Semiparametric Distributions With Estimated Shape Parameters

Abstract

Purpose

To investigate the use of adaptive transformations to assess the parameter distributions in population modeling.

Methods

The logit, box-cox, and heavy tailed transformations were investigated. Each one was used in conjunction with the standard (exponential) transformation for PK and PD parameters. The shape parameters of these transformations were estimated to allow the parameter distributions to more accurately resemble a wider range of parameter distributions. The transformations were tested both in simulated settings where the true distributions were known and in 30 models developed from real data.

Results

In the simulated setting the transformations were better than the standard lognormal distribution at characterizing the true distributions. Improvement could also be seen in objective function value (OFV) and in simulation based diagnostics. In the real datasets, significant model improvement based on OFV could be seen in 22, 18, and 22 out of the 30 models for the three transformations respectively.

Conclusion

Transformations with estimated shape parameters are a promising approach to relax the often erroneous assumption of a known shape of the parameter distribution. They offer a simple and straightforward way of handling and characterizing parameter distributions.

Appendices

Appendix I. Assessment of the Type I Error Rates

When building models using maximum likelihood as in NONMEM, the likelihood ratio test is often used to test if the inclusion of a new parameter that is to be estimated into the model is statistically significant. Under asymptotic conditions the likelihood ratio statistic is χ2 distributed, which implies that a drop in OFV needs to be larger than 3.84 for an inclusion of a new parameter, corresponding to one degree of freedom, to be statistically significant at p < 0.05 (28). Monte Carlo simulations were performed to investigate if the likelihood ratio statististic of the shape parameters estimated in these transformations are χ2 distributed with the same number of degrees of freedom as additional parameters estimated, which would be in accordance with statistical theory. This kind of investigation would also test the robustness of the likelihood ratio test as it has been done with other model misspecifications (4,5).

Datasets were simulated from a model with a constant rate infusion at steady state, thus parameterized only with one structural parameter; CL, modeled with interindividual variability, this simple model was chosen for runtime reasons. Simulations were performed with transformations included, but the shape parameters set to values that would give no change in the shape of the distributions. These values were 0.00001 for box-cox, 0.5 and 4 for the two parameters of the logit transformation, and 0 for HTT. The simulation settings were altered and combined with respect to number of individuals and number of observations. The number of individuals ranged between 25 and 500, and observations ranged between 2 and 19. For each combination, 1000 datasets were simulated. The datasets were then estimated both with the full model with transformation as well as with the reduced model, i.e. the standard lognormal distribution. The OFV values from the estimation with both models were then compared for each simulated dataset, and the drop in OFV which produced an error rate of 5% was calculated. Expected values would be 3.84 for box-cox and HTT and 5.99 for logit because of its two parameters.

The results of the assessment of the type I error rates through simulations showed that the cut-off values for inclusion of a transformation of a parameter distribution were drops in OFV of 7, 4, and 5 for the logit, box-cox, and HT transformations, respectively. These values do not differ to any large extent from the nominal values of 3.84 and 5.99. These error rates were considered to be stable from 50 individuals and up. At lower numbers of individuals (25) the values are slightly elevated (see Table VI, which shows the average cut-off values of from varying the number of observations per individual).

Table VI Average Empirical OFV Cut-off Values Across Different Number of Individuals

Appendix II Examples of NM-TRAN Code

Logit transformation

TVCL=THETA(1)

LGPAR1 = THETA(2)

LGPAR1 = THETA(3)

PHI = LOG(LGPAR1/(1-LGPAR1))

PAR1 = EXP(PHI+ETA(1))

ETATR = (PAR1/(1+PAR1)-TVTH)*LGPAR2

CL=TVCL*EXP(ETATR)

Box-Cox transformation

TVCL=THETA(1)

BXPAR=THETA(2)

PHI = EXP(ETA(1))

ETATR = (PHI**BXPAR-1)/BXPAR

CL=TVCL*EXP(ETATR)

Heavy tailed transformation

TVCL=THETA(1)

HTPAR=THETA(2)

ETATR=ETA(1)*SQRT(ETA(1)*ETA(1))**HTPAR

CL=TVCL*EXP(ETATR)