I have a memory experiment where on each trial, a 7 letter scrambled word is presented, and after a delay the participant is shown the intact word and has to type the previously seen scrambled word exactly as it appeared. They do this for 100 trials, and I collect 50 subjects worth of data. I score each trial from 0 to 7, where 7 means they got the order of letters exactly correct, 5 means they made a mistake on 2/7 letters, and 0 means they completely messed up the order of the letters.

What distribution would make the most sense for modeling this discrete and ordinal data? I'm assuming that it could be best characterized by a mixture of two underlying distributions, a random guessing distribution (randomly inputting letter orderings) and a non-guessing distribution (participant has at least some memory of the correct ordering). I am having trouble figuring out what kind of distribution (e.g., negative binomial, normal gaussian, etc.) would make the most sense to characterize these distributions.

Any thoughts are appreciated!

More details: As an example, it would be like showing COPPRON for a fraction of a second, and then showing POPCORN and asking the subject to correctly type the previously seen scrambled word. I am looking for a distribution that can characterize the 0-7 point scale data I would receive. So for the non-guessing distribution, I want a distribution that can be bounded from 0 to 7 (or 0 to 1 if I normalize the data) where there is likely a peak centered around the most common score that the subject showcased (probably 5 or 6 depending on the difficulty). For the random guessing distribution, it's the same, but it's the distribution that results if we assume a subject always randomly typing the ordering of the letters.

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    $\begingroup$ There's nothing in this question that provides any useful information about what the distribution of scores would be: isn't that why you're doing the experiment? Indeed, what is its purpose and how do you hope to employ such a distribution in your analysis? $\endgroup$ – whuber Jul 12 '19 at 18:04
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    $\begingroup$ Re the edit: the distribution of scores based on random typing depends on the patterns of repeated letters in the word. As an extreme example, the score associated with the word "AAAAAAA" would always be zero--unless you wish to account for inserting other letters. But in that case we're back to the beginning: there's no information available about the rates at which a subject might insert letters not from the original word. $\endgroup$ – whuber Jul 12 '19 at 18:08
  • $\begingroup$ Sorry if I was too confusing/unspecific! I added some more details. I am hoping to use a hierarchical model to characterize two hypothesized underlying distributions that would make up the 0-7 point scale data that I would receive. I am having trouble figuring out what type of prior distribution is appropriate for me to use in my model specification. (I am also assuming in this example that all words have the same number of letters and all have unique letters, and all responses are consisting of the original letters.) $\endgroup$ – Paul Scotti Jul 12 '19 at 18:08
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    $\begingroup$ That suggests you're in an exploratory study, not a confirmatory one, which means you ought to be characterizing the distributions you observe rather than inventing or guessing at distributions that might have nothing to do with how the subjects behave. $\endgroup$ – whuber Jul 12 '19 at 18:09
  • $\begingroup$ I guess what I want to know is what type of distribution may be appropriate, in general, for characterizing discrete, bounded, ordinal data? The exact distribution parameters (like scale or shape) are not what I am interested in, rather the type of probability distribution that might make sense if we assume some sort of bounded distribution with a peak not at the boundaries. $\endgroup$ – Paul Scotti Jul 12 '19 at 18:13

You could a mixed effects models for an ordinal outcome variable. The two most popular models for ordinal data are the proportional odds model and the continuation ratio model.

For the latter you can find a detailed example on how to fit the model and extract the category-specific probabilities in the vignette Mixed Models for Ordinal Data of the GLMMadaptive package.

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