# Orderenergetic, hypoenergetic and equienergetic graphs resulting from some
graph operations

Jahfar T K<EMAIL_ADDRESS>and Chithra A V<EMAIL_ADDRESS>
Department of Mathematics, National Institute of Technology, Calicut, Kerala,
India-673601

###### Abstract

A graph $G$ is said to be orderenergetic, if its energy equal to its order and
it is said to be hypoenergetic if its energy less than its order. Two non-
isomorphic graphs of same order are said to be equienergetic if their energies
are equal. In this paper, we construct some new families of orderenergetic
graphs, hypoenergetic graphs, equienergetic graphs, equiorderenergetic graphs
and equihypoenergetic graphs.

AMS classification: 05C50

Keywords: orderenergetic graphs, equienergetic graphs, hypoenergetic graphs,
equiorderenergetic graphs, equihypoenergetic graphs.

##  1 Introduction

In this paper, we consider simple undirected graphs. Let $G=(V,E)$ be a simple
graph of order $p$ and size $q$ with vertex set
$V(G)=\\{v_{1},v_{2},...,v_{p}\\}$ and edge set
$E(G)=\\{e_{1},e_{2},...,e_{q}\\}$. The adjacency matrix $A(G)=[a_{ij}]$ of
the graph $G$ is a square symmetric matrix of order $p$ whose $(i,j)^{th}$
entry is defined by

$a_{i,j}=\begin{matrix}\begin{cases}1,&\text{if $v_{i}$ and $v_{j}$ are
adjacent, }\\\ 0,&\text{otherwise .}\end{cases}\end{matrix}$

The eigenvalues $\lambda_{1},\lambda_{2},...,\lambda_{p}$ of the graph $G$ are
defined as the eigenvalues of its adjacency matrix $A(G)$. If
$\lambda_{1},\lambda_{2},...,\lambda_{t}$ are the distinct eigenvalues of $G$,
then the spectrum of $G$ can be written as
$spec(G)=\begin{pmatrix}\lambda_{1}&\lambda_{2}&...&\lambda_{t}\\\
m_{1}&m_{2}&...&m_{t}\\\ \end{pmatrix}$, where $m_{j}$ indicates the algebraic
multiplicity of the eigenvalue $\lambda_{j}$, $1\leq j\leq t$ of $G$. The
energy [7] of the graph $G$ is defined as
$\varepsilon(G)=\displaystyle\sum_{i=1}^{p}|{\lambda_{i}}|$. More results on
graph energy are reported in [7, 2]. Two non-isomorphic graphs are said to be
cospetral if they have the same spectrum, otherwise they are known as non-
cospectral. Two non-isomorphic graphs of the same order are said to be
equienergetic if they have the same energy[12]. A graph of order $p$ is said
to be hyperenergetic if its energy is greater than $2(p-1)$, otherwise graph
is non hyperenergetic. Graphs of order $p$ with energy equal to $p$ is called
orderenergetic graphs[1]. The number of graphs whose energy equal to its order
are relatively small. So we are trying to find new families of orderenergetic
graphs. The orderenergetic graphs are studied in [1].
The spectrum of complete bipartite graph $K_{p,p}$ is

$spec(K_{p,p})=\begin{pmatrix}-p&0&p\\\ 1&2p-2&1\\\ \end{pmatrix}.$

Then $\varepsilon(K_{p,p})=2p$, so $K_{p,p}$ is orderenergetic for every $p$.
So our interest is to find the orderenergetic graphs other than $K_{p,p}$. In
2007, I.Gutman et al. [10] introduced the definition of hypoenergetic graphs.
A graph is said to be hypoenergetic if its energy is less than its order,
otherwise it is said to be non hypoenergetic. The properties of hypoenergetic
graphs are discussed in detail [6, 9, 10]. In the chemical literature there
are many graphs for which the energy exceeds the order of graphs. In 1973,
England and Ruedenberg published a paper [5] in which they asked “why does the
graph energy exceed the number of vertices?”. In 2007, Gutman [8] had proved
that if the graph G is regular of any non-zero degree, then G is non
hypoenergetic. The orderenergetic and hypoenergetic graphs have several
applications in theoretical chemistry. A graph is said to be integral if all
of its eigenvalues are integers. The aim of this paper is to construct new
families of orderenergetic, hypoenergetic and equienergetic graphs using some
graph operations.

The complement graph $\overline{G}$ of $G$ is a graph with vertex set same as
that of $G$ and two vertices in $\overline{G}$ are adjacent only if they are
not adjacent in $G$. We shall use the following notations throughout this
paper, $C_{p}$, $K_{p}$, $P_{m}$ and $K_{r,s}$ denotes cycle on $p$ vertices,
complete graph on $p$ vertices, path on $m$ vertices and complete bipartite
graph on $r+s$ vertices respectively. The symbols $I_{m}$ and $J_{m}$ will
stands for the identity matrix of order $m$ and $m\times m$ matrix with all
entries are ones respectively.
The rest of the paper is organized as follows. In Section 2, we state some
previously known results that will be needed in the subsequent sections. In
Section 3, we construct some orderenergetic graphs. In Section 4, some new
families of hypoenergetic graphs are presented. In Section 5, an infinite
family of equienergetic, equiorderenergetic and equihypoenergetic graphs are
given.

##  2 Preliminaries

In this section, we recall the concepts of the $m$-splitting graph, the
$m$-shadow graph and the $m$-duplicate graph of a graph and list some
previously established results.

######  Definition 2.1.

[4] The Kronecker product of two graphs $G_{1}$ and $G_{2}$ is a graph
$G_{1}\times G_{2}$ with vertex set $V(G_{1})\times V(G_{2})$ and the vertices
$(x_{1},x_{2})$ and $(y_{1},y_{2})$ are adjacent if and only if
$(x_{1},y_{1})$ and $(x_{2},y_{2})$ are edges in $G_{1}$ and $G_{2}$
respectively.

######  Definition 2.2.

[4] Let $A\in M_{m\times n}(\mathbb{R})$ and $B\in M_{p\times q}(\mathbb{R})$
be two matrices of order $m\times n$ and $p\times q$ respectively. Then the
Kronecker product of $A$ and $B$ is defined as follows

$A\otimes B=\begin{bmatrix}a_{11}B&a_{12}B&a_{13}B&\dots&a_{1n}B\\\
a_{21}B&a_{22}B&a_{23}B&\dots&a_{2n}B\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
a_{m1}B&a_{m2}B&a_{m3}B&\dots&a_{mn}B\\\ \end{bmatrix}.$

######  Proposition 2.1.

[4] Let $A,B\in M_{n}(\mathbb{R})$ be two matrices of order $n$. Let $\lambda$
be an eigenvalue of matrix $A$ with corresponding eigenvector $x$ and $\mu$ be
an eigenvalue of matrix $B$ with corresponding eigenvector $y$, then
$\lambda\mu$ is an eigenvalue of $A\otimes B$ with corresponding eigenvector
$x\otimes y.$

######  Lemma 2.1.

[3]. If $G_{1}$ and $G_{2}$ are any two graphs, then $\varepsilon(G_{1}\times
G_{2})=\varepsilon(G_{1})\varepsilon(G_{2})$.

######  Definition 2.3.

[3]. The join of graphs $G_{1}$ and $G_{2}$, $G_{1}\vee G_{2}$ is obtained
from $G_{1}\cup G_{2}$ by joining every vertex of $G_{1}$ with every vertex of
$G_{2}$.

######  Proposition 2.2.

[4]. If $G_{1}$ is a $r_{1}$-regular graph with $n_{1}$ vertices and $G_{2}$
is a $r_{2}$-regular graph with $n_{2}$ vertices, then the characteristic
polynomial of $G_{1}\vee G_{2}$ is given by

$\phi(G_{1}\vee
G_{2},x)=\frac{\phi(G_{1},x)\phi(G_{2},x)}{(x-r_{1})(x-r_{2})}[(x-r_{1})(x-r_{2})-n_{1}n_{2}].$

######  Definition 2.4.

[15] Let $G$ be a $(p,q)$ graph.Then the $m$-splitting graph of a graph $G$,
$spl_{m}(G)$ is obtained by adding to each vertex $v$ of $G$ new $m$ vertices
say, $v_{1},v_{2},...,v_{m}$ such that $v_{i}$ ,$1\leq i\leq m$ is adjacent to
each vertex that is adjacent to $v$ in G. The adjacency matrix of
$m$-splitting graph of $G$ is

$A(spl_{m}(G))=\begin{bmatrix}A(G)&A(G)&A(G)&\dots&A(G)\\\ A(G)&O&O&\dots&O\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\ A(G)&O&O&\dots&O\\\
\end{bmatrix}_{(m+1)p}.$

######  Proposition 2.3.

[15] Let $G$ be a $(p,q)$ graph. Then the energy of $m$-splitting graph of $G$
is, $\varepsilon(spl_{m}(G))=\sqrt{1+4m}\varepsilon(G).$

######  Definition 2.5.

[15] Let $G$ be a $(p,q)$ graph. Then the $m$-shadow graph $D_{m}(G)$ of a
connected graph $G$ is obtained by taking m copies of $G$ say,
$G_{1},G_{2},...,G_{m}$ then join each vertex $u$ in $G_{i}$ to the neighbors
of the corresponding vertex $v$ in $G_{j},1\leq i\leq m,1\leq j\leq m.$ The
adjacency matrix of m-shadow graph of $G$ is $A(D_{m}(G))=J_{m}\otimes A(G).$
Note that the number of vertices in $D_{m}(G)$ is $pm$.

If $m=2$, then the graph $D_{2}(G)$ is called shadow graph of $G$.

######  Lemma 2.2.

[15] Let $G$ be any graph. Then $\varepsilon(D_{m}(G))=m\varepsilon(G).$

######  Definition 2.6.

[13]. Let $G=(V,E)$ be a $(p,q)$ graph with vertex set $V$ and edge set $E$.
Let $W$ be a set such that $V\bigcap W=\emptyset$, $|V|=|W|$ and
$f:V\rightarrow W$ be bijective $($for $a\in V$ we write $f(a)$ as
$a^{\prime}$ for convenience $)$. A duplicate graph of $G$ is
$D(G)=(V_{1},E_{1})$, where the vertex set $V_{1}=V\cup W$ and the edge set
$E_{1}$ of $D(G)$ is defined as, the edge ab is in $E$ if and only if both
$ab^{\prime}$ and $a^{\prime}b$ are in $E_{1}$.
In general m-duplicate graph $D^{m}(G)$ is defined as
$D^{m}(G)=D^{m-1}(D(G))$.
Note that the $m$-duplicate graph has $2^{m}p$ vertices and $2^{m}q$ edges.

Note 1.[11] Energy of the duplicate graph $D(G)$,
$\varepsilon(D(G))=2\varepsilon(G).$

##  3 Construction of orderenergetic graphs

In this section, it is possible to construct an infinite family of
orderenergetic graphs from the given orderenergetic graphs. Let $G$ and $H$ be
orderenergetic graphs, then $G\cup H$ is orderenergetic. For example, the
graph $K_{p,p}\cup mK_{2}$ is orderenergetic, but this graph is not connected.

The following theorems give some new methods to construct an infinite family
of connected orderenergetic graphs.

######  Theorem 3.1.

Let $G$ be a connected orderenergetic graph of order $p$. Then the m-shadow
graph, $D_{m}(G)$ is a connected orderenergetic graph.

###### Proof.

Since $G$ is orderenergetic, by Lemma 2.2,
$\varepsilon(D_{m}(G))=m\varepsilon(G)=mp.$ So $D_{m}(G)$ is orderenergetic.
Also the $m$-shadow graph of a connected graph is connected. ∎

######  Remark 3.1.

Let $G$ be an orderenergetic graph. Then the $m$-shadow graph of a duplicate
graph, $D_{m}(D(G))$ is orderenergetic.

######  Theorem 3.2.

Let $G$ be an $r$-regular orderenergetic graph of order $p$. Then
$G\vee\overline{K_{n}}$ is orderenergetic if and only if $n=4p-2r.$

###### Proof.

Let $r=\lambda_{1},\lambda_{2},...,\lambda_{p}$ be the eigenvalues of $G.$
Since $G$ is orderenergetic, we have,

$\displaystyle\sum_{i=2}^{p}|{\lambda_{i}}|=p-r.$

From Proposition 2.2, the characteristic polynomial of $G\vee\overline{K_{n}}$
is given by

$\phi(G\vee\overline{K_{n}},x)=x^{n-1}(x-\lambda_{2})(x-\lambda_{3})...(x-\lambda_{p})(x^{2}-rx-
np).$

Let $\alpha$ and $\beta$ be the roots of the equation $x^{2}-rx-np=0$. It is
easy to observe that $\alpha$ and $\beta$ are of opposite sign. Without loss
of generality we assume that $\alpha>0$ and $\beta<0$, then $\alpha+\beta=r$,
$\alpha\beta=-np.$ Thus the spectrum of $G\vee\overline{K_{n}}$ is

$spec(G\vee\overline{K_{n}})=\begin{pmatrix}0&\lambda_{2}&\lambda_{3}&...&\lambda_{p}&\alpha&\beta\\\
n-1&1&1&...&1&1&1\\\ \end{pmatrix}.$

Hence

$\varepsilon(G\vee\overline{K_{n}})=\displaystyle\sum_{i=2}^{p}|{\lambda_{i}}|+|\alpha|+|\beta|=\displaystyle\sum_{i=2}^{p}|{\lambda_{i}}|+\alpha-\beta.$

If $G\vee\overline{K_{n}}$ is orderenergetic, then

$\displaystyle\varepsilon(G\vee\overline{K_{n}})=p+n$
$\displaystyle\Longleftrightarrow p-r+\alpha-\beta=p+n$
$\displaystyle\Longleftrightarrow\alpha-\beta=n+r$

Also $\alpha+\beta=r$ and $\alpha-\beta=n+r$ implies that
$\alpha=\frac{n+2r}{2}$ and $\beta=-\frac{n}{2}.$

$\displaystyle\alpha\beta=-np$
$\displaystyle\Longleftrightarrow\alpha\beta=\frac{-n^{2}-2nr}{4}$
$\displaystyle\Longleftrightarrow-np=\frac{-n^{2}-2nr}{4}$
$\displaystyle\Longleftrightarrow 4np=n^{2}+2nr$
$\displaystyle\Longleftrightarrow n=4p-2r.$

. ∎

######  Example 3.1.

Let $G=C_{4}$. Then $G\vee\overline{K_{12}}$ is orderenergetic.

$spec(G)=\begin{pmatrix}-2&0&2\\\ 1&2&1\\\ \end{pmatrix}.$
$spec(G\vee\overline{K_{12}})=\begin{pmatrix}-6&-2&0&8\\\ 1&1&13&1\\\
\end{pmatrix}.$

$\varepsilon(G\vee\overline{K_{12}})=16$ and order of $G\vee\overline{K_{12}}$
is 16.

######  Example 3.2.

Consider $K_{2}\vee\overline{K_{6}},$

Figure 1: Graph $K_{2}\vee\overline{K_{6}}$

$spec(K_{2}\vee\overline{K_{6}})=\begin{pmatrix}-3&-1&0&4\\\ 1&1&5&1\\\
\end{pmatrix}.$

$\varepsilon(K_{2}\vee\overline{K_{6}})=8$ and order of
$K_{2}\vee\overline{K_{6}}$ is 8.

######  Theorem 3.3.

Let $G$ be an orderenergetic graph with $p$ vertices. Then the $2$-splitting
graph of $G$ , $spl_{2}(G)$ is orderenergetic.

In[14], D.Stevanovic introduced the graph superpath
$SP(a_{1},a_{2},...,a_{m})$ obtained by replacing each vertex $v_{i}$ of the
path $P_{m}$ with totally disconnected graph $\overline{K}_{a_{i}}$. Two
vertices $u\in\overline{K}_{a_{i}}$ and $w\in\overline{K}_{a_{j}}$ are
adjacent in $SP(a_{1},a_{2},...,a_{m})$ if $v_{i}$ and $v_{j}$ are adjacent in
$P_{m}$, $i,j\in\\{1,2,...,m\\}$. The order of $SP(4,1,3,2,2,3,1,4)$ is
$m(m+1).$

For example, the superpath $SP(4,1,3,2,2,3,1,4).$

Figure 2: Graph $SP(4,1,3,2,2,3,1,4).$

Note that the maximum degree $\bigtriangleup$ of $SP(m,1,m-1,2,...2,m-1,1,m)$
is $2m-1$.

######  Theorem 3.4.

[14] The superpath $SP(m,1,m-1,2,...,2,m-1,1,m)$ is integral for each natural
number m. Its spectrum consists of the simple eigenvalues $\pm
m,\pm(m-1),\pm(m-2),...,\pm 1$ and the eigenvalue 0 with multiplicity
$m(m-1)$.

From this theorem, we can say that the eigenvalues of
$SP(m,1,m-1,2,...,2,m-1,1,m)$ are consecutive integers $\pm 1,\pm 2,\pm
3,...,\pm m$.

######  Corollary 3.5.

The energy of $SP(m,1,m-1,2,...2,m-1,1,m)$ is $m(m+1).$

The following corollary gives the existence of orderenergetic graph of maximum
degree $2m-1$.

######  Corollary 3.6.

The graph $SP(m,1,m-1,2,...2,m-1,1,m)$ is orderenergetic for every $m$.
Observation 1. The graph $SP(m,1,m-1,2,...,2,m-1,1,m)$ is a graph with least
maximum degree in the collection of all orderenergetic graphs having $m^{2}+m$
vertices.

Observation 2. Let $G$ be an orderenergetic graph. Then $G$ is an integral
graph.

##  4 Hypoenergetic graphs

In 2007, I.Gutman et al.[10] introduced the definition of hypoenergetic
graphs. In this section, we present some techniques for constructing sequence
of hypoenergetic graphs.

######  Proposition 4.1.

Kronecker product of two hypoenergetic graphs is hypoenergetic.

###### Proof.

Let $G_{1}$ and $G_{2}$ be two hypoenergetic graphs with order $n_{1}$ and
$n_{2}$ respectively. Then $\varepsilon(G_{1})<n_{1}$ and
$\varepsilon(G_{2})<n_{2}$. By Lemma 2.1, $\varepsilon(G_{1}\times
G_{2})=\varepsilon(G_{1})\varepsilon(G_{2})<n_{1}n_{2}$. Thus Kronecker
product of $G_{1}$ and $G_{2}$, $G_{1}\times G_{2}$ is hypoenergetic graph. ∎

The following theorem enable us to construct infinitely many hypoenergetic
graphs.

######  Proposition 4.2.

Let $G_{1}$ be an orderenergetic graph and $G_{2}$ be a hypoenergetic graph.
Then Kronecker product of $G_{1}$ and $G_{2}$, $G_{1}\times G_{2}$ is
hypoenergetic graph .

###### Proof.

Let $G_{1}$ and $G_{2}$ be two graphs with order $n_{1}$ and $n_{2}$
respectively. Since $G_{1}$ is orderenergetic, $\varepsilon(G_{1})=n_{1}$ and
$G_{2}$ is hypoenergetic, $\varepsilon(G_{2})<n_{2}$. By Lemma 2.1,
$\varepsilon(G_{1}\times
G_{2})=\varepsilon(G_{1})\varepsilon(G_{2})<n_{1}n_{2}$. ∎

######  Example 4.1.

Let $G=K_{p,p}\times K_{1,3}.$ Then $|V(K_{p,p}\times K_{1,3})|=8p$ and
$\varepsilon(K_{p,p}\times K_{1,3})=4\sqrt{3}p<8p$. So $K_{p,p}\times K_{1,3}$
is hypoenergetic for every $p$ .

######  Proposition 4.3.

Let $G$ be a hypoenergetic graph of order $p$. Then the $m$-shadow graph of
$G$, $D_{m}(G)$ is hypoenergetic for every $m$ .

######  Example 4.2.

Let $G=K_{r,s}$ $r\neq s$. Then $|V(K_{r,s})|=r+s$ and
$\varepsilon(D_{m}(K_{r,s}))=2m\sqrt{rs}<m(r+s)=|V(D_{m}(K_{r,s}))|$ .

######  Remark 4.1.

Let $G$ be a hypoenergetic graph. Then the $m$-shadow graph of duplicate
graph, $D_{m}(D(G))$ is hypoenergetic.

######  Proposition 4.4.

Let $G$ be a hypoenergetic graph of order $p$. Then the $m$-splitting graph of
$G$, $spl_{m}(G)$ is hypoenergetic for $m>2$ .

###### Proof.

Since $G$ is a hypoenergetic graph, $\varepsilon(G)<p$. Also
$|V(spl_{m}(G))|=p(m+1)$.
As $m>2$,

$\displaystyle m(m-2)>0$ $\displaystyle\Longrightarrow(m+1)^{2}>1+4m$
$\displaystyle\Longrightarrow\sqrt{1+4m}\varepsilon(G)<p(m+1)$
$\displaystyle\Longrightarrow\varepsilon(spl_{m}(G))<p(m+1).$

Thus $spl_{m}(G))$ is hypoenergetic graph. ∎

Let $G$ be a graph of order $p$ and we denote $G_{r}^{s}=K_{r,s}\times
G,r,s\in N$.

It is very interesting to construct hypoenergetic graphs from non
hypoenergetic graphs.

The following theorem describes a construction of hypoenergetic graphs from
the complete graph.

######  Theorem 4.1.

Let $G=K_{p}$ be a complete graph on $p$ vertices and $m\geq 14$. Then the
graph $G_{1}^{m}$ is hypoenergetic.

###### Proof.

The energy of complete graph is $2(p-1)$ and $|V(G_{1}^{m})|=p(1+m)$.
As $m\geq 14$,

$\displaystyle m(m-14)+1>0$ $\displaystyle\Longrightarrow m^{2}+2m+1>16m$
$\displaystyle\Longrightarrow 4\sqrt{m}<(m+1)$ $\displaystyle\Longrightarrow
4\sqrt{m}(p-1)<p(m+1)$ $\displaystyle\Longrightarrow
2\sqrt{m}\varepsilon(K_{p})<p(m+1)$
$\displaystyle\Longrightarrow\varepsilon(G_{1}^{m})<p(1+m).$ (4.0.1)

Thus $G_{1}^{m}$ is hypoenergetic whenever $m\geq 14$. ∎

######  Remark 4.2.

The graph $G=K_{p}$ is non hypoenergetic but $G_{1}^{m}$ is hypoenergetic for
$m\geq 14$.

Since $4\sqrt{m}>(m+1)$ for $m<14$, so inequality (4.0.1) is satisfied for
every $p\leq k$, where $k=\lfloor\frac{4\sqrt{m}}{4\sqrt{m}-(m+1)}\rfloor$,
$\lfloor x\rfloor$ is floor of x. Thus $G_{1}^{m}$, ($m<14$) is hypoenergetic
whenever $p\leq k$.

######  Corollary 4.2.

Let $G$ be any non hyperenergetic graph. Then $G_{1}^{m}$ is hypoenergetic for
every $m\geq 14$.

##  5  Equienergetic graphs

In this section, we construct some new pairs of equienergetic graphs.

######  Proposition 5.1.

Let $G$ be a $(p,q)$ graph. Then the graphs $D_{m}(D(G))$ and $D_{2m}(G)$ are
non-cospectral equienergetic graphs.

######  Proposition 5.2.

Let $G$ be a $(p,q)$ graph and $D^{m}(G)$ be the m-duplicate graph of $G$.
Then $\varepsilon(D^{m}(G))=2^{m}\varepsilon(G)$.

######  Proposition 5.3.

The graphs $D^{m}(G)$ and $D_{2^{m}}(G)$ are non-cospectral equienergetic
graphs for all $m$.

######  Proposition 5.4.

Let $G$ be a simple $(p,q)$ graph. Then $G$ is integral if and only if its
$m$-duplicate graph $D^{m}(G)$ is integral.

The following propositions describes the class of equiorderenergetic graphs.

######  Proposition 5.5.

Let $G$ be an orderenergetic graph of order $p$. Then the the graphs
$spl_{2}(G)$ and $D_{3}(G)$ are equiorderenergetic graphs.

Figure 3: Graph $SP(2,1,1,2).$

Figure 4: Graph $D_{3}(SP(2,1,1,2)).$

Figure 5: Graph $spl_{2}(SP(2,1,1,2)).$

The graphs $D_{3}(SP(2,1,1,2))$ and $spl_{2}(SP(2,1,1,2))$ are
equiorderenergetic graphs.

######  Proposition 5.6.

Let $G$ be a hypoenergetic graph. Then the graphs $spl_{2}(G)$ and $D_{3}(G)$
are equihypoenergetic graphs.

### Conclusion

In this paper, we construct some family of orderenergetic graphs from the
known orderenergetic graphs. Also, some new families of hypoenergetic graphs
are derived by using some graph operations. Moreover, the problem for
constructing equienergetic graphs are discussed. In addition to that a new
class of equiorderenergetic and equihypoenergetic graphs are obtained.

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http://dx.doi.org/10.1145/3430895.3460139 rightsretained

# Should College Dropout Prediction Models Include Protected Attributes?

Renzhe Yu

Hansol Lee

René F. Kizilcec

University of California, Irvine  Irvine, CA, USA<EMAIL_ADDRESS>Cornell
University  Ithaca, NY, USA<EMAIL_ADDRESS>Cornell University  Ithaca, NY,
USA<EMAIL_ADDRESS>
###### Abstract

Early identification of college dropouts can provide tremendous value for
improving student success and institutional effectiveness, and predictive
analytics are increasingly used for this purpose. However, ethical concerns
have emerged about whether including protected attributes in these prediction
models discriminates against underrepresented student groups and exacerbates
existing inequities. We examine this issue in the context of a large U.S.
research university with both residential and fully online degree-seeking
students. Based on comprehensive institutional records for the entire student
population across multiple years (N = 93,457), we build machine learning
models to predict student dropout after one academic year of study and compare
the overall performance and fairness of model predictions with or without four
protected attributes (gender, URM, first-generation student, and high
financial need). We find that including protected attributes does not impact
the overall prediction performance and it only marginally improves the
algorithmic fairness of predictions. These findings suggest that including
protected attributes is preferable. We offer guidance on how to evaluate the
impact of including protected attributes in a local context, where
institutional stakeholders seek to leverage predictive analytics to support
student success.

###### keywords:

Dropout prediction; Predictive analytics; Higher education; Online learning;
Algorithmic fairness

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[500]Applied computing Law, social and behavioral sciences [500]Applied
computing Education

##  1 Introduction

With the rapid development of learning analytics in higher education, data-
driven instructional and learning support systems are increasingly adopted in
classroom settings, and institution-level analytics systems are used to
optimize resource allocation and support student success on a large scale. A
common objective of these systems is the early identification of at-risk
students, especially those likely to drop out of college. This type of
prediction has significant policy implications because reducing college
attrition has been a central task for institutional stakeholders ever since
higher education was made accessible to the general public [34]. As of 2018,
fewer than two-thirds of college students in the United States graduated
within six years, and this share is even smaller at the least selective
institutions which serve disproportionately more students from disadvantaged
backgrounds [24]. At the same time, the supply of academic, student affairs,
and administrative personnel is insufficient to provide just-in-time support
to students in need [24]. It is within these resource-strained contexts that
predicting dropouts based on increasingly digitized institutional data has the
potential to augment the capacity of professionals who work to support student
retention and success. Starting with the Course Signals project at Purdue
University, an increasing number of early warning systems have explored this
possibility at the institutional level [2, 26, 19, 13].

Accurately forecasting which students are likely to drop out is essentially
profiling students based on a multitude of student attributes. These
attributes often include socio-demographic information that is routinely
studied in higher education research. Although the analysis of historical
socio-demographic gaps in retention and graduation rates is well established
in higher education research [14], it becomes controversial to use these same
characteristics when making predictions about the future. For example, is it
fair to label a black first-year student as at risk based on the higher
dropout rate among black students in previous cohorts? The answer may be
equivocal [38]. On the one hand, the observed historical gaps capture
systematic inequalities in the educational environment of different student
groups, which may well apply to future students from the same groups and
therefore contribute to similar gaps. In this sense, explicitly using socio-
demographic data can result in more accurate predictions and improve the
efficiency of downstream interventions and actions based on those algorithmic
decisions [35]. On the other hand, from an ethics and equity perspective, the
inclusion of socio-demographic variables may lead to discriminatory results if
predictive models systematically assign differential predicted values across
student groups based on the records of their historical counterparts. When
these results are used for decision-making, stigmas and stereotypes could
carry over to future students and reproduce existing inequalities [27, 5].

In this paper, we investigate the issue of using protected attributes in
college dropout prediction in real-world contexts. Protected attributes are
traits or characteristics based on which discrimination is prescribed as
illegal, such as gender, race, age, religion, and genetic information. We
examine students in a residential college setting as well as students in fully
online degree programs, which have been increasingly represented in formal
higher education. In Fall 2018, 16.6% of postsecondary students in the United
States were enrolled in exclusively online programs, up from 12.8% in Fall
2012 [36, 39]. The absence of a residential experience exposes students to
additional challenges to accountability and engagement, and also makes it
harder for faculty and staff members to identify problems with students’ well-
being and provide timely support. The COVID-19 pandemic has forced most
colleges to move instruction online, which will likely increase the importance
of online learning in the future of higher education [33]. Predictive
analytics are therefore just as useful for online higher education as they are
for residential settings for supporting student achievement and on-time
graduation. Our findings in both residential and online settings offer
practical implications to a broad range of stakeholders in higher education.

By systematically comparing predictive models with and without protected
attributes in two higher education contexts, we aim to answer the following
two research questions:

  1. 1.

How does the inclusion of protected attributes affect the overall performance
of college dropout prediction?

  2. 2.

How does the inclusion of protected attributes affect the fairness of college
dropout prediction?

This research contributes to the literature on predictive modeling and
algorithmic fairness in (higher) education on several dimensions. First, we
present one of the largest and most comprehensive evaluation studies of
college dropout prediction based on student data over multiple years from a
large public research university. This offers robust insights to researchers
and institutional stakeholders into how these models work and where they might
go wrong. Second, we apply the prediction models with the same features to
both residential and online degree settings, which advances our understanding
of generalizability across contexts, such as in which environment it is easier
to predict dropout and to what degree key predictors differ. Third, we
contribute some of the first empirical evidence on how the inclusion of
protected attributes affects the fairness of dropout prediction, which can
inform equitable higher education policy around the use of predictive
modeling.

##  2 Related Work

###  2.1 College Dropout Prediction

Decades of research have charted the ecosystem of higher education as a
complex journey with "a wide path with twists, turns, detours, roundabouts,
and occasional dead ends that many students encounter" and jointly shape their
academic and career outcomes [29]. Among the variety of factors that influence
students’ journey, background characteristics such as demographics, family
background, and prior academic history are strong signals of academic, social,
and economic resources available to a student before adulthood, which are
substantially correlated with college success [12]. For example, ethnic
minorities, students from low-income families, and first-generation college
students have consistently suffered higher dropout rates than their
counterparts [14, 10], and students who belong to more than one of these
groups are even more likely to drop out of college. In addition to these
largely immutable attributes at college entry, students’ experiences in
college such as engagement and performance in academic activities are major
factors for success. In particular, early course grades are among the best
predictors of persistence and graduation, even after controlling for
background characteristics [29].

With the advent of the "datafication" of higher education [37], there has been
an increasing thrust of research to translate the empirical understanding of
dropout risk factors into predictive models of student dropout (or success)
using large-scale administrative data [4, 15, 26, 16, 6, 7, 25]. These
applications are usually intended to facilitate targeted student support and
intervention programs, and the extensive research literature on college
success has facilitated feature engineering grounded in theory. For example,
Aulck and colleagues [4] used seven groups of freshman features extracted from
registrar data to predict outcomes for the entire student population at a
large public university in the US. The model achieved an accuracy of 83.2% for
graduation prediction and 95.3% for retention. In a more application-oriented
study as part of the Open Academic Analytics Initiative (OAAI), Jayaprakash
and colleagues [26] developed an early alert system that incorporated
administrative and learning management system data to predict at-risk students
(those who are not in good standing) at a small private college, and then
tested the system at four other less-selective colleges.

While the recent decade has seen a steady growth in prediction-focused studies
on college dropout, a large proportion of them are focused on individual
courses or a small sample of degree programs [22]. Most of them investigate
dropouts at brick-and-mortar institutions. Our study pushes these research
boundaries by examining dropout prediction for multiple cohorts of students
across residential and exclusively online degree programs offered by a large
public university. The breath of our sample is rare in the dropout prediction
literature and promises to offer more generalizable insights about the utility
and feasibility of predictive models.

###  2.2 Algorithmic Fairness in Education

A central goal of educational research and practice has been to close
opportunity and achievement gaps between different groups of students. More
recently, algorithmic fairness has become a topic of interest as an increasing
number of students are exposed to intelligent educational technologies [27].
Inaccuracies in models might translate into severe consequences for individual
students, such as failing to allocate remedial resources to struggling
learners. It is more concerning if such inaccuracies disproportionately fall
upon students from disadvantaged backgrounds and worsen existing inequalities.
In this context, the fairness of algorithmic systems is generally evaluated
with respect to protected attributes following legal terms. The specific
criteria of fairness, however, vary and largely depend on the specific
application(s) [40].

In the past few years, a handful of papers have brought the fairness framework
to real-world learning analytics research. Most of these studies audit whether
supervised learning models trained on the entire student population generate
systematically biased predictions of individual outcomes such as correct
answers, test scores, course grades, and graduation [42, 30, 21, 25, 17, 32].
For example, Yu and colleagues [42] found that models using college-entry
characteristics to predict course grades and GPA tend to predict lower values
for underrepresented student groups than their counterparts. Other studies
have examined biases encoded in unsupervised representations of student
writing [3], or go further to refine algorithms for at-risk student
identification under fairness constraints [23]. Overall, this area of research
is nascent and in need of systematic frameworks specific to educational
contexts to map an agenda for future research.

When it comes to strategies to improve algorithmic fairness, a contentious
point is whether protected attributes should be included as predictors
(features) in prediction models. Most training data from the real world are
the result of historical prejudices against certain protected groups, so
directly using group indicators to predict outcomes risks imposing unfair
stereotypes and reproduce existing inequalities [5]. In educational settings,
it may be considered unethical to label students from certain groups as "at
risk" from day one, when in fact, these students have demonstrated an
exceptional ability to overcome historical obstacles and might therefore be
more likely to succeed [38]. This concern motivated the research effort to
“blind” prediction models by simply removing protected attributes (i.e.
fairness through unawareness) or more complicated statistical techniques to
disentangle signals of protected attributes from other features due to their
inherent correlation [9]. In contrast, recent work has advocated for
explicitly using protected attributes in predictive models (i.e. fairness
through awareness) [18]. In particular, Kleinberg and colleagues [28] showed
in a synthetic example of college admission that the inclusion of race as a
predictor of college success improves the fairness of admission decisions
without sacrificing efficiency. Given the well-documented relationship between
student background and their educational outcomes, a recent review also
suggests that predictive models in education should include demographic
variables to ensure that algorithms are value-aligned, i.e., all students have
their needs met [35].

To our knowledge, however, there is only limited empirical evidence to support
either side of this debate. Our study therefore presents an in-depth
examination of the consequences of including or excluding protected attributes
on algorithmic fairness of a realistic, large-scale dropout prediction model.

##  3 Methodology

Table 1: Features used for dropout prediction. Category |  Features
---|---
Protected attributes |  Gender (binary), first-generation college student (binary), underrepresented minority (URM; binary; defined as not Asian or White), high financial need (binary; FASFA-based expected family contribution under $5,500)
Incoming attributes |  Age, high school GPA, math and verbal SAT/ACT scores, transfer student (binary), transferred credits, transfer GPA
Program information |  Part-time student (binary), major, minor, STEM major (binary)
Course performance |  Total courses enrolled, total units enrolled, percentage of courses that are required, credits received from different types of courses (lecture, seminar, etc.), levels of courses (100, 200, etc.), term GPA, mean and variance in course grades within each session during the term, percentage distribution of letter grades

###  3.1 Dataset

We analyze de-identified institutional records from one of the largest public
universities in the United States. This broad-access research university
serves nearly 150,000 students with an 86% acceptance rate and 67% graduation
rate. Its student population is representative of the state in which it is
located, which makes it a Hispanic-serving institution (HSI). The university
has offered many of the same undergraduate degree programs fully online to
over 40,000 students. The dataset we use in this study focuses on
undergraduate students and contains student-level characteristics and student-
course-level records for their first term of enrollment at the university,
including transfer students (except for those who transfer into their senior
year). For our prediction task, we only keep students whose first term was in
the Fall along with their course-taking records in their first term, including
terms between 2012-18 (residential) and 2014-18 (online).

This sample comprises a total of 564,104 residential course-taking records for
93,457 unique students and 2,877 unique courses, and 81,858 online course-
taking records for 24,198 unique students and 874 unique courses. The course-
taking records include both a student’s letter grade and course-level metadata
(subject, course number, units, required for major, etc.). Student-level
information includes socio-demographic information (age, gender,
race/ethnicity, first-generation status, etc.), prior academic achievement
(high school GPA, standardized test scores), enrollment information (transfer
student status, part-time status, academic major and minor, etc.). These data
are representative of what most higher education institutions routinely manage
in their student information systems (SIS) [4].

###  3.2 Prediction Target and Feature Engineering

The primary goal of a dropout prediction model is to alert relevant
stakeholders to currently enrolled students who are at risk of dropping out of
a degree program so that they can reach out and offer support at an early
stage. While the general framework of dropout prediction is well established,
the exact definition of dropout, or attrition, varies based on the specific
context [34]. In our context, we define dropout as not returning to school a
year from the first time of enrollment. We only analyze students who first
enrolled in Fall, so dropout means not returning in the following Fall. This
final operationalization aligns well with retention, one of the two standard
metrics of post-secondary student success in national reports of the United
States [39, 24].111The other standard metric is graduation within 100% or 150%
of the normative time (i.e. 4 or 6 years for four-year institutions). We do
not examine this metric because the span of our dataset is only six years and
we do not observe graduation outcomes for all student cohorts.

We use students’ background characteristics and academic records in the first
enrolled term (Fall) to predict dropout, because it would be beneficial to
identify risks as early as possible and institutional records are usually
updated and available at the end of each term. Informed by existing research
in higher education and learning analytics (see Related Work), we construct 58
features from the dataset for both residential and online students. Table 1
summarizes these feature by four categories. We include four protected
attributes, which are the most commonly used dimensions along which to examine
educational inequalities and set equity goals in policy contexts [10, 11, 24].

Table 2 depicts the student profile in our analysis. The statistics reaffirm
that, regardless of format, the institution serves a large proportion of
students from historically disadvantaged groups. There are also major
differences across formats. In line with the national statistics of
exclusively online programs [39], the online sample has a higher concentration
of transfer and non-traditional (older, part-time) students, and also higher
dropout rates compared to residential students. These characteristics validate
that the current analysis is performed on student populations who are most in
need of institutional support and allow us to scrutinize the generalizability
of our findings across two distinct contexts of higher education.

Table 2: Comparison of online and residential student populations. | Online | Residential
---|---|---
N | 24,198 | 93,457
Dropout | 40.7% | 16.9%
Female | 60.9% | 47.9%
First-gen | 42.4% | 33.6%
URM | 33.1% | 34.6%
High need | 61.9% | 51.3%
Transfer | 85.2% | 31.8%
Part-time | 77.2% | 12.9%
Average age | 27.1 | 19.7

###  3.3 Dropout Prediction

To investigate the consequences of using protected attributes in dropout
prediction models, we generate two feature sets: the AWARE set includes all
features shown in Table 1, while the BLIND set excludes the four protected
attributes from the AWARE set. For convenience, we will refer to a specific
model by the feature set it uses in the remainder of this paper. Given our
binary target variable, the dropout prediction task is formalized as a binary
classification problem. As we focus on identifying the effect of including
protected attributes, we experiment with two commonly used algorithms –
logistic regression (LR) and gradient boosted trees (GBT). We choose LR
because it is a linear additive and highly interpretable classifier that can
achieve reasonable prediction performance with well-chosen features. The
choice of GBT, on the other hand, is for its ability to accommodate a large
number of features, efficiently handle missing values, and automatically
capture non-linear interactions between features.

We predict dropping out separately for online and residential students. For
each format, we split the data into a training set and a test set based on
student cohort: the last observed cohort (6,939 online and 14,275 residential
students entering in Fall 2018) constitutes the test set and the remaining
cohorts make up the training set (17,259 online and 79,182 residential
students). There are two reasons for doing the train-test split by student
cohorts. Practically, this split aligns with the real-world application where
stakeholders rely on historical data to make predictions for current students
[26]. Technically, this approach alleviates the issue of data contamination
between the training and test set [20], as the features we use, especially the
first-semester records, might be highly correlated within the same cohort but
much less so across cohorts.

There are a few additional technical details about model training. First, we
tune hyperparameters of the two algorithms by performing grid search over a
specified search space and evaluating the hyperparameters using 5-fold cross-
validation. Second, we add indicator variables for missing values in course
grades, standardized test scores, and academic majors and minors. Third, we
apply robust scaling to training features to regulate the influence of
outliers. Fourth, because the class imbalance in both datasets can bias the
model learning towards the majority class (i.e. non-dropout), we adjust the
sample weights to be inversely proportional to class frequencies during the
training stage.

The trained classifiers are then applied to the test set to evaluate the
performance. The immediate output of each classifier is a predicted
probability of dropping out for each student. To make a final binary
prediction of dropout, we use dropout rates in the training data to determine
the decision thresholds for the test set, such that the proportion of
predicted dropouts in the test set matches the proportion of observed dropouts
in the training set [7]. Compared to the default of 0.5, this choice of
threshold is more reasonable when we rely on the observed history to predict
the unknown future in practice.

###  3.4 Performance Evaluation

We evaluate prediction performance based on three metrics: accuracy, recall,
and true negative rate (TNR). In the context of dropout prediction, recall is
the proportion of actual dropouts who are correctly identified, whereas TNR
quantifies how likely a student who persists into the second year of college
is predicted to persist. To examine the effects of including protected
attributes on overall performance, we compute these metrics separately for
each model and test whether each metric significantly changes from BLIND to
AWARE models, using two proportion $z$-tests.

We operationalize fairness as the independence between prediction performance,
measured by the three metrics above, and protected group membership. This
definition of fairness with respect to the three metrics corresponds to the
established notions of overall accuracy equality, equal opportunity, and
predictive equality, respectively [27]. Specifically, to quantify the fairness
of a given model with regard to a binary protected attribute, such as URM, we
compute the differences in each of the three metrics between the two
associated protected groups, URM and non-URM students. We then compare how
much these differences change between BLIND and AWARE models in order to
quantify the effect of including protected attributes as predictors on
fairness.

##  4 Results

###  4.1 Overall Prediction Performance

Table 3: Overall prediction performance of AWARE and BLIND models trained with gradient boosted trees (GBT) and logistic regression (LR). | GBT | LR
---|---|---
Metric | AWARE | BLIND | $\Delta$ | AWARE | BLIND | $\Delta$
Online (Non-dropout: 59.3%)
Accuracy | 75.8 | 75.6 | 0.2 | 75.2 | 75.4 | -0.2
Recall | 67.3 | 67.1 | 0.2 | 66.7 | 66.8 | -0.1
TNR | 82.4 | 82.3 | 0.1 | 81.9 | 82.0 | -0.1
Residential (Non-dropout: 83.1%)
Accuracy | 83.9 | 83.9 | 0.0 | 83.6 | 83.6 | 0.0
Recall | 54.1 | 54.1 | 0.0 | 53.2 | 53.3 | -0.1
TNR | 89.1 | 89.1 | 0.0 | 88.9 | 88.9 | 0.0
Note: none of the $\Delta$ values is statistically significant with $p<0.1$.

Figure 1: Distribution of predicted dropout probability

We first illustrate the effects of including protected attributes on overall
prediction performance. Table 3 reports the overall performance of AWARE and
BLIND models, trained with GBT and LR algorithms, on the test dataset. The
last column under each algorithm reports the percentage point differences in
performance between the two models (from BLIND to AWARE). The main finding is
that including or excluding protected attributes does affect the performance
of the dropout prediction in either context. None of the performance metrics
(accuracy, recall, TNR) differs significantly between the BLIND and AWARE
models. Additionally, while the more sophisticated GBT algorithm performs
better than the simple LR on all metrics, the advantage is comparatively small
(less than one percentage point on all metrics). Because of this, we restrict
the following analysis to GBT-based models.

Compared to a naïve baseline which simply predicts every student to be the
majority class (non-dropout) and achieves an accuracy equal to that majority’s
share, the predictive models can accurately predict online dropouts with a
decent margin. However, the accuracy margin for predicting residential
dropouts is fairly small. The other two metrics, which describe the accuracy
among dropouts and non-dropouts respectively, achieve a higher value when the
corresponding group has a larger share and vice versa. Specifically, the
models are able to identify 67.3% of online dropouts and 54.1% of residential
dropouts. This latter value is somewhat lower but still comparable to the
recall performance in recent prior work on dropout prediction in residential
programs [7, 16].

To take a closer look at the model predictions, beyond the three aggregate
performance metrics, we examine whether including protected attributes alters
the distribution of predicted dropout probabilities. As shown in Figure 1, the
distributions are highly similar across the models which further validates the
limited marginal impact of protected attributes. An additional insight from
these plots is that dropouts might be much more heterogeneous than non-
dropouts in terms of the features in Table 1, as their predicted probabilities
are highly spread out, especially in residential settings where the majority
of dropouts are assigned a small dropout probability. This pattern is
consistent with the lower recall performance shown in Table 3.

This finding appears to conflict with prior research that demonstrates the
critical role of demographic and background characteristics for student
success in higher education [29]. In an effort to better understand our
result, we explore two mutually compatible hypotheses inspired by the
algorithmic fairness literature. One hypothesis is that dropping out, the
prediction target, is not sufficiently correlated with protected attributes,
and thus adding the latter to a dropout prediction model would not improve
performance much. To test this, we fit separately for each enrollment format
in the test data a logistic regression model that predicts dropout using all
the possible interaction terms between the four protected attributes. We find
that, even though a few coefficients are statistically significant, the
adjusted McFadden’s $R^{2}$ is as small as 0.006 for either format, lending
support to our hypothesis.

The second hypothesis is that protected attributes are already implicitly
encoded in the BLIND feature set, and adding them directly does not add much
predictive power. We test this by fitting four logistic regressions for each
format which use the BLIND feature set to predict each of the four protected
attributes. Based on the adjusted McFadden’s $R^{2}$, we find that only gender
can plausibly be considered encoded in the other features (0.159 for online
and 0.187 for residential). This lends partial support to our second
hypothesis.

###  4.2 Fairness of Prediction

Figure 2: Fairness of AWARE and BLIND models in terms of accuracy (left),
recall (middle), and TNR (right). Positive group differences (y-axis) indicate
higher values for the listed groups compared to their corresponding reference
groups. Group differences closer to zero reflect higher levels of fairness.
Error bars indicate 95% confidence intervals.

We further examine how the inclusion of protected attributes might affect the
fairness of dropout predictions. As mentioned in the previous section, for
each of the four protected attributes, we first measure fairness by the group
difference in a chosen performance metric. For example, a prediction model
that achieves the same accuracy on male and female students is considered fair
in terms of accuracy (0% difference). Following this construction, Figure 2
visualizes these fairness results of the AWARE and BLIND models for each of
the four protected attributes in terms of the three metrics. Each bar in a
subplot depicts the difference in that metric between the labeled group and
their counterpart (e.g., male - female). The closer the bar is to zero, the
fairer that model prediction is. Overall, the figure shows that both the AWARE
and BLIND models are unfair for some protected attributes and some metrics,
but fair for others. This lack of universal fairness is expected given the
many dimensions of protected attributes, models, and metrics. However, for
residential students, the model consistently exhibits unfairness across all
protected attributes and metrics, especially in terms of recall. The inclusion
or exclusion of protected attributes does not in general lead to different
levels of fairness in terms of any metric in any enrollment format, as all
adjacent error bars in the figure exhibit a high degree of overlap.

Figure 3: Distribution of change in individual ranking of predicted dropout
probability from AWARE to BLIND. One unit increase means going up by one place
in the AWARE model compared to the BLIND model.

While the aggregated group fairness metrics do not differ with vs. without
protected attributes, we take a step further to explore how individual-level
changes in model predictions can shed light on the overall change in fairness.
We examine changes in the individual ranking of predicted dropout probability
among all predicted students (test set) from BLIND to AWARE model. Figure 3
plots the distribution of this ranking change for each protected group, where
higher values represent moving up in the assigned risk leaderboard when
protected attributes are included for prediction.

We find that overall the ranking change is centered around zero, but there are
observable group differences in certain cases. In the online setting, the
AWARE model tends to move up females and students without a high financial
need on the dropout risk leaderboard simply based on their identity.
Similarly, continuing-generation college students are moved up more in
residential settings compared to their first-generation counterparts. We argue
that these group differences suggest improved fairness if the group going up
more in the ranking spectrum has lower dropout rates in reality, and vice
versa. To formally evaluate this reasoning, we conduct a series of $t$-tests
between pairs of protected groups on their ranking change. We also compute
Cohen’s $d$ to gauge the standardized effect size. Comparing Table 5 which
describes these results and Table 4 which presents the actual dropout rates of
each group, we find that moving from BLIND to AWARE causes students from
advantaged (lower dropout rates) groups to be assigned relatively higher risk
rankings compared to their disadvantaged (higher dropout rates) reference
groups, and that this effect size is larger when the two paired groups have
larger gaps in dropout rates. Thus, adding protected attributes to the model
is working against existing inequities to a marginal extent instead of
reinforcing them.

Table 4: Dropout rates among different protected groups in the test set. | Dropout rate
---|---
| Online | Residential
Overall | 40.7 | 16.9
Male | 49.5 | 15.6
Female | 40.7 | 14.0
URM | 46.6 | 16.8
Non-URM | 42.4 | 13.7
First-gen | 43.7 | 17.9
Continuing-gen | 44.1 | 13.5
High need | 45.8 | 17.0
Low need | 40.5 | 12.8
Table 5: Welch two-sample t-test results and Cohen’s d effect size of individual ranking change. Positive ranking change means increased predicted dropout risks from BLIND to AWARE model. Group (Avg. ranking change) |  Rank $\Delta$ |  Cohen’s $d$
---|---|---
Online
Female (50.5) | Male (-88.4) | 138.9*** | 0.71
Non-URM (0.5) | URM (-0.9) | 1.4 | 0.01
Continuing-gen (-13.9) | First-gen (18.6) | -32.5*** | 0.16
Low need (104.5) | High need (-60.0) | 164.5*** | 0.87
Residential
Female (15.9) | Male (-15.1) | 31.0*** | 0.13
Non-URM (13.0) | URM (-22.8) | 35.8*** | 0.15
Continuing-gen (27.2) | First-gen (-62.1) | 89.3*** | 0.38
Low need (6.8) | High need (-7.0) | 13.8*** | 0.06
Significance levels: *** p<0.001; ** p<0.005; * p<0.01

##  5 Discussion and Conclusion

We set out to answer a simple question: Should protected attributes be
included in college dropout prediction models? This study offers a
comprehensive empirical examination of how the inclusion of protected
attributes affects the overall performance and fairness of a realistic
predictive model. We demonstrate this finding across two large samples of
residential and online undergraduate students enrolled at one of the largest
public universities in the United States. Our findings show that including
four important protected attributes (gender, URM, first-generation student,
high financial need) does not have any significant effect on three common
measures of overall prediction performance when commonly used features
(incoming attributes, enrollment information, academic records) are already in
the model. Even when used alone without those features, the group indicators
defined by the protected attributes are not highly predictive of dropout,
although the actual dropout rates are somewhat higher among minoritized
groups. In terms of fairness, we find that including protected attributes only
leads to a marginal improvement in fairness by assigning dropout risk scores
with smaller gaps between minority and majority groups. However, this trend is
not sufficiently large to systematically change the final dropout predictions
based on the risk scores, and therefore the formal fairness measures are not
significantly different between models with and without protected attributes.

In short, our results suggest limited effects of including protected
attributes on the performance of college dropout prediction. This does not
point to a clear answer to our normative question and prompts us to further
reflect on the focal issue of using protected attributes. Recent work in the
broader machine learning community has been in favor of “fairness through
awareness” [18], and has specifically suggested that race-aware models are
fairer for student success prediction because they allow the influence of
certain features to differ across racial groups [28]. Our findings resonate
with these existing studies around fairness but only to a marginal extent.
Notably, student groups with historically higher dropout rates are slightly
compensated by being ranked lower in predicted dropout risks when protected
attributes are used. This compensating effect, however, does not accumulate to
statistically significant changes in predicted labels, possibly because the
group differences in dropout rates were not sizeable in the past at the
institution we study (see Table 4). In other words, protected attributes might
have more to contribute to the fairness of prediction in the presence of
substantial existing inequalities. Still, the existence of a weak compensating
instead of segregating effect justifies the inclusion of these attributes.
After all, a major argument for race-aware models, and more generally socio-
demographic-aware models, is to capture structural inequalities in society
that disproportionately expose members of minoritized groups to more adverse
conditions. In addition, the deliberate exclusion of protected attributes from
dropout prediction models can be construed as subscribing to a “colorblind”
ideology, which has been criticized as a racist approach that serves to
maintain the status quo [8].

Another contribution of this work lies in our approach to fairness evaluation.
The analyses and visualizations we present are the result of many iterations
to arrive at simple yet compelling ways to communicate fairness at different
levels of aggregation and across many protected attributes. These methods can
be used by those who seek to evaluate model fairness for research and
practice. Prior research has mostly focused on evaluating one protected
attribute at a time, but in most real-world applications we care about more
than one protected attribute. We recommend comparing AWARE against BLIND
models in terms of the individual ranking differences by group (Figure 3) as
well as the group difference plots for multiple performance metrics and
protected attributes (Figure 2). This approach offers a sensitive instrument
for diagnosing fairness-related issues in various domains of application,
which could easily be implemented in a fairness dashboard that evaluates
multiple protected attributes, models, and performance metrics [41]. This will
remain a promising line of our future work.

This research has broader implications for using predictive analytics in
higher education beyond its contributions to algorithmic fairness. With a
common set of institutional features, we achieve 76% prediction accuracy and
67% recall on unseen students in online settings, that is, correctly
identifying 67% of actual dropouts with their first-term records. For
residential students, we achieve a higher accuracy of 84% but a lower recall
of 54%. These performance metrics may seem somewhat lower than in prior
studies of dropout prediction, but this might be because most existing studies
examine a smaller sample of more homogeneous students, such as students in the
same cohort or program [16, 15]. This highlights the general challenge of
predicting college dropout accurately. As suggested by the large variance in
predicted probabilities for dropouts (Figure 1), widely used institutional
features might not perform well in capturing common signals of dropout. This
may point to important contextual factors that our institutional practices are
presently overlooking. We view this as a limitation and important next step
that will require both an interrogation of the theoretical basis for
predictors and close collaboration with practitioners.

Further directions for future research in this area include exploring
counterfactual notions of fairness in this context by testing how predictions
would differ for counterfactual protected attributes, all else being equal.
This would benefit the contemporary education system which relies increasingly
on research that provides causal evidence. We would also like to move from
auditing to problem-solving by evaluating correction methods for any pre-
existing unfairness in predictions to see how the AWARE relative to the BLIND
model responds [31]. We hope that this study inspires more researchers in the
learning analytics and educational data mining communities to engage with
issues of algorithmic bias and fairness in the models and systems they develop
and evaluate.

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