The occurrence of a biased sample is especially prevalent when a researcher is conducting medical studies. A good example of when such an incident happens is when research is sampling a group of people and one of them refuses to respond to questions. A well-known example of such an occurrence happens when there is a case-control study (Reilly & Pepe, 1995). Here, the status obtained from the population is considered and covariates are analyzed for their potential among different independent groups.
Notably, the sampling becomes stratified, where the likelihood is dependent on the distribution of covariate values. Nonetheless, if sampling was done unconditionally, the likelihood would be given by another expression altogether, which represents a regression model. This is because research has shown that through the use of a regression model, the maximum possible values for the likelihood are given. Therefore, the case control can be treated as a prospective matter. Additionally, further research has shown that it is possible to extend the intercepts to have a multiplicities intercept model. However, it is regression models that can hold without g (x) that are of interest. This is the likelihood normally used for problems that are response-selective in nature (Lee, Scott, & Wild, 2010). As a result, they fall within the area of interest since they can provide answers for biased data.
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Methods
There are several methods that one can consider when ignoring the sample scheme and considering only complete units. Since this method does not lead to the efficient analysis of data, different methods have been proposed in going about it:
Weighted likelihood – in this case, only the complete observed data is considered, while incomplete sets of data are ignored. Each unit is therefore weighted against its probability of selection for full observation. Despite its robust nature, it is inefficient.
Conditional likelihood – it improves efficiency by offering an alternative method since the method gives a dependency on G. nonetheless, this dependence can be avoided by conditioning the likelihood of x .
Maximum likelihood method – the most efficient methods can be seen here, where the y value is used. In some special cases, full efficiency can be achieved.
Fully efficient estimators have been developed for the purpose of profiling full likelihood. Maximization is considered on the values to provide the researcher with accurate likelihood values (Lawless, Kalbfleisch, & Wild, 1999). In effect, fully efficient methods become useful since they can solve multivariate case control problems, by finding the discrete values of x and y. additionally, they could obtain discrete covariates with continuous Reponses.
Objectives
Therefore, the purpose of this research seeks to unify the results of various pieces of research in this area and describe situations where conditioning is better than weighting, especially in scenarios where both methods achieve full efficiency.
References
Lawless, J. F., Kalbfleisch, J. D., & Wild, C. J. (1999). Semiparametric methods for response‐selective and missing data problems in regression. , . Journal of the Royal Statistical Society: Series B (Statistical Methodology), 61(2) , 413-438.
Lee, A. J., Scott, A. J., & Wild, C. J. (2010). Efficient estimation in multi-phase case-control studies. Biometrika, 97(2) , 361-374.
Reilly, M., & Pepe, M. S. (1995). A mean score method for missing and auxiliary covariate data in regression models. Biometrika, 82(2) , 299-314.
