# Longitudinal Fairness with Censorship

Wenbin Zhang and Jeremy C. Weiss

###### Abstract

Recent works in artificial intelligence fairness attempt to mitigate
discrimination by proposing constrained optimization programs that achieve
parity for some fairness statistic. Most assume availability of the class
label, which is impractical in many real-world applications such as precision
medicine, actuarial analysis and recidivism prediction. Here we consider
fairness in longitudinal right-censored environments, where the time to event
might be unknown, resulting in censorship of the class label and
inapplicability of existing fairness studies. We devise applicable fairness
measures, propose a debiasing algorithm, and provide necessary theoretical
constructs to bridge fairness with and without censorship for these important
and socially-sensitive tasks. Our experiments on four censored datasets
confirm the utility of our approach.

## Introduction

With the rise of big data, artificial intelligence (AI)-based decision making
systems are used in a growing number of applications, including many high-
impact areas such as healthcare, employment, credit lending and criminal
justice (Beutel et al. 2019; Meyer 2018; Liu et al. 2021). There is concern
that automated decisions made in this fashion encode and even exacerbate
existing real-world disparities, inflicting harm to certain individuals or
social groups (Vasudevan and Kenthapadi 2020; Quy et al. 2022). This issue has
motivated a number of approaches to quantify and mitigate algorithmic
unfairness and has given rise to an active field of research deemed AI
fairness (Mehrabi et al. 2021). The vast majority of existing studies tackle
the problem by taking the existing notions of algorithmic fairness that focus
on either individual level fairness, e.g., disparate treatment, to guarantee
similar people are treated similarly (Dwork et al. 2012), or group level
fairness, e.g., disparate impact, seeking approximate parity of some statistic
across different groups (Zhang and Ntoutsi 2019). Moreover, most of the work
in this literature studies how to make machine learning algorithms fair in the
presence of a class label—where the fairness notions are defined based on the
class label, and where the predictive model is trained contingent upon them.
Surprisingly, little attention has been given to _censorship_ settings in
which the time to an event of interest could be inaccessible to the learner
(Turner et al. 2022), thus making existing fairness notions and approaches
inapplicable. In practice, _e.g._ , evaluating the retention rates of each
marketing channel in marketing analytics, and judging defendant’s criminal
recidivism for bail and sentencing in recidivism prediction instruments, the
latter is often the case.

In this work, we consider such a censorship setting where the true time to
event might be unknown to the learner, while fulfilling the requirements of
fair and accurate predictions. Addressing unfairness and censorship
simultaneously presents unique challenges, and transferring from respective
domains is not straightforward. In contrast to previous works (Verma and Rubin
2018), our goal here is to prevent cases of unequal treatment according to
their certain characteristics or sensitive attributes (_e.g._ , gender or
race) at the individual and group level. In addition, unlike previous works
limited to categorical sensitive attributes that are binary (Zhang and Weiss
2021), our goal also generalizes to an k-way categorical and continuous
variables. Our definitions of fairness can therefore be thought of versatile
notions focusing on individual and group levels, and inclusive of data with
censored individuals.

Our formulation of fairness is motivated by the following observation: in
healthcare, the impressive human-level or even surpassing human-level
performance of AI systems is balanced against a plethora of observed
discriminatory incidents (Rajkomar et al. 2018). As an example, a Propublica
report found that state-of-the-art clinical prediction models underperformed
on black patients even when a treatment was aimed at a particular type of
cancer that disproportionately impacted them (Chen et al. 2020). Such
observations also go beyond the medicinal domain with examples in marketing
analytics (Chang 2021), actuarial (Frezal and Barry 2019) and recidivism
prediction instruments (Angwin et al. 2016), where the common challenge, other
than ethical concerns, is the individual’s true time to event might be
unknown. We remark that thus far the fairness-in-AI community has primarily
focused on _no censorship_ settings with clearly defined class labels of
instances, despite these prevailing censorship scenarios.

Armed with this broader observation of AI unfairness, care must be taken to
ensure that an automated decision making system is fair or independent of
harmful and sensitive attributes-based stereotypes in the presence of
censorship. This motivates the definitions of fairness involving censored
individuals and a corresponding algorithm to address discrimination with
censorship that we propose in this work. More specifically, the novelty of
this research comes from four aspects: i) We study a novel problem of
longitudinal fairness with censorship, which is commonly rooted in socially
sensitive applications but remains highly under-explored. ii) Corresponding
fairness notions explicitly considering censorship are devised to measure bias
in the presence of censorship, as well as a respective fair learner to ensure
accurate predictions while also preserving a low discrimination score in
longitudinal censorship settings. iii) We theoretically establish the
connections of fairness in censored and non-censored settings, offering
greater understanding and explanation for AI fairness. iv) Detailed
experimental evaluations validating our model with regards to both fairness
and accuracy on four real-world censored and discriminated datasets.

## Background and Related Work

### Longitudinal Biased Data with Censorship

In the typical AI fairness settings, the biased data $X$ normally consists of
a sequence of feature represented instances $x_{1},x_{2},\cdots,x_{n}$. Among
the feature representation, a special attribute $G$ is referred as the
_sensitive attribute_ and its attribute values distinguish the discriminated
community, i.e., the deprived group, from the privileged community, i.e., the
favored group. In addition, instances are also described by their
corresponding class labels $y_{1},y_{2},\cdots,y_{n}$. However, class labels
can become inaccessible in the presence of censorship.

The discriminated and censored data, in contrast to the typical data
representation, therefore further contains the survival time $T$ and an event
indicator $\delta$ in addition to the observed features $x$, typically
represented in the form of ($x$, $T$, $\delta$). If the event of interest has
occurred, $T$ is the actual time from the individual entered the study till
the time of the event occurring, and $\delta$ becomes 1 indicating certainty
on the event observation; otherwise $T$ corresponds to the elapsed time
between individual entered the study and last follow-up with the individual,
and the event indicator $\delta$ = 0, i.e., the survival time is censored
(Wang et al. 2021).

Compared with AI fairness in supervised settings, addressing discrimination
bias in censoring settings leads to censorship on $y_{1},y_{2},\cdots,y_{n}$
which limits the applicability of the existing fairness notions. In addition,
the uncertainty on $y_{1},y_{2},\cdots,y_{n}$ could also further accompany and
complicate the biased decision regions. Given the discriminated and censored
data $X$, the aim of longitudinal AI fairness with censorship is then to model
a fair survival function $H(\cdot)$ which makes accurate predictions based on
$X$ but also does not discriminate with respect to $G$ for the discriminated
and censored datasets.

### AI Fairness

While artificial intelligence is increasingly permeating facets of life,
significant concerns on the unfair and discriminatory manner of AI-based
systems have been voiced and observed (Beutel et al. 2017). The AI community
has responded by proposing a growing body of fairness notions to measure the
level of discrimination along with a number of approaches to mitigate bias in
order to provide fair decision making systems (Hajian, Bonchi, and Castillo
2016; Mehrabi et al. 2021; Zhang et al. 2022).

The broad set of existing mathematical formulations of fairness can be
typically divided into two main families, _individual fairness_ and _group
fairness_. The former aims to ensure that similarly situated individuals are
treated similarly (Dwork et al. 2012) while the latter asks for group level
approximate parity of some statistic over class labels (Verma and Rubin 2018).
Although a vast of fairness notions exist, most of them formulate fairness
depend on class label thus limiting their applicability in censorship
settings. Kamrun et al. (Keya et al. 2020) directly extend the existing
fairness notions to the application with censoring problems, and it is the
only relevant work to the best of our knowledge. However, their definitions
exclude the censorship information when measuring discrimination which could
introduce substantial bias as censored information can be of importance and
cannot simply be ignored (Clark et al. 2003).

The aforementioned fairness definitions could be directly used or slightly
modified as a constraint or a regularizer to enforce fairness, leading to
three categories of debiasing mechanisms: _pre-processing approaches_ (Kamiran
and Calders 2009; Žliobaitė 2017), _in-processing solutions_ (Zhang and
Ntoutsi 2019; Babaioff, Ezra, and Feige 2021), and _post-processing
techniques_ (Hardt et al. 2016; Fish, Kun, and Lelkes 2016). The critical
limitation of these methods as well as other existing fairness works is that
they are in need of class label for their unfairness formulations and
algorithmic solutions, and fairness when some class labels are unknown has not
been well explored (Keya et al. 2020). Our work seeks to alleviate such
limitation by jointly addressing bias reduction and censoring management.

### Survival Analysis

The critical challenge of the main outcome under assessment could be unknown
for a portion of the study group, deemed censorship, hinders the use of many
methods of analysis. This motivates the study of _survival analysis_ to
address the problems of partial survival information access from the study
cohort (Clark et al. 2003). The censored data, also known as _survival data_ ,
are generally considered and modeled in terms of two quantitative terms,
namely the hazard function and the survival function. The former models the
instantaneous rate of event occurs at a specified time $t$ conditioned on
surviving to $t$:

$h(t|x)=\lim\limits_{\bigtriangleup t\rightarrow
0}\frac{Pr(t<T<t+\bigtriangleup t|T\geq t,x)}{\bigtriangleup t}$ (1)

The latter is the probability that the event does not occur up to time $t$ and
can be determined from the hazard function (and vice versa):

$S(t|x)=exp(-H(t|x)),~{}H(t|x)=\int_{0}^{t}h(t|x)dt$ (2)

Given the ubiquity of censored data in real-world applications, survival
analysis has gained its popularity in various applications ranging from
medicine to customer and actuarial analytics to predictive maintenance in
mechanical operations (De Angelis et al. 1999). Among the various methods
proposed for modeling censored data, the Cox proportional hazards model (CPH)
(Cox 1972) is the most commonly used in which the multiplicative relation
between the risk, as expressed by the baseline hazard function, and covariates
is described. More recently, deep neural network structure has also been
extended to model the feature interactions of survival data. For example,
DeepSurv (Katzman et al. 2018) employs the loss function of CPH with L2
regularization to train the networks. Another line of effort is the tree based
methods (Bou-Hamad et al. 2011), particularly random forests due to its
superior capabilities in handling nonlinear effect of variables and avoiding
restrictive assumptions such as that of proportional hazards (Ishwaran et al.
2008). A comprehensive literature survey covering recent censored data
modeling effort is provided in (Wang, Li, and Reddy 2019).

With the popularity of survival models, care must be taken to ensure their
fairness, the same as other AI approaches. Our work situates in this under-
explored research direction to tackle fairness in the presence of censorship.
Our in-processing approach incorporates a pairwise-comparison fairness notion
in the algorithm design to guide a accuracy-driven as well as fairness-
oriented learning procedure. Relevantly, the survival model is modified to
ensure fair risk predictions as in (Keya et al. 2020). Three key differences
are that our model: i) does not necessitate a distance metric to be specified,
ii) explicitly considers survival information to address discrimination in the
presence of censorship, and iii) enjoys the merit of free from hyperparameter
tuning.

## Our Approach

### Defining Bias with Censorship

#### Concordance Imparity

The presence of censorship in data limits the applicability of commonly used
fairness definitions introduced in the existing AI fairness studies. To fill
this gap, we introduce _Concordance Imparity (CI)_ to specifically account for
model unfairness in the presence of censorship. Specifically, CI first
considers individual level pairwise comparison based on the consistency
between model prediction and true outcomes, then measures, at the group level,
whether the discriminative ability of the model is fairly distributed across
different groups. Different from the previous definitions (Keya et al. 2020),
the survival time and survival information are explicitly involved in CI to
avoid important information loss and introducing substantial bias. The sketch
of Concordance Imparity is shown in Algorithm 1.

Algorithm 1 Concordance Imparity

0: Censored and biased dataset $D$, risk scores $r$, sensitive attribute $G$

0: CI score

1: if G is continuous then

2: Discretize G according to Equation (6)

3: end if

4: for each instance $d_{i}$ in $D$ do

5: for each instance $d_{j}$ in $D$ & $d_{j}$ $\neq$ $d_{i}$ do

6: if $t_{i}<t_{j}$ & $\delta_{i}==0~{}|~{}t_{j}<t_{i}$ & $\delta_{j}==0~{}|$
($t_{i}$ == $t_{j}$ & ($\delta_{i}$ == 0 & $\delta_{j}$ == 0)) then

7: continue

8: else

9: $P_{G(d_{i})==g}$ = $P_{G(d_{i})==g}$\+ 1

10: end if

11: if $t_{i}<t_{j}$ then

12: if $r(d_{i})>r(d_{j})$ then

13: $C_{G(d_{i})==g}$ = $C_{G(d_{i})==g}$\+ 1

14: else if $r(d_{i})=r(d_{j})$ then

15: $C_{G(d_{i})==g}$ = $C_{G(d_{i})==g}$\+ 0.5

16: end if

17: else if $t_{i}>t_{j}$ then

18: if $r(d_{i})<r(d_{j})$ then

19: $C_{G(d_{i})==g}$ = $C_{G(d_{i})==g}$\+ 1

20: else if $r(d_{i})=r(d_{j})$ then

21: $C_{G(d_{i})==g}$ = $C_{G(d_{i})==g}$\+ 0.5

22: end if

23: else if $t_{i}==t_{j}$ then

24: if $\delta_{i}==1~{}\&~{}\delta_{j}==1$ then

25: if $r(d_{i})==r(d_{j})$ then

26: $C_{G(d_{i})==g}$ = $C_{G(d_{i})==g}$\+ 1

27: else

28: $C_{G(d_{i})==g}$ = $C_{G(d_{i})==g}$\+ 0.5

29: end if

30: else if $\delta_{i}==0~{}\&~{}\delta_{j}==1~{}\&~{}r(d_{i})<r(d_{j})$ then

31: $C_{G(d_{i})==g}$ = $C_{G(d_{i})==g}$\+ 1

32: else if $\delta_{i}==1~{}\&~{}\delta_{j}==0~{}\&~{}r(d_{i})>r(d_{j})$ then

33: $C_{G(d_{i})==g}$ = $C_{G(d_{i})==g}$\+ 1

34: else

35: $C_{G(d_{i})==g}$ = $C_{G(d_{i})==g}$\+ 0.5

36: end if

37: end if

38: end for

39: end for

40: CF(G=g) = $C_{g}$/$P_{g}$

41: return CI= $\max_{g,g^{\prime}\in G~{}\&~{}g\neq
g^{\prime}}|CF_{(g)}-CF_{(g^{\prime})}|$

The concordance imparity measurement starts with deciding whether the
sensitive attribute $G$ is an continuous attribute (line 1) and discretizes G
according to the proposed fair survival difference to be discussed in the
following section if so (line 2). Next, CI forms all possible pairs of
comparison for each individual and omits those incomparable pairs, _i.e._ ,
the shorter time is censored, and both-censored pairs with identical survival
time (line 4-7). The remaining are the permissible pairs across different
demographic groups (line 8-9). Among them CI checks three possibilities: 1) if
the individual under consideration, $d_{i}$, has a shorter survival time,
$t_{i}$, than the compared individual’s survival time $t_{j}$, then the
concordance count of respective demographic group, $C_{G(d_{i})==g}$, that
$d_{i}$ belongs to increments by 1 if the model actually assigns a higher risk
score to $d_{i}$ and by 0.5 if predicted outcomes are tied (line 11-16). 2)
Line 17-22 checks the opposite scenario that $d_{i}$’s survival time is longer
than $d_{j}$’s and counts are incremented similarly. 3) When identical
survival times observed (line 23) and neither are censored (line 24),
$d_{i}$’s respective concordance count will be added by 1 on the condition
that the predicted outcomes are tied, and by 0.5 otherwise (line 25-29); When
the survival times are still the same but not both are censored,
$C_{G(d_{i})==g}$ increments by 1 if the non-censored individual has a higher
predicted risk score and by 0.5 otherwise (line 30-36). Line 40 then evaluates
_concordance fraction (CF)_ , the group-wise correct pairwise ordering, and
the final CI score is measured as the largest deviation of discriminative
abilities across different demographic groups of the model (line 41). The
lower the concordance imparity score the fairer the model.

Note that in comparison to existing fairness notions mainly focus on binary
protected categorical attributes (Verma and Rubin 2018), other than the
explicit inclusion of censorship information, concordance imparity also looks
at the generalization of measuring the level of discrimination to k-way
categorical attributes. This is done by reformulating the general
discrimination measurement to consider the largest difference among sub-
community, which is equivalent to the typical fairness definition when $k=2$.
In addition, CI also considers the discrimination in regards to continuous
attribute domain. Specifically, the allowing test (Han, Kamber, and Pei 2011)
is first used to explore potential binary split candidates, then the allowed
split with the largest merit achieved is selected as the threshold for
splitting. The merit is gauged according to the proposed fair survival
difference to be discussed hereafter. The calculation of CI can then proceed
the way as the categorical attribute after such a discretization. This gives
us a generalized definition extending CI to fair regression tasks, enabling an
inclusive discrimination evaluation consisting of censored individuals.

#### Fair Calibration

To further exploit the semantic information of survival probabilities produced
by the model which are also labels for individuals (Haider et al. 2020), we
propose _fair calibration (FC)_ to measure whether the model under
consideration creates probability value based disparity systematically.

In summary, fair calibration of survival probabilities was assessed by: i)
plotting group-wise (i.e., $\forall g\in G$) observed proportions versus
predicted probabilities and ii) by calculating corresponding fair calibration
validity. The first step examines the agreement between the predicted
probabilities of the model with the observed outcome. To do so, FC sorts and
splits the predicted probabilities for a particular time $t$ for each
demographic group into deciles, then checks whether these probabilities are
sufficiently close to the observed proportions. Note that smaller subgroups
will lead to greater statistical uncertainty about within-group predictive and
fairness performance, but provide more similarity of instances within
subgroups. Fewer subgroups result in the opposite. Convention is to use
deciles, which is what we follow. In addition, the observed proportions could
be unknown due to the censorship. To this end, the Kaplan-Meier (KM) curve
estimate (Ranstam and Cook 2017) is employed and the significance of these
group-wise results respect to these bins are defined by the Hosmer-Lemeshow
(HL) goodness-of-fit test statistic (Hosmer and Lemesbow 1980):

$HL_{g}(S(t|x))=\sum_{i=1}^{B}\frac{({KM_{i}}_{g}-\bar{p_{i}}_{g})^{2}{n_{i}}_{g}}{\bar{p_{i}}_{g}(1-\bar{p_{i}}_{g})}$
(3)

where $B$ represents the number of bins, ${KM_{i}}_{g}$ is the KM estimated
probability in the $i$th decile at time $t$, $\bar{p_{i}}_{g}$ is the
predicted probability for individuals in the $i$th decile and ${n_{i}}_{g}$ is
the number of observations in decile $i$. Note that all of them are group-
wise, i.e., $G=g$.

Based on the group-wise examined level of agreements, the second step of FC
evaluates consistency across different demographic groups as shown in Equation
(4), involving the first two fair calibrated scenarios and the third biased
calibrated scenario otherwise: i) representation consistency: the predicted
probabilities are representative of the actual probabilities; the p-value of
each group’s HL statistic passes the test with a value not smaller than 0.05,
ii) difference consistency: representation inconsistent but the difference
between predicted probabilities and actual probabilities (i.e., $\triangle
p_{g}/\triangle p_{g^{\prime}}$) is accordant across subgroups; the p-value of
each difference test among subgroups evaluated by Wilcoxon signed-rank test
(Woolson 2007) is greater than 0.5, iii) Neither representation nor difference
is consistent.

$FC=\begin{cases}\text{fair~{}calibrated,}&p(HL_{g}(S(t|x)))\geq 0.05\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\forall
g\in G\\\ \text{fair~{}calibrated,}&wilcoxon(\triangle p_{g},\triangle
p_{g^{\prime}})>0.5\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\forall
g,g^{\prime}\in G\\\ \text{biased~{}calibrated,}&\text{otherwise}\end{cases}$
(4)

### Mitigating Bias with Censorship

Armed with the afore established fairness statistics, measuring unfairness in
the presence of censorship becomes feasible. This section then serves to
fulfill the subsequent bias mitigation amidst censorship.

The proposed approach follows the general idea of random forests (RF) by
constructing an array of base learners to improve the predictive ability. In
particular to censored data, RF is also nonparametric while enjoying the
merits of nonlinear interactions modeling (Wang, Li, and Reddy 2019). However,
such ensemble methods aim to optimize for data encoding for predictive
performance, and fairness, which we desired to add, is imperceptible (Ishwaran
et al. 2008). In this work, to jointly optimizing for censored data encoding
and debiasing, we propose _Fair Survival Random Forests (FSRF)_ which extends
the RF model in two ways: i) by introducing a new splitting criterion that
jointly considers the reduction of an attribute split w.r.t. impurity and also
w.r.t. discrimination, ii) by illustrating the way to provide fair risk
predictions amidst censorship.

The information gain and Gini impurity, when censorship is absent, are
commonly used splitting criteria to guide the induction of the tree for
classification performance (Han, Kamber, and Pei 2011). However, the presence
of censorship leads to inaccessibility of class label thus making their
computation impractical. The survival difference can be instead used to
measure the impurity reduction for candidate splitting evaluation. In FSRF,
such survival difference between different groups are evaluated by the
_logrank test_ (Bland and Altman 2004):

$SD=\frac{\sum_{j=1}^{k}(O_{j}-E_{j})}{\sqrt{\sum_{j=1}^{k}V_{j}}}\sim N(0,1)$
(5)

where $O_{j}$ and $E_{j}$ represent the observed number of events and the
expected number of events, respectively with $V_{j}$ being the variance of
$O_{j}$. The candidate with a larger logrank test therefore leads to more
similarity within child nodes but also more dissimilarity among child nodes if
it is being selected for splitting.

We then combine concordance imparity and survival difference as a conjunctive
criterion that takes both predictive performance and fairness into
consideration. We define the conjunctive criterion _fair survival difference
(FSD)_ as:

$FSD=\begin{cases}\log SD-\log CI&\text{if CI $\neq$ 0}\\\
+\infty&\text{otherwise}\end{cases}$ (6)

Intuitively, FSD closely ties SD and CI. When the candidate attributes that
are free of discrimination, i.e., CI equals 0, FSD becomes positive infinite
to prioritize fair splitting.

In practice, the calculation of CI depends on the distribution that a
potential splitting could lead to and the associated risk predictions based on
the distribution. FSRF employs the cumulative hazard function $H(t|x)$ to
predict such risk score for the sake of having a direct interpretation of the
expected number of events, as well as serving the intermediate function
between hazard and survival functions for direct derivation when needed.
Formally, the risk score is estimated by the Nelson-Aalen estimator (Borgan
2014) as:

$H(t|x)=\sum_{j\leq t}\frac{d_{j}}{n_{j}}$ (7)

where $d_{j}$ and $n_{j}$ represent the number of individuals experiencing
events and have not experienced the event at time $j$ respectively, and $t$ is
evaluated as the last event time. Similar to the same node sharing identical
class label in non-censoring trees, all individuals within the node of FSRF
share an identical risk score when evaluating splitting but also predicting
final risk.

### Bridging Fairness in the Presence and Absence of Censorship

The previously proposed fairness definitions and debiasing algorithm
explicitly consider the indispensable survival information and lay the
groundwork for AI fairness in the presence of censorship. In addition, it is
also desire to understand the connections of AI fairness in the presence and
absence of censorship to build fundamental theoretical frameworks for AI
fairness. So are of practitioners and policy makers’ interests to help them
have an additional layer of understanding of AI fairness for fair decision
making navigation and customization. This section serves to fill this gap.

As previously discussed, the standard AI fairness techniques, such as the most
widely used statistical parity, are not suitable in the presence of censoring
in the data (Verma and Rubin 2018). Here, we take the devised concordance
imparity as the illustrative example to connect AI fairness in the presence
and absence of censorship, so as to facilitate the study of fairness with
censorship. Recall that CI first measures subgroup-wise fraction of concordant
pairs, then gauges concordance difference between different subgroups defined
by the sensitive attributes. The first part of CI can therefore be thought of
as an extension of the standard concordance index (C-index) (Li et al. 2016)
in subgroup-wise and then along with the second part as the weighted subgroup
based area under the receiver operating curve (AUC) difference in no censoring
settings. We will next elaborate this connection.

Charac. | SUPPORT | ROSSI | COMPAS | KKBOX
---|---|---|---|---
Sample # | 8,873 | 432 | 10,325 | 2,814,735
Censored% | 0.320 | 0.736 | 0.732 | 0.347
Feature # | 14 | 9 | 14 | 18
| Sensitive
---
Attribute
gender | race | race | gender
| Sensitive
---
Value
female |  | African
---
American
| African
---
American
female
Table 1: An overview of the datasets.

Starting from the standard C-index, it is a “global” index for validating the
predictive ability of the model. Specifically, it is the fraction of pairs
within the whole group, where the observation experienced the event of
interest had a higher risk score than an observation who experienced the event
later or had not experienced the event, representing the global assessment of
the model’s discrimination power. By definition, the C-index is a
generalization of the Wilcoxon-Mann-Whitney statistics (Austin and Steyerberg
2012) and thus of the AUC with equivalence in binary classification in the
absence of censorship. The difference between the concordance part of CI and
C-index is that the concordance of CI measures subgroup-wise fraction of
concordant pairs in comparison to the whole group-wise concordant probability
of C-index. Note that the subgroup-wise concordance calculation of CI compares
an observation with both intra group and inter group observations, i.e., with
all remaining observations other than itself. The concordance of CI can
therefore be regarded as a subgroup based AUC, abbreviated as ${s}$AUC, in
contract to the standard global AUC of C-index. Next, based on the obtained
${s}$AUC from different subgroups, the imparity part of CI gauges concordance
difference among them. Finally, CI can be interpreted as the largest deviation
among weighted ${s}$AUC, and CI’s counterpart of the original formulation
(e.g., line 41 in Algorithm 1) in the absence of censorship is:

$CI=\max_{g,g^{\prime}\in G~{}\&~{}g\neq
g^{\prime}}|{s}AUC_{(g)}-{s}AUC_{(g^{\prime})}|$ (8)

where ${s}AUC_{(G=g)}$ is the weighted subgroup based AUC from subgroup that
defined by $g$ and is mathematically represented as:

${s}AUC_{(g)}=\frac{n_{g}{s}AUC_{g}+\sum_{j=1}^{|\\{G-g\\}|}n_{g^{\prime}}{s}AUC_{g^{\prime}}}{\sum_{j=1}^{|G|}n_{j}}$
(9)

where ${s}AUC_{g}$ and ${s}AUC_{g^{\prime}}$ are AUC values when comparing
with intra- and inter-subgroup observations, respectively, and $n_{j}$
represents respective number of comparable pairs of each subgroup.

Datasets |  Method Metrics | CI% | Fair Calibration | C-index% | Brier Score% | Time-dependent AUC%
---|---|---|---|---|---|---
SUPPORT | IDCPH | 19.12 | Not fair calibrated | 69.08 | 31.16 | 76.17
GDCPH | 13.12 | Fair calibrated | 75.12 | 24.46 | 78.21
CPH | 17.45 | Not fair calibrated | 74.11 | 21.21 | 80.02
RSF | 20.11 | Not fair calibrated | 75.18 | 16.64 | 81.01
DeepSurv | 18.65 | Not fair calibrated | 75.65 | 16.11 | 80.68
FSRF | 9.21 | Fair calibrated | 76.17 | 13.23 | 82.86
ROSSI | IDCPH | 15.31 | Not fair calibrated | 52.28 | 18.73 | 77.32
GDCPH | 9.32 | Fair calibrated | 59.34 | 22.87 | 78.51
CPH | 11.43 | Not fair calibrated | 64.24 | 17.67 | 77.12
RSF | 16.53 | Not fair calibrated | 65.56 | 15.12 | 79.32
DeepSurv | 12.32 | Not fair calibrated | 66.67 | 14.71 | 77.17
FSRF | 8.92 | Fair calibrated | 69.02 | 12.69 | 79.65
COMPAS | IDCPH | 25.18 | Not fair calibrated | 62.16 | 25.03 | 63.78
GDCPH | 11.77 | Fair calibrated | 72.16 | 16.32 | 66.21
CPH | 22.43 | Not fair calibrated | 69.24 | 20.35 | 65.15
RSF | 25.32 | Not fair calibrated | 72.61 | 15.62 | 71.76
DeepSurv | 16.72 | Not fair calibrated | 75.12 | 13.42 | 71.83
FSRF | 9.63 | Fair calibrated | 76.24 | 13.81 | 72.33
KKBOX | IDCPH | 17.79 | Not fair calibrated | 72.61 | 21.23 | 69.73
GDCPH | 14.98 | Fair calibrated | 79.45 | 19.92 | 73.03
CPH | 18.91 | Not fair calibrated | 80.02 | 18.17 | 72.95
RSF | 21.14 | Not fair calibrated | 82.32 | 14.24 | 78.18
DeepSurv | 20.66 | Not fair calibrated | 83.01 | 14.33 | 80.71
FSRF | 14.42 | Fair calibrated | 82.43 | 13.13 | 82.16
Table 2: Performance comparison of all methods on various datasets. The best
results are marked in bold.

## Experimental Results

### Dataset Description

We validate our model on four real-world censored datasets with socially
sensitive concerns: i) The _SUPPORT_ dataset is from a large study to
understand prognoses preferences outcomes and risks of treatment by analyzing
the survival time of inpatients (Knaus et al. 1995). ii) The _ROSSI_ dataset
pertains to predict the reoffending risk score of convicted criminals from
Maryland state prisons, who were followed up for one year after release (Fox,
Carvalho et al. 2012). iii) The landmark algorithmic unfairness _COMPAS_
dataset to predict recidivism from Broward County (Angwin et al. 2016). iv)
The _KKBOX_ dataset from the WSDM-KKBox’s Churn Prediction Challenge (Kvamme,
Borgan, and Scheel 2019) to study users’ risk scores of canceling their
subscription from KKBOX. Table 1 is a summary description of them. Note that
survival time and censoring information are explicitly included in our study
to specifically account for censorship.

### Comparison Methods

We compare FSRF against five baselines to evaluate its theoretical design: i)
two recent proposed fair survival models _IDCPH_ and _GDCPH_ (Keya et al.
2020), which are the most competitive approaches among several variants
proposed therein, and are the only works for fair survival analysis problem to
the best of our knowledge, ii) along with the baseline therein, the most
commonly used survival analysis tool _CPH_ (Cox 1972), iii) the state of the
art random forests based non-linear survival model _RSF_ (Ishwaran et al.
2008), and iv) the most recent deep model on survival analysis _DeepSurv_
(Katzman et al. 2018). We do not compare with other fairness baselines due to
their inapplicability in the presence of censorship.

Datasets |  | C-index%
---
discriminated
| C-index%
---
privileged
| Brier Score%
---
discriminated
| Brier Score%
---
privileged
| Time-dependent%
---
AUC discriminated
| Time-dependent %
---
AUC privileged
FSRF- | FSRF | FSRF- | FSRF | FSRF- | FSRF | FSRF- | FSRF | FSRF- | FSRF | FSRF- | FSRF
SUPPORT | 60.05 | 69.64 | 80.16 | 78.85 | 25.54 | 17.65 | 8.87 | 10.14 | 73.45 | 79.67 | 87.52 | 85.92
ROSSI | 57.71 | 63.78 | 74.24 | 72.7 | 21.03 | 16.36 | 9.87 | 10.22 | 69.98 | 73.77 | 84.87 | 82.21
COMPAS | 54.82 | 68.88 | 80.14 | 78.51 | 18.76 | 16.71 | 7.66 | 11.67 | 62.81 | 65.65 | 77.62 | 75.31
KKBOX | 64.53 | 70.22 | 85.67 | 84.64 | 18.87 | 14.97 | 7.16 | 9.12 | 72.31 | 78.03 | 85.87 | 84.52
Table 3: Prediction performance confusion matrix for FSRF.

### Performance Comparison

Due to the presence of censoring in the data, the standard evaluation metrics
of AI fairness such as accuracy and statistical parity are not suitable for
measuring the performance in AI fairness with censorship (Verma and Rubin
2018). Instead, in addition to the previously tailored fairness measures
considering censorship, the typical survival accuracy metrics, including the
C-index, Brier score and Time-dependent AUC, are utilized to evaluate the
predictive performance of our model and other baselines. The _C-index_
(Harrell et al. 1982) evaluates a model’s discrimination power in terms of
correct pairwise ordering, and is a generalization of the area under ROC curve
(AUC) in the presence of censorship. The _Brier score_ (Brier and Allen 1951)
is roughly the mean squared difference of the probability estimations assigned
to possible outcomes and the actual outcome. Different from C-index, the lower
the Brier score the merrier. Finally, the _Time-dependent AUC_ (Chambless and
Diao 2006) tests how well a model can distinguish individuals who experienced
the event of interest from those have not prior to or at time $t$, and thus
the model with a higher Time-dependent AUC score is desired. Table 2 provides
detailed 5-fold cross validation results of all methods on various
longitudinal biased data with censorship.

(a) SUPPORT

(b) ROSSI

(c) COMPAS

(d) KKBOX

Figure 1: The fair calibration plots of the discriminated community. The color
and height of the bar represent different methods and corresponding
probabilities. The true probability observed by KM is marked in red while
FSRF’s is in dark.

The results in Table 2 show that the proposed model wins almost every metric
across all of the datasets. This demonstrates that our proposed method has a
superior debiasing capability while maintaining competence in predictive
performance in the presence of censorship. Specifically, our new FSRF
dominates all other methods when diminishing discrimination with censorship.
This result is especially important when we contrast to other fair models
proposed for censoring settings, which verifies the necessity of including
survival information and survival time to mitigate bias with censorship as
well as the drawback of involving task-specific similarity metric. In
addition, our model, on all datasets tested, does not suffer from performance
instability as other methods do, indicating FSRF is a more robust approach to
building fair model with censorship. In terms of prediction performance, FSRF
outperforms other models on almost all metrics in all of the datasets. This
reflects that FSRF is able to handle and utilize both censored and uncensored
instances when building fair model with censorship. What’s more, FSRF, in
contrast to other fair survival baselines, performs no parameter tuning, thus
benefiting end user with simplicity while making fair decisions in the
presence of censorship.

From Table 2 we also note that our fairness regularizer is able to actually
improve predictive performance. To have a better understanding of this
phenomena, we further analyze the predictive performance confusion matrix of
FSRF, according to the sensitive attribute that defines the discriminated
community and privileged community as well as with and without our fairness
constraints, represented by FSRF and FSRF-. Table 3 summarizes the results. As
one can see, improvement on the characterization of discriminated community is
indeed achieved by including fairness attention in our method. What’s more,
the improved overall prediction performance also demonstrates the potential
additional rewards of the debiasing design of FSRF.

Table 2 additionally shows the risk prediction of FSRF is fairly calibrated as
each demographic group’s p-value passes its significance test, suggesting
FSRF’s predicted probabilities are representative of corresponding community’s
true probabilities. To see the full picture behind the unrankable p-values,
Figure 1 graphs the predicted probabilities by FSRF in comparison to the true
probabilities observed by KM (fair calibration plots of privileged community
are omitted due to space constraints). In the visualization, the heights of
dark bars are always close to the red bars’ while bars in other colors do not
follow this pattern which conclusively match with the results of fair
calibration, suggesting FSRF is indeed an effective fair risk predictor.

## Conclusion

Despite the increasing attention on AI fairness, existing studies have mainly
focused on no censorship settings. This paper tackles fairness with censorship
which is particularly prevalent in many real-world socially sensitive
applications. To accomplish this objective, we devised generalized censored-
specific fairness notions to quantify unfairness along with a unified
debiasing algorithm to mitigate discrimination in the presence of censorship.
The results on real biased datasets with censorship show our propose
techniques are versatile in censoring settings. This work studies a new
research problem and opens possibilities for future work on AI fairness with a
broader applicability to practical scenarios concerning fairness.

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