# Tools to identify linear combination of prognostic factors which maximizes area under receiver operator curve

- Nicolae Todor†
^{1}Email author, - Irina Todor†
^{2}and - Gavril Săplăcan†
^{3}

**4**:10

https://doi.org/10.1186/2043-9113-4-10

© Todor et al.; licensee BioMed Central Ltd. 2014

**Received: **15 April 2014

**Accepted: **19 June 2014

**Published: **4 July 2014

## Abstract

### Background

The linear combination of variables is an attractive method in many medical analyses targeting a score to classify patients. In the case of ROC curves the most popular problem is to identify the linear combination which maximizes area under curve (AUC). This problem is complete closed when normality assumptions are met. With no assumption of normality search algorithm are avoided because it is accepted that we have to evaluate AUC n^{d} times where n is the number of distinct observation and d is the number of variables.

### Methods

For d = 2, using particularities of AUC formula, we described an algorithm which lowered the number of evaluations of AUC from n^{2} to n(n-1) + 1. For d > 2 our proposed solution is an approximate method by considering equidistant points on the unit sphere in R^{d} where we evaluate AUC.

### Results

The algorithms were applied to data from our lab to predict response of treatment by a set of molecular markers in cervical cancers patients. In order to evaluate the strength of our algorithms a simulation was added.

### Conclusions

In the case of no normality presented algorithms are feasible. For many variables computation time could be increased but acceptable.

## Keywords

## Background and previous results

In oncology one of the most used endpoint is treatment response. Let’s denote by *D* the associated variable. There are two possible values: *D* = 1 if the patient responds to treatment and *D* = 0 if the patient has no response.

Let’s suppose that there are two prognostic factors and let’s denote by *X*_{1} and *X*_{2} the random variable associated. *X*_{1} and *X*_{2} could be numeric or ordinal and the patient is getting better or worse as the value is smaller or bigger. For simplicity of the talk we suppose that both are numeric.

*X*is one of

*X*

_{1}or

*X*

_{2}and

*c*is a value from the range of

*X*then the sensitivity (

*Se*) or value "true positive" (TP) of variable

*X*for

*c*value is the probability that

*X*>

*c*for the patients which have a positive response to treatment:

*Sp*) is the probability that

*X*≤

*c*for the patients which have no response "true negative" (TN):

*FP*) value defined by

*X*, "receiver operating characteristics" (ROC) curve [1] is the curve formed with the points

for all possible values of *c*.

Area under curve (AUC) "measures" the potential influence of the random variable on treatment response. AUC values are between 0.5 and 1 and if they are in the proximity of 1 the variable is more important in the process of response prediction.

*c*

_{1}<

*c*

_{2}< … <

*c*

_{ n }, the ROC curve is formed by joining the points

For continuous variables with unknown distributions the simplest way to evaluate AUC is to take a random sample and to build the polygonal line as for discrete variables.

The theory is similar if the signs > and < are changed each other in previous definitions. In practice it is chosen an increasing sequence *c*_{1} < *c*_{2} < … < *c*_{
n
}or a decreasing sequence *c*_{1} > *c*_{2} > … > *c*_{
n
} so that AUC > 0.5.

Major interest is to test equality of AUC with 0.5.

If we have a unique random variable from all studied variables which has AUC > 0.5, at chosen significance level, than we can use this variable as prediction instrument.

If exists multiple variables with AUC > 0.5 emerges the problem of multivariate prediction counting on all variables.

Let’s suppose first that we have only two random variables. First natural variant is to choose a linear combination of the two variables as a global instrument of response prediction.

*α*

_{1},

*α*

_{2}) so that global random variable

induces a maximal AUC.

*X*

_{1},

*X*

_{2}) there are

*n*distinct observed values denoted by

*i*;

denote the whole number of patients without response, with response respectively.

The ideea to solve frontal the problem without supplementary hyoptheses was generally rejected because at first sight the algorithms that evaluate AUC for all possible cases are complicated and this needs longer times to solve even for lower values of *n* and even with the help of computers.

Usually this problem is solved adding supplementary conditions or hypoteses to variables X_{1} and X_{2} [2–8]. In [9, 10] there are two comprehensive surveys. The problem is completly solved only when normality is supposed for variables X_{1} and X_{2}. As sofware we have to mention SAS solution of [11] for normality case.

Present paper for a pair of variables (*X*_{1}, *X*_{2}) shows a reasonable algorithm which evaluates AUC for at most *n*(*n* - 1) + 1 times where *n* is the number of distinct values of the sample. For more than two variables it is proposed an algorithm which produces well aproximate solutions.

Firstly we prove some properties of linear combinations of two variables which are the basis of our algorithm. Next paragraph introduces an approximate solution for the case of two and extends the algorithm to more than two variables. An example occured in the cancer reaserch of our lab is presented subsequently. The example is solved with programs showed in Additional file 1. For each program short explanations or comments are inserted. The paper end with a summary of a simmulation on 20 studies with 200 observations each in order to evaluate the reliability of altgorithms.

## Results

### Properties of AUC evaluated for variables formed by linear combinations of two variables

The algorithm from next section is based on some elementary proprieties derived from the calculus formula of AUC.

Let’s suppose that there are two real values *α*_{1}, *α*_{2} fixed and we try to evaluate AUC for the linear combination

*Z* = *α*_{1}*X*_{1} + *α*_{2}*X*_{2} with the observations shown at (1) and (2).

the sample values of *Z* variable.

*Z*is

with *z*_{1}, *z*_{2}, …, *z*_{
n
} sorted ascending. In practice it is chosen ascending or descending order of *z*_{1}, *z*_{2}, …, *z*_{
n
} so that AUC ≥ 0.5 but the results are similar.

#### Property 1

For (*α*_{1}, *α*_{2}) fixed, ROC curve depends only by the order (increasing or decreasing) in which values *z*_{1}, *z*_{2}, …, *z*_{
n
} are.

**Proof** For fixed *α*_{1}, *α*_{2} let’s denote *T*(*α*_{1}, *α*_{2}) = {*z*_{
i
} = *α*_{1}*x*_{1i} + *α*_{2}*x*_{2i}|*i* = 1, …, *n*}

If *T*(*α*_{1}, *α*_{2}) has *m* distinct elements *t*_{1} < *t*_{2} < … < *t*_{
m
} and if *I*_{
t
} denotes the set of indexes so that the variable *Z* takes value *t*: *I*_{
t
} = {*i*|*z*_{
i
} = *t*} then from (5) and (6) ROC curve depends only by the set $\mathit{M}\left({\mathit{\alpha}}_{1},{\mathit{\alpha}}_{2}\right)=\left\{{\mathit{I}}_{{\mathit{t}}_{1}},\dots ,{\mathit{I}}_{{\mathit{t}}_{\mathit{m}}}\right\}$.

#### Property 2

Each point located on a line through origin determines same ROC curves.

**Proof** For a fixed pair*α*_{1}, *α*_{2}, {(*λα*_{1}, *λα*_{2})|*λ* real, *λ* ≠ 0} is the line through origin. It produces same ROC curve due to fact that *M*(*λα*_{1}, *λα*_{2}) = *M*(*α*_{1}, *α*_{2}) for any *λ*.

#### Property 3

Each pair of values *i*_{1} ≠ *i*_{2} with ${\mathit{z}}_{{\mathit{i}}_{1}}={\mathit{z}}_{{\mathit{i}}_{2}}$ determines a line trough origin and the points of this line generate same ROC curve.

**Proof**Let’s suppose that at least two values from the set

*T*(

*α*

_{1},

*α*

_{2}) are equal. Let’s denote

*i*

_{1},

*i*

_{2}two indexes with

*i*

_{1}≠

*i*

_{2}and ${\mathit{z}}_{{\mathit{i}}_{1}}={\mathit{z}}_{{\mathit{i}}_{2}}$ that is

In the plane *α*_{1}0*α*_{2}, (7) is the equation of a line that passes through origin.

#### Property 4

The set of points (*α*_{1}, *α*_{2}) where ROC curve has same value is convex.

**Proof** The forth property shows that if $\mathit{M}\left({\mathit{\alpha}}_{1},{\mathit{\alpha}}_{2}\right)=\mathit{M}\left({\mathit{\alpha}}_{{}_{1}}^{\text{'}},{\mathit{\alpha}}_{{}_{2}}^{\text{'}}\right)$ for $\left({\mathit{\alpha}}_{1},{\mathit{\alpha}}_{2}\right)\ne \left({\mathit{\alpha}}_{{}_{1}}^{\text{'}},{\mathit{\alpha}}_{{}_{2}}^{\text{'}}\right)$ then $\mathit{M}\left({\mathit{\alpha}}_{{}_{1}}^{"},{\mathit{\alpha}}_{{}_{2}}^{"}\right)=\mathit{M}\left({\mathit{\alpha}}_{1},{\mathit{\alpha}}_{2}\right)=\mathit{M}\left({\mathit{\alpha}}_{{}_{1}}^{\text{\'}},{\mathit{\alpha}}_{{}_{2}}^{\text{\'}}\right)$ for any point $\left({\mathit{\alpha}}_{{}_{1}}^{"},{\mathit{\alpha}}_{{}_{2}}^{"}\right)$ located on the segment determined by (*α*_{1}, *α*_{2}) and $\left({\mathit{\alpha}}_{{}_{1}}^{\text{'}},{\mathit{\alpha}}_{{}_{2}}^{\text{'}}\right)$. The proof comes from the observation that for any point $\left({\mathit{\alpha}}_{{}_{1}}^{"},{\mathit{\alpha}}_{{}_{2}}^{"}\right)$ on the segment (*α*_{1}, *α*_{2}) and $\left({\mathit{\alpha}}_{{}_{1}}^{\text{'}},{\mathit{\alpha}}_{{}_{2}}^{\text{'}}\right)$ there is a real number *λ* ∊ [0, 1] so that ${\mathit{\alpha}}_{{}_{1}}^{"}=\mathit{\lambda}{\mathit{\alpha}}_{1}+\left(1-\mathit{\lambda}\right){\mathit{\alpha}}_{{}_{1}}^{\text{\'}}$ and ${\mathit{\alpha}}_{{}_{2}}^{"}=\mathit{\lambda}{\mathit{\alpha}}_{2}+\left(1-\mathit{\lambda}\right){\mathit{\alpha}}_{{}_{2}}^{\text{\'}}$. We show that the order of values *z*_{1}, *z*_{2}, …, *z*_{
n
} remains unchanged also for $\left({\mathit{\alpha}}_{{}_{1}}^{"},{\mathit{\alpha}}_{{}_{2}}^{"}\right)$.

*i*,

*j*with

*z*

_{ i }<

*z*

_{ j }for both (

*α*

_{1},

*α*

_{2}) and $\left({\mathit{\alpha}}_{{}_{1}}^{\text{'}},{\mathit{\alpha}}_{{}_{2}}^{\text{'}}\right)$ we compute the values of

*z*

_{ i }and

*z*

_{ j }for $\left({\mathit{\alpha}}_{{}_{1}}^{"},{\mathit{\alpha}}_{{}_{2}}^{"}\right)$:

Further if the order of *z*_{1}, *z*_{2}, …, *z*_{
n
} is unchanged for $\left({\mathit{\alpha}}_{{}_{1}}^{"},{\mathit{\alpha}}_{{}_{2}}^{"}\right)$, the ROC curves are identical.

### Algorithm to identify the linear combination of two variables which maximizes AUC

We have to identify in plane (*α*_{1}, *α*_{2}) the regions where AUCs are constants. From previous section we know that these regions are infinite triangles with the peak in origin. These triangles can be defined by the lines coming from (7). The whole number of them is ${\mathit{C}}_{\mathit{n}}^{2}=\frac{\mathit{n}\left(\mathit{n}-1\right)}{2}$ and they divide the plane in maximum *C*_{
n
}^{2} + 1 distinct regions. From the last property *M*(*α*_{1}, *α*_{2}) is constant if (*α*_{1}, *α*_{2}) are in the same region. Now we have to compute AUC for a point from each region and for a point from each line through origin that split two regions. The maximum number of AUC evaluations are *n*(*n* - 1) + 1.

To finish we need a strategy to chose the points where AUC will be evaluated. Our proposition consists of building up an auxiliary line that intersects all lines (7). The intersections with lines (7) generates maximum *C*_{
n
}^{2} - 1 finite segments and two infinite segments. For the finite segments we have chosen the margins and the middles as points to evaluate AUC. For the infinite segments we have chosen points located at distance of one unit from the fixed margin.

and the line passes through the point (0, 1). This slope is lower than all slopes derived from equations (7) so that the intersection points are certain.

Supplementary the points where we evaluate AUC can be chosen normalized conform to second property on the unity circle so that *α*_{1}^{2} + *α*_{2}^{2} = 1.

### Approximate methods to identify the linear combination with maximal AUC

*α*

_{1},

*α*

_{2}) with

*α*

_{2}≠ 0 in the expression

*α*

_{1}

*X*

_{1}+

*α*

_{2}

*X*

_{2}is reduced at the identification of

*α*∊ [ - 1, 1] in

*X*

_{1}+

*αX*

_{2}and then the interval [ - 1, 1] is divided in 201 equal segments. The maximal value is from the set of AUC on each segment extremity. Our proposition is to consider on unity circle all the points where AUC is evaluated. Supplementary from symmetry we need to evaluate AUC only in quadrant I and IV. More exactly we evaluate AUC for (

*α*

_{1},

*α*

_{2}) with

The precision can be improved by dividing quadrant I and IV in more and more regions subsequently. Practically we divide the quadrant I and IV till the divisions are smaller than an apriori limit.

This view permits easy extension when we have more than two prognostic factors.

*X*

_{1},

*X*

_{2}, …,

*X*

_{ f }prognostic factors, with

*f*> 2, extension consists in a method to highlight or to move on the unit sphere in space with

*f*dimensions. Our proposal is to consider for

*α*

_{1},

*α*

_{2}, …,

*α*

_{ f }the following values:

Of course if we want to increase the precision we can increase the number points inside the interval $\left[-\frac{\mathit{\pi}}{2},+\frac{\mathit{\pi}}{2}\right]$.

The authors have a program in Additional file 1 which was used to solve the example from next section.

### Example

In [13] there is an interim result of a study for several molecular markers in relation to response to treatment for cervix cancers. Endpoint was considered the patient status found at 30 days after the end of treatment. We have *D* = 1 or *D* = 0 as the patient presented complete remission or residual tumor at 30 days. It were 14 patients with *D* = 1 and 12 patients with *D* = 0.

From univariate analysis were retained: Vascular Endothelial Growth Factor Receptor (VEGFR) (AUC = 0.74, p = 0.02), dimesion of tumor (AUC = 0.73, p = 0.001) and age (AUC = 0.67, p = 0.06). Logistic model for multivariate analysis [14] did not validate any linear combination of these factors.

Due to this failure we built a program associated to the method described in paragraph 3 (see Additional file 1).

We started by dividing quadrant I and IV in 50 parts. Linear combination that maximizes the AUC for this division has solution:

{0.998027, -0.0608178, 0.0156154}

and AUC = 0.815476.

Dividing the I-st and IV-th quadrant in 100 parts yields the following solution

{0.998027, -0.0602973, 0.017518},

{0.998027, -0.0608178, 0.0156154},

{0.995562, -0.0939226, 0.00590911}

and AUC = 0.815476.

For 150 parts the solution is

{0.998027, -0.0604775, 0.0168856},

{0.998027, -0.0608178, 0.0156154},

{0.996493, -0.0835127, 0.00525418}

and AUC = 0.815476

For 200 parts the solution is

{0.996917, -0.0753438, 0.0218894},

{0.996917, -0.0756783, 0.0207032}

and AUC = 0.821429.

For 300 parts the solution is

{0.997314, -0.0694128, 0.02336},

{0.997314, -0.0696537, 0.0226318},

{0.997314, -0.0698868, 0.0219012},

{0.997314, -0.0701124, 0.0211682},

{0.997314, -0.0714744, 0.0159764},

{0.997314, -0.0716378, 0.0152271}

and AUC = 0.821429.

As can be seen increased number of divisions for 50, 100 and 150 does not change the maximum of AUC but increases the number of points where maximum AUC value is reached.

For 200 and 300 divisions the same area under the curve with very small increase for AUC of 0.00595238 makes us believe that we are close to global solution.

Note that both scores have the values of p highly significant, and we propose the solution that has higher AUC.

Furthermore criteria of classification from ROC curve analysis [9] tells us for this choice that patients with score higher than 1.425782 are patients from whom we expect a better result (Se = 0.71, Sp = 0.92).

### Simulation

As previous example has a small number of observations we have made a simulation for 20 studies with 200 observations each with three prognostic factors. For the first factor, cases were selected from a pseudonormal variable with mean 1 and standard deviation of 3 and controls from a pseudonormal variable with mean 3 and standard deviation 3.5. The second and third prognostic factor, also come from a pseudonormal variable with standard deviation of 3 and 3.5 respectively for cases and controls and with averages of 4 and 6 for controls respectively 6 and 6.5 for cases.

**Results of 20 simulations with 200 observations**

Crt.Nb. | Time | AUC50 | Time | AUC100 | Time | AUC200 | AUC100 - AUC50 | AUC200 - AUC100 |
---|---|---|---|---|---|---|---|---|

1 | 1314s (0H 21M 54 s) | 0.7091 | 5033 s (1H 23M 53 s) | 0.7098 | 20354 s (5H 39M 14 s) | 0.7098 | 0.0007 | 0.0000 |

2 | 1283s (0H 21M 23 s) | 0.6589 | 5154 s (1H 25M 54 s) | 0.6589 | 31636 s (8H 47M 16 s) | 0.6589 | 0.0000 | 0.0000 |

3 | 1501s (0H 25M 1 s) | 0.6406 | 5842 s (1H 37M 22 s) | 0.6412 | 23352 s (6H 29M 12 s) | 0.6412 | 0.0006 | 0.0000 |

4 | 1173s (0H 19M 33 s) | 0.6862 | 4681 s (1H 18M 1 s) | 0.6862 | 25012 s (6H 56M 52 s) | 0.6867 | 0.0000 | 0.0005 |

5 | 1277s (0H 21M 17 s) | 0.6629 | 10790 s (2H 59M 50s) | 0.6633 | 12321 s (3H 25M 21 s) | 0.6638 | 0.0004 | 0.0005 |

6 | 1353s (0H 22M 33 s) | 0.6715 | 4574 s (1H 16M 14 s) | 0.6717 | 15292 s (4H 14M 52 s) | 0.6726 | 0.0002 | 0.0009 |

7 | 1342s (0H 22M 22 s) | 0.6761 | 5132 s (1H 25M 32 s) | 0.6772 | 18625 s (5H 10M 25 s) | 0.6773 | 0.0011 | 0.0001 |

8 | 1297s (0H 21M 37 s) | 0.6944 | 6988 s (1H 56M 28 s) | 0.6953 | 18813 s (5H 13M 33 s) | 0.6954 | 0.0009 | 0.0001 |

9 | 1070s (0H 17M 50s) | 0.6988 | 5399 s (1H 29M 59 s) | 0.6990 | 19498 s (5H 24M 58 s) | 0.6994 | 0.0002 | 0.0004 |

10 | 536 s (0H 8M 56 s) | 0.6638 | 3022 s (0H 50M 22 s) | 0.6640 | 18556 s (5H 9M 16 s) | 0.6646 | 0.0002 | 0.0006 |

11 | 1329s (0H 22M 9 s) | 0.6900 | 4766 s (1H 19M 26 s) | 0.6902 | 20419 s (5H 40M 19 s) | 0.6906 | 0.0002 | 0.0004 |

12 | 1288s (0H 21M 28 s) | 0.6946 | 5086 s (1H 24M 46 s) | 0.6948 | 20573 s (5H 42M 53 s) | 0.6948 | 0.0002 | 0.0000 |

13 | 637 s (0H 10M 37 s) | 0.6873 | 2454 s (0H 40M 54 s) | 0.6875 | 21271 s (5H 54M 31 s) | 0.6875 | 0.0002 | 0.0000 |

14 | 513 s (0H 8M 33 s) | 0.7031 | 2025s (0H 33M 45 s) | 0.7032 | 20139 s (5H 35M 39 s) | 0.7040 | 0.0001 | 0.0008 |

15 | 952 s (0H 15M 52 s) | 0.7200 | 2082s (0H 34M 42 s) | 0.7202 | 21224 s (5H 53M 44 s) | 0.7204 | 0.0002 | 0.0002 |

16 | 1176s (0H 19M 36 s) | 0.7401 | 4923 s (1H 22M 3 s) | 0.7413 | 27836 s (7H 43M 56 s) | 0.7413 | 0.0012 | 0.0000 |

17 | 796 s (0H 13M 16 s) | 0.7398 | 4332 s (1H 12M 12 s) | 0.7399 | 18213 s (5H 3M 33 s) | 0.7405 | 0.0001 | 0.0006 |

18 | 1296s (0H 21M 36 s) | 0.6635 | 2534 s (0H 42M 14 s) | 0.6638 | 20165 s (5H 36M 5 s) | 0.6644 | 0.0003 | 0.0006 |

19 | 797 s (0H 13M 17 s) | 0.7041 | 3407 s (0H 56M 47 s) | 0.7045 | 20313 s (5H 38M 33 s) | 0.7045 | 0.0004 | 0.0000 |

20 | 1420s (0H 23M 40s) | 0.6825 | 5532 s (1H 32M 12 s) | 0.6826 | 15051 s (4H 10M 51 s) | 0.6826 | 0.0001 | 0.0000 |

Average | 1117s (0H 18M 37 s) | 4687 s (1H 18M 7 s) | 20433 s (5H 40M 33 s) |

Average execution time for 50, 100 and 200 segments was 18 minutes, 1 hour and 18 minutes, 5 hours and 40 minutes which is an acceptable time for a practical problem.

## Discussions and conclusions

Multivariate analysis is used largely in any medical paper. However testing the hypotheses in modeling is not a very simple task and this is the reason for trying a lot of potential models and choose the model best suited to observations. The papers of [4, 6, 15–19] prove that there is a large basis to use linear combinations of variables in ROC analysis. If we do not have solid condition to apply for example one of the cited models, the method from our paper produces always a score for which we have maximal AUC or an approximate.

On the other hand in a classical model of regression it is known that the numerical methods used to identify the model parameters not always provide a global maximum and depends heavily on the initial values of the algorithm. The solution presented we believe could be used there as a baseline for these algorithms.

The main advantage of presented algorithms is that it always provides a solution. However for many prognostic factors and observations, time of the calculation could be a problem.

Certainly, approximate method is more appropriate in this last case despite the fact that it does not guarantee a global solution. However it is guaranteed to yield a solution with AUC higher than each variable taken separately.

Our algorithm can be used in any medical paper as an alternate method for multivariate analysis.

The presented algorithms have major advantage to provide always a solution with no supplementary constraints.

For many variables computation time is high but not high enough as not to accept this cost.

## Notes

## Declarations

### Acknowledgements

The authors thank to Nagy Viorica for the data comming from his cited study. The authors thank to Company for Applied Informatics with his executive director Nas Sorin for their’s upport of research.

## Authors’ Affiliations

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