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Context

I'm working on a project where I need to understand the impact of some variables on satisfaction (y). My y variable is an NPS measure, ranging from 0 to 10 and does not have float values, only integers.

I plotted a histogram of that variable, here is the graph:

enter image description here

Questions

  • What distribution is this?
  • How can I test which distribution fits better on my data?

Any help or insights would be greatly appreciated. Thank you!

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    $\begingroup$ Like almost every distribution you observe, it doesn't have a specific name. If you treat the "satisfaction" values as categories (rather than being at least ordinal, or more than just ordinal if it's a sum of items), then you might argue that it's a special case of the multinomial -- but that's saying essentially nothing. $\endgroup$
    – Glen_b
    Commented Jan 17 at 2:16
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    $\begingroup$ What's NPS? Net promoter score, or some other thing? Undefined initialisms are usually best avoided even when they're statistical terms (well, ANOVA is probably safe) but even more so when they're from some other area -- you really should spell them out on first use. With variables - even when the name is spelled out - it's best to define explicitly how they're constructed when trying to discuss some model for them. Was it just responses to "How likely on a scale from 0 to 10 would you be to recommend <whatever> to a friend?" Were responses limited to integers? $\endgroup$
    – Glen_b
    Commented Jan 17 at 2:22
  • $\begingroup$ Thank you for your answer and sorry to don't define a new term. NPS is Net Promoter Score, a simple question that is supposed to measure customer loyalty and satisfaction. You are right! The answer is limited to integers that go from 0 to 10. I am tryng to build a model (maybe GLM - Generalized linear model) and I need to select a function that fits better to this distribution. $\endgroup$ Commented Jan 17 at 2:53
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    $\begingroup$ Could you explain what you hope would be accomplished by fitting a distribution to these data? $\endgroup$
    – whuber
    Commented Jan 17 at 20:06
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    $\begingroup$ This seems to be an xy problem. There might be an underlying problem that can be interesting, but the two questions 'what is this?' and 'how do I test it?' are too much unclear. (Well, the first question is clear and can be easily answered. We can straightforwardly say what it is, namely a 'categorical distribution'. But to be any more specific than that, some more information is needed) $\endgroup$ Commented Jan 18 at 21:53

2 Answers 2

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If you're looking to fit a GLM like model, the marginal distribution of the response is not informative (certainly not by itself), since the model assumption relates to the conditional distribution of the response. That is the plotted histogram is not helpful in thinking about a model. For example, it may well be that the distribution at each predictor combination is unimodal, even though when you combine responses across all the data the marginal distribution is bimodal.

If the net promoter score (NPS) is an answer to a question of the form "How likely on a scale from 0 to 10 would you be to recommend this to a friend?" then it's arguably ordinal in scale for which you might consider proportional odds (ordinal logit/ordered logit) models, which is a form of GLM. There's more possibilities, however, such as ordered probit.

If you had only 2 groups & no other IVs (predictors) in the GLM, proportional odds models would correspond to using a Wilcoxon-Mann-Whitney.

Quite a few posts on site discuss this model (there's also a Wikipedia article), you might try the site's search function (or if you want to find comments you might try a google search with the site set to here).

On the other hand if you decide to treat it as having the points on the 11 point scale being equispaced (or near enough to it that it would not be consequential to act as if it were) then there's no exponential family member I'm aware of that's exactly suitable for this circumstance.

In some circumstances ordinary regression might work well enough (particularly if the mean doesn't approach the boundaries too closely). In other circumstances you might want some form of curved link function to keep the fitted conditional mean of responses within bounds (such as the logit or probit). In that situation a scaled binomial (quasi-binomial) might perhaps work well enough. Another possibility for an approximate model might be a scaled beta regression, perhaps, though exactly where you would scale the endpoints to might be a source of some consideration; I don't have good advice about whether it's better to just divide by 10 (and use 0-1 inflated beta) or to bring the 0 and 10 endpoints of the discrete scale slightly inside the bounds of the (continuous) beta model.

A multinomial model ignores the ordering in the response, which is (a) throwing out a lot of information, and (b) very likely to be leaving you answering quite a different question from the one your research question is asking.

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  • $\begingroup$ Dear Glen_b I want to extend my heartfelt thanks for your detailed and enlightening response. Your explanation of the difference between marginal and conditional distributions, and the suggestions for appropriate NPS models, were immensely helpful. $\endgroup$ Commented Jan 20 at 3:53
  • $\begingroup$ I am excited to test these different models in my research to see which best fits my data and objectives. I also appreciate the additional recommendations for research and considerations regarding the limits and nature of the data. Thank you once again for your valuable contribution and the time you dedicated to assisting. $\endgroup$ Commented Jan 20 at 3:54
  • $\begingroup$ I made a small change to the last paragraph $\endgroup$
    – Glen_b
    Commented Jan 20 at 5:12
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A simple solution is to use a multinomial distribution with a prior/parametrization on the probabilities.

That is, the observation is $M(n, k=10, p)$ and when fitting the model, add a penalty $\lambda \lVert p_k - p_{k-1}\rVert$. This assumptions forces adjacent probabilities closer to each other which is a way to impose an ordinal relationship. The loglikelihood/posterior becomes

$$ \ell(p) = \log(n!) - \sum_{j=1}^{k} \log(x_k!) + \sum_{j=1}^kx_j\log(p_j) -\lambda \sum_{j=2}^k(p_j-p_{j-1})^2. $$ After removing terms independent of $p$, this gives $\sum_{j=1}^kx_j\log(p_j) -\lambda \sum_{j=2}^k(p_j-p_{j-1})^2 $ which is easy to maximize numerically. ($k\cdot n$ is probably a good guess for $\lambda$.)

Note that the prior can be selected in any way. It could be flat or, as it seems in your data, skewed U-shaped. To achieve this, penalize only the values 2-7.

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