When the Dependent Variable is Missing

To truly account for left-censored data requires a likelihood approach that defines the total likelihood as the sum of the likelihoods for the observed data and the missing data and then maximizes the total censored and uncensored likelihood with respect to the model parameters. In the simplest case with n independent observations that are not longitudinal in nature, m of which are below the LLOQ, the likelihood equals 

It should be noted that in the case of right-censored data the likelihood is simply 

---

Blunders must be eliminated, and all specified data must be collected. The efficiency of these experimental designs has another side effect any missing or defective data has a disproportionate effect relative to the amount of information that can be extracted from the final data set. When simpler experimental designs are used, where each piece of data is collected for the sole purpose of determining the effect of one variable, loss of that piece of data results in the loss of only that one result. When the more efficient statistical experimental designs are used, each piece of data contributes to more than one of the final results, thus each one is used the equivalent of many times and any missing piece of data causes the loss of all the results that are dependent upon it. 

A totally different situation arises when covariates are missing because of the value of the observation, not because the covariate wasn t measured. In such a case the value is censored, which means that the value is below or above some critical threshold for measurement. On the other hand, a covariate may be censored from above where the covariate reported as greater than upper limit of quantification (ULOQ) of the method used to measure it. In such a case the covariate is reported as >ULOQ, but its true value may lie theoretically between ULOQ and infinity. The issue of censored covariates has not received as much attention as the issue of censored dependent variables. Typical solutions include any of the substitution or imputation methods described for imputed missing covariates that are not censored. 

A weakness with the standard mediod for simplex optimisation is a dependence on the initial step size, which is defined by the initial conditions. For example, in Figure 2.37 we set a very small step size for both variables this may be fine if we are sure we are near the optimum, but otherwise a bigger triangle would reach the optimum quicker, the problem being that the bigger step size may miss the optimum altogether. Another method is called the modified simplex algorithm and allows the step size to be altered, reduced as the optimum is reached, or increased when far from the optimum. 

---