2 Non-Euclidean differential geometry

In this section, we will explain briefly the basic notions of differential geometry for statistical models. Interested readers can consult various sources for more information (see, e.g., Amari, Reference Amari2016; Boothby, Reference Boothby1986; Lee, Reference Lee2012; Tu, Reference Tu2010). The relation between statistics and differential geometry is more deeply elaborated in Amari (Reference Amari2016), Calin and Udrişte (Reference Calin and Udrişte2014), Kass (Reference Kass1989), and Kass and Vos (Reference Kass and Vos2011). Two noteworthy papers from quantitative psychology with nice introductions into some of the material used in this article are Ly et al. (Reference Ly, Marsman, Verhagen, Grasman and Wagenmakers2017) and Segert and Davis-Stober (Reference Segert and Davis-Stober2019).

A parametric statistical model can be viewed as a collection $\mathcal {M}$ of probability distributionsFootnote ^{3 ,} Footnote ⁴ (with outcome space $\mathcal {S}$ ) indexed by a parameter $\theta $ : $\mathcal {M}= \left \{ p(y|\theta ) \, | \, \theta \in \Theta \right \}$ (McCullagh, Reference McCullagh2002). The parameter $\theta $ can be a scalar (as in the 3PL model from Equation (1)) or, more generally, a vector of dimension k. The set $\Theta \subseteq \mathbb {R}^k$ is called the parameter space. The tools in this article require that we are dealing with well-behaving statistical models. By well-behaving, we mean that the model should be well-identified (i.e., $\theta \neq \theta ^{\prime }$ implies that $p(y|\theta ) \neq p(y|\theta ^{\prime })$ ) and satisfy general regularity conditions. In addition, if the model is multidimensional, then the Fisher information matrix should be of full rank for all $\theta $ . The most common statistical models belong to this class of regular statistical models (e.g., normals with unknown location and scale, exponential family models, etc.).

A regular parametric statistical model $\mathcal {M}$ constitutes a smooth (or differentiable) manifold of distributions, which means that an (infinitely) differentiable and invertible map exists from the manifold to the parameter space $\Theta \subseteq \mathbb {R}^k$ (such a map is called a chart map or chart). The dimensionality k of the map is the dimensionality of the manifold. For instance, a two-dimensional statistical manifold is illustrated graphically in Figure 3. A smooth manifold of regular statistical distributions is called here a statistical manifold.

Figure 3

Graphical illustration of a smooth two-dimensional statistical manifold $\mathcal {M}$ together with a coordinate or parameter space $\Theta $ . Each location on the manifold corresponds to a distribution of some family and $(\theta _1,\theta _2)$ is a pair of coordinates indexing the distributions. The mapping between $\mathcal {M}$ and the parameter space is represented by the black lines.

Figure 3 Long description

Graphical illustration of a smooth two-dimensional statistical manifold together with a coordinate or parameter space . Each location on the manifold corresponds to a distribution of some family and is a pair of coordinates indexing the distributions. The mapping between and the parameter space is represented by the black lines.

A simple example is the family of normal distributions $ \mathcal {M} = \left \{ p(y|\mu ,\sigma ) = N(\mu ,\sigma ) | (\mu ,\sigma )\in \Theta \right \}$ , $\Theta = \mathbb {R}\times (0,\infty ).$ Here, each pair $(\mu ,\sigma )$ determines one probability distribution, so the model is indexed by two parameters and therefore has dimension $k=2$ . The mapping that assigns to each distribution in $\mathcal {M}$ its parameter values $(\mu ,\sigma )$ is a chart, and it provides a coordinate system for the manifold. Since the density of the normal distribution depends smoothly on $\mu $ and $\sigma $ (all derivatives of $p(y|\mu ,\sigma )$ with respect to $\mu $ and $\sigma $ are continuous), this family forms a smooth statistical manifold. In this case, the manifold can be understood as the collection of all normal distributions, while $(\mu ,\sigma )$ gives one particular coordinate system for describing them.

Different parametrizations of the model provide different coordinate systems on the same statistical manifold. For instance, consider the statistical manifold for the univariate normal distribution, which can be parametrized with $(\mu ,\sigma )$ (mean and standard deviation), but also with $(\mu ,\sigma ^2)$ (mean and variance), or with $(\mu ,1/\sigma ^2)$ (mean and precision, as is common in Bayesian statistics). These different parametrizations provide different coordinate systems to distinguish the distributions of the normal statistical manifold, in the same way as there exist different types of 2D geographical maps that distinguish between actual places on the earth. This comparison also makes clear that the manifold (be it the earth or a statistical manifold) and its properties (e.g., curvature and distance between objects) does not depend on the coordinate system or parametrization that is used.

The statistical manifold of the 3PL model with known item parameters from Equation (1) is a one-dimensional manifold. The most common parametrization is based on $\theta \in \Theta =\mathbb {R}$ . However, as explained in Section 1, any smooth strictly monotone transformation of $\theta $ can be used as well: $\psi = e^{\frac {\theta }{2}}$ , $\omega =\sqrt [3]{\theta }$ , etc.

A particular choice of parametrization of the statistical manifold has to be made in order to perform calculations on the statistical manifold. For example, finding the best fitting model (i.e., the point on the statistical manifold best representing the data) requires a particular parametrization that allows for parameter estimation. However, the arbitrariness of the parametrization requires a researcher to think about which quantities are invariant under reparametrization and which are not. As shown in Figure 1, the difference between two persons’ $\theta $ scores is not parametrization-invariant. Thus, we seek to derive a notion of distance between two persons that is intrinsic to the model’s geometric structure rather than depending on the chosen coordinate system or parametrization.

A first step is to find an expression for an infinitesimal distance between probability distributions. Consider a parametric family of probability densities $p(y \mid \theta )$ , where $\theta \in \Theta $ (with dimension k), and two nearby points $\theta $ and $\theta + d\theta $ in parameter space. In the Euclidean parameter space, the squared distance between these points is given by

$$ \begin{align*}\delta^2(\theta,\theta+d\theta) = \sum_i d\theta_i^2 = d\theta^{\top} d\theta. \end{align*} $$

This expression, however, uses the notion of a distance in a Euclidean space and it does not reflect how easily one can distinguish between different probability distributions in the close vicinity of $\theta $ (for an elaboration on this, see, e.g., Myung et al., Reference Myung, Balasubramanian and Pitt2000). Taking this into account (see the derivation in Appendix A), we see that the squared infinitesimal distance between two nearby distributions becomes

(3) $$ \begin{align} (ds)^2 = d\theta^\top g(\theta)\, d\theta, \end{align} $$

where $g(\theta )$ is the Fisher information matrix. This squared distance is called the squared line element, or also the Fisher–Rao metric. Equation (3) can be considered a generalization of the well-known Pythagorean theorem for non-Euclidean geometry. In general, $(ds)^2$ is not equal to the squared distance in the (Euclidean) parameter space (i.e., $\delta ^2(\theta ,\theta +d\theta )$ ), unless the statistical manifold itself is a “flat” Euclidean space, in which case $g(\theta )=I_k$ (the $k\times k$ identity matrix) for all $\theta $ . In the univariate case, the Fisher–Rao metric is defined as

(4) $$ \begin{align} (ds)^2 = g(\theta)\, (d\theta)^2. \end{align} $$

Remark. The squared line element $(ds)^2$ informs us about the local distinguishability between $p(y|\theta )$ and its nearby distributions. A more common quantity to assess differences between distributions is the Kullback–Leibler divergence $D_{KL}$ . Using $D_{KL}$ , the difference of $p(y|\theta ')$ from $p(y|\theta )$ is

$$ \begin{align*}D_{KL}\left( p(y|\theta) \Vert p(y|\theta') \right) = \int_S p(y|\theta) \log \left( \frac{p(y|\theta)}{p(y|\theta')} \right) dy. \end{align*} $$

Although $D_{KL} \geq 0$ (with equality when $p(y|\theta )=p(y|\theta ')$ for all y), it is not symmetric, nor satisfies the triangle inequality, and thus the $D_{KL}$ divergence is not a distance. However, when $\theta '=\theta +d\theta $ , the quadratic approximation to $D_{KL}$ results in

$$ \begin{align*}D_{KL}\left( p(y|\theta) \Vert p(y|\theta+d\theta) \right) \approx \frac{1}{2} (ds)^2,\end{align*} $$

which means that for nearby distributions, the $D_{KL}$ divergence is actually carrying the same information as the local distinguishability $(ds)^2$ . (This ends the remark.)

Returning to the main argument, the squared line element $(ds)^2$ is called the Fisher–Rao metric because Rao (Reference Rao1945) was the first to show that Fisher’s information matrix $g(\theta )$ is in fact a so-called metric tensorFootnote ⁵ that is used to measure distances on a smooth manifold equipped with this metric. A differentiable manifold endowed with such a metric is called a Riemannian manifold in differential geometry. An important property of $(ds)^2$ is stated in the proposition below.

Proof. We will prove this for the univariate case. For a general proof for the parameter vector case, we refer to Calin and Udrişte (Reference Calin and Udrişte2014).Footnote ⁶

Assume that $\phi = \phi (\theta )$ is a smooth reparametrization. This implies that $\phi (\cdot )$ is differentiable, one-to-one, and invertible. The inverse transformation is $\theta =\theta (\phi )$ .

Let us denote the distribution under the new $\phi $ -parametrization as $p^{\ast}(y|\phi )=p(y|\theta (\phi ))$ and the Fisher information as $g^{\ast}(\phi )$ . Then, we have to show that $(ds)^2=(ds^{\ast})^{2}$ .

Because $g^{\ast}(\phi )=E\left [ \left ( \frac {d\log p^{\ast}(y|\phi )}{d\phi }\right )^2 \right ]$ , we need an expression for $\frac {d}{d\phi }\log p^{\ast}(y|\phi )$ . Using the chain rule, we find that $\frac {d\log p^{\ast}(y|\phi )}{d\phi } = \frac {d\log p(y|\theta (\phi ))}{d\phi } = \left. \frac {d\log p(y|\theta )}{d\theta } \right |_{\theta =\theta (\phi )} \frac {d\theta }{d\phi }$ .

Inserting this in the definition of the Fisher information gives (we write an integral over the sample space $\mathcal {S}$ , but if $\mathcal {S}$ is discrete, the integral should be understood as a summation):

(5) $$ \begin{align} g^{\ast}(\phi) &= \int_{\mathcal{S}} p^{\ast}(y|\phi) \left( \frac{d\log p^{\ast}(y|\phi)}{d\phi}\right)^2 dy \nonumber \\ &= \int_{\mathcal{S}} p(y|\theta(\phi)) \left( \left. \frac{d\log p(y|\theta)}{d\theta} \right|_{\theta=\theta(\phi)} \frac{d\theta(\phi)}{d\phi}\right)^2 dy \nonumber \\ &= \int_{\mathcal{S}} p(y|\theta(\phi)) \left( \left. \frac{d\log p(y|\theta)}{d\theta} \right|_{\theta=\theta(\phi)} \right)^2 dy \ \ \left( \frac{d\theta(\phi)}{d\phi}\right)^2\nonumber \\ &= g \big(\theta(\phi) \big) \left(\frac{d\theta(\phi)}{d\phi}\right)^2, \end{align} $$

or in shorter (but somewhat sloppier) notation: $g^{\ast}(\phi ) = g (\theta ) \left (\frac {d\theta }{d\phi }\right )^2$ . Because the differential $d\phi $ transforms as well (but in the opposite way), $d\phi =\frac {d\phi }{d\theta }d\theta $ , the squared line element $(ds^{\ast})^{2}$ becomes

$$ \begin{align*}(ds^{\ast})^{2} = g^{\ast}(\phi)\, (d\phi)^2 = g(\theta)\left( \frac{d\theta}{d\phi} \right)^2\, \left( \frac{d\phi}{d\theta} \right)^2\, (d\theta)^2 = g(\theta)\, (d\theta)^2 = (ds)^2, \end{align*} $$

so the (squared) line element (i.e., the Fisher–Rao metric) does not change.

In the multivariate case, the result generalizes to $(ds^{\ast})^{2}= d\phi ^\top \, g^{*}(\phi )\, d\phi = d\theta ^\top \, g(\theta )\, d\theta = (ds)^2$ .

A distance on the manifold $\mathcal {M}$ between two arbitrary (not necessarily nearby) distributions $p_a=p(y|\theta _a)$ and $p_b=p(y|\theta _b)$ can then be calculated by defining a path between them and then computing the length of the path (i.e., an arc length on the statistical manifold). In general, the parameter $\theta $ has k components. Assume a smooth parametrized curve $\theta (t)= (\theta _1(t), \dots , \theta _k(t))$ with $t \in [0,1]$ such that $\theta (0)=\theta _a$ and $\theta (1)=\theta _b$ . The arc length on the manifold $\mathcal {M}$ is found by assuming a fine-grained grid along the path and then accumulating the many small line elements $ds$ :

(6) $$ \begin{align} d_{\theta(t)}(p_a,p_b) &= d_{\theta(t)}( p(y|\theta_a), p(y|\theta_b) ) \nonumber \\ &= \int_{\text{along path } \theta(t)} ds \nonumber \\ &= \int_{0}^{1} \sqrt{ \sum_{i=1}^k \sum_{j=1}^k g_{ij}(\theta(t)) \frac{d\theta_i(t)}{dt} \frac{d\theta_j(t)}{dt} }\,dt. \end{align} $$

Note that for the distance or arc length on the manifold, we use as a symbolFootnote ⁷ $d_{\theta (t)}$ to denote that in general the distance depends on the path taken by the curve $\theta (t)$ . Another path $\tilde {\theta }(t)$ will likely lead to another arc length.

One particular important path is the one with the shortest possible arc length $d_{\min }$ , called the geodesic distance:

$$ \begin{align*}d_{\min}( p_a, p_b ) = \min_{\theta(t)} d_{\theta(t)}( p(y|\theta_a), p(y|\theta_b) ). \end{align*} $$

The corresponding path is called the geodesic curve, denoted here as $\theta _{\min }(t)$ :

$$ \begin{align*}\theta_{\min}(t) = \arg \min_{\theta(t)} d_{\theta(t)}( p_a, p_b ). \end{align*} $$

The geodesic distance is invariant under smooth parametrizations. Moreover, it is a genuine distance (using p, q, and r for distributions): (1) $d_{\min }(p,q)\geq 0$ with $d_{\min }(p,p)=0$ (non-negativity), (2) $d_{\min }(p,q) = d_{\min }(q,p)$ (symmetry), and (3) $d_{\min }(p,q)+d_{\min }(q,r) \geq d_{\min }(p,r)$ (triangle inequality) (Lee, Reference Lee2012).

Finding the geodesic curve is in most situations not straightforward and requires the solution to a variational problem (see, e.g., Atkinson & Mitchell, Reference Atkinson and Mitchell1981; Calin & Udrişte, Reference Calin and Udrişte2014; Gelman & Meng, Reference Gelman and Meng1998). For a number of common distributions, these geodesic distances have been derived and can be found in various sources (see, e.g., Atkinson & Mitchell, Reference Atkinson and Mitchell1981; Calin & Udrişte, Reference Calin and Udrişte2014; Miyamoto et al., Reference Miyamoto, Meneghetti, Pinele and Costa2024).

3 Geodesic ability in the 3PL model

Let us now return to the 3PL model. Consider a single person (with unknown ability) taking a test of m items (and known item parameters $\alpha _j, \beta _j$ , and $c_j$ ). The probability of a particular $y_j$ response on a single item j is given by $\pi _j(\theta )^{y_j} (1-\pi _j(\theta ))^{1-y_j}$ and the joint probability for the m items is

(7) $$ \begin{align} p(y|\theta) = \prod_{j=1}^m \pi_j(\theta)^{y_j} (1-\pi_j(\theta))^{1-y_j}, \end{align} $$

where $\pi _j(\theta )$ is defined in Equation (1). As mentioned above, with known item parameters, the 3PL model is a one-dimensional statistical manifold and thus the Fisher information matrix is a scalar (Lord, Reference Lord1980):

$$ \begin{align*} g(\theta) &= E\left[ \left( \frac{d \log p(y|\theta)}{d \theta} \right)^2 \right] \\ &= \sum_{j = 1}^m \frac{\alpha_j^2(1-c_j)e^{2\alpha_j(\theta - \beta_j)}}{(c_j + e^{\alpha_j(\theta - \beta_j)})( 1 + e^{\alpha_j(\theta - \beta_j)})^2}. \end{align*} $$

In this context, the Fisher information is also called the test information function (because it gives the amount of information provided about $\theta $ by the particular test of m items).

Proposition 2. For the one-dimensional 3PL model with known item parameters, the geodesic distance between two persons located at $\theta _P$ and $\theta _Q$ (with $\theta _Q<\theta _P$ ) is

(8) $$ \begin{align} d(Q,P)=\int_{\theta_Q}^{\theta_P}\sqrt{g(\theta)}\, d\theta = \int_{\theta_Q}^{\theta_P}\sqrt{\sum_{j = 1}^m \frac{\alpha_j^2(1-c_j)e^{2\alpha_j(\theta' - \beta_j)}}{(c_j + e^{\alpha_j(\theta' - \beta_j)})( 1 + e^{\alpha_j(\theta' - \beta_j)})^2} } d\theta'. \end{align} $$

Proof. Consider Equation (6). On a one-dimensional statistical manifold with squared line element $(ds)^2=g(\theta )\, (d\theta )^2$ , the path as the image of the parametric curve $\theta (t)$ between $\theta _Q=\theta (0)$ and $\theta _P=\theta (1)$ has length

$$\begin{align*}L=\int_0^1 \sqrt{g(\theta(t))}\, \left| \frac{d\theta(t)}{dt} \right| \, dt. \end{align*}$$

Because non-monotone curves will always lead to larger lengths compared to monotone curves, we will only consider the latter and thus $\frac {d\theta (t)}{dt}>0$ . The image of these monotone curves is the path connecting $\theta _Q$ and $\theta _P$ . Moreover, it is safe to state that for the statistical manifold of the 3PL model, there is only one path connecting the points $\theta _Q$ and $\theta _P$ .

Now, we can do a change of variable $\theta ' = \theta (t)$ (so that $d\theta ' = \frac {d\theta (t)}{dt}dt$ ), the length is therefore

$$\begin{align*}d(Q,P)=\int_{0}^{1}\sqrt{g(\theta(t))} \, \frac{d\theta(t)}{dt} \, dt = \int_{\theta_Q}^{\theta_P}\sqrt{g(\theta')}\, d\theta'.\\[-34pt] \end{align*}$$

Remark. Following Atkinson and Mitchell (Reference Atkinson and Mitchell1981), an alternative derivation is possible in the one-dimensional situation: Find a transformation $\phi = \phi (\theta )$ , such that the metric of $\phi $ is that of a Euclidean space (i.e., $(ds)^2=(d\phi )^2$ with $g^{\ast}(\phi )=1$ ). Such a map only exists in general for one-dimensional manifolds, but not for higher-dimensional ones, which makes this method a “trick” that cannot be applied in general.

Thus, we are looking for a parameter $\phi $ such that

$$ \begin{align*}(ds)^2 = g^{\ast}(\phi)\, (d\phi)^2 = (d \phi)^2, \end{align*} $$

because $g^{\ast}(\phi ) =1$ for all $\phi $ . In that case, the geodesic distance between two test takers can be simply computed as $\phi _P-\phi _Q$ . Expressing this difference as a function of the familiar $\theta $ then gives the solution.

To find the map from $\theta $ to $\phi $ (i.e., $\phi = \phi (\theta )$ ) so that the result is a Euclidean metric, we start with the requirement that $g^{\ast}(\phi ) = g(\theta )\, \left ( \frac {d\theta }{d\phi }\right )^2 = 1$ . Then, taking the square root of both sides and solving for $\phi $ gives the result:

$$ \begin{align*} \phi = \int \sqrt{g(\theta')} \, d\theta' + C. \end{align*} $$

The integration constant C will cancel when taking differences between two $\phi $ values. (This ends the remark.)

Now that we have an expression for the geodesic distance between two persons P and Q (such that $\theta _Q<\theta _P$ ), we can also replace Q with a reference person that has the lowest possible trait score (i.e., $\theta _Q \rightarrow -\infty $ ). Using Equation (8), gives a corresponding distance defined relative to a fixed origin at $-\infty $ :

(9) $$ \begin{align} A(P)=\int_{-\infty}^{\theta_P}\sqrt{g(\theta')}\, d\theta' = \int_{-\infty}^{\theta_P}\sqrt{\sum_{j = 1}^m \frac{\alpha_j^2(1-c_j)e^{2\alpha_j(\theta' - \beta_j)}}{(c_j + e^{\alpha_j(\theta' - \beta_j)})( 1 + e^{\alpha_j(\theta' - \beta_j)})^2} } d\theta'. \end{align} $$

We call $A(P)$ the geodesic ability of person P for the given test. To the best of our knowledge, there exists no closed-form solution to this integral.

The interpretation of $A(P)$ differs from that of the conventional latent trait value $\theta _P$ . Equal increments in $\theta $ do not in general correspond to equal statistical separation between persons, because the local geometry varies with $g(\theta )$ . By contrast, equal increments in geodesic ability correspond to an equal distance along the 3PL model manifold.

Proof. Let $\phi =\phi (\theta )$ be a smooth transformation. Using Equation (5), the Fisher information expressed as a function of $\phi $ satisfies

$$\begin{align*}g^{\ast}(\phi)=g(\theta(\phi))\,\left(\frac{d\theta}{d\phi}\right)^2. \end{align*}$$

Therefore,

$$\begin{align*}\int \sqrt{g^{\ast}(\phi)}\, d\phi = \int \sqrt{g(\theta(\phi))}\left|\frac{d\theta}{d\phi}\right|\, d\phi = \int \sqrt{g(\theta(\phi))}\frac{d\theta}{d\phi}\, d\phi = \int \sqrt{g(\theta)}\, d\theta, \end{align*}$$

where the absolute value has been removed because the transformation is monotone. This shows that the distance is unchanged.

Comparing the geodesic ability $A(P)$ (Equation (9)) and Ramsay’s arc length $L(P)$ (Equation (6)), it can be concluded that the equations are quite similar, but not the same. The difference comes from the fact that in our derivation, we have taken into account the metric of the statistical manifold. However, more detailed comparisons can be made for the Rasch model.

4 Geodesic ability in the Rasch model

We will now consider the special case of the Rasch model ( $\alpha _j = 1, c_j = 0$ for all j). The geodesic ability becomes

(10) $$ \begin{align} A(P) &= \int_{-\infty}^{\theta_P}\sqrt{\sum_{j = 1}^m \frac{e^{\theta' - \beta_j}}{\left(1 + e^{\theta' - \beta_j)}\right)^2} }d\theta'. \end{align} $$

This can be compared to Ramsay’s arc length for the Rasch model:

$$ \begin{align*}L(P) = \int_{-\infty}^{\theta_P}\sqrt{\sum_{j = 1}^m \left[ \frac{e^{\theta' - \beta_j}}{\left(1 + e^{\theta' - \beta_j)}\right)^2} \right]^2 } d\theta'. \end{align*} $$

The formulas are even more similar than for the 3PL model. The only difference is that the terms of the sum under the radical in Ramsay’s arc length are squared.

As for the 3PL model, there exists no closed-form expression for the geodesic ability in the Rasch model. However, we can get more insight by considering a further simplification: Assume that the m items all have difficulty equal to 0. In this case, the geodesic distance between two persons P and Q reduces to (using the subscript 0 to refer to the special case):

$$ \begin{align*} d_{0}(Q,P) &= \sqrt{m} \int_{\theta_Q}^{\theta_P} \sqrt{\frac{e^{\theta'}}{\left( 1+e^{\theta'} \right)^2}} d\theta' \\ &= 2\sqrt{m} \left( \tan^{-1}\left( e^{\frac{\theta_P}{2}}\right) - \tan^{-1}\left( e^{\frac{\theta_Q}{2}}\right)\right). \end{align*} $$

Note that this result is actually equal to the geodesic distance between two binomial distributions as derived by Atkinson and Mitchell (Reference Atkinson and Mitchell1981). This should not come as a surprise because, in the equal item difficulty case, the Rasch model with m items reduces to a binomial model (because the success probability is constant).

Similarly to Equation (9), we can set $\theta _Q \to -\infty $ to obtain the geodesic ability for person P, denoted now with a subscript 0 as $A_0(P)$ . Thus, it simplifies to

$$ \begin{align*} A_0(P) &= 2\sqrt{m} \tan^{-1}\left( e^{\frac{\theta_P}{2}} \right). \end{align*} $$

It can be seen that $\lim _{\theta _P \rightarrow -\infty } A_0(P) = 0$ and $\lim _{\theta _P \rightarrow +\infty } A_0(P) = \pi \sqrt {m}$ .

Figure 4 graphically illustrates $A(P)$ and $A_0(P)$ as functions of $\theta _P$ for various choices of $\beta $ and number of items m. Several observations can be made from this plot. First, the plots confirm that the geodesic ability starts at 0 and is positive. Second, the major factor that determines where the function levels off is the number of items. Third, if the items are placed symmetrically around zero, $A_0(P)$ serves as a decent approximation to $A(P)$ .

Figure 4

The geodesic ability $A(P)$ (or $A_0(P)$ ) as a function of $\theta _P$ . The sets of lines with the same color correspond to the same test length: $m=3$ (red), $m=5$ (blue), and $m=10$ (cyan). The dotted lines stand for $A_0(P)$ (thus, all item difficulties equal to zero). The solid and dashed lines correspond to tests with item difficulties located symmetrically or asymmetrically around zero, respectively. More specifically, for the solid lines from bottom to top, the following $\beta $ values (denoted as $a(b)c$ , such that a sequence starts at a, ends at $c,$ and has b units between consecutive values) are being used: $(-1(1)1)$ , $(-1(0.5)1)$ , and $(-1(2/9)1)$ . The dashed lines have asymmetrically placed $\beta $ values: $(-1(2)3)$ , $(-1(1)3)$ , and $(-1(4/9)3)$ .

Figure 4 Long description

The geodesic ability (or ) as a function of . The sets of lines with the same color correspond to the same test length: (red), (blue) and (cyan). The dotted lines stand for (thus all item difficulties equal to zero). The solid and dashed lines correspond to tests with item difficulties located symmetrically or asymmetrically around zero, respectively. More specifically, for the solid lines from bottom to top, the following values (denoted as , such that a sequence starts at a, ends at c and has b units between consecutive values) are being used: , , . The dashed lines have asymmetrically placed values: , , .

Suppose one has to estimate the geodesic ability of a person based on a response vector and known (or calibrated) item difficulties. The standard IRT software can be used to obtain an estimator of $\theta $ , denoted as $\hat {\theta }$ . Plugging this estimate into Equation (10) gives the estimated geodesic ability: $\widehat {A(P)}$ . It is also important to quantify the uncertainty of this estimator. To study this, let us assume again a test of m items with all $\beta _j=0$ . Thus, we want to assess the uncertainty of $\widehat {A_0(P)}$ . From statistical theory, we know that $\hat{\theta}-\theta \xrightarrow{D} N\left( 0, g^{-1}(\theta) \right)$ , where $g^{-1}(\theta )$ is the reciprocal of the Fisher information (or, in the multiparameter case, it is the inverse of the Fisher information matrix). Next, we apply the delta method by first computing $A_0'(P)=\frac {dA_0(P)}{d\theta } = \sqrt {m}\frac {e^{\frac {\theta _P}{2}}}{1+e^{\theta _P}}$ . This then gives

$$ \begin{align*} \widehat{A_0(P)} - A_0(P) &\xrightarrow{D} N\left( 0, g^{-1}(\theta) \left( A_0'(P) \right)^2 \right) \\ &\xrightarrow{D} N(0,1), \end{align*} $$

because if all $\beta _j=0$ , then $g(\theta ) = m\frac {e^\theta }{\left ( 1+e^\theta \right )^2}$ . Consequently, $SE\left ( \widehat {A_0(P)} \right ) \approx 1$ . Hence, we have an estimator with a constant uncertainty. This will not hold exactly for the general $\widehat {A(P)}$ , but it can be expected to hold approximately.

It is interesting to note that this result coincides with the findings by Hougaard (Reference Hougaard1982). Based on earlier work, Hougaard (Reference Hougaard1982) studied alternative parametrizations with certain properties. More specifically, for an exponential family model,Footnote ⁸ the following class of transformations of the canonical parameter $\theta $ is considered:

$$ \begin{align*}B_{\delta}(\theta) = \int_L^{\theta} \left[ \frac{d^2}{d\theta^{'2}} \Psi(\theta') \right]^{\delta} d\theta', \end{align*} $$

where L is the lower bound for the parameter $\theta $ (in our case, $L=-\infty $ ) and $\delta $ is a constant. The choice of $\delta $ determines a new parametrization with a particular property. For example, when $\delta = \frac {1}{3}$ , this results in a normal likelihood. The $\delta =\frac {1}{2}$ leads to a parametrization with stability of variance. In fact, for $\delta =\frac {1}{2}$ , it becomes clear that $B_{\frac {1}{2}}(\theta ) = A(P)$ for the Rasch model. Moreover, we have shown the constant variance for the geodesic ability.