bimodal.Rmd

# Bimodal: Extreme missingness in bivariate normal data {#bimodal}

```{r bimodal_setup,message=FALSE,cache=FALSE}
library("rstan")
library("tidyverse")
library("stringr")
```

Simple methods for dealing with missing data can run into trouble given pernicious patterns of missingness.  A famous artificial data set designed to highlight this point was created by Gordon Murray, to show how an EM algorithm can run into problems [@Murray1977a,@DempsterLairdRubin1977a].

```{r Bimodal}
Bimodal <- tribble(
  ~ x1, ~ x2,
     1,    1,
     1,   -1,
    -1,    1,
    -1,   -1,
     2,   NA,
     2,   NA,
    -2,   NA,
    -2,   NA,
    NA,    2,
    NA,    2,
    NA,   -2,
    NA,    -2
)
```

```{r}
Bimodal
```

Assume bivariate normality, and that the means of the two variables are both
zero, but the variances and covariance are unknown.  Inference about the
correlation coefficient  $r$  between these two variables is not trivial in
this instance.  The marginal complete-data likelihood for $r$  is not unimodal,
and has a saddle-point at zero, and two local maxima close to -1 and 1.  A
Bayesian analysis (with uninformative priors) similarly recovers a bimodal
posterior density for the correlation coefficient; e.g., [@Tanner1996a,
@Congdon2007a].

```{r bimodal_mod,message=FALSE,results='hide',cache.extra=tools::md5sum("stan/bimodal.stan")}
bimodal_mod <- stan_model("stan/bimodal.stan")
```

```{r echo=FALSE}
bimodal_mod
```

You can ignore the **rstan** warning,

  DIAGNOSTIC(S) FROM PARSER:
  Warning (non-fatal):
  Left-hand side of sampling statement (~) may contain a non-linear transform of a parameter or local variable.
  If it does, you need to include a target += statement with the log absolute determinant of the Jacobian of the transform.
  Left-hand-side of sampling statement:
      X[i] ~ multi_normal(...)

since the left hand side is a simple linear relationship and no
Jacobian adjustment is needed.
All we did was replace index values in the transformed parameter.

```{r bimodal_data}
X_mat <- as.matrix(Bimodal)
X <- X_mat %>%
  as_data_frame() %>%
  mutate(.row = row_number()) %>%
  gather(.col, value, -.row) %>%
  mutate(.col = as.integer(str_replace(.col, "x", "")))

X_obs <- filter(X, !is.na(value))
X_miss <- filter(X, is.na(value))
bimodal_data <- within(list(), {
  N <- nrow(X_mat)
  x_obs <- X_obs$value
  x_obs_row <- X_obs$.row
  x_obs_col <- X_obs$.col  
  N_obs <- nrow(X_obs)
  x_miss_row <- X_miss$.row
  x_miss_col <- X_miss$.col
  N_miss <- nrow(X_miss)
  df <- 100
})
```

```{r bimodal_fit,message=FALSE,results='hide'}
bimodal_fit <- sampling(bimodal_mod, data = bimodal_data,
                        chains = 4)
```

```{r}
bimodal_fit
```

This example is derived from Simon Jackman, "Bimodal: Extreme missingness in
bivariate normal data",
[URL](https://web-beta.archive.org/web/20070724034055/http://jackman.stanford.edu:80/mcmc/bimodal.odc).