Philippe Jehiel

Investment strategy and selection bias: An equilibrium perspective on overoptimism

Introduction

A key aspect of entrepreneurial activity consists in deciding whether and when to make investments on the basis of the ex ante impressions decision makers get from the projects that reach them. In a Bayesian framework, the impression would be modelled as a signal received by the decision maker. Together with the knowledge of how signals, returns and costs are jointly distributed, the decision maker would be able to make the optimal investment decision using the standard Bayesian updating machinery. Yet, it is not clear how decision makers would know the joint distribution of signals, returns and costs.

Instead, decision makers are likely to look at implemented projects and consider the realized returns observed in those projects so as to build statistics (even informally) about what impressions imply in terms of the distribution of returns. [START_REF] Bénabou | The economics of motivated beliefs[END_REF] A natural approach consists for the decision maker in aggregating the returns observed in those implemented projects that look similar to the current project of interest in the sense of delivering the same impression to the decision maker from an ex ante viewpoint.

The decision maker would invest in a current project associated with some impression a if the returns observed over the implemented projects associated to the same impression a for him are su¢ ciently high in a statistical sense that may in general depend on the risk tolerance of the decision maker as well as his estimate of the costs. For example, if the decision maker is risk neutral and knows the cost c -an assumption that is maintained for simplicity in the rest of the paper-he would invest if the observed mean return exceeds the cost c in the considered pool, and he would not invest otherwise.

Given a pool of implemented projects, the heuristics just proposed would give rise to a new distribution of implemented projects for the current generation of decision makers, and this new distribution would itself be used for the derivation of the investment strategy of the next generation of investors. Assuming stationarity in the arrival of new projects and decision makers, I am interested in understanding the properties of the steady states generated by such dynamic systems, which I will refer to as equilibria with sampling investors.

When analyzing such equilibria, I will be assuming that the impressions obtained by investors about a given project need not be the same across investors. More precisely, every investor's impression about a given project will be modelled as an independent signal realization drawn from a distribution assumed to be common to all investors and that typically depends on the return of the project. I will also be assuming that a higher realization of investors' signals is more representative of a higher return in the sense that the joint distribution of the return and investors' signals satis…es the monotone likelihood ratio property. A canonical illustration (routinely considered in …nance models, see for example Grossman and Stiglitz, 1976) stipulates that conditional on the return x, the impression of investor i takes the form a i = x + " i where " i is the investor-speci…c realization of a normal distribution with mean 0 and variance . While such a speci…cation will be used to illustrate the main …ndings, I am allowing for more general speci…cations of the distribution of impressions and returns.

The main …ndings of the paper are as follows. In equilibrium, no matter what impression investors get from their project, sampling investors have an overly optimistic assessment of the expected returns of their project. As a result, there is more investment in the equilibrium with sampling investors than what the optimal strategy based on the knowledge of the distribution of signals and returns would dictate. Modelling the impression as a noisy signal about the return as described above, I note that the overoptimism bias is more pronounced for intermediate realizations of the impressions, and that the welfare loss induced by the excessive investment is lowest either when the noise is small or when it is very signi…cant so that the biggest welfare loss is obtained for intermediate levels of informativeness of the impressions. Finally, when investors are either rational (making the optimal investment decisions) or sampling, I note that the overoptimism and overinvestment biases of the sampling investors are all the more severe that the share of rational investors is bigger, thereby illustrating a negative externality that rational investors impose on sampling investors.

The overoptimism and overinvestment biases identi…ed in this paper are the consequences of two ingredients: the selection neglect implicit in the heuristics used by sampling investors and the hypothesis that the impressions given by the same project are not the same across investors. [START_REF] Bénabou | Self-con…dence and personal motivation[END_REF] Each e¤ect is key for the derivation of the biases.

Clearly, as a direct implication of the law of large numbers, if decision makers could have access also to non-implemented projects, the heuristics of sampling investors (now allowing for the aggregation of both implemented and non-implemented projects according to the delivered impression) would lead them to have the correct estimate of the expected return for each possible impression. But, the samples considered by sampling investors are biased in that they include only projects that were implemented, and these are not randomly drawn projects. The sampling heuristics can be viewed as re ‡ecting a kind of selection neglect in that it treats the biased samples as if they were not biased. Selection neglect has been documented in a number of psychological studies (Nisbett and Ross, 1980), and even if decision makers were aware of selection bias, it would be very hard to fully adjust for it. [START_REF] Woo | Entrepreneurs'perceived chances of success[END_REF] But, selection neglect alone is not enough to explain the overoptimism and overinvestment biases arising in the equilibrium with sampling investors. If all investors were to get the same impression about any given project, there would be no pro-investment bias in equilibrium. Biases would not arise in this case because assuming investment occurs with positive probability for some impression, the sample of implemented projects associated with that common impression would be unbiased. The dispersion of beliefs even when exposed to the same objective facts has been documented in various …elds in particular in experts'surveys about in ‡ation expectations (Mankiw et al., 2004) or in relation to traders'reactions to public announcements (Kandel and Pearson (1995)).

Likewise, it is most likely at work regarding investors'impressions about objectively similar projects.

An illustrative example:

To illustrate some of the main …ndings, consider a setting in which the returns can take two equally likely values x, x with the cost c lying in between these two values, i.e.

x < c < x. Assume decision makers can get three possible impressions labelled Good, M edium and Bad, and when the return is high (resp. low), the decision maker gets an impression that is either Good (resp. Bad) or M edium each with probability half. Thus, when the impression is Good, it is optimal to invest, since a Good impression can only come form a high return project. Similarly, when the impression is Bad, it is optimal to not invest, since a Bad impression can only come from a low return project. Assuming that c > x+x 2 , it is optimal to not invest when the impression is M edium, since given the symmetry of the problem, Bayesian updating would then tell the decision maker that the two returns x and x are equally likely.

Consider a sampling investor who would observe in his pool only projects handled by rational investors. Since rational investors invest only when their impression is Good, the pool of implemented projects would all have high returns. Thus, a sampling investor would choose to invest when getting impression M edium, since half of the implemented projects would give him impression M edium and all of them would correspond to a high return. In the equilibrium with sampling investors only, decision makers invest more than in the rational case, but potentially less than a sampling investor would do when surrounded with rational investors only. The reason why the investment decisions of sampling investors may be altered is that the presence of sampling investors results in the presence of low return projects in the pool of implemented projects, and such a compositional e¤ect reduces the pro-investment bias, even if it does not eliminate it, as implied by the main result of the paper. More precisely, within the proposed example, in the equilibrium with sampling investors only, when c < 2x+x 3 decision makers invest when they get impressions Good or M edium, but when c > 2x+x 3 , only a fraction of projects associated to impression M edium is implemented by sampling investors and the perceived expected return associated to that impression coincides exactly with the cost c in equilibrium. [START_REF] Daniel | Overcon…dent investors, predictable returns, and excessive trading[END_REF] Related literature:

This paper can be viewed as bringing together ideas from the literatures on over-con…dence, bounded rationality, and econometrics combining them in a novel way. The econometrics literature has been discussing at length selection bias but essentially from the viewpoint of the analyst, assuming economic agents are perfectly rational (see Heckman, 1979). By contrast, this paper assumes that sampling investors are subject to selection neglect, and it analyzes the consequences this may cause on the e¢ ciency of the decision making. The literature on overcon…dence has documented that entrepreneurs tend to be overly optimistic about their projects (see for example Cooper et al. (1988) or Malmendier and Tate (2005)), which has generally been used to justify that investors rely on subjective priors or attach excessive precision to the signals they receive (see for example Xiong (2013) or Daniel and Hirshleifer (2015) for such a use in …nance models). By contrast, this paper derives the overoptimism bias from selection neglect, assuming only the data about implemented projects are available to investors. As mentioned above, the equilibrium approach pursued in this paper allows me to relate the degree of overoptimism to the informativeness of the objective signals received by the investors, which can in principle be tested (and would not be implied by the subjective prior approach). [START_REF] Denrell | Vicarious learning, undersampling of failure, and the myths of management[END_REF] Finally, the literature on bounded rationality has developed various solution concepts allowing for misspeci…ed expectations (see in particular, the analogy-based expectation equilibrium (Jehiel, 2005), and the cursed equilibrium (Eyster and Rabin, 2005)), and it has sometimes connected such equilibrium approaches to selection bias (see in particular the behavioral equilibrium (Esponda, 2008)). This paper adopts an equilibrium perspective in line with the solution concepts just mentioned, but unlike the previous approaches it applies it to pure decision problems and derives the overoptimism bias in such non-strategic contexts. [START_REF] Gruber | Survivorship bias and mutual fund performanc[END_REF] 5 Theoretical approaches to overcon…dence that complement the one discussed in this paper include: 1) Rabin and Schrag (1999) who derive overcon…dence from another psychological bias, the con…rmatory bias that leads agents to sometimes behave as if they had not made observations that go against their current beliefs, 2) Van den Steen (2004) who de…nes overcon…dence as the subjective belief that one performs better than others, which Van den Steen derives from a revealed preference argument in a subjective prior world -Van de Steen's insight will be further discussed later, and 3) several studies that derive overcon…dence from motivated cognition purposes among which Bénabou and Tirole (2002), Köszegi (2006) or Bénabou (2015). [START_REF] Gruber | Survivorship bias and mutual fund performanc[END_REF] Viewing nature as a player, I suggest later how the equilibrium with sampling investors can be viewed 6

Relating erroneous judgements to selection neglect appears in other studies. For example, Denrell (2003) discusses how one can wrongly believe that risky projects are associated with high mean performance if data on failed projects are not accessible or less visible (essentially because large negative shocks tend not to be recorded in the accessible data). [START_REF] Esponda | Behavioral equilibrium in economies with adverse selection[END_REF] However, to the best of my knowledge, this paper is the …rst to consider an equilibrium approach to this in the sense of viewing the pool of data agents have access to as resulting from the erroneous judgments agents make based on their observations.

The endogeneity discussed here is essential for example for the understanding of why the presence of extra rational investors makes the overoptimism bias worse (which is consistent with the empirical …nding of Lerner and Malmendier (2013) who observe that a higher share of entrepreneurial peers decreases entrepreneurship, see subsection 4.3 for elaborations).

The rest of the paper is organized as follows. Section 2 presents the investment problem. Section 3 analyzes the overoptimism and overinvestment biases arising in the equilibrium with sampling investors. Section 4 develops further insights in particular allowing for a mix of rational and sampling investors, and adding dynamic considerations in relation to cycling and the convergence to steady state.

The investment problem

A large number of investors idealized as a continuum is considered. Each investor assumed to be risk neutral has to decide whether or not to invest in one project that is di¤erent for as an analogy-based expectation equilibrium after appropriately de…ning the extensive-form game and the required analogy partitions (see the working paper version Jehiel (2015) for more details). I believe such connections are useful in that they allow to develop a unifying theme of the e¤ects of imperfect learning over di¤erent applications. In a recent paper, Spiegler (2016) discusses how to interpret the equilibrium with sampling investors using the tool of Bayesian networks that he has recently introduced into economics. It should be mentioned that his interpretation requires that decision makers would have access to the joint distribution of signals and returns, and in this case I would argue decision makers should be able to make the optimal investment decisions. [START_REF] Esponda | Behavioral equilibrium in economies with adverse selection[END_REF] In a very di¤erent context, Streufert (2000) discusses the idea that parents of poor neighborhoods may not consider the data of successful youngsters who would leave the neighborhood, and as a result would have a downard biased perception of the returns to schooling. As Denrell though, he does not discuss how the schooling decisions generated by such erroneous perceptions would a¤ect the pool of data from which agents base their estimates. each investor. The cost of every project is c. The return of a project is random and can take various possible values x in a set X R (assumed for simplicity to consist of …nitely many values). Before making his decision, an investor knows the cost c but does not know the return realization x of his project. However, he observes a signal realization a for his project. The signal realization a can be thought of as representing the impression that the investor gets from the project. It takes values in (a; a) with a < a (where I allow that a = 1 and a = +1). [START_REF] Eyster | Cursed equilibrium[END_REF] Based on a, the investor has to decide whether or not to invest. If the project is implemented (i.e., the investor decides to invest), it is accessible by everyone, and the obtained return x is assumed to be freely observable.

Non-implemented projects are not accessible. When an investor has access to a project, he can freely generate an impression similarly to how he generates an impression for his own project. That is, he can observe a signal realization a for every implemented project.

It should be mentioned that investors are not assumed to be observing the impressions of other investors. They only observe their own impressions.

Returns and impressions are generated similarly for all projects and for all investors.

Importantly, I assume that conditional on a return realization, the impressions of two di¤erent investors are statistically independent so as to re ‡ect that impressions are in ‡uenced by e¤ective returns but investors'impressions (about the same objective project) are heterogeneous. Speci…cally, for each project, the probability that the return realization be x 2 X is l(x) 0 with X x2X l(x) = 1. Conditional on the realization x of a project, the signal realization a observed by any investor for this project is assumed to be distributed according to the density f ( j x) with support (a; a), and two di¤erent investors i and i 0 get two independent draws a i , a i 0 from this distribution. Assuming that the distribution of a takes the form of a density will simplify the exposition of the analysis but is not required (both the example in introduction and one example below assume a can take …nitely many values). For concreteness, one may think of the situation in which the impression a would take the form x + " where x is the return and " is the realization of a noise term for example distributed according to a normal distribution with mean 0 and variance . While such a speci…cation will serve as the leading example to illustrate the main results, the analysis developed next allows for much more general speci…cations of the distributions of the impression as a function of the return.

If the investor knew how the signal realization a and the return realization x are jointly distributed, he could adjust the optimal strategy. Given the assumed risk neutrality, the optimal investment strategy requires the investor to invest upon observing a whenever E(x j a) > c and to not invest whenever E(x j a) > c where E(x j a) is derived from l( ) and f ( j x) by Bayes'law, i.e.,

E(x j a) = P x2X l(x)f (a j x) x P x2X l(x)f (a j x) :
In the following, I will assume that the optimal strategy requires that, for some signal realizations a, it is best to invest.

Importantly, I am assuming that neither l( ) nor f ( j ) is a priori known to the investor. Instead, I am embedding the above investment environment into a multi-period framework in which in every period, a new cohort of investors has to make investment decisions similar to the ones just described, and investors design their investment strategy by considering the past data accessible to them. According to the above observability assumptions, an investor in a given period can freely access those past projects that were implemented, observe the corresponding returns and derive the associated signals/impressions according to the process described above. I am assuming that in order to decide whether or not to invest, investors adopt the following heuristic. When getting a signal realization (an impression) a for his current project, the decision maker gathers all projects in the past for which he gets the same signal realization (impression) a. Then he computes the empirical mean return in those projects (this only requires averaging the x observed in those projects having the same a signal realization), and he invests whenever the obtained empirical mean return is above the cost c, and he does not invest otherwise. I will consider the steady states of such a dynamic system and refer to the resulting investment strategies as equilibria with sampling investors (in Section 4, I brie ‡y consider whether and when the dynamic system converges to a steady state). In order to rule out trivial situations in which there would be no investment at all, I will also assume that whatever the observed signal there is a tiny probability (assumed to be the same for all signal realizations) that the decision maker invests.

To de…ne formally an equilibrium with sampling investors, let q(a) denote the (steady state) probability with which an investor observing a would invest, and assume q is bounded away from 0 for some positive measure of signals. The probability of observing an implemented project with return x conditional on the impression a being in A (a; a) would be

Pr(x j a 2 A; implemented; q) = l(x) Pr(a 2 A, implemented j x; q) X x 0 2X l(x 0 ) Pr(a 2 A, implemented j x 0 ; q) where Pr(a 2 A, x j implemented; q) = Pr(a 2 A j x)

Z a a q(b)f (b j x)db
given that a randomly drawn project with return realization x would be implemented with probability Z a a q(b)f (b j x)db (the investor in charge of such a project would receive signal b according to the density f ( j x) and invests then with probability q(b)).

Thus, the empirical mean return of implemented projects for which the investor gets the signal realization a would be b v(a; q) = P x2X l(x)f (a j x)

Z a a q(b)f (b j x)db x P x2X l(x)f (a j x) Z a a q(b)f (b j x)db
as results from the induced proportion of projects with return x in the pool of implemented projects associated to impression a. This leads to the following de…nition: 9

De…nition 1 An investment strategy q( ) over (a; a) that induces some investment with positive probability is an equilibrium with sampling investors if q(a) > 0 implies b v(a; q) c and q(a) = 0 implies b v(a; q) c. [START_REF] Grossman | Information and competitive price systems[END_REF] To present formally the equilibrium with trembles, one may de…ne the set Q n of q such that q(a)

1 n for all a, de…ne an1 n -equilibrium to be q n ( ) such that q n (a) > 1 n implies b v(a; q n ) c and q n (a) = 1 n implies b v(a; q) c; and de…ne an equilibrium to be the limit as n grows large of q n such that q n is an

The strategy just de…ned is the result of a …xed point. The probabilities q(b) with which other investors choose to invest when getting signal b a¤ect the compositions of return x projects in the pool of implemented projects, which in turn a¤ects the probability with which an investor getting signal a is willing to invest. In equilibrium, these two probability mappings should be the same.

Comments. 1) In the working paper version (see Jehiel, 2015), I envision projects as being described by strings of attribute realizations and each individual investor as observing just one attribute realization. Assuming there are as many attributes as there are investors and that each attribute realization has the same distribution conditional on the return realization, one gets a formulation similar to the one developed above. When there are …nitely many attributes, the observations of two di¤erent investors would not be independent conditional on the return realization, which would result in some extra complications (see the section on correlation in Jehiel, 2015). 2) One may interpret an equilibrium with sampling investors as de…ned above as an analogy-based expectation equilibrium (Jehiel, 2005) in which nature would be considered as a player, and the game would let nature …rst select a project describing the return realization and the vector of signal realization for every investor, then the investor would have to decide whether or not to invest on the basis of the observed signal and in case of investment nature would implement the return realization. The analogy partition of a given investor required to support the above equilibrium with sampling investors consists in bundling all the decision nodes of nature regarding the (second) choice of return that correspond to the same signal realization of the considered investor (see Jehiel (2015) for further details in the multi-attribute speci…cation).

3 Overoptimism as a result of selection bias I analyze the above investment environment assuming that a higher signal realization is more representative of a higher return. Such a condition is satis…ed whenever conditional on x, a is distributed according to a normal distribution with mean x and variance , and it is satis…ed for many other speci…cations. In particular, it is without loss of generality whenever there are two possible return realizations x = x, x, since then a can be reordered so that the likelihood ratio f (ajx) f (ajx) is increasing with a. Formally, the following monotone likelihood ratio property is assumed to hold: Assumption (MLRP): For any a 0 > a and x 0 > x, it holds that: f (a 0 jx 0 ) f (ajx 0 ) > f (a 0 jx) f (ajx) . [START_REF] Heckman | Sample selection bias as a speci…cation error[END_REF]

Equilibrium characterization

Proposition 1 Under MLRP, there exists a unique equilibrium with sampling investors.

The equilibrium is such that for some threshold a S , a decision-maker chooses to invest if the observed signal realization a is above a S and to not invest otherwise where a S is uniquely de…ned by

P x2X f (a S j x)[1 F (a S j x)]l(x) x P x2X f (a S j x)[1 F (a S j x)]l(x) = ( c if a S = a c if a S 2 (a; a) (1)
Proof of Proposition 1:

Suppose that, in equilibrium, investment occurs with probability q(a) when a 2 (a; a) is observed. As already highlighted, the perceived expected return of a project with signal realization a would then be:

b v(a; q) = P x2X l(x)f (a j x) R a a q(b)f (b j x)db x P x2X l(x)f (a j x) R a a q(b)f (b j x)db
Since a ! b v(a; q) is increasing (whatever q( )) by MLRP, one can infer that investors must follow a threshold strategy, i.e. for some z, invest if a > z and do not invest if a < z

where z (if interior) is de…ned by b v(z; q) = c. De…ne H(a; z) = P x2X l(x)f (a j x)[1 F (z j x)] x P x2X l(x)f (a j x)[1 F (z j x)] (2)
This is the perceived expected return of an a-project when other investors follow the z-threshold investment strategy. One has:

Lemma 1 Under MLRP, H( ; ) is increasing in a and z.

Proof of Lemma 1. The monotonicity in a has already been noted. The monotonicity in z follows from the observation that under MLRP, the hazard rate f (zjx)

1 F (zjx)

decreases with x (see any textbook or the monotone likelihood ratio entry of wikipedia)

and thus x !

@ @z [1 F (zjx)] 1 F (zjx) = f (zjx) 1 F (zjx) increases with x. Q.E.D.
Given the assumption that there is some investment in the rational case, it follows that H(a; a) > c. An equilibrium must employ a threshold strategy z as already noted (by the monotonicity of H( ; ) in a) and the threshold z must satisfy

H(z; z) = 8 > > < > > : c if z = a c if z 2 (a; a) c if z = a (3)
Given that H(a; a) > c, the monotonicity of H( ; ) in the second argument implies that H(a; a) > c, and thus the latter case can be ignored. Suppose then that H(a; a) < c. The continuity of H ensures that there exists z 2 (a; a) satisfying H(z; z) = c. Hence, there must exist z > a satisfying (3).

Consider now a z 1 > z 2 a. Clearly, H(z 1 ; z 1 ) > H(z 2 ; z 2 ) and ( 3) cannot be simultaneously satis…ed for z = z 1 and z 2 . One concludes that there is only one equilibrium, and that this equilibrium is a threshold equilibrium a S where a S is uniquely de…ned to satisfy (1). Q.E.D.

Given that equilibrium is unique, one can unambiguously speak of the equilibrium subjective value that an investor observing the signal realization a assigns to the expected mean return of the project. It is denoted by v S (a). Using the de…nition (2) of H( ; ), one has v S (a) = H(a; a S ) where a S is as de…ned in Proposition 1.

Comparison to the rational benchmark

A rational investor is de…ned to be one who makes the optimal investment decision based on the true statistical distributions as de…ned by the densities f ( j ) and the probabilities l( ). Accordingly, upon observing the realization a of the signal, a rational investor rightly perceives the expected return of the project to be v R (a) = E(x j a) =

P x2X l(x)f (a j x) x P x2X l(x)f (a j x)
A rational investor invests if this value is above c and does not otherwise.

Let a R 2 (a; a) be uniquely de…ned by [START_REF] Jehiel | Analogy-based expectation equilibrium[END_REF] v R (a R ) =

( c if a R = a c if a R > a
A rational investor invests whenever a > a R and he does not when a < a R .

Using the H( ; ) function introduced in (2), it is readily veri…ed that v R (a) = H(a; a)

(since for all x, F (a j x) = 0), and thus a R is uniquely de…ned by

H(a R ; a) = ( c if a R = a c if a R > a
There are two ways to think of a rational decision maker, as just described. One way is to hold the view that a rational investor knows l( ) and f ( j ) to start with and does the corresponding Bayesian updating when observing a, as already suggested. Another way is to hold the view that a rational investor is an experienced decision maker who has had su¢ ciently many learning opportunities to …nd out the investment strategy (de…ned as a function of the observed signal) that delivers the highest expected payo¤.

Whatever the interpretation of the rational investment strategy, I note that the sampling heuristic used by decision makers to assess the expected return of a-projects in the equilibrium shown in Proposition 1 leads them to have an overly optimistic perception as compared with the rational perception, since H(a; a S ) is bigger than H(a; a) (the rational assessment) due to the monotonicity of H( ; ) in its second argument. This observation and its implication for the volume of investment is summarized in the next Proposition [START_REF] Jehiel | Analogy-based expectation equilibrium[END_REF] Uniqueness comes again from MLRP, which ensures that v R ( ) is increasing in a. The fact that a R < a comes from the assumption that there is investment with positive probability in the optimal solution.

whose complete proof appears in the Appendix.

Proposition 2 Under MLRP, in equilibrium, sampling investors overvalue the expected returns of a-projects as compared with the rational benchmark whatever the signal observation a, i.e. v S (a)

v R (a) for all a. There is at least as much investment in the equilibrium with sampling investors as in the rational benchmark. That is, a S a R .

It is tempting to think that the overoptimism arising in the equilibrium with sampling investors is the mere consequence of the fact that the subset of projects investors have access to is biased toward having higher returns. However, this property alone would not necessarily give rise to the overoptimism bias without the additional statistical structures assumed in the main model. In particular, assume in contrast to the main model that the impressions that di¤erent investors get about a given project are always the same across investors (they may of course di¤er across projects). This can be modeled by assuming that for every project with return x, every investor observes the same signal a drawn from the distribution f ( j x) (unlike in the main model in which two di¤erent investors were assumed to get independent draws from f ( j x)). In this case, letting q(a) denote the steady state probability that there is investment whenever the (common) signal realization is a and assuming that q(a) > 0 (which would always hold true if some trembling behavior were assumed), the subjective assessment of a sampling investor observing the signal realization a would be:

P x2X l(x)f (a j x)q(a) x P x2X l(x)f (a j x)q(a)
:

This expression simpli…es into the rational expression v R (a) shown above, thereby implying that there is no overoptimism bias in the corresponding equilibrium with sampling investors. Note that there is no bias even though the sample investors have access to is biased toward the high return ones (since in equilibrium only those projects for which a > a R would be voluntarily implemented).

Thus, the overoptimism bias identi…ed in Proposition 2 requires the combination of the sampling heuristic and the assumption that the impressions produced by any project are heterogeneous across investors. As it turns out, the MLRP assumption also plays a role in the result. While the MLRP assumption is a very natural one (it is without loss of generality if x can take only two values and it holds whenever a is a noisy signal about

x for many speci…cations of the noise distribution), I note that the result of Proposition 2 would not necessarily hold if the MLRP assumption were removed (still assuming as in the main model that two di¤erent investors get independent draws from f ( j x) for a project with return x). This is illustrated through the following example.

Example 1 Returns x can take four values x = 2; 1; 1; 2 which are equally likely (i.e., l(x) = 1=4 for all x); a can take three values a = a 1 ; a 2 ; a 3 ; cost c is 0. The distribution of a given x is summarized in the following table in which the number at the intersection of the a i row and the x column -referred to as p i (x)-is the probability that signal a i is drawn conditional on the return realization being x. The rational investment strategy requires that there is investment when a = a 1 or a 2 but not when a = a 3 .12 Noting that X

x p i (x)(p 1 (x)+p 2 (x))
x is positive for i = 1 but negative for i = 2; 3 implies that if all other investors were following the rational investment strategy, a sampling investor would choose to invest when observing a 1 but not when observing a 2 or a 3 . The equilibrium with sampling investors would then take the following form: invest when observing a 1 , invest with probability such that X x p 2 (x)(p 1 (x) + p 2 (x)) x = 0 when observing a 2 and not invest when observing a 3 . Overall, there would be less investment than in the optimal strategy, and the subjective assessment attached to an a 2 project -which would coincide with c = 0-would be lower than the rational assessment.

Informativeness, overoptimism, and overinvestment

In order to illustrate the …ndings obtained so far, it may be useful to consider the following simple scenario. The cost c is normalized to 0. Return x can take two values x = 1 or x = 1 with the same probability (l(x) = 1=2 for x = 1 and 1). Conditional on x, the signal a takes the form a = x + " where " is the realization of a normal distribution with mean 0 and variance . In such a symmetric setting, the optimal strategy requires to invest if a > 0 and to not invest otherwise. That is, a R = 0 (whatever ). After simple arrangements, the threshold a S that arises in the equilibrium with sampling investors is characterized by

P DF (a S + 1)(1 CDF (a S + 1)) = P DF (a S 1)(1 CDF (a S 1))
where P DF and CDF stand respectively for the density and the cumulative of the normal distribution with mean 0 and variance . Clearly, a S < 0 given that P DF (1) = P DF ( 1)

and CDF (1) > CDF ( 1), which con…rms the general result of Proposition 2. In the next graph, the overoptimism bias v S (a) v R (a) known to be positive by Proposition 2 is depicted as a function of a, assuming the variance of the normal distribution is 1. [START_REF] Kandel | Di¤erential interpretation of public signals and trade in speculative markets[END_REF] 1.

FIGURE 1

13 By way of comparison, v R (0) = 0 and v R (+1) = 1.

Interestingly, one observes that the bias is maximal for intermediate values of a and goes to 0 as a approaches 1 or +1. This is no coincidence. When a approaches either a = 1 (respectively, a = +1), the likelihood that the return is x = x (resp.

x) becomes very large. [START_REF] Köszegi | Ego utility, overcon…dence, and task choice[END_REF] More generally, there would be no bias for extreme realizations of a (for a close to a or a) whenever x can take two values x = x or x, x < c < x and

f (ajx)
f (ajx) approaches 0 (resp. +1) as a approaches a (resp. a). This is because as a mere consequence of extreme signals being very informative, an investor observing such a signal realization would mostly see projects with homogeneous return realizations in his sample, thereby making the selection bias negligible.

Another quantity I consider now is the welfare loss induced in the equilibrium with sampling investors as compared with the rational benchmark. In the context of the example, the welfare loss can be expressed as As can be seen, the welfare loss is single-peaked, converges to 0 as goes to 0 or +1, and is thus maximal for intermediate values of the overall informativeness of the signal as parameterized here by . Such properties would hold more generally. Indeed, when the signal is very informative, the e¤ect of the selection bias on the performance of the decision made by sampling investors becomes negligible as already noted, and thus when signals are globally informative with very high probability (as is the case when is large), the aggregate welfare loss is small. At the other extreme, when the signals are poorly uninformative, in generic situations, either investors would always invest or they would never invest, and when there is investment it would have to be optimal. [START_REF] Malmendier | CEO overcon…dence and corporate investment[END_REF] Hence, beyond the canonical example considered here, the maximal welfare loss would arise for intermediate levels of the informativeness of the signals.

W L = 1 2
As illustrated above, the degree of overoptimism arising in the equilibrium with sampling investors and the welfare consequences of it depend on the informativeness of signals. [START_REF] Mankiw | Disagreement about in ‡ation expectations[END_REF] Such a dependence would not necessarily arise in the subjective prior approach to overoptimism, which typically puts no structure on how investors from their subjective prior and thus on how overoptimism varies with the primitives of the model. It may be mentioned here that within the subjective prior paradigm, Van den Steen (2004) makes the interesting and simple observation that as a consequence of a revealed preference argument, no matter how subjective priors are modelled, others' decisions always look (weakly) suboptimal from the subjective viewpoint of any agent, thereby leading to the systematic subjective belief that one performs better than others. Such a relative overoptimism bias would also arise in my setting. If one were to ask any given investor at an ex ante stage (i.e., before he receives a signal realization for his project) whether he thinks he would perform better than other investors, he would be a¢ rmative. Relative overoptimism arises here for the very same reason highlighted by Van den Steen that the investor believes (based on his subjective perception of the mapping between his perception and assesses the mean return of a project with signal realization a , and it requires in equilibrium that if a is interior, this perceived mean return should be equal to the cost c.

The di¤erence with the main model is that when a sampling investor makes an observation of another project, with probability she is facing a rational investor who invests only if the signal realization observed by the rational investor is larger than the rational threshold a R (as de…ned in subsection 3.2), and with probability 1 she is facing another sampling investor who invests if the signal realization he observes is larger than a .

Given that conditional on the return realization, the signals are independently distributed across investors, the above expression follows.

Denote the above threshold a by a S ( ). One has previously seen that when there is no rational investor around ( = 0), it holds that a S (0) a R . The e¤ect of on a S ( ) is unambiguously given by: Proposition 3 Under MLRP, the higher the share of rational investors, the more severe the pro-investment bias of sampling investors. That is, a S ( ) is weakly decreasing in , and for all , a S ( ) a S (0) a R .

The intuition behind Proposition 3 whose detailed proof appears in the Appendix is simple. If an investor is surrounded with more rational decision makers, the decisions made by others are better, and thus when sampling from these to form an assessment regarding the pro…tability of the project it appears to the investor that the project is even more pro…table. The selection bias is more severe, which leads the investor to make a poorer decision. In some sense, rational investors exert a negative externality on those investors who follow the sampling heuristic.

It is natural to consider the e¤ect of an increase of on welfare. Given Proposition 3, an increase of deteriorates the welfare of sampling investors, but at the same it increases the share of rational investors whose welfare is larger. Aggregating these two e¤ects leads to ambiguous comparative statics in general. When the share of rational investors is su¢ ciently large, an increase of always enhances expected welfare (essentially because there are too few sampling investors who su¤er from the negative externality imposed by rational investors). When the share of rational investors is su¢ ciently away from 1, the negative e¤ect on sampling investors of increasing may dominate for some distributional assumptions resulting in an overall negative impact of increasing the share of rational investors on expected welfare. [START_REF] Nisbett | Human inference: Strategies and shortcomings of social judgment[END_REF]

Convergence to steady state

Embedding the above framework into a dynamic setting in which new cohorts of investors sample from previous cohorts of investors naturally leads to asking when we should expect to see convergence to steady state as considered in the main analysis. Another legitimate concern is whether the overoptimism bias identi…ed in the main analysis would still arise in case there would be no convergence. To model the dynamics most simply, consider within the MLRP scenario discussed above a sequence of time periods t = 1; 2; ::: Assume that in every period t > 1 there is a new cohort of investors of the same mass who sample from the implemented projects handled by the cohort of investors living in period t 1, and assume to …x ideas that in the …rst period investors choose to invest whatever signal they observe.

In such a dynamic setting, investors in period t would adopt a threshold strategy z t specifying to invest if the observed signal realization a is above z t and to not invest otherwise where the sequence of z t would be characterized inductively by z 1 = a (since the …rst generation of investors was assumed to invest always) and for all t > 1, the threshold z t+1 would be uniquely de…ned by H(z t+1 ; z t ) = c (assuming H(a; z) < c < H(a; z) for all [START_REF] Nisbett | Human inference: Strategies and shortcomings of social judgment[END_REF] To illustrate these two points, consider a two return x, x scenario with x < c < x and l(x) = l(x) = 1=2. Simple calculations yield that the marginal e¤ect of on global welfare dW d can be written as A+B

2 where A = Z a R a S ( ) (f (a j x)(c x) f (a j x)(x c))da B = (1 ) da S ( ) d (f (a S ( ) j x)(c x) f (a S ( ) j x)(x c))
A (resp B) is shown to be postive (resp. negative) using the MLRP property, f (a R j x)(c x) = f (a R j x)(x c), a S ( ) < a R and da S ( ) d < 0. When is close to 1, B becomes negligible and thus dW d > 0. When is away from 1, A can be made small relative to B by having a su¢ ciently small probability that signal realizations a fall in (a S ( ); a R ) (this is consistent with MLRP which only requires that f (ajx) f (ajx) is increasing in a, but puts no restriction on how likely the various a are). In such cases, dW d < 0 holds. In the leading example in which conditional on x, a = x + " where " is drawn from a normal distribution with variance = 1, one can show that dW d > 0 for all . z). It appears that z 2 coincides with a R , and using the monotonicity of H, it can be shown by induction that the sequence (z 2k+1 ) k 1 is weakly decreasing and satis…es z 2k+1 a S for all k while the sequence (z 2k ) k 1 is weakly increasing and satis…es z 2k a S for all k where a S is the equilibrium threshold de…ned in Proposition 1. Thus, (z 2k+1 ) k 1 converges to z and (z 2k ) k 1 converges to z with z a S z . If z = z = a S the system converges to the steady state described in Proposition 1. If z < a S < z , the system converges to a limit two-period cycle in which in odd periods there is less activity as dictated by the threshold strategy z and in even periods there is more activity as dictated by the threshold strategy z . Whether the system converges or cycles depends on how the slope @H=@z @H=@a compares to 1. When it is uniformly lower than 1, (as is the case for the leading example with variance = 1), there is convergence. When it is larger than 1 in the neighborhood of a = z = a S , the two-period limit cycle prevails. [START_REF] Rabin | First impressions matter: A model of con…rmatory bias[END_REF] It should be noted that in the above dynamics whether or not there is convergence, the overoptimism and overinvestment biases hold in every period (this follows from the monotonicity of H and the observation that H(a R ; a) = c). Moreover, since z t a R for all t and z 2 = a R , the monotonicity of H implies that the smallest z t which corresponds to the most biased investment strategy is obtained in period 3 when the samples considered by the current cohort consist of projects handled by rational investors. In all subsequent periods, because sampled investors adopt suboptimal strategies, the sampling heuristic leads to less severe biases.

Cycling with heterogeneous investors

It is natural to combine dynamics as just considered with the possibility that investors could vary in their degree of sophistication, some of them being rational and others being subject to selection neglect as proposed in the main model. A full- ‡edged dynamic model along these lines would aim at endogenizing entry and exit of entrepreneurs, assuming for example entrepreneurs' sophistication vary with their experience. Analyzing such a model is clearly beyond the scope of this paper. Yet, in order to illustrate that some rich dynamics can be expected, consider the following stylized setting. In each period t = 1; 2; ::: a new cohort of agents decides whether or not to become entrepreneur. Every entrepreneur faces the same distribution of projects as described above but agents may have di¤erent outside options assumed to be drawn independently across agents from a distribution with cumulative G. In every period, the share of rational agents is and the share of sampling agents is 1 . Let w R denote the expected payo¤ a rational investor gets by becoming an entrepreneur (i.e., w R = E(max v R (a) c; 0)), and let w S ( ) denote the expected payo¤ a sampling investor subjectively expects to get when facing a share (resp. 1 ) of rational (resp. sampling) investors. [START_REF] Spiegler | Data monkeys: A model of naive extrapolation from partial statistics[END_REF] Rational agents become entrepreneur whenever their outside option falls below w R , i.e. with probability G(w R ). Sampling agents who would sample from a mix of rational investors and 1 of sampling investors would become entrepreneur with probability G(w S ( )). Thus assuming the cohort of (sampling) agents in period t samples from the implemented projects in period t 1, the share t of rational investors in period t would follow the dynamic: 1)G(w S ( t 1 )) :

t = G(w R ) G(w R ) + (
As can be inferred from the above analysis, w S ( ) is increasing in . Thus, a higher share of rational investors in period t would lead more sampling agents to become entrepreneurs in period t + 1, which would result in a lower share of rational investors in period t + 1.

Depending on the shape of G, such a dynamic system may either converge to a limit share of rational investors or lead to long term cycling between high and low shares (away and respectively above and below ) of rational investors, corresponding respectively to low and high levels of entrepreneurial activity. [START_REF] Streufert | The e¤ect of underclass social isoçlation on schooling choice[END_REF] The prediction that there is more chance to become entrepreneur if one is exposed to fewer (better skilled) entrepreneurial cases should be subject of further empirical investigation, but it seems in agreement with Lerner 1)G(w S ( )) and if G has su¢ cient mass around w S ( ) one should expect cycling to emerge.

  (a + 1) P DF (a 1))da representing the aggregate loss due to the suboptimal implementation of projects with signal realizations a falling in (a S ; 0). The following graph depicts how W L varies with the variance .[START_REF] Lerner | With a little help from my (random) friends: Success and failure in post-business school entrepreneurship[END_REF]

FIGURE 2

 2 FIGURE 2

and 1 )( 1 F([ 1 )( 1 F

 1111 Malmendier's (2013) …nding that a higher share of entrepreneurial peers decreases[START_REF] Spiegler | Data monkeys: A model of naive extrapolation from partial statistics[END_REF] With the notation previously introduced, w S ( ) = E[max(H(a; a S ( )) c; 0)] where the density ofa is f (a) = P x2X f (ajx)[((a S ( )jx))+ (1 F (a R jx))]l(x) P x2X (a S ( )jx))+ (1 F (a R jx))]l(x) R ) G(w R )+(

The very reason why statistics would in general rely excusively on such data is that data on nonimplemented projects are typically not accessible.

The monotone likelihood ratio property also plays a role as illustrated later through an example.

From a theoretical viewpoint, correcting the bias would require some structural knowledge about how impressions are generated and how other investors process the data. While dispensing from a parametric knowledge, the knowledge that the impressions of di¤erent investors are iid conditional on the return realization would still be required and such a knowledge cannot be inferred from the data (see[START_REF] Manski | Measuring expectations[END_REF] for related discussions as to why correcting the selection bias requires a lot of knowledge). From an empirical viewpoint, Elton et al.(1996) show that the bias persists in the context of assessing the performance of funds even though everyone there is aware that funds have a tendency to disappear when they perform poorly and thus that the sample consisting of still alive funds is not representative of all funds.

Observe that if the impressions obtained by di¤erent investors were always the same, investment when the impression is M edium would not be possible. By contradiction, if investment sometimes occured with the M edium impression, a sampling investor would rightly get at the conclusion when the impression is M edium that there is an equal chance that the return is high or low (unlike in the dispersed impression scenario in which some projects delivering impression M edium had been decided by investors who had got a Good impression), and thus he would not invest.

In the sequel, for any continuous function g( ), I will refer to lim a!a g(a) (resp. lim a!a g(a)) as g(a) (resp. g(a)).

n -equilibrium. Our assumption that the optimal investment strategy involves non-trivial investment will imply that all equilibria must result in positive probability of investment.

Assuming f ( j x) is smooth, this can be formulated as requiring that @f (ajx)=@a f (ajx) is increasing in x.

This follows because Xx p i (x) x > 0 for i = 1; 2 for not for i = 3.

This is so because P DF (a S + 1)=P DF (a S 1) ! +1 (resp. 1) as a ! 1 (resp. +1).

By way of comparison, the value of the expected maximal welfare is 0:5 when = 0, 0:34 when = 1, and 0 when = 1.

The considered example is non-generic in the sense that if investors receive no signal, they are indi¤erent as to whether or not to invest, but the conclusion still holds true in this case, since whether or not they invest it is equally good.

This can in principle be the subject of empirical tests. In contexts in which market conditions are very clear or very unclear (so that signals are either very informative or very uninformative), one should observe less overoptimism bias than in contexts in which market conditions are partially predictable.

If investors were sampling from all previous cohorts rather than just the most recent one, I suspect the convergence scenario would be made more likely (because such a sampling device would smoothen the reaction to previous behaviors), but more work is needed to establish this formally.

Lerner and Malmendier also note that the reduction is driven by a reduction in unsuccessful entrepreneurial ventures, which can be related to our …nding that when sampling from a more active cohort the investment decisions get closer to the rational ones.

the investment decision) that he can screen projects better than others. [START_REF] Manski | Measuring expectations[END_REF] 4 Further insights 4.1 When rational investors exert negative externalities Suppose the population of investors is mixed. A share 1 of investors (referred to the sampling investors) proceeds as described in the main model: They observe a signal realization a for their project, sample all implemented projects in which they get the same signal realization a, and invest if the observed empirical mean return exceeds the cost c. A share of investors (referred to as rational investors) makes the optimal investment decision based on the observation of the signal realization. Signals and returns are distributed as in the main model.

It should be noted that in the sample considered by the sampling investors, there are both projects held by sampling investors and by rational investors. Since the decision rule is not the same for sampling and rational investors, the selection bias is typically a¤ected by the heterogeneity of the population of investors. The purpose of the next Proposition is to investigate the e¤ect of cognitive heterogeneity on the performance of sampling investors (rational investors are una¤ected by the presence of sampling investors given that they face a decision problem and they behave optimally).

To pave the way toward the main result of this subsection, observe that sampling investors follow in equilibrium a threshold strategy that consists in investing in a project with signal realization a only if a exceeds a where a is de…ned by

The left hand-side of this expression represents how a sampling investor subjectively [START_REF] Manski | Measuring expectations[END_REF] Formally, his perceived ex ante payo¤ would be given by E(v S (a) c j a > a S ) Pr(a > a S ) whereas his perception of other investors'performance would be correctly given by E(v S (a)) c where the density of a is the one arising in the pool of implemented projects, i.e., f S (a) = X

) is increasing and v S (a S ) = c, the former expression is larger than the latter, con…rming Van den Steen's relative overoptimism insight in the present setting.

Appendix

Proof of Proposition 2: The …rst part, v S (a) v R (a), is proven noting that v S (a) = H(a; a S ), v R (a) = H(a; a) and using the monotonicity of H in z (see Lemma 1). As for the second part, whenever a R < a, one has that H(a R ; a) c and thus H(a R ; a R ) c

by the monotonicity of H in its second argument. This implies that a S a R , as desired.

Q.E.D.

Proof of Proposition 3. De…ne H(a; z; ) =

Lemma Under MLRP, H is increasing in a and z. It is decreasing in for z a R .

Proof. H increasing in a follows directly from MLRP.

H increasing in z follows from the observation that

) is decreasing in x, which is proven in the same way as the decreasing hazard rate property. [START_REF] Van Den Steen | Rational overoptimism (and other biases)[END_REF] H decreasing in for z a R follows because

1 F (zjx) is decreasing in x (which follows from the fact MLRP implies the …rst order stochastic dominance property noting that F (ajx) 1 F (zjx) is the cumulative of F conditional on x and a being no smaller than z and that MLRP still holds when we truncate the support of a). Q. E. D.

Proving that a S ( ) is smaller than a R follows by noting that H(a R ; a R ; ) H(a R ; 0; 0).

Proving that a S ( ) is decreasing follows by noting that for an interior solution H(a S ( ); a S ( ); ) = c, and thus if 0 > , H((a S ( ); a S ( ); 0 ) c (by the monotonicity of H in ), which implies that a S ( 0 ) a S ( ) (by the monotonicity of H in a and z). Q.E.D. [START_REF] Van Den Steen | Rational overoptimism (and other biases)[END_REF] Speci…cally, integrate f (a 1 j x 1 )f (a 0 j x 0 ) f ((a 0 j x 1 )f (a 1 j x 1 ) (which holds for all a 1 a 0 , x 1 x 0 ) in a 1 from a 0 to a and multiply by 1 and integrate in a 1 from a B to a and multiply by to obtain that f (a j x 0 ) ( 1)(1 F (a B j x 0 ) + (1 F (a j x 0 ) f (a j x 1 ) ( 1)(1 F (a B j x 1 ) + (1 F (a j x 1 )

as required.


Ana Silva

Tiago Oliveira

Vicente Julian

José Neves

Paulo Novais

A Mobile and Evolving Tool to Predict Colorectal Cancer Survivability

In this work, a tool for the survivability prediction of patients with colon or rectal cancer, up to five years after diagnosis and treatment, is presented. Indeed, an accurate survivability prediction is a difficult task for health care professionals and of high concern to patients, so that they can make the most of the rest of their lives. The distinguishing features of the tool include a balance between the number of necessary inputs and prediction performance, being mobile-friendly, and featuring an online learning component that enables the automatic evolution of the prediction models upon the addition of new cases.

Introduction

The colorectal cancer, is a subtype of cancer, which affects the lower portion of the gastrointestinal tract and develops in the cells lining the colon and rectum [START_REF] Vachani | All about rectal cancer[END_REF]. It can be further divided according to the site where the pathology develops. Colon and rectum cancers are, in fact, different pathologies, with different associated genetic causes and different progressions according to distinct molecular pathways [START_REF] Yamauchi | Colorectal cancer: a tale of two sides or a continuum?[END_REF]. Statistics show that colorectal cancer is the most common form of cancer in the digestive system, the third most common and the fourth deadliest cancer overall [START_REF] Ferlay | Globocan 2012: Estimated cancer incidence, mortality and prevalence worldwide in 2012[END_REF].

The use of machine learning (ML) techniques has been growing in cancer research [START_REF] Kourou | Machine learning applications in cancer prognosis and prediction[END_REF]. The accurate prediction of survivability in patients with cancer remains a challenge namely due to the heterogeneity and complexity of the disease. However, accurate survivability prediction is important for patients with cancer so that they can make the most of the rest of their lives. It is also important to help clinicians to make the best decisions, when palliative care is an essential component of the process. Given that colon and rectal are the most common cancers of the digestive system, one would expect the existence of numerous tools for ascertaining the likelihood of a patient surviving this disease. Although there are some tools for this task, few provide predictions for both colon and rectal cancer, and none of them apply ML techniques in order to build evolving predictive models. Furthermore, their digital support may hinder their consultation at care delivery.

The objective of this work is to present an easy to use tool that provides survivability predictions of colon and rectal cancer patients for 1, 2, 3, 4 and 5 years after diagnosis and treatment. Due to the ubiquitous presence of mobile devices in everyday life and the ease with which one is able to consult these devices and use their applications, we chose to develop this tool as a mobile application. The underlying model for survivability prediction was obtained through ML techniques applied to the data from the Surveillance, Epidemiology, and End Results (SEER) program [START_REF]Surveillance, epidemiology and end results program[END_REF], a large cancer registry in the United States, and arguably the most complete cancer database in the world. The dataset includes records of patients diagnosed with different types of cancer from 1973 to 2012. The focus of this paper will be placed on the mobile solution developed for survivability prediction, but a part of the paper will be dedicated to briefly describing its underlying ML model so as to provide a better comprehension of the work as a whole.

The paper is structured as follows. Section 2 presents related work featuring survivability prediction tools for colon and rectal cancer, with an analysis of their main strengths and limitations. Section 3 describes the selected requirements for the tool, its underlying ML-based predictive model, architecture and a comprehensive use case. Section 4 provides a reflection about the strengths and limitations of our approach. Finally, Section 5 presents the conclusions drawn so far and future work considerations.

Related Work

Existing tools for colon or rectal cancer survivability prediction are mostly available as web applications. Table 1 shows a summary of their main features, namely: (1) whether the application is used for colon or rectal cancer; (2) the number of features necessary to get a prediction; (3) the data set that its underlying model is based on; (4) the technique used to construct the predictive model; [START_REF] Benson | NCCN Clinical Practice Guideline in Oncology Rectal Cancer[END_REF] the type of target prediction it produces; and (6) a measure of performance in the form of a concordance index (C-index). The C-index corresponds to the probability of giving a correct response in a binary prediction problem. It is considered to be numerically equivalent to the area under the ROC curve (AUC) [START_REF] Hanley | The meaning and use of the area under a receiver operating characteristic (roc) curve[END_REF].

There is a disparity in the number of features used in each tool. However, twelve [START_REF] Renfro | Accent-based web calculators to predict recurrence and overall survival in stage iii colon cancer[END_REF] or even nine [START_REF] Bush | Derivation : Nodes + PrognosticFactors Equation for Colon Cancer accuracy of the Nodes + PrognosticFactors equation[END_REF][START_REF] Valentini | Nomograms for Predicting Local Recurrence, Distant Metastases, and Overall Survival for Patients With Locally Advanced Rectal Cancer on the Basis of European Randomized Clinical Trials[END_REF] features may be too much information for a physician to input on-the-fly. Furthermore, there are cases in which the increased number of features does not necessarily translate into a better performance, as can be seen in the direct comparison between the works in [START_REF] Weiser | Predicting survival after curative colectomy for cancer: Individualizing colon cancer staging[END_REF] and in [START_REF] Renfro | Accent-based web calculators to predict recurrence and overall survival in stage iii colon cancer[END_REF].

All the underlying models are based on statistical modelling, most notably on Cox regression analysis [START_REF] Bradburn | Survival Analysis Part II: Multivariate data analysis -an introduction to concepts and methods[END_REF]. This is the dominant multivariate approach used in survivability prediction and corresponds to a multiple linear regression of the hazard on a set of variables. This indicates that the use of soft computing techniques, namely ML, in survivability prediction, especially in colon and rectal cancer, has yet to be fully explored. Since one of the advantages of ML is having more discriminative power in identifying patterns in data and finding nuances that may escape statistical modelling, its usage for survivability prediction may result in models with better performances [START_REF] Kourou | Machine learning applications in cancer prognosis and prediction[END_REF]. As such, ML was chosen as the modelling approach for this work.

Most of the target predictions, either for colon or for rectal cancer, cover a 5-year span [START_REF] Wang | An interactive tool for individualized estimation of conditional survival in rectal cancer[END_REF][START_REF] Weiser | Predicting survival after curative colectomy for cancer: Individualizing colon cancer staging[END_REF][START_REF] Renfro | Accent-based web calculators to predict recurrence and overall survival in stage iii colon cancer[END_REF]. Even though there are models that cover a wider time span [START_REF] Bush | Derivation : Nodes + PrognosticFactors Equation for Colon Cancer accuracy of the Nodes + PrognosticFactors equation[END_REF][START_REF] Chang | Practical application of a calculator for conditional survival in colon cancer[END_REF][START_REF] Valentini | Nomograms for Predicting Local Recurrence, Distant Metastases, and Overall Survival for Patients With Locally Advanced Rectal Cancer on the Basis of European Randomized Clinical Trials[END_REF][START_REF] Bowles | An individualized conditional survival calculator for patients with rectal cancer[END_REF], the five year barrier is an important goal for a colorectal cancer patient to overcome, and is used throughout clinical practice guidelines [START_REF] Benson | NCCN Clinical Practice Guideline in Oncology Colon Cancer[END_REF][START_REF] Benson | NCCN Clinical Practice Guideline in Oncology Rectal Cancer[END_REF] as a turning point for follow-up procedures, in which the vigilance over the patient is lightened, and for the assessment of the recurrence risk. For this reason, the present work will also have a target prediction of five years. Another noteworthy observation is that only two of the tools feature conditional survivability predictions.

To determine if the tools are suitable for mobile devices, the applications were analysed using the mobile-friendly test tool from Google 3 . The results showed that, except for the tools reported in [START_REF] Weiser | Predicting survival after curative colectomy for cancer: Individualizing colon cancer staging[END_REF] and [START_REF] Renfro | Accent-based web calculators to predict recurrence and overall survival in stage iii colon cancer[END_REF], all the others are unsuitable for mobile access. The test revealed that the text was too small to read, the mobile viewport was not set, links were too close to each other and usually the content was wider than the screen. Therefore, few of these applications had a mobile-friendly design. Another goal is to address this by developing a crossplatform tool that is available to users in a practical and intuitive way, through a smartphone or tablet. Throughout the last decade, mobile phones have gone from being simple phones to being handheld pocket-sized computers. Their capabilities, namely the processing and on-board computing capacity incite the development of applications [START_REF] Boulos | How smartphones are changing the face of mobile and participatory healthcare: an overview, with example from ecaalyx[END_REF]. According to data from the International Data Corporation (IDC) Worldwide Quarterly Mobile Phone Tracker, the Android of Google and iOS of Apple are the two most popular smartphone operating systems [START_REF] Corporation | Idc: Smartphone os market share 2015[END_REF].

For the health care industry, mobile applications yielded new boundaries in providing better care and services to patients. Moreover, it is making a revolution in the way information is managed and made available [START_REF] Siau | Mobile healthcare informatics[END_REF]. The portability of mobile applications can increase the productivity of health care professionals. It grants a rapid access to information and multimedia resources, allowing health care professionals to make decisions more quickly with a lower error rate, increasing the quality of patient documentation and improved workflow patterns [START_REF] Ventola | Mobile devices and apps for health care professionals: uses and benefits[END_REF]. This work discloses an assistive tool to help physicians to improve their practice. The problem it addresses is predicting the survivability of colorectal cancer patients in an individualized manner.

Requirements for the Survivability Prediction Tool

Several functionalities were delineated to achieve a solution that covers the limitations mentioned in Section 2 and, at the same time, is able to help physicians to improve their practice. These functionalities are summarized in the following functional requirements for the prediction tool: allow the user to select the cancer type (either colon or rectal) for which he seeks a prediction; allow the user to provide inputs for a set of selected features, based on which the underlying models generate survivability predictions; allow the user to choose the value of an input for a feature from a set of pre-determined values; provide a survivability prediction, according to the inputs, for 1, 2, 3, 4 and 5 years after the diagnosis and treatment; provide a likelihood value for the prediction of each year; to allow the visualization of the predictions and likelihood values in a chart; and allow the insertion of new patient registries into the case database, thus increasing the number of cases for the periodic recalculation of the prediction models.

Additional requirements for the tool are that it should be made available in the two main mobile platforms (iOS and Android) and it should be able to recalculate the prediction models upon the addition of a significant number of new patient registries. This confers a dynamism to the prediction models and should ensure their evolution over time.

Colon and Rectal Cancer Survivability Prediction Models

Survivability prediction was approached as a binary classification problem. The goal was to produce predictions for 1, 2, 3, 4 and 5 years after treatment of colon or rectal cancer. Each classification label (there were five representing years 1, 2, 3, 4 and 5) could only have two values: survived ornot survived. As such, it was necessary to build five survivability prediction models (one per year) for each type of cancer. The created models were based on the SEER dataset. The criteria for selecting patient registries was the same for both colon and rectal cancer. Only patients with age greater than or equal to 18 years old were selected. Patients who were alive at the end of the data collection whose survival time had not yet reached 60 months (five years) and those who passed away of causes other than colon or rectal cancer were sampled out. After preprocessing, 38,592 cases were isolated for colon cancer and 12,818 cases were considered for rectal cancer. From the isolated cases for each pathology, 10% were selected for testing sets. After filtering cases with "unknown" values, the colon cancer testing set had 2,221 cases and the training set had 20,061 cases. The testing set for rectal cancer had 551 cases and the training set had 4,962 cases. In total, the training set had 61 attributes representing possible classification features.

All the phases, from preprocessing to evaluation, were executed using Rapid-Miner 4 , an open source data mining software chosen for its workflow-based interface and an intuitive application programming interface (API).

Using the Optimize Selection [START_REF]RapidMiner: Rapidminer documentation: Optimize selection[END_REF] operator for feature selection with the classification labels as target, a total of 6 features were obtained from a feature selection phase for each cancer type. Their name and description are shown in Tables 2 and3. The training sets for colon and rectal cancer with their respective selected features were used in the learning of multiple prediction models using different ML ensemble methods such as bagging, adaboost, bayesian boosting, stacking, and voting. The accuracy, the AUC and the F-measure were used as performance measures in order to evaluate the models developed for colon and rectal cancer. The accuracy is the percentage of correct responses among the examined cases [START_REF] Bradley | The use of the area under the ROC curve in the evaluation of machine learning algorithms[END_REF]. The F-measure is a combination of precision (a form of accuracy, also known as positive predictive value) and recall (also known as sensitivity) [START_REF] Powers | What the F-measure doesn't measure[END_REF]. The AUC can be interpreted as the percentage of randomly drawn data pairs of individuals that have been accurately classified in the two populations [START_REF] Hanley | The meaning and use of the area under a receiver operating characteristic (roc) curve[END_REF]. These measures were calculated using the training data set and 10-fold cross validation. By applying the testing sets to the models, we calculated the percentage of incorrectly classified cases. The stacking5 [START_REF] Džeroski | Is combining classifiers with stacking better than selecting the best one?[END_REF], using k-NN, decision tree, and random forest classifiers as base learners and a naive bayes classifier as a stacking model learner, was the best performing model for both colon and rectal cancer. Upon prediction, the model is capable of providing a confidence value that represents the likelihood of the prediction. Table 4 shows the performance values of the best model developed, for both cancer types.

As the intent with this paper is to present the features of the developed tool and describe its inner workings, it was considered that an exhaustive description of the ML process was out of scope. The clinically evident regional lymph nodes

AJCC Stage

The grouping of the TNM information combined Primary Site Identification of the site in which the primary tumor originated

Regional Nodes Examined

The total number of regional lymph nodes that were removed and examined by the pathologist

Architecture

The CRCPredictor is a hybrid mobile application targeting smartphones and tablets. The back-end of this tool includes two web services: one to give the survivability prediction responses for colon or rectal cancer to the user and another to recalculate the survivability prediction models. Figure 1 shows the architecture of the CRCPredictor system.

The Survival Prediction App was developed using a hybrid approach, between a web and a native methodology. This allows an abstraction from the native Fig. 1: Architecture of the CRCPredictor system. language of the target operating system while retaining the core features of a native app. A hybrid application is developed by applying web technologies (mainly, HTML5, CSS and JavaScript) and is executed inside a native container on the mobile device. It is suitable for multiple platforms and is distributable through an application store, just like native applications. This type of approach can have an inferior performance compared with native applications. However, nowadays mobile devices have powerful capabilities and the performance gap is hardly noted. The application was developed using AngularJS, Ionic Framework, and Cordova. Cordova wraps the HTML/JavaScript app into a native container which can access the device functions of several platforms [1]. These functions are exposed via a unified JavaScript API, for an easy access to the full native functionalities.

The Survival Prediction Model Server Application was developed to cover the need of an individualized system, able to respond according to a particular set of patient characteristics. It exposes a set of RESTful web services. This service architecture was chosen for being light-weight, easy to access and scalable [START_REF] Zhao | Towards automated restful web service composition[END_REF]. The web services were developed in Java with the Java API for RESTful Web Services (JAX-RS) [2]. The data is sent over the HTML POST method when the health care professional submits the values for the prediction features on the Survival Prediction App. The RESTful web service, using the RapidMiner API, receives the values and feeds them to the corresponding models, encoded in XML files. The response with the survivability predictions for the five years is returned in a JSON format.

The Online Learning Server Application also follows a REST architecture. It handles newly submitted patient data. The outcomes are added to a database for a posterior recalculation of all the models, which keeps them up-to-date. The data is inserted into a NoSQL database and, for each 1000 new registries, the models for the five years are recalculated, generating five new XML files for the type of cancer that just got the thousandth new case. The 1000 mark was arbitrarily defined and can be subject to adjustment.

Use Case

Figure 2a shows the first screen that appears when the Survival Prediction App of the CRCPredictor is initiated. By clicking on the menu (Figure 2b), all options available in this application become visible. A typical use case is getting a prediction for colon cancer survivability. Supposing a physician is treating a patient diagnosed with colon cancer, once the type of cancer in the home screen is set (as shown in Figure 2b), the health care professional inserts the values for the selected features (Figure 3a). All features, except for the age of the patient, are filled in by choosing the value from a list of available options. By submitting a case of a patient with 55 years old, having a positive/elevated carcinoembryonic antigen value, with clinical assessment of regional lymph nodes of not clinically evident, with the primary site of the cancer being in the sigmoid colon, with stage 0 and with 5 as the number of regional nodes examined, the values are sent to the Survival Prediction Model Server Application and the outcome is calculated. The prediction is always provided in the form of confidence values for a positive prediction, i.e., the confidence that the patient will survive. This is displayed in a new screen in the form of a bar chart (Figure 3b). For the stage of the patient, the physician can choose between the TNM system or the grouped stage, known as American Joint Committee on Cancer (AJCC) stage. The results show that, while the model was able to predict with 100% confidence that the patient will survive the first three years, the confidence of his surviving the fourth and fifth years is 0%. To predict the survivability of a patient diagnosed with rectal cancer, the procedure is similar to the one used for colon cancer.

Analysis and Discussion

In terms of inputs the constructed prediction models, for both colon and rectal cancer, require only the input of six selected features. Comparing with the related tools in section 2, this number is inferior to the number of features used in the underlying models of the tools described in [START_REF] Bush | Derivation : Nodes + PrognosticFactors Equation for Colon Cancer accuracy of the Nodes + PrognosticFactors equation[END_REF], [START_REF] Renfro | Accent-based web calculators to predict recurrence and overall survival in stage iii colon cancer[END_REF], [START_REF] Valentini | Nomograms for Predicting Local Recurrence, Distant Metastases, and Overall Survival for Patients With Locally Advanced Rectal Cancer on the Basis of European Randomized Clinical Trials[END_REF], and [START_REF] Bowles | An individualized conditional survival calculator for patients with rectal cancer[END_REF], and is closer to the number of inputs of the remaining prediction tools. The number of input features may be crucial to the adoption or the rejection of a tool, as it may become difficult to use it on-the-fly if too much information is needed. Another aspect to note is that, apart from the age at diagnosis and the AJCC Stage, the feature selection produced two very different sets for colon and for rectal cancer, which is in line with the notion that the two, although having aspects in common, are different pathologies. Regarding the colon cancer features, the age at diagnosis and the AJCC stage are present in most colon cancer prediction tools [START_REF] Chang | Practical application of a calculator for conditional survival in colon cancer[END_REF][START_REF] Weiser | Predicting survival after curative colectomy for cancer: Individualizing colon cancer staging[END_REF][START_REF] Renfro | Accent-based web calculators to predict recurrence and overall survival in stage iii colon cancer[END_REF]. The other selected features are not usually present, but they are closely related to the ones that are. For instance, the clinical assessment of regional lymph nodes is a product of a medical evaluation of a feature widely used in the existing tools [START_REF] Weiser | Predicting survival after curative colectomy for cancer: Individualizing colon cancer staging[END_REF][START_REF] Renfro | Accent-based web calculators to predict recurrence and overall survival in stage iii colon cancer[END_REF] that is the number of lymph nodes found to have cancer out of the lymph nodes isolated during surgery. The same can be said about the selected features for rectal cancer, i.e., they are, at the very least, closely related to the ones used in other prediction tools.

In [START_REF] Al-Bahrani | Colon cancer survival prediction using ensemble data mining on seer data[END_REF], the use of ML ensemble models to develop survival prediction models for colon cancer is described. The modelling component of our work is similar to that approach; therefore, it is possible to compare the performance of our selected model with theirs. The classification accuracies reported in [START_REF] Al-Bahrani | Colon cancer survival prediction using ensemble data mining on seer data[END_REF] for years 1, 2, and 5 were 90.38%, 88.01%, and 85.13%. The reported AUCs were 0.96, 0.95, and 0.92, respectively. As shown in Table 4, we were able to improve the classification results with our models. This direct comparison is not possible for the rectal cancer model as it was not possible to find such a closely related work in the literature. However, it is possible to verify that the rectal cancer model performs worse than the colon model in every metric, possibly due to the smaller size of the training set used in the learning process. At the same time, both the colon and rectal cancer models showed low classification errors on the randomly selected test data sets. Additionally, when comparing the AUCs of the generated models in Table 4 with the C-indexes in Table 1, it is possible to conclude that the generated models show a better discriminative power than the currently available models.

Regarding the CRCPredictor system, it fulfils the requirements defined at the beginning of the work. The distinguishing features of the system's architecture are its flexibility and scalability, which make the addition of new features (services) simple and easy. The Survival Prediction App was developed as a mobilefriendly application, enabling the easy access of health care professionals to its functionalities on their mobile devices. Another component that distinguishes this system from established tools is the Online Learning Server Application which ensures the continuous evolution of the prediction models. However, the system does not provide conditional survivability predictions, which makes it less appealing when compared with the works in [START_REF] Chang | Practical application of a calculator for conditional survival in colon cancer[END_REF], [START_REF] Valentini | Nomograms for Predicting Local Recurrence, Distant Metastases, and Overall Survival for Patients With Locally Advanced Rectal Cancer on the Basis of European Randomized Clinical Trials[END_REF], and [START_REF] Bowles | An individualized conditional survival calculator for patients with rectal cancer[END_REF], as this is a type of information that health care professionals generally like to know.

Conclusions and Future Work

The main contribution of this work is a survivability prediction tool for colon and rectal cancer. Its distinguishing features are a balance between the number of necessary inputs and prediction performance, being mobile-friendly, and featuring an online learning component that enables the automatic recalculation and evolution of the prediction models upon the addition of new cases. The goal with this tool is to facilitate the access of health care professionals to instruments capable of enriching their practice and improving their results. Future work on the tool includes the development of conditional survivability models that allow the user to get a prediction knowing that the patient has already survived a number of years after diagnosis and treatment. Additionally, we intend to conduct experiments to assess how well the tool fulfils the needs of health care professionals and identify aspects to improve. Additionally, the models presented herein will be considered for inclusion in a guideline-based decision support system, described in [START_REF] Novais | Moving towards a new paradigm of creation, dissemination, and application of computer-interpretable medical knowledge[END_REF], as a dynamic knowledge complement to the static recommendations of clinical practice guidelines. Since colon and rectal cancer affect mostly the elderly, this survivbility prediction apllication can be used within a technological environment, such as the one disclosed in [START_REF] Costa | Integration and Innovation Orient to E-Society[END_REF], to provide better support to this population group.

Fig. 2 :

 2 Fig. 2: Home screen and menu of the Survival Prediction App of the CRCPredictor.

  (a) Survivability Features of Colon Cancer. (b) Results for Colon Cancer.

Fig. 3 :

 3 Fig. 3: Colon Cancer Survivability Calculator.

Table 1 :

 1 Characteristics of applications for colon and rectal cancer survivability prediction.

	Characteristics	Bush and Michaelson (2009) [10]	Chang et al. (2009) [11]	Weiser et al. (2011) [29]	Renfro et al. (2014) [23]	Wang et al. (2011) [28]	Valentini et al. (2011) [26]	Bowles et al. (2013) [7]
	(1)	Cancer Type	colon	colon	colon	colon	rectal	rectal	rectal
	(2)	Number of Features	9	6 ‡	2/3/7	12	5 ‡	9	7 ‡
	(3) Data set	SEER	SEER	SEER	Adjuvant Colon Cancer End Points (ACCENT)	SEER	five European randomized trials	SEER
	(4) Model	regression based	regression based	regression based	regression based	regression based	regression based	regression based
				1 -10 years					1 -10 years
				(disease					(disease
				specific					specific
	(5) Target	0 -15 years	survivability)	5 years	5 years	0 -5 years	1-10 years	survivability)
				0 -5 years					0 -5 years
				(conditional					(conditional
				survivability)					survivability)
	(6)	Performance C-index	-	0.816	0.61/0.63/0.68	0.66	0.75	0.70	-
	‡ Including months which the patient has already survived (for conditional survivability calculation).

Table 2 :

 2 Features obtained by feature selection and used for colon cancer models.

	Attribute	Description
	Age at diagnosis	The age (in years) of the patient at time of diagnosis
	Carcinoembryonic Antigen	The interpretation of the highest Carcinoembryonic Antigen test results
	Clinical Assessment of
	Regional Lymph Nodes

Table 3 :

 3 Features obtained by feature selection and used for rectal cancer models.

	Attribute	Description
	Age at diagnosis	*
	Extension of the Tumor Information on extension of the tumor
	Tumor Size	Information on tumor size (in mm)
	AJCC Stage	*
		Describes a surgical procedure that removes and/or
	Surgery of Primary Site	destroys tissue of the primary site performed as part of
		the initial work-up or first course of therapy.
	Gender	The sex/gender of the patient at diagnosis
	* Described in Table 2.

Table 4 :

 4 Performance measures for 10-fold cross validation and the incorrectly classified cases from the test data set of the stacking model.

	Cancer Type	Measure	1 Year 2 Year 3 Year 4 Year 5 Year Average
		AUC	0.982 0.985 0.987 0.989 0.987	0.986
	Colon	Accuracy F-Measure	95.84% 96.62% 96.79% 97.09% 97.03% 96.67% 90.41% 94.43% 95.55% 97.61% 97.51% 95.10%
		Incorrectly Classified (%) 3.24% 3.02% 2.97% 2.84% 3.11% 3.03%
		AUC	0.961 0.973 0.973 0.975 0.969	0.970
	Rectal	Accuracy F-Measure	95.61% 95.81% 93.81% 94.82% 94.32% 94,87% 80.56% 87.32% 85.85% 96.63% 96.20% 89.31%
		Incorrectly Classified (%) 3.81% 3.99% 6.17% 4.36% 4.36% 4.54%

Software available at https://rapidminer.com.

Stacking combines base classifiers of different types. Each base classifier generates a model using the training set, then a meta-learner integrates the independently learned base classifier models into a high level classifier by re-learning a meta-level training set.

Acknowledgements

This work has been supported by FCT -Fundação para a Ciência e Tecnologia within the Project Scope UID/CEC/00319/2013. The work of Tiago Oliveira is supported by a FCT grant with the reference SFRH/BD/85291/ 2012.