Link Domains And Inverse-Link Ranges

Domains and Ranges in Link Transformations

A link connects a mean-scale value to a linear-predictor-scale value. To use that transformation correctly, it is necessary to know which values may be supplied to the link and which values can be produced by its inverse. These are related questions, but they are not the same: the domain of a link concerns its inputs, while the range of an inverse link concerns its outputs.

This distinction is concrete for the identity, logarithmic, and logit transformations. The identity link accepts any real input. The logarithmic link requires a positive input, while the logit link requires an input strictly between zero and one. Their inverse transformations produce outputs in the corresponding domains.

The domain is determined by the mathematical expression defining the link. For example, a logarithm is defined only for positive arguments. The range of the inverse is determined by what its formula can output. For instance, an exponential is always positive. These facts give the following summary.

Identity Link: No Input Restriction From the Formula

The identity link leaves its input unchanged: \(g(\mu)=\mu\). The formula imposes no restriction on a real-valued input, so its domain is \(\mathbb R\). Solving \(\eta=\mu\) for \(\mu\) gives \(g^{-1}(\eta)=\eta\). Since the inverse returns its input unchanged, every real input produces a real output, and every real number is obtainable.

Logarithmic Link: Positive Inputs and Positive Outputs

The logarithmic link is \(g(\mu)=\log\mu\), where \(\log\) denotes the natural logarithm. Its input must be positive: \(\mu\in(0,\infty)\). If \(\mu=0\) or \(\mu<0\), the real logarithm is not defined. To find the inverse, solve \(\eta=\log\mu\) for \(\mu\), obtaining \(\mu=\exp(\eta)\).

To verify the result, first note that \(\log\mu\) is a real number for every \(\mu>0\). Conversely, for any real \(\eta\), \(\exp(\eta)>0\), and \[ \log\{\exp(\eta)\}=\eta. \] Also, for every \(\mu>0\), \[ \exp\{\log(\mu)\}=\mu. \] These identities verify that the two formulas undo one another on the stated domains. Finally, every positive value \(x\) is in the inverse link's range: choosing \(\eta=\log x\) gives \(\exp(\eta)=x\). Therefore the range is all of \((0,\infty)\), not merely a subset of positive values.

Logit Link: Inputs Between Zero and One

The logit link is \[ g(\mu)=\log\left(\frac{\mu}{1-\mu}\right). \] For this expression to be a real logit, both \(\mu\) and \(1-\mu\) must be positive. Those two inequalities require \(0<\mu<1\). Thus the link's input domain is the open interval \((0,1)\).

To obtain the inverse, start with \(\eta=\log\{\mu/(1-\mu)\}\) and exponentiate: \[ \exp(\eta)=\frac{\mu}{1-\mu}. \] Multiplying both sides by \(1-\mu\) and collecting the terms involving \(\mu\) gives \[ \exp(\eta)-\exp(\eta)\mu=\mu, \qquad \exp(\eta)=\mu\{1+\exp(\eta)\}. \] Since \(1+\exp(\eta)>0\), division yields the inverse formula \[ g^{-1}(\eta)=\frac{\exp(\eta)}{1+\exp(\eta)}. \]

For any real \(\eta\), let \(q=\exp(\eta)\). The exponential is positive, so \(q>0\). Consequently, the inverse-logit output satisfies \[ 0<\frac{q}{1+q}<1. \] The first inequality follows from \(q>0\) and \(1+q>0\); the second follows because \(q<1+q\). This proves that every inverse-logit output lies strictly between zero and one.

To show that the inverse produces every value in that interval, take any \(\mu\in(0,1)\) and set \(\eta=\log\{\mu/(1-\mu)\}\). The ratio is positive, so \(\eta\) is real. Substitution gives \(\exp(\eta)=\mu/(1-\mu)\), and therefore \[ g^{-1}(\eta) =\frac{\mu/(1-\mu)}{1+\mu/(1-\mu)} =\frac{\mu/(1-\mu)}{1/(1-\mu)} =\mu. \] Thus every value strictly between zero and one is attained, completing the range argument. \(\square\)

Keeping the Two Questions Separate

When checking a transformation, ask two questions in order. First, what values may enter the link? Second, what values can emerge from the inverse link? For the logarithmic link these answers are \((0,\infty)\) and \((0,\infty)\); for the logit link they are \((0,1)\) and \((0,1)\). The intervals happen to match within each pair, but the roles are different: the first restricts the link input, and the second describes the inverse's possible outputs.

The linear predictor \(\eta\) is the input to the inverse link, not the input to \(g\). For all three inverse links above, \(\eta\) may be any real number. In particular, a negative \(\eta\) does not violate the input restriction of the logarithmic link: the logarithmic link acts on \(\mu\), whereas the inverse link acts on \(\eta\).

Common Mistakes

• Applying the link's restriction to the predictor. The log link requires a positive \(\mu\), but its inverse accepts every real \(\eta\). For example, \(\exp(-1.2)>0\) is a valid inverse-link output.
• Using a closed interval for the logit. The logit link's domain is \((0,1)\), not \([0,1]\): at either endpoint its defining ratio does not give a finite real logarithm.
• Assuming an exponential can output zero or a negative number. For every real \(\eta\), \(\exp(\eta)>0\), so the inverse-log range is \((0,\infty)\).
• Stopping after showing that outputs are in an interval. Showing that an inverse logit is between zero and one proves its outputs are contained in \((0,1)\). To establish that this is the full range, also show that every \(\mu\in(0,1)\) can be obtained.