# Statistical reliability

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In statistics, reliability is the consistency of a set of measurements or measuring instrument. Reliability does not imply validity. That is, a reliable measure is measuring something consistently, but not necessarily what it is supposed to be measuring. For example, while there are many reliable tests, not all of them would validly predict job performance.

In experimental sciences, reliability is the extent to which the measurements of a test remain consistent over repeated tests of the same subject under identical conditions. An experiment is reliable if it yields consistent results of the same measure. It is unreliable if repeated measurements give different results.

## Estimation

Reliability may be estimated through a variety of methods that fall into two types: Single-administration and multiple-administration. Multiple-administration methods require that two assessments are administered. In the test-retest method, reliability is estimated as the Pearson product-moment correlation coefficient between two administrations of the same measure. In the alternate forms method, reliability is estimated by the Pearson product-moment correlation coefficient of two different forms of a measure, usually administered together. Single-administration methods include split-half and internal consistency. The split-half method treats the two halves of a measure as alternate forms. This "halves reliability" estimate is then stepped up to the full test length using the Spearman-Brown prediction formula. The most common internal consistency measure is Cronbach's alpha, which is usually interpreted as the mean of all possible split-half coefficients.

Each of these estimation methods is sensitive to different sources of error and so might not be expected to be equal. Also, reliability is a property of the scores of a measure rather than the measure itself and are thus said to be sample dependent. Reliability estimates from one sample might differ from those of a second sample (beyond what might be expected due to sampling variations) if the second sample is drawn from a different population because the true reliability is different in this second population. (This is true of measures of all types--yardsticks might measure houses well yet have poor reliability when used to measure the lengths of insects.)

Reliability may be improved by clarity of expression (for written assessments), lengthening the measure, and other informal means. However, formal psychometric analysis, called the item analysis, is considered the most effective way to increase reliability. This analysis consists of computation of item difficulties and item discrimination indices, the later index involving computation of correlations between the items and sum of the item scores of the entire test.

• R(t) = 1 - F(t).
• R(t) = e^-ʎ.(t). (where ʎ is the failure rate)

## Classical test theory

In classical test theory, reliability is defined mathematically as the ratio of the variation of the true score and the variation of the observed score. Or, equivalently, one minus the ratio of the variation of the error score and the variation of the observed score:

${\rho}_{xx'}=\frac{{\sigma}^2_T}{{\sigma}^2_X}=1-\frac{{{\sigma}^2_E}}{{{\sigma}^2_X}}$

where ${\rho}_{xx'}$ is the symbol for the reliability of the observed score, X; ${\sigma}^2_X$, ${\sigma}^2_T$, and ${\sigma}^2_E$ are the variances on the measured, true and error scores respectively. Unfortunately, there is no way to directly observe or calculate the true score, so a variety of methods are used to estimate the reliability of a test.

Some examples of the methods to estimate reliability include test-retest reliability, internal consistency reliability, and parallel-test reliability. Each method comes at the problem of figuring out the source of error in the test somewhat differently.

## Item response theory

It was well-known to classical test theorists that measurement precision is not uniform across the scale of measurement. Tests tend to distinguish better for test-takers with moderate trait levels and worse among high- and low-scoring test-takers. Item response theory extends the concept of reliability from a single index to a function called the information function. The IRT information function is the inverse of the conditional observed score standard error at any given test score. Higher levels of IRT information indicate higher precision and thus greater reliability.