False rejection decision error

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. 

Suppose that the true mean concentration p is llOmg/kg, but the sample mean concentration is 90mg/kg. In this case, the null hypothesis is true (H0 llOmg/kg > lOOmg/kg). However, basing our decision on the sample data, we reject it in favor of the alternative hypothesis (Ha 90mg/kg< lOOmg/kg) and make a false rejection decision error. 

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)... 

Example 2.2 describes lead-contaminated soil with the baseline condition stating that the true mean concentration of lead in soil exceeds the action level. If a false rejection decision error has been made, contaminated soil with concentrations of lead exceeding the action level will be used as backfill, and will therefore continue to pose a risk to human health and the environment. On the opposite, as a consequence of a false acceptance decision error, soil with lead concentrations below the action level will not be used as backfill and will require unnecessary disposal at an additional cost. 

In this example, the two decision errors have quite different consequences a false rejection decision error has consequences that will directly affect the environment, whereas a false acceptance decision error will cause unnecessary spending. Recognizing a decision error with more severe consequences is a pivotal point in the DQO process and a dominating factor in optimizing the data collection design. 

The consequence of a false rejection decision error will be a more severe one because of the unmitigated threat to the environment. The consequence of a false acceptance decision error will be a more severe one because of the violation of the NPDES permit. 

The probability curve is the same as in Figure 2.3, but the gray region in this case is situated on the opposite side of the action level, and the false acceptance and false rejection decision errors have changed places on the probability curve. With low probabilities of decision error, soil with sample mean concentrations significantly below 100 mg/kg will be accepted as backfill, soil with sample mean concentrations clearly exceeding 100 mg/kg will be rejected. However, if the true mean concentration is slightly above the action level, for example, 110 mg/kg, and the sample mean concentration is less than 100 mg/kg (e.g. 90 mg/kg), then the project team will be likely to make a false acceptance decision error. A consequence of false acceptance... 

The two diagrams illustrate how the formulation of the baseline condition effects the outcome of the decision process, as quite different decisions may be made for the same data set depending on how the baseline condition has been stated. That is why the baseline condition should be re-evaluated and, if necessary, restated after the consequences of false acceptance and false rejection decision errors have been evaluated. If the most severe decision error occurs above the action level, the baseline condition should assume that the mean concentration exceeds the action level (H0 fi>CJ. 

The planning team is now prepared to assign decision error limits to false acceptance and false rejection decision errors. A decision error limit is the probability that an error may occur when making a decision based on sample data. The probability curve tells us that the highest probability of error exists in the gray region this error goes down as the mean concentrations move away from either side of the action level. The probability curve reflects our level of tolerance to uncertainty associated with a decision or, conversely, level of confidence with which a decision will be made. 

The EPA recommends the most stringent decision error limits (0.01 for false acceptance and false rejection decision errors or x=/J=0.01) as a starting point in selecting the appropriate decision error limits (EPA, 2000a). The more practical limits are the less stringent ones x=0.05 for false rejection decision errors and j> 0.20 for false acceptance decision errors (EPA, 1996b). 

The false acceptance and the false rejection decision error rates are assigned as follows a = 0.10 / = 0.10. Because of the non-probabilistic sampling approach, this step is not applicable. 

If an analytical test results in a lower value x, < x0, then the customer may reject the product as to be defective. Due to the variation in the results of analyses and their evaluation by means of statistical tests, however, a product of good quality may be rejected or a defective product may be approved according to the facts shown in Table 4.2 (see Sect. 4.3.1). Therefore, manufacturer and customer have to agree upon statistical limits (critical values) which minimize false-negative decisions (errors of the first kind which characterize the manufacturer risk) and false-positive decisions (errors of the second kind which represent the customer risk) as well as test expenditure. In principle, analytical precision and statistical security can be increased almost to an unlimited extent but this would be reflected by high costs for both manufacturers and customers. 

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

As suggested above it is customary to work at the 95 percent or sometimes at the 99 percent probability level. The 95 percent probability level, which gives a 5 percent chance of a Type I error, represents the usual optimum for minimizing the two types of statistical error. A Type I error is a false rejection by a statistical test of the null hypothesis when it is true. Conversely, a Type II error is a false acceptance of the null hypothesis by a statistical test. The probability level at which statistical decisions are made will obviously depend on which type of error is more important. 

One can generally say that a and (3 are risks of accepting false hypotheses. Ideally we would prefer a test that minimized both types of errors. Unfortunately, as a decreases, (3 tends to increase, and vice versa. Apart from the terms mentioned we should introduce the new term power of a test. The power of a test is defined as the probability of rejecting H0 when it is false. Symbolically it is power of a test = 1- 3 or probability of making a correct decision. 

For a given sample size n, alpha and beta errors are inversely related in that, as one reduces the a error rate, one increases the j3 error rate, and vice versa. If one wishes to reduce the possibility of both types of errors, one must increase n. In many medical and pharmaceutical experiments, the alpha level is set by convention at 0.05 and beta at 0.20 (Sokal and Rohlf, 1994 Riffenburg, 2006). The power of a statistic (1 — /3) is its ability to reject both false alternative and null hypotheses that is, to make correct decisions. 

Our decision to accept or reject a null hypothesis may be a correct one that is, we may accept a true null hypothesis or reject a false null hypothesis or it may be a wrong one. Rejecting a true null hypothesis is called committing a Type I error. Accepting as true a false hypothesis is called committing a Type II error. These possibilities can be summarized in a truth table. 

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