Simulation-based Inference

Inference for Linear Regression in R

Jo Hardin

Professor, Pomona College

A scatter plot of twins' IQS.

Twin data

A table of twins IQs. Each row corresponds to a pair of twins. The first column contains the IQ of the twin raised by foster parents, and the second column contains the IQ of the twin raised by biological parents.

Permuted twin data

The table of twins IQs has had each column permuted, so that pairs of twins no longer appear together on the same row.

Permuted data (1) plotted

Original data

Permuted data (1)

Permuted data (2) plotted

Original data

Permuted data (2)

Permuted data (1) and (2)

Permuted data (1)

Permuted data (2)

twins %>%
   specify(Foster ~ Biological) %>%
   hypothesize(null = "independence") %>%
   generate(reps = 10, type = "permute") %>%
   calculate(stat = "slope")

A tibble: 10 x 2
   replicate          stat
       <int>         <dbl>
 1         1  0.0007709302
 2         2 -0.0353592305
 3         3 -0.0278627974
 4         4 -0.0072547982
 5         5 -0.1252761541
 6         6 -0.1669869287
 7         7 -0.2610519170
 8         8 -0.0157695494
 9         9  0.0581361900
10        10  0.1598471947

Many permuted slopes

perm_slope <- twins %>%
   specify(Foster ~ Biological) %>%
   hypothesize(
     null = "independence"
     ) %>%
   generate(reps = 1000, 
            type = "permute") %>%
   calculate(stat = "slope") 

ggplot(data = perm_slope, aes(x = stat)) + 
   geom_histogram() +
   xlim(-1,1)

Permuted slopes with observed slope in red

obs_slope <- lm(Foster ~ Biological,
                data = twins) %>%
   tidy() %>%   
   filter(term == "Biological") %>%
   select(estimate) %>%   
   pull()
obs_slope

0.901436

ggplot(data = perm_slope, aes(x = stat)) + 
   geom_histogram() +
   geom_vline(xintercept = obs_slope, color = "red") 
   + xlim(-1,1)

Let's practice!

Inference for Linear Regression in R