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I would like to model a percentage response. This percentage is not quite a probability or ratio of 2 quantities. It is reported by customers and expresses they pleasure of shopping. They report it using 10% intervals, I mean: 0%, 10%, 20%, 30%, .... 100%.

I want to analyse the relationship between this and a few predictors: percentage = x1 + x2

The response is not an ordinal variable, but rather discretized continuous. Fractional predictions, like 22% or 67.25% would make perfect sense to me.But it cannot go lower than 0 and more than 100%.

I think I have three options:

  1. linear regression, where the interpretation would be very easy. But I was told my response is a truncated one, and I don't want to play with -10% or 120%. Also, I was told that percentages and proportions may expose problems with variance, which may depend on the mean, so the standards errors will be incorrect. I am not sure, if my response is a proportion? Just a number from range 0-1, but well, yes, it's a % of the range.

  2. fractional logistic regression - but I don't understand the outcomes, so let's skip this.

  3. beta regression. I will handle the 0 and 100% by adding/subtracting 0.001 from it. I was told to use the logit link. But now I don't understand the outcome.

But how to read the print out? What does it mean, say, exp(x1) = 1.2? I know, that it usually means "change in log odds", but what does the "odds" correspond to? I don't have any "categories", to which my customers belong, just percentages.

So, how can I read the output of my beta regression, where exp(x1) is 1.2? That "unit change in x1" multiplies my percentages by 1.2?

So, if x1 changes by 1, and the percentage was 10%, will it be 12% now?

I will be happy if you can propose me another methods of analysis (I heard about truncated linear regression), but first of all, kindly please, refer to the beta regression.

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  • $\begingroup$ Would you please post (or link to) the data? $\endgroup$ Dec 31, 2019 at 1:49
  • $\begingroup$ The data is confidential. I only need the interpretation of exp(x1) = 1.2 in beta regression with logit link. Isn't it possible to provide an explanation without the data? Like for the linear regression: "unit change in 1, ceteris paribus, increases the response by K units". In beta regression with logit link I don't understand the change in odds ratios, if my dependent variable is like this: "10%, 20%, 20%, 10%, 60%, 20%, 50%, 20%, 10%, 90%, 100%, 0%, 50%, 70%", etc. (0r 0.1, 0.2, 0.2, 0.1, 0.6 ...etc) $\endgroup$
    – Natalie
    Dec 31, 2019 at 1:59
  • $\begingroup$ Another very closely related answer regarding the beta regression coefficient interpretation: stats.stackexchange.com/questions/297659/… $\endgroup$
    – Stefan
    Dec 31, 2019 at 5:40
  • $\begingroup$ Stefan, thank you very much. Indeed, it's a very useful and clear answer. $\endgroup$
    – Natalie
    Dec 31, 2019 at 15:40

1 Answer 1

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I assume that you have scaled your percentages to values in [0,1] (or in the open interval (0,1) after you've added/subtracted 0.001; you might want to consider adjusting by smaller values, as values very close to 0/1 will be extreme on the log-odds or logit scale; e.g. logit(0.001) = -6.9, logit(0.01) = -4.59). I'll call these numbers "pleasure indices".

But how to read the print out? What does it mean, say, exp(x1) = 1.2? I know, that it usually means "change in log odds", but what does the "odds" correspond to? I don't have any "categories", to which my customers belong, just percentages.

If $\beta_1$ is the coefficient associated with x1 and $\exp(\beta_1) = 1.2$ and $\hat p$ is the predicted pleasure index that means that an increase of 1 unit in x1 increases $\hat p/(1-\hat p)$ (the "odds" of the pleasure index) by a factor of 1.2. How much $\hat p$ increases depends on the baseline value of $\hat p$; this is one of the inevitable, annoying aspects of dealing with the log-odds scale (or with pretty much any scale that is bounded on [0,1]).

So, how can I read the output of my beta regression, where exp(x1) is 1.2? That "unit change in x1" multiplies my percentages by 1.2?

No. It multiplies the "odds of the pleasure index" by 1.2

So, if x1 changes by 1, and the percentage was 10%, will it be 12% now?

Not exactly. If originally $\hat p$ was 0.1, then $\hat p/(1-\hat p)$ = 1/9, so the new "odds" is $1/9 \times 1.2 = 2/15 \approx 0.1333$.

If $q = p/(1-p)$ then $p=q/(1+q)$, so the new $$ \hat p = (2/15)/(1+2/15) = (2/15)/(17/15) = 2/17 \approx 0.117 $$

In this case you are actually about right, but that's because for small baseline $\hat p$ values, the log-odds scale acts like the log-probability scale. The answer would be different if you had specified the baseline value as 0.5 or 0.9, e.g. see here.


An alternative approach to analyzing data with this response variable (discrete, ordered outcomes {0, 10%, 20%, ..., 100%}) would be to fit an ordinal model, e.g. with MASS::polr() in R.

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  • $\begingroup$ Dear Ben, I knew that you are a brilliant statistician and author of R packages, but you are also a great teacher. Thank you very much. Got it! Happy new year! $\endgroup$
    – Natalie
    Dec 31, 2019 at 2:16
  • $\begingroup$ thanks, just picking the low-hanging fruit :-) $\endgroup$
    – Ben Bolker
    Dec 31, 2019 at 2:24
  • $\begingroup$ Thank you for updating your answer, proposing the proportional odds model. My percentages, are, however, normal numbers on interval scale. I'm afraid I will lose something by treating discrete continuous variable like just ordered categories. Here the unit change is defined, I mean 80%-79% = 2%-1%. Ideally I would use linear or truncated regression, if only the distributional assumptions of residual were met. $\endgroup$
    – Natalie
    Dec 31, 2019 at 2:53

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