Core and nor core conditions: ambiguity in Parcimonious Solution

[Gabriel Lourenço]

Dear all,

When the QCA-IS yields only one solution (M1) and the QCA-PS yields two outcomes (M1 and M2), how can I determine the core and peripheral conditions? Should I consider only one solution from QCA-PS?

Please find the example attached.

Thank you so much!

Best regards,

Gabriel

[Pedro Carmona]

Hello,

I also find it confusing when there are multiple solutions, such as M1 and M2. I don't understand this concept. Could anybody please explain the meaning of the content within parentheses?

For example, in Adrian's fantastic book – specifically Chapter 8 – there's a certain example, but he doesn't provide any additional insights:

M1: ~DEV*~URB*LIT*~IND + DEV*LIT*IND*STB + (DEV*~URB*LIT*STB) -> SURV
M2: ~DEV*~URB*LIT*~IND + DEV*LIT*IND*STB + (~URB*LIT*~IND*STB)  -> SURV 

Best regards,

Pedro

[Adrian Dușa]

Hello Pedro,

The solution terms are the so-called "prime implicants", that is the end result of the minimization process, which is the first step of the entire process.

The second step is solving the prime implicants chart (PI chart), which has the PIs on the rows and the positive truth table configurations on the columns.

As there are many ways to solve this chart, there are multiple combinations of prime implicants that cover the columns equally well.

These are the models: all possible combinations of PIs that equally cover the chart.

The simplified form of the PI chart you mention looks like this (where OC means observed configuration):

       OC1 OC2 OC3 OC4 OC5

PI1    x       x

PI2                     x       x

PI3                              x      x

PI4             x      x

Here, PI1 and PI3 are essential: PI1 is the only one that covers OC1, and PI3 the only one that covers OC5.

Without them, the PI chart cannot be covered on all columns. In model notation, you would see:

M1: P1 + PI3 + (P2)

M2: P1 + PI3 + (P4)

The PI2 in the brackets are not essential, that is what they mean.

Hope this answer your question,

Adrian

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[tocfo...@gmail.com]

Hi Gabriel,

model ambiguity, here taking the form of two parsimonious models, is a thorny issue. My preferred default is to treat both models on equal ground because both are logically equally valid summaries of the data (put simply). This means you have to determine the core and peripheral conditions twice. Once for the intermediate solution and parsimonious model 1, once for the intermediate solution and parsimonious model 2. For presentational purposes, you may emphasize one parsimonious model over the other, for example on the basis of theory, but I would recommend to always report the second parsimonious model in a footnote or the appendix.

Kind regards

Ingo

[Adrian Dușa]

I think this deserved a little bit of discussion.
One has to understand that model ambiguity appears only in R, this is not an issue with Charles Ragin's software called fs/QCA.

That happens because in his software, solving the PI chart involves a manual (and conscious) action of selecting the most relevant PIs to solve the chart. And if I understand it correctly, the most relevant PIs are selected based on theoretical grounds. In other words, the solution is or should be theoretically relevant.

The R package, on the other hand, outputs all possible ways to solve the PI chart, sometimes yielding hundreds of alternative solution models (by comparison, two models are just a breeze).

But the final task is still the same: instead of selecting the most (theoretically) relevant PIs, one has to select the most theoretically relevant model, since all of them are equally valuable.

This would be my best guess as to what to do in these situations. The R package QCA has a function called modelFit(), which might help selecting theoretically relevant models, based on their intersection with an expected theoretical explanatory model. The final choice is still the researcher's, with proper justifications in the methodological section of the paper.

I hope this helps,

Adrian

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[Gabriel Lourenço]

Excellent, Ingo.

Thank you so much for the clarification.

Best,

Gabriel

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