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I can't seem to find this anywhere on the internet or in my books. I'm putting together a construct for an underlying variable.

Is it acceptable to have a construct with two variables if that produces a bigger Cronbach alpha than the construct with 3 variables in a professional statistical analysis?

Using 2 variables seems a bit on the low side, but it produces better results.

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    $\begingroup$ If alpha is low when you have 3 items, that's an indication that your items don't go together that well. If it is much higher when you drop one item, that's an indication of which item was problematic. Some latent variables are hard to measure; but maybe you can come up with a third variable that works well. If you give us context, maybe someone here can help. $\endgroup$ – Peter Flom May 6 '12 at 11:16
  • $\begingroup$ The contruct originally consisted of these three variables. Apparently it doesn't measure what I want all that well. Can I still use it as a construct if it just has two variables? Is it common for a construct to only consist out of 2 variables? $\endgroup$ – J. Maes May 6 '12 at 12:23
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    $\begingroup$ Are you saying that the research that supported a 3 item scale did not hold up when you replicated it? If so, why not. What Ns were involved in the two research studies? In any case, I would not discard a two item scale out of hand. Do you know if this two item scale correlates with any other variables of interest (i.e., construct validity or criterion-related validity)? $\endgroup$ – Joel W. May 6 '12 at 15:16
  • $\begingroup$ @JoelW. asks good questions. $\endgroup$ – Peter Flom May 6 '12 at 16:51
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The unstandardized Cronbach's alpha on two items does not make much sense, see e.g. The reliability of a two-item scale: Pearson, Cronbach, or Spearman-Brown? (International Journal of Public Health 58(4):637-642). You can also think in terms of of item correlation with the total score when correcting for overlap, see, e.g., W. Revelle's textbook on psychometrics (chapter 7). As for a good reference, I would recommend Test theory: A unified approach, by R.P. McDonald (1999), who suggested 3 items per dimension for uncorrelated factor models, IIRC, which is also handy when it comes to estimate the parameters of a CFA, and Psychometric Theory, by Nunally and Bernstein (1994).

Beware also that using a statistic like "alpha if item is deleted" without cross-validation is proned to over-fitting (the alpha is already a lower bound for reliability, but sample reliability may appear higher than population-level reliability when further capitalizing on chance), notwithstanding the fact that constuct coverage (or validity) may be quite poor when using only two items.

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