# Can we use fractional regression for a dependent variable that is made of continuous numerator and denominator?

I have a dependent variable that is a ratio, which takes values between 0 and 1. Some 30% of values are 1s. The dependent variable measures the distribution of funds and is calculated as amount of distributed money / total amount of proposed money. So the ratio comes from continuous data (the amounts of distributed and proposed money).

I am building a model to see what factors influence the disbursement rate.

I was told that a fractional logit regression could be appropriate in such a case? Here is a source that talks about fractional regression.

What puzzles me is that this source mentions "raw counts" in the introduction:

"Introduction

It is sometimes the case that you might have data that falls primarily between zero and one. For example, these may be proportions, grades from 0-100 that can be transformed as such, reported percentile values, and similar. If you had the raw counts where you also knew the denominator or total value that created the proportion, you would be able to just use standard logistic regression with the binomial distribution. Similarly, if you had a binary outcome (i.e. just zeros and ones), this is just a special case, so the same model would be applicable. Alternatively, if all the target variable values lie between zero and one, beta regression is a natural choice for which to model such data. However, if the variable you wish to model has values between zero and one, and additionally, you also have zeros or ones, what should you do?"

Do you know if this suggests that a dependent variable must necessarily come from count data if we are to use fractional regression, or can it come from continuous data like in my example?

Appreciate any useful sources and comments.

It appears to me that it is not necessary that the dependent variable would come from count data. Please see the source one and source two, which gives examples of studies that have used fractional regression with fractional dependent variables that are based on various types of data.

Of course, this does not necessarily mean that these studies have been conducted correctly. Thus, I am open to hearing more opinions, arguments, and articles on that.

• Aside: ratios are not not defined by taking "the values between 0 and 1." E.g., the risk ratio and odds ratio both take values between $0$ & $\infty$, and other ratios derived from numerators and denominators which can take positive and negative values may take values between $-\infty$ and $\infty$. Commented Jun 1, 2021 at 16:21
• Note that regarding money as continuous is a convention arising because the number of distinct values you might have is rather large. At some level, you are counting cents, or whatever is the smallest currency unit, and your variable is discrete. Demographers and others move smoothly between remembering that people are individuals and doing mathematics as if population can be treated as if continuous. Commented Jun 1, 2021 at 16:52
• @NickCox: True & germane; but doubtless not intended to tempt any readers into modelling the reimbursement of £3,000 out of £7,000 as 300,000 'successes' in 700,000 independent trials, as 'raw counts' might suggest in this context. Commented Jun 2, 2021 at 9:48
• @Scortchi-ReinstateMonica I was just reacting to the OP's stress on their data being continuous, to which the shortest and pedantic comment is Not strictly and any longer comment is, more or less, we know what you mean and fair enough. Commented Jun 2, 2021 at 10:10
• As I guess we both know, you just need careful coding to call up appropriate standard errors there. $Y = \exp(Xb)$ could well be a suitable functional form. Commented Jun 2, 2021 at 11:27

Yes. Imagine the continuous fraction $$p$$ as a counted fraction $$\frac{\sum_{j=1}^m y_j}{m}$$ (choosing any numerator & denominator that maintain the recorded precision of $$p$$). If $$y_j$$ is taken to be a Bernoulli variate, the contribution to the log-likelihood of a single observation in a generalized linear model with link function $$g(\cdot)$$, predictors $$\vec x$$, & coefficients $$\vec\beta$$, is

\begin{align} & \sum_{j=1}^m \left[y_j \log \left(g^{-1}(\vec x \vec\beta)\right) + (1-y_j) \log\left( 1- g^{-1}(\vec x \vec \beta)\right)\right] \\ =& \left(\sum_{j=1}^m y_j \right) \log \left(g^{-1}(\vec x \vec\beta)\right) + \left(m - \sum_{j=1}^m y_j\right) \log\left( 1- g^{-1}(\vec x \vec\beta)\right) \\ =& m \left[p \log \left(g^{-1}(\vec x \vec\beta)\right) + (1-p) \log \left(1- g^{-1}(\vec x \vec\beta)\right)\right] \end{align}

As the value of $$m$$ makes no difference to the maximum-likelihood estimate of $$\vec\beta$$, in a fractional logit regression you simply maximize

$$p \log \left(g^{-1}(\vec x \vec\beta)\right) + (1-p) \log \left(1- g^{-1}(\vec x \vec\beta)\right)$$

This gives a consistent estimate for the conditional mean $$\operatorname E P$$ under the assumption that

$$g(\operatorname{E} P)= \vec x \vec \beta$$

The procedure doesn't know or care whether the fractions modelled originate from raw counts or not.

In fine, you can consider the continuous fractions 0.734, 0.642, & 0.149 to be 734, 642, & 149 'successes' out of 1000 trials, or 7340, 6420, & 1490 out of 10 000; & you'll get the same answer, at least as far as parameter estimates go (you'd typically be using robust variance estimates (or perhaps estimating a dispersion parameter) for standard errors in any case).