A Study Comparing the Performance of the Proportional Odds Model to that of the Differential Drug Effect Model for Cumulative Logits

MC Kjellsson (1), PH Zingmark (1, 2), EN Jonsson (1), MO Karlsson (1)

(1) Division of Pharmacokinetics and Drug Therapy, Department of Pharmaceutical Biosciences, Uppsala University, Uppsala, Sweden(2); Department of Clinical Pharmacology, AstraZeneca R&D Södertälje, Sweden

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Introduction: In 1994, Lewis Sheiner proposed the use of the proportional odds model, a special case of the cumulative logit model, for the analysis of ordered categorical data within the area of PK/PD mixed effects modelling [1]. The model (Eq. 1 for the j^th observation in the i^th individual of an m-categorical effect Y) assumes that for all monotonically increasing drug effects (f_D), the relative increase in probability increases with increasing categorical level regardless of the exposure (C). This constrain of the proportional odds model is expressed in Eq. 2.

 P(Y_ij<=Y_m|eta_i) = e^{f(alpha, C, eta)} / (1+e^{f(alpha, C, eta)}) 

f(alpha, C, eta) = SUM_n=1^m-1alpha_n + f_D + eta_i                                      (1)

 where eta_i=N(0,omega), and alpha are the fixed effects for the intercepts.

 P(Y_m|C<0)/P(Y_m|C=0) / P(Y_m-1|C<0)/P(Y_m-1|C=0) > 1                          (2)

 While this may be a valid assumption for many types of ordered categorical data, it may not apply to all. In an extension to the proportional odds model, here called differential drug effect model for cumulative logits, an additional, category-specific, factor, f_diff multiplies the drug effect (Eq. 3). This relaxes the assumption made in the proportional odds model, but by its constraint to be in the 0-1 interval, retains the order of the cumulative logits.

 f_diff = PRODUCT_n=2^m-1 e^{beta n} / (1+e^{beta n})                                            (3)

The objective with this study was to evaluate the relative performance of the proportional odds model and the differential drug effect model for cumulative logits.

Data: The differential drug effect model was applied to four datasets: 

1. 4-Category simulated data. The simulated dataset consisted of 1000 patients evenly divided into 4 dose groups (0, 1, 2 and 4 units of drug), with each patient having 1 baseline observation and 3 after study drug intervention. Each trial was replicated 1000 times. This simulation was performed to investigate the Type I error rate of the differential drug effect model compared to the proportional odds model.
2. 3-Category T-cell data following a phase I clinical study of an antibody, where the response represents categorized continuous data of T-cell receptor density measured by FACS [2].
3. 6-Category data on sedation as a side-effect and natural consequence of stroke in a placebo-controlled trial of clomethiazole as a neuroprotective drug in stroke patients [3].
4. 5-Category data on diarrhea severity as a side-effect of anti-cancer treatment with irinotecan [4].

 Datasets 2-4 have been previously published with a proportional odds model for the drug effect. The drug effects, f_D, have been described with step (sedation [3]), linear (diarrhea [4], simulations) and sigmoid Emax (T-cell [2]) models.

Method: The proportional odds model and the differential drug effect model are nested and the statistical significance was assessed using the likelihood ratio test, based on the difference in objective function value (deltaOFV).

Results: The model for sedation data was significantly improved (p<0.0005, deltaOFV=74, df=5) using the differential drug effect model. This improvement of the fit was also evident on visual inspection. In addition, the PD model, f_D, changed from a step effect to a more mechanistically plausible graded effect.

Neither for the diarrhea data nor for the T-cell data were there a significant (p>0.05) improvement for including the differential drug effect model. For the simulated data, the Type I error rate was as expected, 4.9% and 0.86% at a nominal value of 5% and 1% (998 successful minimizations of 1000). 

Discussion: In line with expectations [5], the differential drug effect model did not improve the fit to the simulated data or to the categorized continuous T-cell data. As opposed to the T-cell data, the sedation data do not represent a categorization of an underlying continuous variable, but a clinical interpretation of a state, likely to be multifactorial in its origin, using different stimuli (observation, communication and pain infliction) in the assessment. In such a situation, it is not surprising that a drug may have differential effects on the different levels of sedation. For the diarrhea data the categories appear to approximately represent a categorization of an underlying continuous scale.

Conclusion: The differential drug effect model appeared to have the desirable properties of not being indicated when not necessary, but to provide meaningful improvement in the description of ordered categorical clinical pharmacodynamic data when needed.

References:
[1]   Sheiner L.B. Clin Pharmacol Ther 1994, 56; 309-22.
[2]   Zingmark P-H et al. Br J Clin Pharmacol 2004, 58; 378-89.
[3]   Xie R. et al. Clin Pharmacol Ther 2002, 72; 265-75.
[4]   Zingmark P-H. et al. Br J Clin Pharmacol 2003, 56; 173-83.
[5]   Agresti A. Categorical Data Analysis. Wiley. 2^nd Ed. 2002, Chapter 7.

Reference: PAGE 14 (2005) Abstr 780 [www.page-meeting.org/?abstract=780]

Oral Presentation: Lewis Sheiner Student Session