An analysis of the sensitivity and stability of patients’ preferences can lead to more appropriate medical decisions

Abstract

Patients are faced with multiple alternatives when selecting the preferred method for colorectal cancer screening, and there are multiple criteria to be considered in the decision-making process. We model patients’ choices using a multicriteria decision model, specifically an Analytic Hierarchy Process-based model, and propose a new approach for characterizing the idiosyncratic preference regions for individual patients. The new approach involves the development of a personalized sensitivity and stability analysis of preferences that provides pertinent insights about the changes that occur in a patient’s preferences, as he/she learns additional relevant medical information. We show how the insights derived from the sensitivity and stability of patients’ preferences could be used by a healthcare provider within the medical decision-making process.

Appendices

Appendix 1: Patient preference elicitation using an AHP-based model

The process of eliciting patient preferences regarding the set of colorectal cancer screening options available followed the three principles described in Sect. 3.1.

• I. First, the problem of choosing the best colorectal cancer screening was decomposed by Dolan et al. (2013) and structured using the AHP-based model depicted in Fig. 3. Level 1 of the model, the Goal, was to “Choose the best colorectal cancer screening option”, Level 2, the Criteria, were identified based on the medical guidelines and Level 3, the Alternatives, were the ten screening options available in the U.S. Preventive Services Task Force (2008). Table 3 summarizes the six criteria and ten alternatives used in the model.

• II. Patient preferences were measured using the 1–9 absolute scale by pairwise comparing the elements within each level of the model with respect to the level above. For example, at the Criteria level, the patient was asked to pairwise compare the four criteria and three sub-criteria with respect to the Goal. The judgments elicited are presented in the following table (Sava et al. 2018). Similarly, the ten screening options have been pairwise compared with respect to each of the six criteria and sub-criteria (Table 13).

  Table 13 Patient’s judgments with respect to the criteria and sub-criteria

• III. The patient’s preferences with respect to the set of the six criteria and ten alternatives were aggregated in the supermatrix presented in Table 4. The judgments are further synthesized by raising the supermatrix to powers until it converged to the alternatives priorities presented in Table 5.

Appendix 2: Proof of Theorem 1

The space of perturbations X is the product space $\left[ { - 1,1} \right]^{m}$, where m is the number of elements perturbed. By definition, X is compact. In addition, it is simply connected, because each perturbation induces a supermatrix W. By Perron–Froebenius theory (Keener 1993), because W is a stochastic matrix, there always exists a real positive eigenvalue equal to 1 and a corresponding real nonnegative eigenvector w. Wilkinson (1965) showed that small continuous perturbations of the entries of W induce small continuous perturbations of its eigenvectors, and, specifically, of w. Thus, the space of perturbations is mapped, via a continuous function, into the space of priorities represented by the principal eigenvector of the supermatrix W. In addition, the limiting priorities obtained from the supermatrix W add to unity. Hence, the space of perturbations is mapped, via a continuous function, into the hyperplane $e^{T} w = 1$. Because the image of a simply connected space via a continuous function is also simply connected, the space of priorities resulting from the space of perturbations is also simply connected. In addition, it is also compact.

The space of perturbations X may be written as the union of subspaces $\varvec{X}\left( i \right),i = 1, \ldots ,n\}$. Thus, ${\mathbf{X}} = {\bigcup }_{i = 1}^{n} {\mathbf{X}}\left( i \right)$, but not all the preference regions have non-null intersections. Because the space X is compact and a subset of $R^{n}$, it is closed and bounded. The subspaces $\varvec{X}\left( i \right)$ must also be closed and bounded, because, otherwise, there would be holes between them. If the $\varvec{X}\left( i \right)$ were open sets instead of closed ones, then there would be holes in X, and X would not be simply connected. Of course, the $\varvec{ X}\left( i \right)$’s could be open sets if two adjacent regions have a non-null intersection, i.e., they overlap. However, an overlap would imply that perturbations could yield non-unique priorities, i.e., there could be perturbations that would yield multiple most preferred alternatives. That is not possible, unless the perturbations are exactly on the boundaries of the most preferred alternatives, and, in such a case, the priorities would be the same as for the most preferred alternatives. Thus, there are no holes in X, and because it is compact and simply connected, all the $\varvec{X}\left( i \right)$’s must also be compact and simply connected. □

Appendix 3: An algorithm to generate the piecewise linear approximations for the two-dimensional case

To approximate $\Delta \left( {i,j} \right) \subseteq \varvec{X}\left( i \right) \cap \varvec{X}\left( j \right)$ with a piecewise linear function of order s (i.e., s segments), we proceed as follows:

• (1) Select (s + 1) points in $\Delta \left( {i,j} \right)$, $\left\{ {x_{k} = \left( {\delta_{ik} ,\delta_{jk} } \right), k = 0,1, \ldots ,s} \right\}$.

• (2) Order the points selected $\left\{ {x_{k} = \left( {\delta_{ik} ,\delta_{jk} } \right), k = 0,1, \ldots ,s} \right\}$ according to the proximity of adjacent points, as measured by the Euclidean distance, i.e., find a point $x_{k + 1} = \left( {\delta_{ik + 1} ,\delta_{jk + 1} } \right)$ that is the closest point to $x_{k} = \left( {\delta_{ik} ,\delta_{jk} } \right)$.

• (3) Minimize the number of segments required, by determining if several points lie in a straight line:

  • a. Select the first three adjacent points and compute the area of the triangle formed by them. If they are in a straight line, then select the first and third point.

  • b. Select another adjacent point and repeat step 3(a) until no more points are aligned for the segment being constructed.

• (4) Construct the linear segments:

  • a. If the point $x_{k} = \left( {\delta_{ik} ,\delta_{jk} } \right)$ is the closest point to the point $x_{h} = \left( {\delta_{ih} ,\delta_{jh} } \right)$, then join them with a line:

    $a_{j} \left( {h,k} \right)\delta_{i} + b_{i} \left( {h,k} \right)\delta_{j} = c_{ij} \left( {h,k} \right)$

    where: $a_{j} \left( {h,k} \right) = \delta_{j,k} - \delta_{j,h}$, $b_{i} \left( {h,k} \right) = \delta_{i,h} - \delta_{i,k}$, $c_{ij} \left( {h,k} \right) = \delta_{j,k} \delta_{i,k} - 2\delta_{j,k} \delta_{i,h} + \delta_{j,h} \delta_{i,h}$.

  • b. Select the last point of the segment constructed in (4.a) and proceed as in (3) to build the next segment.

• (5) Stop when the last point is reached.

Example 1

Consider the supermatrix depicted in Table 14 that was randomly generated based on an ANP model with two criteria and six alternatives (Sava 2016).

Table 14 A randomly generated supermatrix

We perturb all the entries of rows 2 and 3 of the Table 14 supermatrix, using relation (3), with a perturbation increment $\alpha = 0.002$. The number of perturbation levels is $m = 999$ for each of the two criteria, so, for the given value of the increment, the matrix of perturbations has 998,001 rows. The preference regions for the six alternatives within the perturbation space [− 1, 1]² are presented in Fig. 4.

Fig. 4

Preference regions within the perturbation space

Given that the perturbation space [− 1,1]² is partitioned into six regions, we need to determine the piecewise linear boundaries between the following pairs of alternatives: A1–A6; A2–A3; A2–A4; A2–A6; A3–A4; A3–A5; A3–A6; A4–A5 and A5–A6. To obtain those boundaries, we only need the points classified as boundary points. Figure 5 shows the boundary points between the six regions obtained from the sample of 998,001 perturbations, using a cut-off value of $\varepsilon = 10^{ - 5}$, i.e., we consider that a point belongs to the boundary between two regions i and j if the priorities of the alternatives $A_{i}$ and $A_{j}$ satisfy $\left| {v\left( i \right) - v\left( j \right) < \varepsilon = 10^{ - 5} } \right|$.

Fig. 5

Boundary points between pairs of alternatives for $\varepsilon = 10^{ - 5}$

The boundaries between the preference regions are obtained using the algorithm described above. To exemplify its applicability, we go over the steps necessary to obtain the separating boundary between the preference regions A1 and A6.

• (1) Select the boundary points discriminating between the two regions. Based on the numerical cut-off points considered, there are 130,112 boundary points characterizing the boundary between the regions A1 and A6, varying from

  $\left\{ {\left( {0; - 0.514} \right),\left( {0; - 0.516} \right) \ldots \left( { - 0.998; - 0.996} \right),\left( { - 0.998; - 0.998} \right) } \right\}$.

• (2) Euclidean distances are calculated between any two perturbation points, to order the pairs from the closest to the further point.

• (3) For each triplet of perturbation points, we calculate the area of a triangle using the three points as the coordinates of the vertices. Consider the triplet $\left( { - 0.998; - 0.994} \right), \left( { - 0.998; - 0.996} \right) {\text{and}} \left( { - 0.998; - 0.998} \right)$. The area of the triangle formed by these three points is 0, so they are in a straight line. The process continues until we reach a triplet of points that are not in a straight line.

• (4) Using the minimum number of points determined in step 3, that are not on a straight line, we construct the following set of inequalities that define the piecewise linear boundary between regions A1 and A6. If a given point satisfies the inequalities, then it will be characterized as being part of region A1, if not it will be in region A6.

  $$ \left\{ {\begin{array}{*{20}c} {0.327x - y \ge 0.412,} & {x \in [ - \;0.998; - \;0.9]} \\ {0.3x - y \ge 0.436,} & {x \in ( - \;0.9; - \;0.8]} \\ {0.28x - y \ge 0.452,} & {x \in ( - \;0.8; - \;0.7]} \\ {0.245x - y \ge 0.477,} & {x \in ( - \;0.7; - \;0.602]} \\ {0.229x - y \ge 0.486,} & {x \in ( - \;0.602; - \;0.506]} \\ {0.208x - y \ge 0.497,} & {x \in ( - \;0.506; - \;0.4]} \\ {0.191x - y \ge 0.503,} & {x \in ( - \;0.4; - \;0.306]} \\ {0.173x - y \ge 0.509,} & {x \in ( - \;0.306; - \;0.202]} \\ {0.157x - y \ge 0.512,} & {x \in ( - \;0.202; - \;0.1]} \\ {0.115x - y \ge 0.516,} & {x \in ( - \;0.1;0.004]} \\ { - \;0.725x - y \ge 0.513,} & {x \in (0.004;0.106]} \\ { - \;0.941x - y \ge 0.49,} & {x \in (0.106;0.276]} \\ { - \;1.083x - y \ge 0.451,} & {x \in (0.276;0.3]} \\ { - \;1.18x - y \ge 0.422,} & {x \in (0.3;0.4]} \\ { - \;1.303x - y \ge 0.373,} & {x \in (0.4;0.466]} \\ \end{array} } \right. $$

Appendix 4: An algorithm to generate the piecewise triangular approximation for the three-dimensional case

To approximate $\Delta \left( {i,j} \right) \subseteq \varvec{X}\left( i \right){\bigcap }\varvec{X}\left( j \right)$ with a piecewise triangular mesh, we proceed as follows:

• 1. Select (s + 1) points in $\Delta \left( {i,j} \right)$, with $\left\{ {x_{h} = \left( {\delta_{ih} ,\delta_{jh} ,\delta_{kh} } \right),h = 0,1, \ldots ,s} \right\}$.

• 2. Determine the triplets of boundary points from $\Delta \left( {i,j} \right)$ which can form a triangle within the three-dimensional perturbation space, using the Delaunay triangulation algorithm as developed by de Berg et al. (2008).

• 3. The output matrix M gives the number of possible combinations of vertices for the triangular mesh, based on the order of the points in the original input for Step (2):

  $M = \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} {point \,no. x_{i1} } & {point\,no. x_{j2} } & {point\,no. x_{k3} } \\ \end{array} } \\ \ldots \\ \ldots \\ \end{array} } \right)$,

  where $i = 1, \ldots ,s + 1;j = 1, \ldots ,s;k = 1, \ldots ,s - 1$

  and $M^{\prime} = \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} { \left( {\delta_{i1} ,\delta_{j1} ,\delta_{k1} } \right)} & {\left( {\delta_{i2} ,\delta_{j2} ,\delta_{k2} } \right)} & {\left( {\delta_{i3} ,\delta_{j3} ,\delta_{k3} } \right)} \\ \end{array} } \\ \ldots \\ \ldots \\ \end{array} } \right)$ is the matrix containing the corresponding Cartesian coordinates for each triplet.

• 4. Construct the separating hyperplane generated by any set of three vertices from M. We use the first point in M’ for illustrative purposes. Let $x_{1} = \left( {\delta_{i1} ,\delta_{j1} ,\delta_{k1} } \right)$, $x_{2} = \left( {\delta_{i2} ,\delta_{j2} ,\delta_{k2} } \right)$ and $x_{3} = \left( {\delta_{i3} ,\delta_{j3} ,\delta_{k3} } \right)$,

  • a. Determine the direction vectors $\overrightarrow {{x_{1} x_{2} }} = \left( {\delta_{i2} - \delta_{i1} } \right)\hat{i} + \left( {\delta_{j2} - \delta_{j1} } \right)\hat{j} + \left( {\delta_{k2} - \delta_{k1} } \right)\hat{k}$ and $\overrightarrow {{x_{1} x_{3} }} = \left( {\delta_{i3} - \delta_{i1} } \right)\hat{i} + \left( {\delta_{j3} - \delta_{j1} } \right)\hat{j} + \left( {\delta_{k3} - \delta_{k1} } \right)\hat{k}$.

  • b. Calculate the normal vector

    $$ \overrightarrow {{x_{1} x_{2} }} \times \overrightarrow {{ x_{1} x_{3} }} = \left| {\begin{array}{*{20}c} {\hat{i}} & {\hat{j}} & {\hat{k}} \\ {\delta_{i2} - \delta_{i1} } & {\delta_{j2} - \delta_{j1} } & {\delta_{k2} - \delta_{k1} } \\ {\delta_{i3} - \delta_{i1} } & {\delta_{j3} - \delta_{j1} } & {\delta_{k3} - \delta_{k1} } \\ \end{array} } \right| $$

  • c. The equation of the hyperplane is given by $ax + by + cz + d = 0$, where

    $$ a = \left| {\begin{array}{*{20}c} {\delta_{j2} - \delta_{j1} } & {\delta_{k2} - \delta_{k1} } \\ {\delta_{j3} - \delta_{j1} } & {\delta_{k3} - \delta_{k1} } \\ \end{array} } \right|,\,b = - \left| {\begin{array}{*{20}c} {\delta_{k2} - \delta_{k1} } & {\delta_{i2} - \delta_{i1} } \\ {\delta_{k3} - \delta_{k1} } & {\delta_{i3} - \delta_{i1} } \\ \end{array} } \right|, $$

    $c = \left| {\begin{array}{*{20}c} {\delta_{i2} - \delta_{i1} } & {\delta_{j2} - \delta_{j1} } \\ {\delta_{i3} - \delta_{i1} } & {\delta_{j3} - \delta_{j1} } \\ \end{array} } \right|, \,{\text{and}}\,d = - \left( {a\delta_{i1} + b\delta_{j1} + c\delta_{k1} } \right).$

    Set the conditions for any given point $x_{h} = \left( {\delta_{ih} ,\delta_{jh} ,\delta_{kh} } \right)$ to be in the interior of the triangle $\Delta x_{1} x_{2} x_{3}$, using the barycentric coordinates approach developed by Shirley et al. (2009): $\alpha = \frac{{\left| {\overrightarrow {{x_{h} x_{2} }} \times \overrightarrow {{ x_{h} x_{3} }} } \right|}}{{\left| {\overrightarrow {{x_{1} x_{2} }} \times \overrightarrow {{ x_{1} x_{3} }} } \right|}}$, $\beta = \frac{{\left| {\overrightarrow {{x_{h} x_{3} }} \times \overrightarrow {{ x_{h} x_{1} }} } \right|}}{{\left| {\overrightarrow {{x_{1} x_{2} }} \times \overrightarrow {{ x_{1} x_{3} }} } \right|}}$, $\gamma = 1 - \alpha - \beta$ and $0 \le \alpha ,\beta ,\gamma \le 1$, where:

    $\left| {\overrightarrow {{x_{1} x_{2} }} \times \overrightarrow {{ x_{1} x_{3} }} } \right| = det\left| {\begin{array}{*{20}c} {\hat{i}} & {\hat{j}} & {\hat{k}} \\ {\delta_{i2} - \delta_{i1} } & {\delta_{j2} - \delta_{j1} } & {\delta_{k2} - \delta_{k1} } \\ {\delta_{i3} - \delta_{i1} } & {\delta_{j3} - \delta_{j1} } & {\delta_{k3} - \delta_{k1} } \\ \end{array} } \right|$

    $$ \left| { \overrightarrow {{x_{h} x_{2} }} \times \overrightarrow {{ x_{h} x_{3} }} } \right| = det\left| {\begin{array}{*{20}c} {\hat{i}} & {\hat{j}} & {\hat{k}} \\ {\delta_{i2} - \delta_{ih} } & {\delta_{j2} - \delta_{jh} } & {\delta_{k2} - \delta_{kh} } \\ {\delta_{i3} - \delta_{ih} } & {\delta_{j3} - \delta_{jh} } & {\delta_{k3} - \delta_{kh} } \\ \end{array} } \right| $$

    and $\left| { \overrightarrow {{ x_{h} x_{3} }} \times \overrightarrow {{ x_{h} x_{1} }} } \right| = det\left| {\begin{array}{*{20}c} {\hat{i}} & {\hat{j}} & {\hat{k}} \\ {\delta_{i3} - \delta_{ih} } & {\delta_{j3} - \delta_{jh} } & {\delta_{k3} - \delta_{kh} } \\ {\delta_{i1} - \delta_{ih} } & {\delta_{j1} - \delta_{jh} } & {\delta_{k1} - \delta_{kh} } \\ \end{array} } \right|$.

• 5. Stop when the last triplet from matrix M is reached.

Example 2

Consider the randomly generated supermatrix depicted in Table 15 which is associated to an ANP model with three criteria and three alternatives (Sava 2016).

Table 15 A randomly generated supermatrix

We perturb simultaneously all the entries of rows 2, 3 and 4 of the supermatrix in Table 15, using relation (3), with a perturbation increment $\alpha = 0.02$. The level of perturbation $\alpha$ was chosen to result in fewer than one million perturbed matrices. The number of perturbation levels is $m = 99$ for each of the three criteria perturbed, so the matrix of perturbations has 970,300 rows. Different perspectives of the perturbation regions for each of the three alternatives most preferred within the perturbation space $\left[ { - 1;1} \right]^{3}$ are represented in Fig. 6.

Fig. 6

Preference regions within the three-dimensional perturbation space

The three-dimensional perturbation space $\left[ { - 1;1} \right]^{3}$ is partitioned into three regions, thus we need to determine the piecewise triangular meshes that approximate the nonlinear boundaries between regions A1–A2 and A2–A3, using only the boundary points between the pairs of regions. Figure 7 shows the boundary points used as input to the algorithm described at the beginning of the appendix. The points were obtained from the initial sample of 970,300 perturbations, using, as a selection criterion, $\left| {\varvec{v}\left( i \right) - \varvec{v}\left( j \right)} \right| < \varepsilon = 10^{ - 4}$. To obtain “enough” points for generating good approximations to the triangular meshes, we used a more lenient threshold for the three-dimensional case.

Fig. 7

Boundary points between pairs of alternatives for $\varepsilon = 10^{ - 4}$

The algorithm described at the beginning of this section was implemented in MATLAB to generate the Delaunay triangulation, and consequently the boundaries, between the preference regions. Figure 8 shows multiple perspectives of the approximated piecewise triangular boundaries, within the three-dimensional perturbation space, between regions A1–A2 and A2–A3, respectively.

Fig. 8

Approximated piecewise triangular boundaries within the three-dimensional perturbation space

An example of the combinations of boundary points generating the triangles within the piecewise triangular boundary between preference regions A1–A2, and the associated planar equations defining the boundary is presented below:

Boundary A1–A2

$$ M_{12} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\varvec{V}1} & {\varvec{V}2} & {\varvec{V}3} \\ {208} & {282} & {263} \\ {171} & {157} & {147} \\ \end{array} } \\ \cdots \\ {\begin{array}{*{20}c} {468} & {482} & {428} \\ {483} & {493} & {482} \\ \end{array} } \\ \end{array} } \right] \to \left\{ {\begin{array}{*{20}c} {0.0016x + 0.0024y + 0.0032z + 0.00104 \le 0} \\ { - 0.066x + 0.0288y + 0.0056z + 0.0547 \le 0} \\ \ldots \\ { - 0.0036x + 0.0032y + 0.0028z + 0.00399 \le 0} \\ { - 0.0004x + 0.0004y + 0.0004z + 0.00041 \le 0} \\ \end{array} } \right. $$

The total number of planar equations describing the boundary between A1–A2 is 922.