can think of.

Collider bias from conditioning on prisoners

The second scenario is where only looking at the prison population induces a collider bias. A lot of the data used to infer the relationship between crime and immigration status come from the prison population and convictions data. Specifically, violent crime and immigration status might both increase the chances of being in prison. The first statement is obvious, the second is more tentative. I don't have data on this personally, but The Guardian article states that available data indicates offending rates might be higher in foreign nationals than British nationals. A prior article also wrote that:

While non-citizens made up a higher proportion of people convicted for sexual and theft offences over the past decade, researchers cautioned that a lack of data for offenders’ age and sex complicated the picture and meant it was hard to make clear assertions.

— The Guardian (October 1st, 2025)

I'd take this with a grain of salt, for the exact reasons I am writing this post, but it seems plausible enough. For instance, immigrants might be more likely to engage in other offences due to experiencing poorer socio-economic conditions or mental health problems. In any case, to reiterate, my interest here is purely the ways in which statistical inference can lead to erroneous views of causality.

Here's a simulation of this scenario. Notice how I am filtering the crime and immigration variables for only those individuals who are in prison.

immigration = rng.binomial(1, 0.5, size=N)
crime = rng.binomial(1, 0.5, size=N)
prison = rng.binomial(1, ilogit(immigration + 2 * crime))

immigration_in_prison = immigration[prison == 1]
crime_in_prison = crime[prison == 1]

with pm.Model() as model:
  alpha = pm.Normal("alpha", 0, 10)
  beta = pm.Normal("beta", 0, 2)
  eta = pm.math.invlogit(
    alpha + 
    beta * (immigration_in_prison - 0.5)
  )
  likelihood = pm.Binomial(
    "crime", 
    n=1, 
    p=eta, 
    observed=crime_in_prison
  )
  idata = pm.sample(1000, tune=1000, random_seed=rng)

In this scenario, we find the odds of violent crime decreases with immigration status by an average of 25% (95% HDI: [15, 34]). If we filter for the prison population only, we condition on the collider, opening up the path between violent crime and immigration status. If we don't filter, then we get an odds ratio of 1.01% (95% HDI: [0.90, 1.12]), and we'd conclude that there is no direct effect.

Final thoughts

Causal inference is fascinating, but it's also very troublesome. Anyone leading analyses on these complicated topics are in unenviable positions of lacking clean or exhaustive-enough data and the pressures of finding patterns that could help inform policy. I only hope that the individuals working on these topics are trained in causal inference, and have the humility to know that the posterior distribution of fucking-it-up even with correct data might not include zero.