Correction for attenuation - AbsoluteAstronomy.com

Correction for attenuation is a statistical procedure, due to Spearman

Charles Spearman

Charles Edward Spearman, FRS was an English psychologist known for work in statistics, as a pioneer of factor analysis, and for Spearman's rank correlation coefficient...

(1904), to "rid a correlation

Correlation

In statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence....

coefficient from the weakening effect of measurement error" (Jensen, 1998), a phenomenon also known as regression dilution

Regression dilution

Regression dilution is a statistical phenomenon also known as "attenuation".Consider fitting a straight line for the relationship of an outcome variable y to a predictor variable x, and estimating the gradient of the line...

. In measurement

Measurement

Measurement is the process or the result of determining the ratio of a physical quantity, such as a length, time, temperature etc., to a unit of measurement, such as the metre, second or degree Celsius...

and statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, it is also called disattenuation. The correlation

Correlation

between two sets of parameters or measurements is the estimated a manner that accounts for measurement error contained within the estimates

Estimator

In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule and its result are distinguished....

of those parameters.

Background

Correlations between parameters are diluted or weakened by measurement error. Disattenuation provides for a more accurate estimate of the correlation between the parameters by accounting for this effect.

Derivation of the formula

Let

and

be the true values of two attributes of some person or statistical unit

Statistical unit

A unit in a statistical analysis refers to one member of a set of entities being studied. It is the material source for the mathematical abstraction of a "random variable"...

. These values are regarded as random variables by virtue of the statistical unit being selected randomly from some population

Statistical population

A statistical population is a set of entities concerning which statistical inferences are to be drawn, often based on a random sample taken from the population. For example, if we were interested in generalizations about crows, then we would describe the set of crows that is of interest...

. Let

and

be estimates of

and

derived either directly by observation-with-error or from application of a measurement model, such as the Rasch model

Rasch model

Rasch models are used for analysing data from assessments to measure variables such as abilities, attitudes, and personality traits. For example, they may be used to estimate a student's reading ability from answers to questions on a reading assessment, or the extremity of a person's attitude to...

. Also, let

where

and

are the measurement errors associated with the estimates

and

.

The correlation between two sets of estimates is

which, assuming the errors are uncorrelated with each other and with the estimates, gives

where

is the separation index of the set of estimates of

, which is analogous to Cronbach's alpha

Cronbach's alpha

Cronbach's \alpha is a coefficient of reliability. It is commonly used as a measure of the internal consistency or reliability of a psychometric test score for a sample of examinees. It was first named alpha by Lee Cronbach in 1951, as he had intended to continue with further coefficients...

; this is, in terms of Classical test theory

Classical test theory

Classical test theory is a body of related psychometric theory that predict outcomes of psychological testing such as the difficulty of items or the ability of test-takers. Generally speaking, the aim of classical test theory is to understand and improve the reliability of psychological...

is analogous to a reliability coefficient. Specifically, the separation index is given as follows:

where the mean squared standard error of person estimate gives an estimate of the variance of the errors,

. The standard errors are normally produced as a by-product of the estimation process (see Rasch model estimation

Rasch model estimation

Estimation of a Rasch model is used to estimate the parameters of the Rasch model. Various techniques are employed to estimate the parameters from matrices of response data. The most common approaches are types of maximum likelihood estimation, such as joint and conditional maximum likelihood...

).

The disattenuated estimate of the correlation between two sets of parameters or measures is therefore

That is, the disattenuated correlation is obtained by dividing the correlation between the estimates by the square root of the product of the separation indices of the two sets of estimates. Expressed in terms of Classical test theory, the correlation is divided by the square root of the product of the reliability coefficients of two tests.

Given two random variable

Random variable

In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

and

, with correlation

Correlation

, and a known reliability for each variable,

and

, the correlation between

and

corrected for attenuation is

.

How well the variables are measured affects the correlation of X and Y. The correction for attenuation tells you what the correlation would be if you could measure X and Y with perfect reliability.

If

and

are taken to be imperfect measurements of underlying variables

and

with independent errors, then

measures the true correlation between

and

External links

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Background

Derivation of the formula

See also

External links