A test (for example, psychological) should be designed in a manner to get valid information, i.e. it must be valid instrument. Is there any relevance of reliability in constructing a test, too? What is the difference between reliability and validity? They both seem to be often based on correlations, so it is easy to mix the concepts.
Solved – the difference between reliability and validity
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A reliable result that has low validity can't really be redeemed. In your example, it would be like recording something silly like people's height to determine how much money they spent instead of just recording how much money they spent.
An unreliable result that has high validity can be redeemed because it's a problem of measurement error. In your example, suppose you made a numerical mistake in converting currency. In less silly examples, you can integrate measurement error in your model.
I think your conceptual understandings of reliability (via Cronbach's $\alpha$) and convergent validity are correct. However, I believe that the way you have defined evidence for convergent validity is mistaken. Your reflective model implies that these six items are manifestations (i.e., caused by) of your latent construct; to then use these same indicators as "...other measures that it [your latent variable that is presumably causing these indicators] is theoretically predicted to correlate with" seems very circular. How can the variables be considered manifestations of your latent variable, and "other measures" at the same time? Instead, I think you should be establishing convergent validity via inter-construct correlations, much like you would discriminant validity.
Two other quick thoughts:
1) I've not often seen Cronbach's $\alpha$ calculated for latent variables. Rather, Cronbach's $\alpha$ is often calculated for scale scores (averages, sums) that are observed. You might be interested in calculating construct (or sometimes called "composite") reliability (Hatcher, 1994), which can be done with the following formula: ($\Sigma$$\lambda$)$^2$/(($\Sigma$$\lambda$)$^2$+$\Sigma$$\sigma$$^2$) where $\lambda$ is a standardized loading, and $\sigma$$^2$ is a uniqueness.
2) Your AVE seems similar, in concept, to the calculations for how much variance (similar to the previous formula) is explained by a given latent variable. This calculation could be taken as some preliminary evidence of construct validity, as if your latent variable is not explaining a substantial amount of variance in it's indicators (e.g., >.5), then perhaps it is a poorly conceived latent variable: ($\Sigma$$\lambda$$^2$)/(($\Sigma$$\lambda$$^2$)+$\Sigma$$\sigma$$^2$)
Best Answer
Reliability has to with being able to replicate results. So if you apply the same measurment technique multiple times in similar situations you should get similar results. An unreliable measurement adds a lot of random noise to your measurement.
Validity has to do with measuring what you want to measure. This is a much more theoretical concept: if this is a survey you just need to think about what the questions are, how a respondent might interpret that question (and the possible answer categories), the theoretical concept you want to measure, and whether these all match up.
Also see here and here