Modelling Correlation in Random Effects¶
As shown in Inter-Subject Variation (ISV), we can use random effects to account for variability in model parameters between individuals in a population. In that example, we demonstrated the principle for a single parameter whereas we now consider the case where two or more parameters vary between individuals. In particular, we are interested in what happens when pairs of parameters vary in similar ways, for example when they have a common underlying cause.
In the following examples we model inter-subject variability in both m[CL] and m[V] using a combined proportional and additive noise model to generate observations for 200 individuals, and fit various models to the synthesized data.
Uncorrelated Effects¶
Note
See the Diagonal matrix generation diagonal matrix fit using separate univariate normals and Diagonal matrix generation diagonal matrix fit for the Tut Script used to generate results in this section.
We can synthesize observations where parameters are uncorrelated simply by creating, for each parameter, an independent random effect with its own variance:
EFFECTS:
POP: |
f[KA] = 0.3
f[CL] = 3
f[V] = 20
f[PNOISE_STD] = 0.1
f[ANOISE_STD] = 0.05
f[CL_isv] = 0.2
# new: ISV on volume of distribution
f[V_isv] = 0.1
ID: |
c[ID] = sequential(200)
c[AMT] = 100.0
t[RESET] = 0.0
t[DOSE] = 1.0
t[OBS] ~ unif(1.0, 50.0; 4)
r[CL_isv] ~ norm( 0, f[CL_isv] )
# new: ISV on volume of distribution
r[V_isv] ~ norm( 0, f[V_isv] )
Mathematically, this is equivalent to modelling the two parameters jointly via a covariance matrix with the individual variances along the diagonal and zeros everywhere else. In PoPy this structure is enforced using the ~diag_matrix() notation:
EFFECTS:
POP: |
f[KA] = 0.3
f[CL] = 3
f[V] = 20
f[PNOISE_STD] = 0.1
f[ANOISE_STD] = 0.05
# new: Joint ISV on both CL and V
f[CL_isv, V_isv] ~ diag_matrix() [[0.2, 0.1]]
ID: |
c[ID] = sequential(200)
c[AMT] = 100.0
t[RESET] = 0.0
t[DOSE] = 1.0
t[OBS] ~ unif(1.0, 50.0; 4)
# new: Joint ISV on both CL and V
r[CL_isv, V_isv] ~ mnorm( [0,0], f[CL_isv, V_isv] )
for the covariance matrix and using a multivariate normal distribution, ~mnorm(), to model the two random effects as a vector (in this case with zero mean). Throughout this chapter we will continue to use the covariance matrix form rather than independent variables.
With independent inter-subject variability in m[CL] and m[V] the generated curves (Fig. 37) vary widely from individuals with high CL and low V (i.e. a high KE) to individuals with a low CL and high V (i.e. a low KE).
Fig. 37 Simulated prediction curves for a population with independent inter-subject variability on parameters CL and V
In all cases here, we ensure that the generated observations are the same by specifying a fixed rand_seed option in the METHOD_OPTIONS section of the tutorial script:
METHOD_OPTIONS: {py_module: gen, rand_seed: 314159}
Correlated Effects¶
Note
See the Full matrix generation diagonal matrix fit for the Tut Script used to generate results in this section.
We now run the same experiments, only using new observations that have been synthesized from a model where the variability in m[CL] and m[V] is correlated. This is a more realistic example, since both clearance and volume of distribution can increase with weight.
EFFECTS:
POP: |
f[KA] = 0.3
f[CL] = 3
f[V] = 20
f[PNOISE_STD] = 0.1
f[ANOISE_STD] = 0.05
# new: Data in which variability of CL is
# linked to that of V
f[CL_isv, V_isv] ~ spd_matrix() [
[ 0.15 ],
[ 0.05, 0.15 ],
]
ID: |
c[ID] = sequential(200)
c[AMT] = 100.0
t[RESET] = 0.0
t[DOSE] = 1.0
t[OBS] ~ unif(1.0, 50.0; 4)
r[CL_isv, V_isv] ~ mnorm( [0,0], f[CL_isv, V_isv] )
Fig. 38 Simulated prediction curves for a population with correlated inter-subject variability on parameters CL and V.
The generated predictions (Fig. 38) are qualitatively different from those generated without correlation because the resulting KE values have a smaller spread. (Correlated changes in CL and V cancel out to a degree.)