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I am carrying out a logistic regression analysis in entrepreneurship research, in which one of the main independent variables, "innovativeness", is a company trait based on two different questions ("product's/service's potential to change the market" and "product/service novelty"). As these two were internally consistent with a high Cronbachs alpha score, it was decided to create one variable with a 9 point Likert scale instead of the 5 point scale of the individual variables. The final value is the two respective values added together and divided by two ((X+Z)/2).

The problem is that there is a range of missing values for one or the other variable (32), meaning that the entrepreneur "didn't know" or just didn't answer the question. This is unfortunate as the complete sample (182) is not very big in the first place.

Is it legitimate to take one of the values, if the other is missing, and use it as a proxy for the underlying construct (innovativeness)? Or would this create a bias greater than the missing values might potentially lead to?

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    $\begingroup$ Have you investigated multiple imputation? $\endgroup$ – mdewey Mar 16 '17 at 14:15
  • $\begingroup$ Thanks for the suggestion, @mdewey, I will look into it! $\endgroup$ – Lindberg Mar 16 '17 at 15:33
  • $\begingroup$ The key metric in any approach to imputation is the bias in the comparison between the unimputed marginal means and std devs vs the imputed marginals. In general, model-based imputations are less biasing than automatic approaches to imputation such as plugging in averages, sorted "hot deck" imputation, whatever. That said, key constructs in assessing the mechanisms behind the missing values trace back to Rubin and Little and their various books on missing data. They propose several such as MAR (missing at random), MCAR (missing completely at random), etc. Any imputation should factor this in. $\endgroup$ – Mike Hunter Mar 16 '17 at 17:11
  • $\begingroup$ There's a decent overview of how to think about imputation in this thread ... stats.stackexchange.com/questions/266296/… $\endgroup$ – Mike Hunter Mar 17 '17 at 18:08
  • $\begingroup$ Thanks @DJohnson! So, now I've done some research and carried out Little's MCAR Test on my model variables (p-value: 0,054). The amount of data missing is also fairly substantial and I have decided to carry out the multiple imputation (MI). Although I have to say, reading about it makes me hesitant as a stats novice, as there’s a lot to consider (e.g. interaction terms involving the “innovativeness” variable, which should be included in the MI) and not calculated after… If all fails, I will report on a complete case analysis noting the possible bias. $\endgroup$ – Lindberg Mar 17 '17 at 20:28
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The answer will be subjective but surely it will be informed by the magnitude of Cronbach's alpha, or, alternatively, of the linear association between the two survey items that make up the innovation score. It's hard to imagine anyone objecting to your approach if alpha > .90. It's easy to imagine an objection in the face of an alpha < .50. In between, it's really a matter of preference. You may be able to persuade your readers or stakeholders that the benefits of increasing the sample size outweigh whatever imprecision of measurement is introduced by substituting a single-item score for a scale score.

In fact, you could conduct a simulation to assess the potential bias resulting from applying your proposed substitution. That would be useful to report in conjunction with your primary results. As to bias from missing data, that is a larger matter. You haven't mentioned what information you may have as to the differences, on other variables, between those who answered one vs. two innovation items. This is certainly worth evaluating to the extent your data and your situation allow.

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