Mattiullah, AVE is calculated as follows:

(sum of the squared factor loadings) / (sum of the squared factor loadings + sum of error variances)

AVE ranges from 0 to one. For comparing AVE with a correlation, you need to square the correlation.

HTH, Karin

According to popular Fornell Larcker criterion (Fonrnell & Larcher, 1981), you should compare the AVE value with corresponding correlation values with other variables for establishing discriminant validity. However, for variance-based SEM (e.g., PLS), you are advised to use square-root value of AVE to compare with corresponding correlation values with other variables (Hair et al., 2014) because of its sensitivity to overestimate indicator/item loading (Hui and Wold 1982; Lohmöller 1989).

Hope this clarifies your confusion.

Mattiullah,

you may either compare explained variances or correlations. As AVE is a ratio of variances, I would prefer to compare AVE with the squared correlation. But this is not so important. Have a look at Jörg Henseler's publication:

link.springer.com/content/pdf/10.1007%2Fs11747-014-0403-8.pdf

There, Henseler explains both types of comparison, and he also discusses other related topics. Just have a look!

Regards, Karin

To determine discriminant validity in PLS, some researchers use either way:

Dear Mr. Butt,

Discriminant validity refers to the extent to which factors are distinct and uncorrelated. The rule is that variables should relate more strongly to their own factor than to another factor. Two primary methods exist for determining discriminant validity during an EFA. 

When doing a CFA. If your factors do not demonstrate adequate validity and reliability, moving on to test a causal model will be useless - garbage in, garbage out! There are a few measures that are useful for establishing validity and reliability: Composite Reliability (CR), Average Variance Extracted (AVE), Maximum Shared Variance (MSV), and Average Shared Variance (ASV). The video tutorial will show you how to calculate these values. The thresholds for these values are as follows:

Reliability

·         CR > 0.7

Convergent Validity

·         AVE > 0.5

Discriminant Validity

·         MSV < AVE

·         ASV < AVE

·         Square root of AVE greater than inter-construct correlations

If you have convergent validity issues, then your variables do not correlate well with each other within their parent factor; i.e, the latent factor is not well explained by its observed variables. If you have discriminant validity issues, then your variables correlate more highly with variables outside their parent factor than with the variables within their parent factor; i.e., the latent factor is better explained by some other variables (from a different factor), than by its own observed variables.

Usama, it is always best to read the original literature, which is:

Fornell, C. & Larcker, D. F. (1981). Evaluating Structural Equation Models with Unobservable Variables and Measurement Error. Journal of Marketing Research, 18, 39–50.

On p. 45/46 the authors state:

“The average variance extracted … can be calculated as

(sum of the squared factor loadings) / (sum of the squared factor loadings + sum of error variances)”

By the way, Claes Fornell is on ResearchGate, and you may ask for this article if you do not have access to it.

Regards, Karin

Usama, the problem seems to be that researchers often do not understand what the coefficients mean. The result is that different rules-of-thumb are used, and that correlation and squared correlation are mixed.

I was just thinking again about Manuel Heinrich’s formula:

AVE = (7²+ .8² + .9² )/3 = .647

This formula is identical to the formula I used. And this is why: If you use standardized variables, they all have a variance of 1.0. If you look at the denominator of the formula

(sum of the squared factor loadings) / (sum of the squared factor loadings + sum of error variances),

then each squared loading plus the error variance need to be 1.0 for each item, if the model fits the data.

The denominator of this example is:

(.7²+.51 )+ (.8² + .36) + (.9²+ .19)

Obviously, the sum is 3.0, and therefore the formulas are identical.

Regards, Karin

Usama, just by squaring the correlation. If the correlation is .80, then the squared correlation is .64. The squared correlations seem to be necessary in order to compare them with the respective AVEs, they should be smaller than the AVEs. I don't agree with this procedure, but it is often used for determining the discriminant validity.

Regards, Karin

Janab Matiullah Attari sb,

Please see the example of Convergent and discriminant validity. It may help you.

 Conduct Factor analysis of  the predictors of the two latent variables among which the convergent validity is to be calculated.

Calculate average pattern loading of one component extracted. And then for  second component extracted.

Make square of  average pattern loading of one component if the value of the square of average pattern loading of one component is

 Example:

average pattern loading of one component = 0.821025

  for component 1, Square of average pattern loading of one component =  (0.821025)2

                 =0 .758685      is the variance extracted

Here,  0.821025 > 0.758685 , it suggest the convergent validity .

For component 2, average pattern loading of second component = .835655

Square of average pattern loading of second component = .698319 is the variance for second component.

Variance Extracted between component 1 and component 2 = ( Average of variance of component 1 &2 )

Discriminant Validity:

 if  Variance extracted between component 1 and component 2  is greater than the correlation square

Variance extracted  >  (correlation)2

Here, Variance extracted  = 0.728502

Correlation ( as given in component correlation Matrix ) = 0.227

(correlation)2  = 0.051529

Variance extracted  >  (correlation)2

0.728502  >  0.051529  ,  hence discriminant validity established.

Dear All,

I'm searching a citation about "Discriminant Validity established when the MSV < AVE value " condition. Could you suggest a book or an article with page number. 

Thanks and Best Regards,

Binnur

Thanks for your concern Dear @Chalamalla Srinivas 

Is it posible to caluclate AVE with only 2 unobserved variables????

what i really want to do is to know values for fornell larckner criterio

Dear all,

The discussions in this thread were very helpful for me, many thanks!

The vast majority of researchers recommends using factor loadings obtained from a confirmatory factor analysis (CFA) to calculate AVE. However, many tutorials available show how to calculate AVE by using factor loadings obtained from an exploratory factor analysis (EFA), for example, https://www.youtube.com/watch?v=8tyjdfpiAJQ.

Would you recommend using EFA for calculating AVE? Does anyone know some literature to back up the EFA-AVE-procedure?

Best, Benedikt

Very insightful...thank you for providing the details..

Dear

Please visit the website of James Gaskin. He provides very useful video lectures and notes. He can be very helpful. Further he provides stats tool package for free that proves to be very handy in SEM. His link is given below

http://statwiki.kolobkreations.com/

Dear Mattiullah Butt

Please elaborate why would you need to do AVE for Factor Analysis you conducted in SPSS. where AVE, CR are for mostly used in SEM like AMOS or SmartPLS. These values are automatically calculated by the software. It indices the Discriminant and Convergent validity. However, if you wish to learn about the factors you produced through CFA or EFA in SPSS there are certain assumptions that need to be fulfilled like i) Sample size ( Pallant (2005) recommends 10 cases or Tabachnick & Fidell (1989) a sample of 300 cases . ii) Factorability of the correlation matrix: KMO>0.6 iii) Linearity & Outliers :Date should be linear and should not contain any outliers ..Further in results see The Bartlett’s Test of Sphericity it should be within limits, secure a clear pattern of factors and Scree plot should indicate a clear break...

However if you are interested to learn about SEM (AMOS or SmartPLS) please visit the website of James Gaskin. He provides very useful video lectures and notes. He can be very helpful. Further he provides stats tool package for free that proves to be very handy in SEM. His link is given below

http://statwiki.kolobkreations.com/

Please consult hair et al. (2012). It will clear all your doubts.

Fuller, C. M., Simmering, M. J., Atinc, G., Atinc, Y., & Babin, B. J. (2016). Common methods variance detection in business research. Journal of Business Research, 69(8), 3192–3198. https://doi.org/10.1016/j.jbusres.2015.12.008

Construct validity has two sub-categories called convergent validity and discriminant validity. The two measures work together in the sense that if we have evidence for both convergent and discriminant validity, then by definition we have evidence for construct validity. However, one alone is never capable of establishing construct validity.

Convergent validity seeks to show certain measures theoretically supposed to be related to form the same construct are in fact related to each other; that is, it seeks to show a convergence among similar measures. On the other hand, discriminant validity seeks to show two measures that are not theoretically supposed to be related are in fact unrelated; that is, it seeks to discriminate between dissimilar measures.

Researchers of yesteryears used Fornell-Larcker criterion suggested in 1981 for assessing discriminant validity. According to this criterion, if the square root of the AVE of each latent variable is greater than the correlation coefficients between that latent variable and other latent variables in the measurement model, then the model satisfies the discriminant validity criterion. Note that the square roots of the AVEs are reported along the principal diagonal of the correlation matrix of the latent variables.

However Jörg Henseler disapproved of this criterion in the context of variance-based SEM as it was found that it still remained inefficient to establish distinctiveness between constructs. So, Henseler et al. (2015) proposed a new method, Heterotrait-monotrait (HTMT) ratio of correlation, providing evidence for its superior performance by means of Monte Carlo simulation study that showed that HTMT is able to achieve higher specificity and sensitivity rates (97% - 99%) compared with the Fornell-Lacker (20.82%). In general HTMT values close to 1 imply a lack of discriminant validity. In practice, we use a predefined threshold, and if the value of the HTMT is higher than this threshold, we conclude that there is a lack of discriminant validity. Some authors (for example, Kline 2011) suggest a threshold of 0.85; and some others, a value of 0.90.

Henseler J, Ringle CM, Sarstedt M 2015 “A New Criterion for Assessing Discriminant Validity in Variance-based Structural Equation Modeling”. Journal of the Academy of Marketing Science. 43: 115-35

Kline RB 2011 Principles and Practice of Structural Equation Modeling Third Edition (New York: The Guilford Press).

Your explanations were very helpful.

Discriminant validity refers to the extent to which factors are distinct and uncorrelated. The rule is that variables should relate more strongly to their own factor than to another factor. Two primary methods exist for determining discriminant validity during an EFA. 

When doing a CFA. If your factors do not demonstrate adequate validity and reliability, moving on to test a causal model will be useless - garbage in, garbage out! There are a few measures that are useful for establishing validity and reliability: Composite Reliability (CR), Average Variance Extracted (AVE), Maximum Shared Variance (MSV), and Average Shared Variance (ASV). The video tutorial will show you how to calculate these values. The thresholds for these values are as follows:

Reliability

·         CR > 0.7

Convergent Validity

·         AVE > 0.5

Discriminant Validity

·         MSV < AVE

·         ASV < AVE

·         Square root of AVE greater than inter-construct correlations

If you have convergent validity issues, then your variables do not correlate well with each other within their parent factor; i.e, the latent factor is not well explained by its observed variables. If you have discriminant validity issues, then your variables correlate more highly with variables outside their parent factor than with the variables within their parent factor; i.e., the latent factor is better explained by some other variables (from a different factor), than by its own observed variables.

AVE supports calculating discriminant validity for each construct matched with the parallel Squared Inter-construct Correlation (SIC), and the AVE estimate is to be larger than SIC estimate.

Your AVE should be greater than 0.5

I agree with Syed Shah. AVE, and CR values are automatically calculated by the SEM softwaress (AMOS or SmartPLS, etc...).

- You should square root the AVE then compare the results with the correlations

- The correlation that you should take in SPSS is the Pearson correlation value as suggested by the video in this link:

SPSS: Fornell-Larcker condition - YouTube

Discriminant validity of the construct is achieved when the square root of the AVE is greater than the correlation between the constructs or MSV.

In the case of a uni-dimensional scale, no Correlation Matrix or MSV can be calculated. Hence, discriminant validity can't be established in the case of a uni-dimensional scale.