Average recorded variance

Congeneric measurement model

The average recorded variance ( DEV ; English : average variance extracted , short: AVE ) is in multivariate statistics a measure of the quality of how a single latent variable (construct ) explains its indicators ( ). The variance of each indicator is broken down into the variance ( i.e. the squared regression coefficient ), explained by the latent variable, and an error variance ( ), not explained by the construct. ${\displaystyle \eta }$ ${\displaystyle y_{i}}$ ${\displaystyle i=1,\dotsc ,k}$ ${\displaystyle \lambda _{i}^{2}}$ ${\displaystyle \operatorname {Var} (e_{i})}$

calculation

The DEV can be calculated as follows:

${\displaystyle DEV={\frac {\sum _{i=1}^{k}\lambda _{i}^{2}}{\sum _{i=1}^{k}\lambda _{i}^{2}+\sum _{i=1}^{k}\operatorname {Var} (e_{i})}}}$.

interpretation

The DEV can have values ​​between zero and one. It is considered acceptable if , i. H. On average , more than 50% of the variance of each indicator should be explained by the latent variable or at least half of the total variance of all indicators is explained by this variable (and thus more than by the error variances). ${\displaystyle DEV>0{,}5}$

Significance for the measurement of discriminant validity

If the DEV of a construct is higher than any squared correlation with another construct, discriminant validity can be assumed at construct level. This quality measure is known as the Fornell-Larcker criterion, named after the authors who proposed DEV. (It should be noted that the error-corrected correlations between constructs from the CFA model are used instead of the correlations taken from the raw data.) However, this measure of quality has been used in simulation models for variance-based structural equation models (e.g. partial least-squares estimation ) proved to be not very reliable, but in the case of covariance-based structural equation models (e.g. Amos) at the construct level it is very reliable. A more recent method for determining discriminant validity was published by Henseler et al. (2014) and is known as the heterotrait-monotrait ratio (HTMT). It provides reliable results for both variance-based and covariance-based structural equation models. Voorhees et al. (2015) propose a combination of both methods for the latter, with a value of 0.85 being suggested as the cutoff for HTMT.

Related coefficients

In addition to the tau-equivalent reliability (also called Cronbach's alpha , English : tau-equivalent reliability ) and the congeneric reliability ( English : congeneric reliability , outdated also composite reliability ), the average recorded variance is one of the most important variables for testing the reliability of a measuring scale. ${\displaystyle \rho _{T}}$ ${\displaystyle \rho _{C}}$

Mathematical derivation

Based on the measurement model

${\displaystyle y_{i}=\lambda _{i}\eta +e_{i}}$

follows

${\displaystyle \operatorname {Var} (y_{i})=\lambda _{i}^{2}+\operatorname {Var} (e_{i})}$

on the condition of and independence from and . This is e.g. B. given when calculating the coefficients with the help of factor analysis . ${\displaystyle \operatorname {Var} (\eta )=1}$ ${\displaystyle \eta }$ ${\displaystyle e_{i}}$ ${\displaystyle \lambda _{i}}$

The average recorded variance is then the total variance of the indicators explained by the construct:

${\displaystyle DEV={\frac {\sum _{i=1}^{k}\lambda _{i}^{2}}{\sum _{i=1}^{k}\operatorname {Var} (y_{i})}}={\frac {\sum _{i=1}^{k}\lambda _{i}^{2}}{\sum _{i=1}^{k}\left(\lambda _{i}^{2}+\operatorname {Var} (e_{i})\right)}}={\frac {\sum _{i=1}^{k}\lambda _{i}^{2}}{\sum _{i=1}^{k}\lambda _{i}^{2}+\sum _{i=1}^{k}\operatorname {Var} (e_{i})}}.}$

If the coefficients are carried out with the help of factor analysis on the basis of the correlation matrix, the measurement model is actually: ${\displaystyle \lambda _{i}}$

${\displaystyle z_{i}=\lambda _{i}\eta +e_{i}}$

with the standardized variables and the arithmetic mean and the standard deviation of the variables . The following applies to the standardized variable: ${\displaystyle z_{i}=(y_{i}-{\bar {y}}_{i})/s_{y_{i}}}$ ${\displaystyle {\bar {y}}_{i}}$ ${\displaystyle s_{y_{i}}}$ ${\displaystyle y_{i}}$

${\displaystyle 1=\operatorname {Var} (z_{i})=\lambda _{i}^{2}+\operatorname {Var} (e_{i})\Longrightarrow \operatorname {Var} (e_{i})=1-\lambda _{i}^{2}}$.

Individual evidence

1. ^ Claes Fornell, David F. Larcker: Evaluating Structural Equation Models with Unobservable Variables and Measurement Error . In: Journal of Marketing Research . tape 18 , 1981, p. 39-50 , JSTOR : 3151312 .
2. ↑ ^{a b c} Henseler, J., Ringle, CM, Sarstedt, M., 2014. A new criterion for assessing discriminant validity in variance-based structural equation modeling. Journal of the Academy of Marketing Science 43 (1), 115-135, doi : 10.1007 / s11747-014-0403-8 .
3. ↑ ^{a b c} Voorhees, CM, Brady, MK, Calantone, R., Ramirez, E., 2015. Discriminant validity testing in marketing: an analysis, causes for concern, and proposed remedies. Journal of the Academy of Marketing Science 1-16, doi : 10.1007 / s11747-015-0455-4 .