True positive decision

A new tumor marker is evaluated using the same criteria used for many diagnostic tests (i.e., sensitivity, specificity, and accuracy). The diagnostic sensitivity and specificity are best represented by a receiver operating characteristic (ROC) curve. The ROC curve is constructed with the true-positive rate versus false-positive rate at various decision levels. As a test improves in its diagnostic performance, it shifts upward and to the left as the true-positive rate increases and the false-positive rate decreases. 

Type I error (alpha error) An incorrect decision resulting from rejecting the null hypothesis when the null hypothesis is true. A false positive decision. 

Alternative condition is true (Ha i < Ca) False rejection decision error False positive decision error Type I decision error Probability a Risk, error rate 100 x u. Correct decision The probability of making a correct decision (1 —ft)... 

It is often stated that R D is there to provide the business with options for the future development The final decision to launch a new product or to implement a new process involves considerations beyond the R D dimension. Whilst this is no doubt the true position, it is only when some of those options lead to meaningful business for the company on a regular basis, that R D is going to be held in high regard. In other words, there must be link between the expenditure on R D and the business income. 

The recognition accuracy estimation described above faces one very important problem what is the best choice for the threshold value 0 To solve this problem, statistical decision theory is used. ° The basis for this is an analysis of the so-called the Received Operating Characteristic (ROC) curve. By tradition, ROC is plotted as a function of true positive rate TPj TP + FN) (or sensitivity) versus false positive rate FPj TN+FP) (or 1-Specificity) for all possible threshold values 0. Figure 6.5 presents an example of such a ROC curve for the results obtained with our computer program PASS in predicting antineoplastic activity. 

Figure 15-2 Receiver operating characteristic curve of prostate-specific antigen (PSA). Each point on the curves represents a different decision level.The sensitivity (true-positive rate) and I— the specificity (false-positive rate) can be read for Tests A and B. The true-positive and false-positive rates are demonstrated using 4 and IO Xg/L as decision thresholds.

For the case at hand. Figure 15-2 shows the ROC curve for PSA using Chan s data. The x-axis plots the fraction of nondiseased patients who were erroneously categorized as positive for a specific decision threshold. This false-positive rate is mathematically the same as 1 - specificity. The y-axls plots the true-positive rate (the sensitivity). A hidden third axis is contained in the curve itself the curve is drawn through points that represent different decision cutoff levels. The whole curve is a graphical display of the performance of the test. 

Figure IS-4 Simulated distributions of healthy and diseased populations. Note that the ratio of diseased patients to healthy patients, A to B, is less than I and very different at the point of decision (the likelihood ratio) from the ratio of TP to FP, which is much greater than i. TP, True positives TN, true negatives FP, false positives FN, false negatives.

Fig. 10.1. Flow chart shows decision tree for double reading in mammography screening. True positive interpretations may include all positive scores for cancers at baseline reading (1), cancers diagnosed at assessment only (2), or also including cancers with true positive scores dismissed at consensus (arbitration) meeting and presenting as interval or subsequent round cancers (3)...

For a true-positive system action, both variants are different only in their efficacy. However, the consequences of a false-positive action are considered different regarding their severity A warning may eventually lead to a braking by the driver, but the intensity and duration of the braking is up to the driver. The driver s evaluation and decision loop additionally helps to avoid braking maneuvers as consequence of falsepositive warnings. A false-positive automatic braking lacks this second evaluation of the situation. 

Based on the sample data, we may reject the null hypothesis when in fact it is true, and consequently accept the alternative hypothesis. By failing to recognize a true state and rejecting it in favor of a false state, we will make a decision error called a false rejection decision error. It is also called a false positive error, or in statistical terms, Type I decision error. The measure of the size of this error or the probability is named alpha (a). The probability of making a correct decision (accepting the null hypothesis when it is true) is then equal to 1—a. For environmental projects, a is usually selected in the range of 0.05-0.20. 

The EPA provides further guidance for decision-making in this area when the RPD between two results exceeds 40 percent, the EPA conservatively recommends selecting the higher concentration as a true one (EPA, 1996a). This practice, however, often leads to false positive results and may be the cause of unnecessary site remediation. From a practical perspective, in the absence of matrix interferences for an analyte to be present in the sample the agreement between the two results should be better than 40 percent. If matrix interferences are obvious, a chemist experienced in data interpretation should evaluate the chromatograms and make a decision on the presence or absence of the analyte in the sample. 

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