I have a zero-inflated and extremely positively-skewed outcome variable - lottery wins in dollars. Thus, I use two-part analysis, since adjusting is also needed: one for positive values (lognormal regression) and the second for non-zero probability (logistic regression/bernoulli).

I have found that presenting the results of these two regressions in one plot makes the interpretation really simple: you can see the probability of winning (x-axis) together with the received amount of money (in case you won).

The plot looks like this:

  • y-axis outcome variable values come from a lognormal model

  • x-axis outcome variable values come from a logistic regression/bernoulli model (I report them as probabilities, not odds-ratios).

enter image description here

It's well known practice that crude non-normal data should be reported as median (IQR/min-max/quantiles/percentiles). However, if these values come from regression analysis, should I report them as means or medians in this graph? Does regression "convert" these variables to "normally distributed" variables?

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    $\begingroup$ Why should skewed data be reported using median instead of mean? I do think that people often mean the halfway point (median) when they talk about an average value, but if you’re doing technical work involving means, you probably mean an expected value or estimated expected value. $\endgroup$ – Dave Sep 29 at 11:04
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    $\begingroup$ How does the two axes relate to each other? What exactly are the models you mention? "Means or medians" or what? Why can't you just label those as "regression predictions"? $\endgroup$ – Tim Sep 29 at 12:05
  • $\begingroup$ I edited the beginning of the initial post a bit to make things a bit more clear. And the question remains the same: should I use means or medians. $\endgroup$ – st4co4 Sep 29 at 12:45
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    $\begingroup$ The only grounds I can see for talking about medians here are if you use quantile regression to model how median outcomes vary with the predictors. $\endgroup$ – Nick Cox Sep 29 at 12:57
  • $\begingroup$ Thanks! Thus, it would be fine to report Y and X values as means? $\endgroup$ – st4co4 Sep 29 at 13:17

Coefficients from logit models are usually interpreted as changes in odd ratios. That is, the coefficient is not directly interpreted in changes of the median or mean. You can estimate marginal effects on the mean if you specify the positon, where you want to take the derivative.

I'm not familiar with log-normal models, but if you estimate your log-normal model using OLS you can interpret the coefficient as change of the mean.

In order to interpret an estimates as changes in medians one needs additional assumptions.

For example, if the underlying latent distribution of SWB across groups in an ordered >probit or logit model is symmetric, the median and therefore the mean of SWB scores >are the same. So we can reinterpret the usual ordinal model’s estimates as the >effects of the explanatory variables on the median rather than the mean. We emphasise >here that heteroscedasticity should be explicitly accommodated to avoid assuming FOSD >a priori. (https://voxeu.org/article/subjective-wellbeing-focus-median-not-mean)

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