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Chapter 6
Conclusions and Future Works
6.1 Conclusions
In this study, we find that the necessary condition of the linear combination maximiz-ing the pAUC shares a similar form with the best linear combination maximizmaximiz-ing the AUC, proposed by Su and Liu (1993) under the normality assumption. In addition, the pAUC maximizer may not be unique, and sometimes local maximizers exist, see Figure 2.1. Hence, the multiple-initial algorithm is proposed. Under the multivariate normality assumption when the population parameters are unknown, the sample pAUC maximizer has the strong consistency property. Furthermore, in order to know whether a biomarker or a biomarker set has a contribution to the disease diagnosis, we develop three tests for detecting their discriminatory power. Afterward, we apply these tests to the biomarker selections, which include the Forward and Backward approaches. In our testing procedure, we use a parametric bootstrapping method to find the critical value(s) of all tests, because using a non-parametric bootstrapping method to generate data is time-consuming work.
Additionally, through simulations and some real examples, the Forward method is often better than the Backward method.
From our empirical study, the proposed multiple-initial algorithm is found to have perform well in finding the optimal linear combination. Additionally, in these real data analyses, both the optimal linear combination of our algorithm and the solutions by Liu et al. (2005) are provided. When the dimension is three or four, solutions found via a
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grid search are also offered. From these analyses, we find that the solutions found via our algorithm and the solutions found via a grid search are almost the same. Besides,we can observe that the linear combination found using our algorithm is always superior to the solution of Liu et al. (2005) in terms of having a greater pAUC. Our algorithm is based on finding the linear combination that maximizes the pAUC, but the linear combinations by Liu et al. (2005) are not.
In a two-biomarker study, we find that not only the mean and variance of each biomarker, but also the correlation between biomarkers, can have a great impact on pAUC. When the correlation occurs in the non-diseased group, the effect is even more apparent. In addition, from the breast tissue example, we find that when the non-diseased population is homogeneous and the diseased population is heterogeneous, there tends to be a larger pAUC value. Furthermore, the correlation between the biomarker with the magnitude of the correspondent coefficient in the optimal linear combination of the full data set and its marginal pAUC may not be positive.
With sample data, the estimated pAUC maximizer and the test for global discrimina-tory power have adequate empirical performance. When a correlation between biomarkers exists, and the mean difference of the differential biomarker is not too small, the Forward method performs better than the Backward method. Otherwise, no one method is uni-formly better than the other in our study.
Before using the Forward and the Backward approaches to biomarker selection, a data standardization is suggested to avoid having order to the coefficients of the best linear combination based on their variances. Different standardized methods bring different co-efficients in the best linear combination and different ordering. However, in our biomarker selection methods, all biomarkers are evaluated by incorporating their sample variances in the testing procedure. Hence, the effect of standardization is minimized. In fact, in the first two real examples of our study, the biomarker selection results with or without standardization are the same. It means that our methods are robust regarding the choice of standardization. In addition, because our biomarker selection methods do not stop after an insignificant test, the effect of the ordering criterion is reduced.
In two-biomarker simulation study, when total sample size is fixed, a balanced design
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can provide a proper result. In addition, we find that when the correlation between biomarkers in the non-diseased group is large, the required sample size for achieving satisfied power can be reduced.
Our biomarker selection methods are different from others in that they are based on statistical tests. In addition, the proposed methods involve many choices of the cutoff in the selection procedure. Hence, our methods are less appropriate in an exploratory study.
Before using the marker selection methods, the application of adequate data filtering for dimension reduction is recommended.
In the two proposed biomarker selection methods, the type I error rate of the test in every step is controlled at the same significance level, α. Obviously, the overall type I error rate will be greater than α. Hence, for a control of the overall type I error rate, a suitable adjustment in the stepwise significance level is required. For example, we may consider using the Bonferroni method. In applying the Forward method to a data set of p biomarkers, it takes p tests, and the stepwise significance level is α/p. Similarly, there are at most p + 1 steps in the Backward method, and thus we use α/(p + 1) as the stepwise significance level for a conservative conclusion. With the adjustment, the overall type I error rate is controlled below α. However, we do not apply any adjustment methods for controlling the overall type I error rate in biomarker selection.
Our research is all based on both the non-diseased and the diseased groups from multivariate normal distributions. Through a simulation study, we find that the proposed biomarker selection methods are sensitive to the distributional assumption. Thus, a non-parametric estimation of the pAUC such as the empirical pAUC, can be used as an alternative method to generalize our method. However, theoretical portions of this study still require the normality assumption. For example, the strong consistency property of the best linear combination. In addition, non-parametric estimation of the pAUC increases the difficulty of computing.
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6.2 Future Works
In clinical studies, the determination of sample sizes for the non-diseased and the diseased groups is one of the major tasks, because their quantities may not be equal. It involves a degree of difficulty in collecting the data, which includes time, money, the frequency of the disease, etc. For example, we can easily obtain the sample of a commonly occurring disease, such as influenza, but the number of cancer patients is often less than the number of normal subjects. Although many researchers use the same number of non-diseased and diseased subjects in their study, we adopt adequate sample sizes which can help us to better display the strength of our method. Hence, in the near future, we will consider developing a method to find adequate sample sizes for the non-diseased and the diseased groups.
Some related researches have been conducted. For example, Obuchowski and McClish (1997) proposed sample size computation for the non-diseased and the diseased groups based on the approximate variance of two parameters under the normality assumption.
Janes and Pepe (2006) used the empirical estimator of ROC/AUC and minimized its variance to find the optimal ratio of the diseased subjects to the non-diseased subjects.
Moreover, Pepe (2004) introduces some methods which use the confidence interval of the ROC/AUC or a given point (1-specificity, sensitivity) to calculate the sample sizes. The articles of Obuchowski (2000) and Blume (2009) can also serve as references.
Calculating the sample sizes of the non-diseased and the diseased groups aside, from the results of the previously discussed breast tissue example, we find that when the non-diseased population is homogeneous and the non-diseased population is heterogeneous, there tends to be a larger pAUC value. However, when the opposite is true, there tends to be a smaller pAUC value. Hence, we consider developing a method based on maximizing the difference between the variance of the diseased group and the variance of the non-diseased group through a weight method for these two variances. However, in the article of Liu et al. (2005), the linear combination based on minimizing the standard error ratio of the non-diseased to diseased under the normality assumption was used. Our idea and the idea of Liu et al. (2005) are similar.
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Next, in biomedical studies, high-dimensional data, such as microarray data used to discover genomic biomarkers are generated. Hence, one important goal is to select lower-dimensional biomarkers to detect the disease. Ma and Huang (2005) proposed a sigmoid approximation to smooth the empirical AUC function and then applied it to biomarker selection by using the threshold gradient descent regularization method (TGDR)(Friedman and Popescu (2004)). Subsequently, Wang and Chang (2010) also used the smooth function to approximate the empirical pAUC function and proposed an algorithm which selected biomarkers that have higher individual sensitivities than those biomarkers selected by AUC-based methods. Alternatively, Komori and Eguchi (2010) adopted the standard normal distribution to approximate the empirical pAUC function and developed an algorithm based on a boosting technique with natural cubic splines.
These algorithms are all distribution-free and focus on high-dimensional data. In this study, our proposed methods are only applicable to lower-dimensional data. Thus, we will continue to develop the efficient methods with pAUC as the objective function for high-dimensional data sets.
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A.1 Proof of Theorem 1
It follows that
Then, Equation (A.1.1) implies that Z ∞ which can be re-written as
Z ∞