Choosing a Model for a Fraction
A response recorded as a fraction can take values throughout the interval from zero to one. Examples include a share of a budget, the proportion of days a patient followed a treatment plan, or the fraction of a land parcel under cultivation. A useful model must keep its conditional mean in the unit interval, but that requirement alone does not determine a unique model. The link function determines how predictors map to the mean; a distributional model makes additional claims about how the observations vary around it.
This distinction separates two approaches. Fractional logit and fractional probit specify a bounded conditional mean and can use a Bernoulli-type criterion even when the observed fractions are not binary. Beta regression specifies a continuous probability distribution for an outcome strictly between zero and one. These models therefore answer related, but not identical, modelling questions.
Fractional Logit and Fractional Probit
For fractional logit, the inverse link is the logistic function, \(\Lambda(\eta)=\exp(\eta)/[1+\exp(\eta)]\). For fractional probit, the inverse link is the standard normal cumulative distribution function, \(\Phi(\eta)\). Thus both models produce fitted conditional means strictly between zero and one for any finite linear predictor, but the functions translate that predictor into a mean differently.
Fractional logit uses the Bernoulli log-likelihood expression as an estimation criterion:
When \(Y_i\) is a fraction rather than a zero-or-one outcome, \(Q(\beta)\) is not generally the log-likelihood of the observed outcome distribution. It is a quasi-likelihood criterion used to estimate the conditional mean. Fractional probit can use the same criterion with \(\mu_i=\Phi(x_i^\top\beta)\). In either case, the key mean-model condition is \(E(Y_i\mid x_i)=\mu_i\), together with suitable sampling and regularity conditions for estimation. Correct specification of the conditional mean is central; the Bernoulli expression does not assert that a fractional observation was generated by a Bernoulli distribution.
The two links differ in shape, not in their basic purpose. The logistic curve has somewhat heavier tails than the standard normal cumulative distribution function. As a result, for a given linear predictor away from zero, the fitted means need not match. Choosing between the links is a choice about the functional form of the conditional mean, not a choice between a continuous outcome and a binary outcome.
Worked Example: Comparing Fitted Means Under the Two Links
Suppose a hypothetical fractional-response analysis gives a linear predictor \(\eta=x_i^\top\beta=0.60\) for a particular profile. Compute its fitted conditional mean under each link.
Fractional logit. The logistic fitted mean is \(\Lambda(0.60)=\exp(0.60)/[1+\exp(0.60)]\). Since \(\exp(0.60)\approx1.822\), \[ \Lambda(0.60)=\frac{1.822}{1+1.822}\approx0.6457. \]
Fractional probit. From the standard normal distribution, \(\Phi(0.60)\approx0.7257\).
Both fitted means are within the required unit interval, but they are not equal: the logit fit is about 0.6457 and the probit fit about 0.7257. These are model-implied conditional means for the stated predictor profile, not observed fractions and not probabilities that a particular observation takes a particular value. This comparison is meaningful if the respective linear predictors use the same specified predictor profile; in practice, the estimated coefficients may also differ between models.
Why the Fractional Criterion Works With Boundary Values
The criterion remains defined when an observed fraction equals zero or one, provided the fitted mean is strictly between zero and one. For \(Y_i=0\), the observation’s contribution is \(\log(1-\mu_i)\); for \(Y_i=1\), it is \(\log(\mu_i)\). Neither expression requires transforming the observed response away from its boundary. This is a useful property of the fractional mean approach, already encountered for fractional logit; fractional probit has the same property when it uses this criterion.
The score contribution also shows how the links enter differently. Write \(\eta_i=x_i^\top\beta\), and let \(\mu_i=h(\eta_i)\), where \(h\) is either \(\Lambda\) or \(\Phi\). For one observation, differentiate its criterion contribution with respect to \(\eta_i\):
The last equality follows by taking the common denominator: the numerator becomes \(h'(\eta_i)[Y_i\{1-h(\eta_i)\}-(1-Y_i)h(\eta_i)]=h'(\eta_i)[Y_i-h(\eta_i)]\). For logit, \(h'(\eta_i)=\mu_i(1-\mu_i)\), so the score with respect to the linear predictor simplifies to \(Y_i-\mu_i\). For probit, \(h'(\eta_i)=\varphi(\eta_i)\), the standard normal density, giving a link-specific weighting factor. This calculation explains how the same fractional-response criterion leads to different estimating equations under the two links.
Worked Example: Calculating a Fractional Criterion Contribution
Suppose one hypothetical observation has \(Y_i=0.40\). Compare its criterion contribution at a fractional-logit fitted mean of \(0.60\) with its contribution at a fractional-probit fitted mean of \(0.70\). These fitted means are assumed to come from models whose conditional-mean specifications and sampling assumptions are appropriate for the analysis.
At \(\mu_i=0.60\), the contribution is \[ 0.40\log(0.60)+0.60\log(0.40) =0.40(-0.5108)+0.60(-0.9163) \approx-0.7541. \]
At \(\mu_i=0.70\), it is \[ 0.40\log(0.70)+0.60\log(0.30) =0.40(-0.3567)+0.60(-1.2040) \approx-0.8651. \]
For this observation alone, the first fitted mean gives a larger criterion contribution because \(-0.7541\) is greater than \(-0.8651\). That does not establish that fractional logit fits the data better overall: model comparison would require evaluating the models on the same data and using an appropriate comparison strategy. In particular, this quasi-likelihood contribution should not be interpreted as the log probability of observing the exact fraction \(0.40\).
Beta Regression Adds an Outcome Distribution
Beta regression takes a different approach by modelling the distribution of a continuous response. In its standard form, the outcome must satisfy \(0<Y_i<1\). A convenient parameterization uses the conditional mean \(\mu_i\) and a positive precision parameter \(\kappa\). The beta shape parameters are \(\mu_i\kappa\) and \((1-\mu_i)\kappa\), so a link such as the logit can relate \(\mu_i\) to predictors while keeping the mean between zero and one.
The precision parameter controls how concentrated the outcome is around its mean: holding \(\mu_i\) fixed, a larger \(\kappa\) produces a smaller conditional variance. Unlike the fractional logit and probit mean specifications, beta regression therefore provides a full conditional distribution under its assumptions. Its likelihood is the beta density evaluated at each observed response. The distributional assumption is substantive: it is not implied merely by the fact that a response is a fraction.
Worked Example: Beta Mean, Variance, and Density
Consider a hypothetical beta-regression model with conditional mean \(\mu_i=0.40\) and precision \(\kappa=10\). Suppose the observed fraction is \(y_i=0.20\). The shape parameters are \(\mu_i\kappa=4\) and \((1-\mu_i)\kappa=6\), so the conditional distribution is Beta\((4,6)\).
The conditional variance is \[ \operatorname{Var}(Y_i\mid x_i) =\frac{0.40(1-0.40)}{1+10} =\frac{0.24}{11}\approx0.02182. \] The corresponding conditional standard deviation is \(\sqrt{0.02182}\approx0.1477\), in fraction units.
For this observation, the beta density is \[ f(0.20\mid x_i) =\frac{0.20^{3}(1-0.20)^{5}}{B(4,6)} =504(0.20^3)(0.80^5) \approx1.321. \] Here \(B(4,6)=1/504\). A density value can exceed one; it is not the probability of observing exactly \(0.20\). The calculation is valid because \(0.20\) is strictly between zero and one and the assumed beta distribution has the stated shape parameters.
Boundary Observations and Model Choice
The standard beta density is defined only in the open interval. It cannot directly assign positive probability to an exact observed zero or one, because those values lie outside its support. This is a support restriction, not simply a numerical inconvenience. A transformation that moves boundary observations into the interval changes their recorded values and requires a defensible rationale; it does not automatically make the standard beta model appropriate.
If exact zeros or ones are possible in the data, one option is a fractional mean model, which can include those observations in its estimation criterion while targeting the conditional mean. If the scientific process suggests separate mechanisms for boundary outcomes and interior fractions, a model that explicitly treats boundary and interior outcomes separately may be more suitable. There are also extensions of beta regression designed for boundary mass, but those are different distributional models and require their own assumptions. The choice depends on how the observations arise and whether the main objective is a conditional mean or a fuller account of the outcome distribution.
Worked Example: Deciding Whether Standard Beta Regression Fits
A hypothetical analyst records a proportion of planned work completed for six projects: \(0,\ 0.25,\ 0.40,\ 0.65,\ 0.80,\ 1\). The analyst wants to model how project size relates to the conditional mean proportion completed.
The response includes exact zeros and ones, so standard beta regression is not directly applicable to all six observations: its density requires \(0<Y_i<1\). Removing the two boundary observations would change the analysed population and could distort the target mean if those projects are part of the question. Replacing them with nearby values would also alter the data and does not resolve the underlying support issue by itself.
A fractional logit or fractional probit mean model can retain all six responses, provided the chosen conditional-mean specification is appropriate. If the analyst instead needs to model the mechanisms producing exact completion failure, partial completion, and full completion, a boundary-aware distributional model may be preferable. The data alone do not decide between these targets; the observation process and research question do.
Common Mistake and Practical Comparison
A frequent mistake is to call fractional logit a Bernoulli model and conclude that it is invalid whenever the response is not binary. The correction is to distinguish the criterion from the outcome distribution: fractional logit uses a Bernoulli-form quasi-likelihood to estimate a conditional mean, but does not require fractional observations to be Bernoulli draws. Conversely, beta regression is not merely another link function. It specifies a continuous distribution with a support restriction and a variance structure.
| Feature | Fractional logit or probit | Standard beta regression |
|---|---|---|
| Main target | Conditional mean of a response in the closed unit interval | Conditional mean and a continuous distribution for the response |
| Mean link | Logit or probit, among possible specifications | Often logit; other suitable links are possible |
| Outcome support | May include zero and one | Strictly between zero and one |
| Distributional commitment | Mean specification; Bernoulli-form criterion is not generally a full outcome likelihood | Beta conditional distribution with a precision parameter |
| Variance | Not determined by the link alone | \(\mu_i(1-\mu_i)/(1+\kappa)\) under the stated parameterization |
Use fractional logit or probit when the central requirement is a bounded conditional mean and the data may contain boundary observations. Use beta regression when the outcome is strictly interior and the beta distribution is a defensible description of conditional variation. The two links within fractional-response models can be compared as alternative mean specifications; neither should be chosen solely because its name sounds familiar. For any approach, assess whether the model’s assumptions match the data collection process and the quantity the analysis is intended to describe.
Check Your Understanding
Answer each question by distinguishing the conditional-mean specification from the outcome distribution.
- For a linear predictor of \(0\), what conditional mean do fractional logit and fractional probit each produce?
- Why does using a Bernoulli-form criterion for a fractional observation not mean that the observation was assumed to be Bernoulli?
- Under the stated beta-regression parameterization, what happens to the conditional variance when \(\kappa\) increases while \(\mu_i\) is fixed?
- Why can standard beta regression not directly model an observed response equal to one?
- A data set includes both exact zeros and interior fractions. Name a mean-model approach that can retain the zeros and explain what additional consideration is needed before choosing it over a boundary-aware distributional model.