I have a variable that is measured using a 13-item, 4-point Likert scale with the exception of one item (yes $=1$, no $=2)$.
How can I calculate the total score for this variable?
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If you're sure your 13 items are all measuring the same latent construct, you could use a partial credit model to account for (potential) differences in response scaling across all 13 items. If the 12 items with four-point Likert scale (polytomous) measurements are all on the exact same scale though, you might be better off dropping the binary item and using a rating scale model of the 12 polytomous items, depending partly on how much unique and valid information you get out of that binary item. John Michael Linacre and Benjamin D. Wright posted some discussions of the differences between partial credit and rating scale models over at rasch.org that might give you a better sense of what you'd be dealing with if you go the item response theory route here.
Some latent variable analysis programs will let you set certain thresholds to be equal across certain items and leave another item's threshold freely estimated. You might be able to blend the partial credit and rating scale models this way by setting your 12 polytomous items' thresholds (each item will have three) to be equal across items, and estimating the binary item's single threshold independently...but I'm not exactly sure this is all you'd need to do to have the best of both worlds.
The simple, "classical test theory" approach that weighs every item equally would probably have you just standardize all the items and average the $z$-scores, but I don't think that's a good idea, because four-point Likert scales may not approximate a continuous dimension well enough (and a binary item definitely won't; it might not even make sense), though the average of 12 polytomous items might be approximately continuous enough. I've seen it suggested that each item's Likert scale should have at least five items to approximate a continuous distribution, and at least five Likert scale items should measure the same scale if their simple sum / average is to approximate a continuous dimension. (Can't remember where, but I can look it up and edit it in if you want a source but can't find one yourself; just comment!)
If you're not sure your items are all measuring the same latent construct, I'm afraid you have other things to worry about; see these questions: