\[ \newcommand{\arctanh}{\operatorname{arctanh}} \]

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 \[\begin{pmatrix} \epsilon_{1} \\ \epsilon_{2} \end{pmatrix} \sim \mathcal{N}\left(0, \mathbf{\Sigma} \right), \quad \mathbf{\Sigma} = \begin{pmatrix} \sigma_1^2 & \rho\sigma_{1}\sigma_{2} \\ \rho\sigma_{1}\sigma_{2} & \sigma_2^2 \end{pmatrix}.\]

library('lava')
m0 <- lvm(y1+y2 ~ x, y1 ~~ y2)
edgelabels(m0, y1 + y2 ~ x) <- c(expression(beta[1]), expression(beta[2]))
edgelabels(m0, y1 ~ y2) <- expression(rho)
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. First we load the necessary libraries:

The model can be specified (here using the pipe notation via magrittr) with the following syntax where the correlation parameter here is given the label ‘r’:

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

coef(m0, labels=TRUE)
#>       m1       m2       p1       p2       p3       p4       p5 
#>     "y1"     "y2"   "y1~x"   "y2~x" "y1~~y1" "y2~~y2"      "r"

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).

d <- sim(m0, 500, p=c(r=0.9), seed=1)
head(d)
#>           y1         y2           x
#> 1  0.6452154  0.8677628  1.13496509
#> 2  1.1098723  0.9579211  1.11193185
#> 3 -2.2072258 -2.3171509 -0.87077763
#> 4  1.5684365  1.0675354  0.21073159
#> 5  0.8752209  1.0845932  0.06939565
#> 6 -1.5113072 -0.7477956 -1.66264885

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.

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

m0 %<>%
  transform(s1 ~ y1, cens1(-2, 'left')) %>%
  transform(s2 ~ y2, cens1(2,  'right'))
d <- sim(m0, 500, p=c(r=0.9), seed=1)
head(d)
#>           y1         y2           x          s1         s2
#> 1  0.6452154  0.8677628  1.13496509   0.6452154  0.8677628
#> 2  1.1098723  0.9579211  1.11193185   1.1098723  0.9579211
#> 3 -2.2072258 -2.3171509 -0.87077763 -2.0000000- -2.3171509
#> 4  1.5684365  1.0675354  0.21073159   1.5684365  1.0675354
#> 5  0.8752209  1.0845932  0.06939565   0.8752209  1.0845932
#> 6 -1.5113072 -0.7477956 -1.66264885  -1.5113072 -0.7477956

Estimation and inference

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

m <- lvm() %>%
     regression(y1 + y2 ~ x) %>%
     covariance(y1 ~ y2)

e <- estimate(m, data=d)
e
#>                     Estimate Std. Error  Z-value  P-value
#> Regressions:                                             
#>    y1~x              0.93300    0.04443 20.99871   <1e-12
#>     y2~x             0.91652    0.04527 20.24500   <1e-12
#> Intercepts:                                              
#>    y1               -0.00541    0.04482 -0.12076   0.9039
#>    y2               -0.02715    0.04566 -0.59457   0.5521
#> Residual Variances:                                      
#>    y1                1.00419    0.06351 15.81139         
#>    y1~~y2            0.91221    0.06130 14.88041   <1e-12
#>    y2                1.04252    0.06593 15.81139

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

estimate(e, function(p) p['y1~~y2']/(p['y1~~y1']*p['y2~~y2'])^.5)
#>        Estimate  Std.Err   2.5%  97.5% P-value
#> y1~~y2   0.8915 0.008703 0.8745 0.9086       0

Alternatively, the correlations can be extracted using the correlation method

correlation(e)
#>       Estimate Std.Err   2.5%  97.5%   P-value
#> y1~y2   0.8915         0.8721 0.9082 3.58e-224

Note, that in this case the confidence intervals are constructed by using a variance stabilizing transformation, Fishers \(z\)-transform lehmann_romano_2005,

\[z = \arctanh(\widehat{\rho}) = \frac{1}{2}\log\left(\frac{1+\widehat{\rho}}{1-\widehat{\rho}}\right)\] where \(\widehat{\rho}\) is the MLE. This estimate has an approximate asymptotic normal distribution \(\mathcal{N}(\arctanh(\rho),\frac{1}{n-3-k})\). Hence a asymptotic 95% confidence interval is given by \[\widehat{z} \pm \frac{1.96}{\sqrt{n-3-k}}\] 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 \(3+k\)) where the estimate and confidence interval are transformed back to the original scale using the back.transform argument.

estimate(e, function(p) atanh(p['y1~~y2']/(p['y1~~y1']*p['y2~~y2'])^.5), back.transform=tanh)
#>        Estimate Std.Err   2.5%  97.5%    P-value
#> y1~~y2   0.8915         0.8732 0.9074 7.445e-249

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.

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 \(\R^{3}\), while we at the same time are guaranteed a proper covariance matrix which is positive definite.

e2 <- estimate(m2, d)
e2
#>                        Estimate Std. Error  Z-value  P-value
#> Regressions:                                                
#>    y1~x                 0.93300    0.04443 20.99871   <1e-12
#>     y2~x                0.91652    0.04527 20.24500   <1e-12
#> Intercepts:                                                 
#>    y1                  -0.00541    0.04482 -0.12076   0.9039
#>    y2                  -0.02715    0.04566 -0.59457   0.5521
#> Additional Parameters:                                      
#>    l1                   0.00418    0.06325  0.06617   0.9472
#>    l2                   0.04164    0.06325  0.65832   0.5103
#>    z                    1.42942    0.04472 31.96286   <1e-12

The correlation coefficient can then be obtained as

estimate(e2, 'z', back.transform=tanh)
#>     Estimate Std.Err   2.5%  97.5%    P-value
#> [z]   0.8915         0.8729 0.9076 5.606e-243
#> 
#>  Null Hypothesis: 
#>   [z] = 0
    Estimate Std.Err   2.5%  97.5%    P-value
[z]   0.8915         0.8729 0.9076 5.606e-243

 Null Hypothesis:
  [z] = 0

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):

m2 <- lvm() %>%
  regression(y1 + y2 ~ x) %>%
  covariance(y1 ~ y2, constrain=TRUE, rname='z')

e2 <- estimate(m2, data=d)
e2
#>                        Estimate Std. Error  Z-value  P-value
#> Regressions:                                                
#>    y1~x                 0.93300    0.04443 20.99871   <1e-12
#>     y2~x                0.91652    0.04527 20.24500   <1e-12
#> Intercepts:                                                 
#>    y1                  -0.00541    0.04482 -0.12076   0.9039
#>    y2                  -0.02715    0.04566 -0.59457   0.5521
#> Additional Parameters:                                      
#>    l1                   0.00418    0.06325  0.06617   0.9472
#>    l2                   0.04164    0.06325  0.65832   0.5103
#>    z                    1.42942    0.04472 31.96286   <1e-12
estimate(e2, 'z', back.transform=tanh)
#>     Estimate Std.Err   2.5%  97.5%    P-value
#> [z]   0.8915         0.8729 0.9076 5.606e-243
#> 
#>  Null Hypothesis: 
#>   [z] = 0
    Estimate Std.Err   2.5%  97.5%    P-value
[z]   0.8915         0.8729 0.9076 5.606e-243

 Null Hypothesis:
  [z] = 0

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:

tanh(confint(e2, 'z', profile=TRUE))
#>       2.5 %    97.5 %
#> z 0.8720834 0.9081964
      2.5 %    97.5 %
z 0.8720834 0.9081964

Finally, a non-parametric bootstrap can be calculated in the following way

set.seed(1)
b <- bootstrap(e2, data=d, R=1000, mc.cores=4)
b
#> Non-parametric bootstrap statistics (R=1000):
#> 
#>      Estimate     Bias         Std.Err      2.5 %        97.5 %      
#> y1   -0.005411914 -0.001172633  0.045641233 -0.095681252  0.084382238
#> y2   -0.027149492 -0.000455828  0.045103714 -0.117924690  0.057497237
#> y1~x  0.933004351 -0.001055459  0.047352547  0.838921807  1.025171887
#> y2~x  0.916518525 -0.001319299  0.047345777  0.824805874  1.005880603
#> l1    0.004184652 -0.007262228  0.062351891 -0.126711581  0.122154125
#> l2    0.041636106 -0.007371904  0.061581789 -0.082031800  0.155479843
#> z     1.429422707  0.001185955  0.042842182  1.347743380  1.508839440
#> v1    1.004193420 -0.005329636  0.062277364  0.880987790  1.129928261
#> v2    1.042515045 -0.005696525  0.063828553  0.921242663  1.168218400
#> c1    0.912209719 -0.004854787  0.059286166  0.796486098  1.031282548
quantile(tanh(b$coef[,'z']), c(.025,.975))
#>      2.5%     97.5% 
#> 0.8735196 0.9067329

Censored observations

Letting one of the variables be right-censored (Tobit-type model) we can proceed in exactly the same way1. 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.

m3 <- lvm() %>%
  regression(y1 + s2 ~ x) %>%
  covariance(y1 ~ s2, constrain=TRUE, rname='z')

e3 <- estimate(m3, d)
e3
#>                        Estimate Std. Error  Z-value  P-value
#> Regressions:                                                
#>    y1~x                 0.93300    0.04443 20.99870   <1e-12
#>     s2~x                0.92402    0.04643 19.90116   <1e-12
#> Intercepts:                                                 
#>    y1                  -0.00542    0.04482 -0.12083   0.9038
#>    s2                  -0.02119    0.04638 -0.45685   0.6478
#> Additional Parameters:                                      
#>    l1                   0.00419    0.06325  0.06618   0.9472
#>    l2                   0.06317    0.06492  0.97311   0.3305
#>    z                    1.42834    0.04546 31.41837   <1e-12
estimate(e3, 'z', back.transform=tanh)
#>     Estimate Std.Err  2.5%  97.5%  P-value
#> [z]   0.8913         0.872 0.9079 1.5e-226
#> 
#>  Null Hypothesis: 
#>   [z] = 0

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

m3b <- lvm() %>%
  regression(s1 + s2 ~ x) %>%
  covariance(s1 ~ s2, constrain=TRUE, rname='z')

e3b <- estimate(m3b, d)
e3b
#>                        Estimate Std. Error  Z-value  P-value
#> Regressions:                                                
#>    s1~x                 0.92834    0.04479 20.72733   <1e-12
#>     s2~x                0.92466    0.04648 19.89513   <1e-12
#> Intercepts:                                                 
#>    s1                  -0.00233    0.04492 -0.05184   0.9587
#>    s2                  -0.02083    0.04641 -0.44874   0.6536
#> Additional Parameters:                                      
#>    l1                  -0.00075    0.06500 -0.01156   0.9908
#>    l2                   0.06425    0.06498  0.98871   0.3228
#>    z                    1.42627    0.04609 30.94319   <1e-12
estimate(e3b, 'z', back.transform=tanh)
#>     Estimate Std.Err   2.5%  97.5%    P-value
#> [z]   0.8909         0.8713 0.9077 9.014e-222
#> 
#>  Null Hypothesis: 
#>   [z] = 0
tanh(confint(e3, 'z', profile=TRUE))
#>       2.5 %    97.5 %
#> z 0.8714569 0.9082484

SessionInfo

sessionInfo()
#> R version 4.0.2 (2020-06-22)
#> Platform: x86_64-pc-linux-gnu (64-bit)
#> Running under: Ubuntu 16.04.6 LTS
#> 
#> Matrix products: default
#> BLAS:   /usr/lib/openblas-base/libblas.so.3
#> LAPACK: /usr/lib/libopenblasp-r0.2.18.so
#> 
#> locale:
#>  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
#>  [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8    
#>  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
#>  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
#>  [9] LC_ADDRESS=C               LC_TELEPHONE=C            
#> [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       
#> 
#> attached base packages:
#> [1] stats     graphics  grDevices utils     datasets  methods   base     
#> 
#> other attached packages:
#> [1] magrittr_1.5    mets_1.2.8      timereg_1.9.7   survival_3.1-12
#> [5] lava_1.6.8     
#> 
#> loaded via a namespace (and not attached):
#>  [1] Rcpp_1.0.5          graph_1.66.0        cpp11_0.2.1        
#>  [4] knitr_1.29          BiocGenerics_0.34.0 splines_4.0.2      
#>  [7] lattice_0.20-41     R6_2.4.1            ragg_0.3.1         
#> [10] rlang_0.4.7         stringr_1.4.0       tools_4.0.2        
#> [13] parallel_4.0.2      grid_4.0.2          xfun_0.17          
#> [16] htmltools_0.5.0     systemfonts_0.3.1   yaml_2.2.1         
#> [19] assertthat_0.2.1    rprojroot_1.3-2     digest_0.6.25      
#> [22] numDeriv_2016.8-1.1 pkgdown_1.6.1       crayon_1.3.4       
#> [25] Matrix_1.2-18       Rgraphviz_2.32.0    codetools_0.2-16   
#> [28] fs_1.5.0            memoise_1.1.0       evaluate_0.14      
#> [31] rmarkdown_2.3       stringi_1.5.3       compiler_4.0.2     
#> [34] desc_1.2.0          backports_1.1.10    stats4_4.0.2       
#> [37] mvtnorm_1.1-1

Bibliography

[lehmann_romano_2005] Lehmann & Romano, Testing statistical hypotheses, Springer (2005).

Footnotes

1 This functionality is only available with the mets package installed (available from CRAN)