Statistics error types

There are two types of measurement errors, systematic and random. The former are due to an inherent bias in the measurement procedure, resulting in a consistent deviation of the experimental measurement from its true value. An experimenter s skill and experience provide the only means of consistently detecting and avoiding systematic errors. By contrast, random or statistical errors are assumed to result from a large number of small disturbances. Such errors tend to have simple distributions subject to statistical characterization. 

Sample sizes for clinical trials are discussed more fully elsewhere in this book and should be established in discussion with a statistician. Sample sizes should, however, be sufficient to be 90% certain of detecting a statistically significant difference between treatments, based on a set of predetermined primary variables. This means that trials utilising an active control will generally be considerably larger than placebo-controlled studies, in order to exclude a Type II statistical error (i.e. the failure to demonstrate a difference where one exists). Thus, in areas where a substantial safety database is required, for example, hypertension, it may be appropriate to have in the programme a preponderance of studies using a positive control. 

As pointed out by Muller (1978), there are well-documented differences between the ages of materials determined by dating with radioisotopes and the ages determined by other means, such as tree-ring counting. In addition to systematic effects, there are statistical errors due to the limited number of atoms observed. Both types of errors can be considered to be fluctuations in n, the number of atoms observed. A relationship can be derived between the magnitude of these fluctuations and the resulting error in the estimation of age of the sample ... 

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. 

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. 

Robust statistical method Significance tests Standard uncertainty True value Type 1 error Type II error Uncertainty j -Residuals... 

In most experimental studies, there are two sources of variation or error. The first arises from the intrinsic variability in individual measurements or observations that are all obtained in the same way. The second arises from errors in the apparatus or procedures which affect all of the measurements in a similar way. This type of variation, which leads to a consistent deviation from the correct or true measurement is termed systematic error. It is important that these two sources of error are clearly distinguished. The first (intrinsic variability) can be analysed statistically, and replicated measurements will give an estimate of the precision of the measurement or procedure. These errors are termed statistical errors. This does not mean, however, that the result obtained is necessarily an accurate or true one. Pipettes may have become uncalibrated, or solutions have deteriorated there are many ways in which systematic errors can lead to false values, and attention to calibration and independent verification of apparatus or procedures is essential to minimise the risk of this happening. Systematic errors are potentially much more problematic in biochemical work, since these can easily occur without the experimenter being aware of them. 

Figure 9. Computation of the free energy difference between the type a interaction (conformation al) and the type b interaction (conformation bl), with the MCTI protocol. Seven computations of increasing length were performed (300 ps, 600 ps, 900 ps, 1.5 ns, 2.25 ns, 3 ns, 4 ns). The first and the last computation (bold lines) display a 12kJmor deviation from each other whereas the statistical error in both calculations is estimated to be < 3 kJ mol". ...

A single topology description should be used for systems in which mutations of atom types are made that are not likely to lead to different conformations. The one-to-one correspondence in a single topology description usually leads to smaller statistical error because at any time during tbe simulation a single set of coordinates is present. 

These studies clearly have to be of sufficient size to avoid type II (statistical) errors and confirm that the two products are truly clinically equivalent. 

A brief review of the basic relationships between error types and statistical power starts with considering each of five interacting factors [3-5] that serve to determine power and define competing error rates. 

Note that accepting or rejecting a hypothesis does not necessarily imply truth, which leads to a discussion around two types of errors in statistics a Type I error accepts the null hypothesis when in fact it is false a Type II error is when the null hypothesis is rejected when in fact it is true. 

This test of hypothesis is subject to statistical errors, namely, t3 pe I and type II errors. 

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