# Errors-in-variables model consistently underestimates sigma

**URL:** <https://discourse.mc-stan.org/t/errors-in-variables-model-consistently-underestimates-sigma/30453>\
**Category:** Modeling\
**Created:** [February 17, 2023, 4:08pm UTC](https://discourse.mc-stan.org/t/errors-in-variables-model-consistently-underestimates-sigma/30453 "2023-02-17T16:08:06Z")\
**Posts on this page:** 1\
**Showing post:** 2

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**Author:** ![Bob\_Carpenter](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/bob_carpenter/32/9230_2.png) [@Bob\_Carpenter](https://discourse.mc-stan.org/u/Bob_Carpenter)\
**Post date:** [March 17, 2023, 9:49pm UTC](https://discourse.mc-stan.org/t/errors-in-variables-model-consistently-underestimates-sigma/30453/2 "2023-03-17T21:49:22Z")

</div>

The problem is the two bugs in your R code:

```plaintext
xobs <- runif(n, -3, 3) + rnorm(n ,sigma)
yobs <- slope * xobs + intercept + rnorm(n ,sigma) 

```

The first problem is that the `rnorm` here uses `sigma` as the location, not the scale.

The second problem is that `yobs` should be based on `x`, not `xobs`.

Going forward, you probably don’t want the same error scale for the measurement error (`x_obs`) and observations (`y_obs`).

Here’s an R program to simulate:

```plaintext
N <- 1000
sigma <- 0.5
alpha <- 1.3
beta <- -0.4
x <- runif(N, -3, 3)
y_obs <- rnorm(N, alpha + beta * x, sigma)
x_obs <- rnorm(N, x, sigma)

library('cmdstanr')

model <- cmdstan_model('foo.stan')
fit <- model$sample(data = list(N = N, x_obs = x_obs, y_obs = y_obs), parallel_chains=4)
print(fit$summary(variables = c("alpha", "beta", "sigma")))

```

and here’s the matching Stan model:

```stan
data{
  int<lower=0> N;
  array[N] real x_obs;
  array[N] real y_obs;
}
parameters {
  real alpha;
  real beta;
  real<lower=0> sigma;
  vector[N] x;
}
model {
  alpha ~ normal(0, 5);
  beta ~ normal(0, 5);
  sigma ~ normal(0, 5);
  x_obs ~ normal(x, sigma);
  y_obs ~ normal(alpha + beta * x, sigma);
}

```

and here’s the fit that the simulation script prints:

```plaintext
All 4 chains finished successfully.
Mean chain execution time: 3.1 seconds.
Total execution time: 3.2 seconds.

# A tibble: 3 × 10
  variable mean median sd mad q5 q95 rhat ess_bulk ess_tail
  <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 alpha 1.32 1.32 0.0173 0.0168 1.29 1.34 1.00 4515. 3011.
2 beta -0.370 -0.370 0.00912 0.00911 -0.384 -0.355 1.00 4293. 2735.
3 sigma 0.498 0.498 0.0114 0.0114 0.480 0.518 1.00 1812. 2185.

```

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