Abstract
In many clinical trials, information is collected on both the frequency of event occurrence and the severity of each event. For example, in evaluating a new anti-epileptic medication both the total number of seizures a patient has during the study period as well as the severity (e.g., mild, severe) of each seizure could be measured. In order to arrive at a full picture of drug or treatment performance, one needs to jointly model the number of events and their correlated ordinal severity measures. A separate analysis is not recommended as it is inefficient and can lead to what we define as “zero length bias” in estimates of treatment effect on severity. This paper proposes a general, likelihood based, marginal regression model for jointly modeling the number of events and their correlated ordinal severity measures. We describe parameter estimation issues and derive the Fisher information matrix for the joint model in order to obtain the asymptotic covariance matrix of the parameter estimates. A limited simulation study is conducted to examine the asymptotic properties of the maximum likelihood estimators. Using this joint model, we propose tests that incorporate information from both the number of events and their correlated ordinal severity measures. The methodology is illustrated with two examples from clinical trials: the first concerning a new drug treatment for epilepsy; the second evaluating the effect of a cholesterol lowering medication on coronary artery disease.
Highlights
In many clinical trials, data on both the frequency of a disease event as well as the severity of each event, often measured by an ordinal score, is recorded
More examples include: depression may occur a random number of times with the severity of each depression episode being measured by an ordinal scale (e.g., 1=mildly depressed, 2=moderately depressed, 3=very depressed); patients suffering from migraine headache may have a random number of headaches with differing levels of pain; patients with cancer may have a random number of tumors of varying sizes, etc
We summarize the proposed joint model here: 1. The distribution of the random length Ki, following Barnhart et al (1999), is modeled as a slightly modified generalized linear model (GLIM) (McCullagh and Nelder, 1989) with three components: (a) The random length variable Ki has a discrete distribution: Pr (Ki = k) = a (k) exp (kφi − b), k = 0, 1, ..., J
Summary
Data on both the frequency of a disease event as well as the severity of each event, often measured by an ordinal score, is recorded. One may want to test whether a drug changes the inhibition process among cancerous tumors (i.e., large tumors suppressing the ability of other tumors to grow near them) or estimate the probability of observing two or more severe events Questions of this type can only be answered by modeling the joint distribution of the number of events and their correlated severity measures. The probit model, assumes a discretization of a latent multivariate normal distribution for the multivariate ordinal responses and may not be flexible enough in many applications These previous methods for multivariate random length data produce estimates of two treatment effects: one for event severity and one for the number of events. The organization of this paper is as follows: in section 3 we present the proposed joint model; in section 4 we discuss estimation issues; in section 5 we present a limited simulation study; and in section 6 we present two examples from two clinical trials–one concerning an anti-seizure medication, the other a cholesterol lowering drug
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