Why Screen by Response Values?
A candidate response distribution should be able to represent the values the measurement process can produce. If an outcome can be negative, a model whose random variable is positive almost surely cannot describe that event. If exact zeros occur, it matters whether the candidate model assigns zero a positive probability or merely has positive density arbitrarily close to zero. These are basic compatibility questions, separate from questions about the mean or variance.
Four commonly considered GLM families illustrate the distinctions. A Gaussian response ranges over the real line. Gamma and inverse-Gaussian responses are strictly positive almost surely. A Tweedie response with power \(1<p<2\) can be exactly zero or positive, but not negative. Comparing these statements requires care: the topological support of a positive continuous distribution includes zero even though the distribution assigns zero probability to the value zero itself.
Three Different Statements About Support and Zero
For example, a positive continuous density can put positive probability in every interval \((0,\epsilon)\) while assigning probability zero to the singleton \(\{0\}\). Thus zero is a limit point of possible outcomes, but an exact draw equal to zero does not occur under that distribution. For continuous observations, individual values generally have probability zero; a likelihood is ordinarily based on a density, not on the probability of the observed singleton. This fact makes it especially important not to infer “zero likelihood” just from the fact that \(P(Y=0)=0\).
Response Ranges of the Four Families
Use the usual GLM parameterization with mean \(\mu\) and dispersion \(\phi\), where \(\phi>0\). The Gaussian response has distribution \(N(\mu,\phi)\), so \(\phi\) is its variance. The gamma response has mean \(\mu>0\) and variance \(\phi\mu^2\); the inverse-Gaussian response has mean \(\mu>0\) and variance \(\phi\mu^3\). For a Tweedie response, take \(1<p<2\), mean \(\mu>0\), and variance \(\phi\mu^p\).
| Family | Topological support | Almost-sure response constraint | Atom at zero? |
|---|---|---|---|
| Gaussian | All real numbers | No sign constraint | No |
| Gamma | \([0,\infty)\) | \(Y>0\) | No |
| Inverse-Gaussian | \([0,\infty)\) | \(Y>0\) | No |
| Tweedie, \(1<p<2\) | \([0,\infty)\) | \(Y\geq 0\) | Yes |
The gamma density, for \(y>0\), is
The inverse-Gaussian density, for \(y>0\), is
Both densities are positive for every \(y>0\), and both distributions assign probability one to positive values. Their topological supports nevertheless include zero. The Gaussian density is positive for every real \(y\), so its topological support is the entire real line. The Tweedie case has an additional feature: its distribution has a genuine point mass at zero.
Proof. The Gaussian density is positive at every real number. Every open interval around any real number therefore has positive probability, and no value outside \(\mathbb{R}\) is relevant; its topological support is \(\mathbb{R}\). The gamma density is positive for every \(y>0\). Any neighborhood of a positive value contains a positive-length interval on which the density is positive. Every neighborhood of zero contains such an interval as well, so it has positive probability. A neighborhood lying wholly below zero has probability zero. This proves that the gamma topological support is \([0,\infty)\). Its density is defined on \(y>0\), and its distribution is continuous with no mass at the boundary, so \(P(Y>0)=1\) and \(P(Y=0)=0\). The inverse-Gaussian density is likewise positive for every \(y>0\), and the same neighborhood argument gives topological support \([0,\infty)\). Its continuous distribution on the positive half-line has no mass at zero, hence \(P(Y>0)=1\) and \(P(Y=0)=0\). \(\square\)
Worked Example: A Negative Clinical Measurement
Suppose a measurement can genuinely be negative, and one observed value is \(y=-0.4\). Consider candidate responses with mean \(\mu=1\) and, where applicable, variance \(\phi=0.25\).
For the Gaussian candidate, \(Y\sim N(1,0.25)\), so the standard deviation is \(0.5\). The probability of a negative response is
The Gaussian model assigns positive density to \(-0.4\), so this value is within its response range. The gamma and inverse-Gaussian candidates instead have \(P(Y>0)=1\); a negative value is outside their almost-sure range. A Tweedie candidate with \(1<p<2\) also has \(P(Y\geq0)=1\), so it cannot represent a negative response either. This screens out those three families on response-range grounds, without claiming that the Gaussian is otherwise a suitable model.
The Tweedie Zero Mechanism
For \(1<p<2\), the Tweedie distribution has a compound Poisson–gamma representation. Let \(N\) be Poisson with mean
Conditional on \(N\), the response is the sum of \(N\) independent gamma jumps. When \(N=0\), the sum is exactly zero; when \(N\geq1\), it is positive. The Poisson probability of no jumps therefore gives the zero mass.
Proof. By definition, a Poisson random variable with mean \(\lambda\) satisfies \(P(N=0)=e^{-\lambda}\). If \(N=0\), the sum of the jumps is an empty sum and equals zero. If \(N\geq1\), each gamma jump is strictly positive almost surely, so their sum is strictly positive almost surely. Consequently the event \(Y=0\) is exactly the event \(N=0\), up to events of probability zero. Substituting the stated value of \(\lambda\) gives the formula. Since \(\mu,\phi>0\) and \(1<p<2\), \(\lambda\) is positive and finite, so the zero mass is strictly between zero and one. \(\square\)
The atom is what separates this Tweedie model from gamma and inverse-Gaussian responses. It is not merely a density that grows near zero: it is positive probability concentrated at the single value zero. The positive part of the Tweedie distribution also places probability in every positive interval, so its topological support is \([0,\infty)\).
Worked Example: Calculating the Tweedie Zero Probability
Take a Tweedie response with \(\mu=1\), \(\phi=0.5\), and \(p=1.5\). Its Poisson mean is
Therefore,
This model assigns about 1.83% probability to an exact zero. The remaining probability, \(1-e^{-4}\approx0.9817\), is assigned to positive responses. A gamma or inverse-Gaussian model has no such zero mass, even though each has positive probability in every interval immediately above zero.
Exact Zeros, Densities, and Measurement Rules
A recorded zero needs interpretation before it can be used to reject a candidate family. If the response is an exact outcome and the data model treats the event \(Y=0\) as observed, then a model with an atom at zero assigns that event positive probability. Gamma and inverse-Gaussian models assign it probability zero because their responses are strictly positive almost surely. This is a meaningful distinction for a response known to have a genuine exact-zero mechanism.
For a continuous observation, however, likelihood calculations use a density relative to a reference measure. The fact that every singleton has probability zero is not by itself a reason to reject a continuous model: it holds for every exact value of a continuous random variable. Nor is the value of a density at a boundary automatically a probability mass. For gamma responses, the density’s limit as \(y\) decreases to zero depends on the shape \(1/\phi\): it tends to zero if \(1/\phi>1\), is finite and positive if \(1/\phi=1\), and diverges if \(1/\phi<1\). None of these cases creates an atom at zero.
The recording rule can change the relevant event. If an instrument rounds small positive values to zero, a recorded zero may mean \(0<Y<\delta\) for some threshold \(\delta>0\), rather than \(Y=0\). Gamma and inverse-Gaussian models assign positive probability to that interval. Censoring at a detection limit likewise requires the likelihood contribution for the censored event, not the density at zero. The observation process should therefore be specified before interpreting a zero as evidence about response support.
Worked Example: Interpreting a Zero Recorded by Rounding
Suppose a laboratory reports measurements to the nearest tenth and records any true value below \(0.05\) as \(0.0\). A reported zero then represents an interval of underlying values, not necessarily an exact zero.
For a gamma model with positive density throughout \(y>0\), the probability of a reported zero is
The inequality holds because the gamma density is positive on the interval’s interior. The inverse-Gaussian model also assigns positive probability to that interval for the same reason. A Tweedie model assigns probability both to its exact-zero atom and to positive values in the interval; its probability of a reported zero depends on the recording rule and includes the atom as well as the relevant positive-range probability. Thus, the rounded report alone does not establish that the underlying response had an exact-zero mechanism.
A Practical Screening Procedure
Use the scientific definition and measurement process to decide whether negative values, exact zeros, and positive values can occur.
Rule out a positive-only response family when genuine negative responses are possible under the intended model. Do not confuse a value in the topological support with an outcome produced with positive point probability.
Distinguish exact zeros from rounding, censoring, or detection-limit values. A Tweedie model with \(1<p<2\) has a genuine zero atom; gamma and inverse-Gaussian models do not.
Support compatibility narrows candidates but does not establish that a family fits the response distribution well. Other evidence is needed to compare families that remain possible.
Common Mistakes
- Calling zero outside gamma or inverse-Gaussian topological support. Every neighborhood of zero contains positive values with positive probability, so zero belongs to the topological support. The correct distinction is that these responses are strictly positive almost surely and have no atom at zero.
- Treating a zero point probability as a zero density. A continuous distribution assigns probability zero to every singleton. This does not make the density at an observed value zero, and a density value is not a point mass.
- Concluding that a reported zero must be an exact zero. Rounding and censoring can turn small positive values into recorded zeros. Model the recording event that actually occurred.
- Choosing a family solely because its support includes the data. Support compatibility is a necessary screening check, not a complete assessment of whether the family describes the response distribution.
Check Your Understanding
Answer each question by distinguishing response range, topological support, and probability mass where relevant.
- Why does zero belong to the topological support of a gamma distribution even though a gamma response is positive almost surely?
- Which of the four families considered here can assign positive probability to a negative response?
- For a Tweedie response with \(1<p<2\), what event in the compound Poisson representation produces an exact zero?
- Why does \(P(Y=0)=0\) alone not show that a continuous density model has zero likelihood for an exact observed value?
- A device reports zero whenever the true response is below a positive detection limit. What event should a model use to calculate the probability of that report?