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title: Estimating partial correlations with lava
author: Klaus Kähler Holst
date: "`r Sys.Date()`"
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```{r  include=FALSE }
mets <- lava:::versioncheck('mets', 1)
```

\newcommand{\arctanh}{\operatorname{arctanh}}

This document illustrates how to estimate partial correlation
coefficients using `lava`.

Assume that $Y_{1}$ and $Y_{2}$ are conditionally normal
distributed given $\mathbf{X}$ with the following linear structure
$$Y_1 = \mathbf{\beta}_1^{t}\mathbf{X} + \epsilon_1$$
$$Y_2 = \mathbf{\beta}_2^{t}\mathbf{X} + \epsilon_2$$
with covariates $\mathbf{X} = (X_1,\ldots,X_k)^{t}$ and measurement errors
$$\begin{pmatrix}
    \epsilon_{1} \\
    \epsilon_{2}
    \end{pmatrix} \sim \mathcal{N}\left(0, \mathbf{\Sigma} \right), \quad \mathbf{\Sigma}
    =
    \begin{pmatrix}
    \sigma_1^2 & \rho\sigma_{1}\sigma_{2} \\
    \rho\sigma_{1}\sigma_{2} & \sigma_2^2
    \end{pmatrix}.$$

```{r  plotload, results="hide",message=FALSE,warning=FALSE }
library("lava")
m0 <- lvm(y1+y2 ~ x, y1 ~~ y2)
edgelabels(m0, y1 + y2 ~ x) <- c(expression(b[1]), expression(b[2]))
edgelabels(m0, y1 ~ y2) <- "r"
plot(m0, layoutType="circo")
```

Here we focus on inference with respect to the correlation parameter $\rho$.


# Simulation

As an example, we will simulate data from this model with a single covariate.

The model can be specified (here using the pipe notation)
with the following syntax where the correlation parameter here is
given the label '`r`':

```{r  m0 }
m0 <- lvm() |>
  covariance(y1 ~ y2, value='r') |>
  regression(y1 + y2 ~ x)
```

To simulate from the model we can now simply use the `sim` method. The
parameters of the models are set through the argument `p` which must be a
named numeric vector of parameters of the model. The parameter names
can be inspected with the `coef` method

```{r  coef }
coef(m0, labels=TRUE)
```

The default simulation parameters are zero for all intercepts (`y1`, `y2`)
and one for all regression coefficients (`y1~x`, `y2~x`) and residual
variance parameters (`y1~~y1`, `y2~~y2`).

```{r  sim }
d <- sim(m0, 500, p=c(r=0.9), seed=1)
head(d)
```

Under Gaussian and coarsening at random assumptions we can also
consistently estimate the correlation in the presence of censoring or
missing data. To illustrate this, we add left and right censored data
types to the model output using the `transform` method.

```{r  defcens }
cens1 <- function(threshold,type='right') {
  function(x) {
    x <- unlist(x)
    if (type=='left')
      return( survival::Surv(pmax(x,threshold), x>=threshold, type='left') )
      return ( survival::Surv(pmin(x,threshold), x<=threshold) )
  }
}

m0 <-
  transform(m0, s1 ~ y1, cens1(-2, 'left')) |>
  transform(s2 ~ y2, cens1(2,  'right'))
```

```{r  sim2 }
d <- sim(m0, 500, p=c(r=0.9), seed=1)
head(d)
```

# Estimation and inference

The Maximum Likelihood Estimate can be obtainted using the `estimate` method:

```{r  est1 }
m <- lvm() |>
     regression(y1 + y2 ~ x) |>
     covariance(y1 ~ y2)

e <- estimate(m, data=d)
e
```

The estimate `y1~~y2` gives us the estimated covariance between the
residual terms in the model. To estimate the correlation we can apply
the delta method using the `estimate` method again

```{r  delta }
estimate(e, function(p) p['y1~~y2']/(p['y1~~y1']*p['y2~~y2'])^.5)
```

Alternatively, the correlations can be extracted using the `correlation` method

```{r  correlation }
correlation(e)
```

Note, that in this case the confidence intervals are constructed
by using a variance stabilizing transformation, Fishers
$z$-transform [@lehmann2023_testing],

$$
z = \arctanh(\widehat{\rho}) =
\frac{1}{2}\log\left(\frac{1+\widehat{\rho}}{1-\widehat{\rho}}\right)
$$

where $\widehat{\rho}$ is the MLE.  This estimate has an approximate
asymptotic normal distribution
$\mathcal{N}(\arctanh(\rho),\frac{1}{n-3})$. Hence a asymptotic 95%
confidence interval is given by

$$\widehat{z} \pm \frac{1.96}{\sqrt{n-3}}$$

and the confidence interval for $\widehat{\rho}$ can directly be calculated by
the inverse transformation:

$$\widehat{\rho} = \tanh(z) = \frac{e^{2z}-1}{e^{2z}+1}.$$

This is equivalent to the direct calculations using the delta method
(except for the small sample bias correction $3$) where the
estimate and confidence interval are transformed back to the original
scale using the `summary` method.

```{r   }
estimate(e, function(p) atanh(p['y1~~y2']/(p['y1~~y1']*p['y2~~y2'])^.5)) |>
  summary(transform=tanh)
```

The transformed confidence interval will generally have improved
coverage especially near the boundary $\rho \approx \pm 1$.

While the estimates of this particular model can be obtained in closed
form, this is generally not the case when for example considering
parameter constraints, latent variables, or missing and censored
observations. The MLE is therefore obtained using iterative
optimization procedures (typically Fisher scoring or Newton-Raphson
methods). To ensure that the estimated variance parameters leads to a
meaningful positive definite structure and to avoid potential problems
with convergence it can often be a good idea to parametrize the model
in a way that such parameter constraints are naturally fulfilled.
This can be achieved with the `constrain` method.

```{r  constraints }
m2 <- m |>
    parameter(~ l1 + l2 + z) |>
    variance(~ y1 + y2, value=c('v1','v2')) |>
    covariance(y1 ~ y2, value='c') |>
    constrain(v1 ~ l1, fun=exp) |>
    constrain(v2 ~ l2, fun=exp) |>
    constrain(c ~ z+l1+l2, fun=function(x) tanh(x[1])*sqrt(exp(x[2])*exp(x[3])))
```

In the above code,  we first add new parameters `l1` and `l2` to hold the log-variance
parameters, and `z` which will be the z-transform of the correlation
parameter.
Next we label the variances and covariances: The variance of `y1` is called `v1`;
the variance of `y2` is called `v2`; the covariance of `y1` and `y2` is called `c`.
Finally, these parameters are tied to the previously defined
parameters using the `constrain` method such that `v1` := $\exp(\mathtt{l1})$
`v2` := $\exp(\mathtt{l1})$ and `z` := $\tanh(\mathtt{z})\sqrt{\mathtt{v1}\mathtt{v2}}$.
In this way there is no constraints on the actual estimated parameters
`l1`, `l2`, and `z` which can take any values in $\mathbb{R}^{3}$, while we at the
same time are guaranteed a proper covariance matrix which is positive
definite.

```{r  estconstraints }
e2 <- estimate(m2, d)
e2
```

The correlation coefficient can then be obtained as

```{r  deltaconstraints }
estimate(e2, "z") |> labels("rho") |> summary(transform=tanh)
```

In practice, a much shorter syntax can be used to obtain the above
parametrization. We can simply use the argument `constrain`
when specifying the covariances (the argument `rname` specifies the
parameter name of the $\arctanh$ transformed correlation
coefficient, and `lname`, `lname2` can be used to specify the parameter
names for the log variance parameters):

```{r  constraints2 }
m2 <- lvm() |>
  regression(y1 + y2 ~ x) |>
  covariance(y1 ~ y2, constrain=TRUE, rname='z')

e2 <- estimate(m2, data=d)
e2
```

```{r  e2backtransform }
estimate(e2, 'z') |> summary(transform=tanh)
```

As an alternative to the Wald confidence intervals (with or without
transformation) is to profile the likelihood. The profile likelihood
confidence intervals can be obtained with the `confint` method:

```{r  profileci, cache=TRUE }
tanh(confint(e2, 'z', profile=TRUE))
```

Finally, a non-parametric bootstrap (in practice a larger number of
replications would be needed) can be calculated in the following way

```{r  bootstrap, cache=TRUE }
set.seed(1)
b <- bootstrap(e2, data=d, R=50, mc.cores=1)
b
```

```{r  cache=TRUE }
quantile(tanh(b$coef[,'z']), c(.025,.975))
```


## Censored observations

Letting one of the variables be right-censored (Tobit-type model) we
can proceed in exactly the same way (note, this functionality is only
available with the `mets` package installed - available from CRAN). The
only difference is that the variables that are censored must all be
defined as `Surv` objects (from the `survival` package which is
automatically loaded when using the `mets` package) in the data frame.

```{r  cache=TRUE, eval=mets }
m3 <- lvm() |>
  regression(y1 + s2 ~ x) |>
  covariance(y1 ~ s2, constrain=TRUE, rname='z')

e3 <- estimate(m3, d)
```

```{r  eval=mets }
e3
```

```{r  cache=TRUE, eval=mets }
summary(estimate(e3, 'z'), transform=tanh)
```

And here the same analysis with `s1` being left-censored and `s2` right-censored:

```{r  cache=TRUE, eval=mets }
m3b <- lvm() |>
  regression(s1 + s2 ~ x) |>
  covariance(s1 ~ s2, constrain=TRUE, rname='z')

e3b <- estimate(m3b, d)
e3b
```

```{r  eval=mets }
e3b
```

```{r  cache=TRUE, eval=mets }
summary(estimate(e3b, 'z'), transform=tanh)
```

```{r  profilecens, cache=TRUE, eval=mets }
tanh(confint(e3b, 'z', profile=TRUE))
```


# SessionInfo

```{r   }
sessionInfo()
```
# Bibliography
