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Ecological Indicators
journal homepage: www.elsevier.com/locate/ecolind
Research paper
A study on the effects of unbalanced data when fitting logistic regression
models in ecology
Christian Salas-Eljatib
a,⁎
, Andres Fuentes-Ramirez
a
, Timothy G. Gregoire
b
, Adison Altamirano
c
,
Valeska Yaitul
a
a
Laboratorio de Biometría, Departamento de Ciencias Forestales, Universidad de La Frontera, Temuco, Chile
b
School of Forestry and Environmental Studies, Yale University, New Haven, CT 065111, USA
c
Laboratorio de Ecología del Paisaje Forestal, Departamento de Ciencias Forestales, Universidad de La Frontera, Temuco, Chile
ARTICLE INFO
Keywords:
Statistical inference
Model prediction
Logit model
Binary variable
Bias
Precision
ABSTRACT
Binary variables have two possible outcomes: occurrence or non-occurrence of an event (usually with 1 and 0
values, respectively). Binary data are common in ecology, including studies of presence/absence, alive/dead,
and change/no-change. Logistic regression analysis has been widely used to model binary response variables.
Unbalanced data (i.e., an extremely larger proportion of zeros than ones) are often found across a variety of
ecological datasets. Sometimes the data are balanced (i.e., same amount of zeros and ones) before fitting the
model, however, the statistical implications of balancing (or not) the data remain unclear. We assessed the
statistical effects of balancing data when fitting a logistic regression model by studying both its statistical
properties of the estimated parameters and its predictive capabilities. We used a base forest-mortality model as
reference, and by using stochastic simulations representing different configurations of 0/1 data in a sample
(unbalanced data scenarios), we fitted the logistic regression model by maximum likelihood. For each scenario
we computed the bias and variance of the estimated parameters and several prediction indexes. We found that
the variability of the estimated parameters is affected, with the balanced-data scenario having the lowest
variability, thus, affecting the statistical inference as well. Furthermore, the prediction capabilities of the model
are altered by balancing the data, with the balanced-data scenario having the better sensitivity/specificity ratio.
Balancing, or not, the data to be used for fitting a logistic regression models may affect the conclusion that can
arise from the fitted model and its subsequent applications.
1. Introduction
Data of occurrence/non-occurrence of a phenomenon of interest are
vastly found across several disciplines (Alberini, 1995; Arana and Leon,
2005; Bell et al., 1994). This type of variable is known as binary or
dichotomous, and it represents whether an event occurs or not. This
event is represented by the random variable Y, and we usually record
occurrence by Y= 1 and non-occurrence by Y= 0. In ecology, binary
variables arise when studying the presence of a species in a geographic
area (Bastin and Thomas, 1999; Phillips and Elith, 2013; Hastie and
Fithian, 2013) or the occurrence of mortality at the tree or forest level
(Davies, 2001; Wunder et al., 2008; Chao et al., 2009; Young et al.,
2017). Meanwhile in landscape ecology, binary variables are used to
represent the occurrence of fire within a given area (Bigler et al., 2005;
Mermoz et al., 2005; Dickson et al., 2006; Vega-García and Chuvieco,
2006; Palma et al., 2007; Bradstock et al., 2010); deforestation (Wilson
et al., 2005; Schulz et al., 2011; Kumar et al., 2014; Hu et al., 2014);
and in general the change from one land use category to another (Seto
and Kaufmann, 2005; Leyk and Zimmermann, 2007; Lander et al.,
2011).
Logistic regression analysis is the most frequently used modelling
approach for analyzing binary response variables. If we need to model a
binary variable, to statistically relate it to predictor variable(s) or
covariate(s), one of the most used approaches for pursuing this task in
ecology is to use logistic regression models (Warton and Hui, 2011).
These models belong to group of the generalized linear models (GLM).
In a GLM, three compartments must be specified (Lindsey, 1997;
Schabenberger and Pierce, 2002): a random component, a systematic
component, and a link function. A logistic regression model uses: a
binomial probability density function as the random component; a
linear predictor function X′β(where Xis a matrix with the covariates
and βis a vector with the parameters or coefficients) as the systematic
component; and a logistic equation as the link function. One of the key
advantages of using logistic regression models in ecology is that the
http://dx.doi.org/10.1016/j.ecolind.2017.10.030
Received 7 April 2017; Received in revised form 9 August 2017; Accepted 16 October 2017
⁎
Corresponding author. Tel.: +56 45 2325652.
E-mail address: christia.salas@ufrontera.cl (C. Salas-Eljatib).
(FRORJLFDO,QGLFDWRUV²
;(OVHYLHU/WG$OOULJKWVUHVHUYHG
0$5.
probability of the binary response variable is directly modelled, thereby
accounting explicitly for the random nature of the phenomenon of in-
terest.
In many applications when dealing with binary data in ecology, it
happens that the number of observations with ones (Y= 1) is much
smaller than the number of observations with zeros (Y= 0) or vice
versa. We simply term this situation as unbalanced data, but other terms
have been also used for this situation, including disproportionate
sampling (Maddala, 1992) or rarity events (King and Zeng, 2001).
Based on our review of scientific applications of logistic regression to
model ecological phenomena, the proportion of zeros in datasets ranges
between 80% and 95%. Therefore, having balanced data (i.e., equal
numbers of observations of zeros and ones) is more the exception than
the rule.
Both unbalanced and balanced data have been used for fitting lo-
gistic regression models. In ecological studies, some researchers have
adopted the practice of balancing the data before carrying out the
analyses (e.g., Vega-García et al., 1995; Vega-García et al., 1999; Lloret
et al., 2002; Brook and Bowman, 2006; Vega-García and Chuvieco,
2006; Jones et al., 2010; Rueda, 2010). Balancing data means to select,
by some rule (usually at random), the same amount of observations
with ones and zeros from the originally available dataset. Therefore, a
balanced dataset or balanced sample is created, where a 50–50% pro-
portion of zero and one values is met. After the balanced dataset is
built, the logistic regression model is fitted (i.e., its parameters are es-
timated) by maximum likelihood (ML). An example of this practice in
ecological applications is the option for balancing data before fitting a
logit model when conducting analyses of land use changes in the soft-
ware IDRISI (Eastman, 2006). On the other hand, it is important to
point out that unbalanced data have been also used in ecological studies
(Wilson et al., 2005; Echeverria et al., 2008; Kumar et al., 2014; Young
et al., 2017). Therefore, unbalanced data in applied ecological studies
has been considered as not having important effects into the models
being fitted. Moreover, to date, no studies have addressed the effect of
balancing data when fitting logistic regression models in ecological
analyses, and just a handful have explored some statistical implications
in ecological applications (Qi and Wu, 1996; Wu et al., 1997; Cailleret
et al., 2016).
The applied statistical implications of unbalanced data in logistic
regression are not well described nor realized for applied researchers.
Although balancing the data seems to be an accepted practice, the
reasons that justify its use are not well explained. The most immediate
effect of balancing the data is to greatly reduce the sample size avail-
able for fitting purposes, therefore decreasing the precision with which
the parameters of the model are estimated. Among the statistical studies
on logistic regression and unbalanced data, we highlight the following:
Schaefer (1983) and Scott and Wild (1986) pointed out that the max-
imum likelihood estimates (MLE) of a logit model are biased only for
small sample sizes. On the other hand, Xie and Manski (1989) stated
that unbalanced data only affect the intercept parameter of a logit
model, specifically being biased estimated according to Maddala
(1992).King and Zeng (2001), advocated that all the MLE of the logit
parameters are biased. Schaefer (1983) and Firth (1993) proposed
correction for the bias of the MLE of the logistic regression model
parameters. McPherson et al. (2004) conducted one of the few related
analysis when fitting presence-absence species distribution models in
ecology, but only focusing in the prediction capabilities of the fitted
models. Maggini et al. (2006) assessed the effect of weighting absences
when modelling forest communities by generalized additive models.
Recently, Komori et al. (2016) indicated that logistic regression suffer
poor predictive performance, and proposed an alternative model to
improve predictive performance. Komori et al. (2016) approach in-
volves to add a new parameter to the original structure of a logistic
regression model, and fitted it in a mixed-effects modelling framework,
therefore their approaches becomes a different type of statistical model.
From above, we can infer that: (a) most of the statistical studies on
logistic regression and unbalanced data have focus on the bias of the
MLE parameters (a topic that has been rarely taking into account in
ecological applications); (b) much less attention has been put into the
prediction performance; and (c) no study has dealt with the effects of
unbalanced data in the variance of the MLE parameters.
In this study we aimed at assessing the effect of using unbalanced
data when fitting logistic regression models by analyzing both the
statistical properties (i.e., bias and variance) of the estimated para-
meters and the predictive capabilities of the fitted model.
2. Methods
2.1. Base model
The binary variable (Y) is the occurrence of a phenomenon of in-
terest, where Y= 1 denotes occurrence and Y= 0 otherwise. In a
modelling framework, we seek to model the probability of the response
variable being Y= 1, given the values of the predictor variables, this is
Pr(Y= 1|X), that we can more easily represent by π
y∣X
.
In our analysis we used a logistic regression equation with five
predictor variables, as a base model for carrying out our analysis. This
model served as a reference for assessing the statistical effects of un-
balanced data on fitting logistic regression models. The binary variable
of forest mortality occurrence (Y), given the analyses of Young et al.
(2017) in the state of California, USA, is modeled as a function of cli-
mate and biotic variables, as follows:
⎡
⎣
⎢−⎤
⎦
⎥===++++
+
π
πYββXβXβXβX
βX
ln 1logit[ 1]
,
yx
yx
i01
1223344
55
ii
ii
iiii
i
(1)
where Y
i
is the occurrence of forest mortality (i.e., 1 for occurrence, 0
for non-occurrence) at the ith pixel), meanwhile the predictor variables
X
1i
,X
2i
,X
3i
,X
4i
, and X
5i
represent the: mean climatic water deficit
(CWD) or simply Defnorm
i
; basal area of live trees (BA
i
);
B
Ai
2
; CWD
anomaly (Defz0
i
); and Defnorm
i
×BA
i
for the ith pixel, respectively. We
have used the nomenclature for the variables as in the study of Young
et al. (2017) and only the available data for year 2012. Notice that we
could more easily represent model (1) as:
⎡
⎣
⎢−⎤
⎦
⎥===
′β
π
πyX
l
n1logit[ 1] ,
yX
yX (2)
where yis the vector with the binary variable, Xis the matrix with the
predictor variables (and a first column of 1), and βis the vector of
parameters
ˆ
β
[
0
,
ˆ
β
1
,
ˆ
β
2
,
ˆ
β3,
ˆ
β
4,
ˆ
β]
5.
In the sequel, we shall use Eq. (2) as the mean function in various
scenarios of unbalanced data. It is important to point out that we are
not interested in finding the best model, but rather on studying the
effects of using several unbalanced data scenarios on a reference model.
Furthermore, we want to remark that we are not pursuing to assess
different alternative statistical models for unbalanced data (e.g. as in,
Warton and Hui, 2011; Hastie and Fithian, 2013). We also want to
mention that the zero-inflated models are those focusing on modelling
count variables (Schabenberger and Pierce, 2002; Zuur et al., 2010),
such as the prediction of the amount of tree mortality (e.g., Affleck,
2006). These models are not part of our study, since we are dealing with
modelling a binomial variable.
2.2. Unbalanced data scenarios
We use data of forest mortality occurrence from Young et al. (2017),
in California during 2012 as our population, containing 11763 total
observations (N), with 2985 cases of mortaltity occurrence (N
1
) and
8778 cases of non-occurrence (N
0
). In order to assess the effects of
unbalanced data on the statistical properties of the logit model (Eq.
C. Salas-Eljatib et al. (FRORJLFDO,QGLFDWRUV²
(2)), we examined different sample strategies from the population,
where each has a different proportion of occurrence and non-occur-
rence of mortality (1 and 0 values, respectively). We fixed the sample
size in n= 1000 in all scenarios, and the number of cases with zeros
and ones for the response variable that the sample should contain,
across scenarios ranging from 10% to 90%. In this way, we constrained
the sample to containing different cases with zeros (n
0
) and ones (n
1
),
but the same sample size (n= 1000). In order to achieve each of the
proportion of 0/1 values, which has a fixed sample size of 0 and 1 (i.e.,
n
0
and n
1
, respectively), we (i) drew a random sample without re-
placement of size (n
0
) from the sub-population (with size N
0
) of cases
containing zero in the response variable; (ii) drew a random sample
without replacement of size n
1
from the sub-population (with size N
1
)
of cases containing ones in the response variable; and (iii) merge the
randomly selected n
0
and n
1
cases in a sample of size n(i.e.,
n=n
0
+n
1
).
2.3. Statistical assessment
We assessed the statistical properties of the fitted logistic regression
model by stochastic simulations (i.e., Monte Carlo simulation). We
carried out S= 100, 000 simulations so that the sampling error of the
simulation itself is negligibly small. A similar analysis to justify the
number of simulation has been conducted by Gregoire and
Schabenberger (1999), in agreement with the amount of simulations
conducted in other statistical simulation studies (e.g. Gregoire and
Salas, 2009; Salas and Gregoire, 2010). For each simulated sample, we
fitted the logistic regression model (Eq. (2)) by maximum likelihood
using the glm function implemented in R (R Development Core Team,
2016).
Based on the simulations, we examined the empirical distribution of
the estimated parameters and prediction indexes. Our assessment was
divided and focused in: (a) the statistical properties of estimated model
parameters, and (b) the accuracy of predictions from the fitted model.
(a) Statistical properties of the estimated parameters. In order to assess
how the accuracy of the estimated parameters is affected by unbalanced
data, we computed the empirical bias (B
MC
) of each parameter being
estimated,
ˆ
θ, as follows:
=−
ˆˆ
θθ θ
B
[] E[]
,
MC (3)
where θis the respective parameter value and
ˆ
θ
E
[]
is the empirical
expected value of the estimated parameter. The former was obtained
from the maximum likelihood estimate (MLE) of θusing the population
available, and the latter is approximated from the average of the S
values of the estimated parameter
ˆ
θ. Notice that
ˆ
θin Eq. (3) is replaced
by each parameter of the model (i.e.,
ˆ
β
0
,
ˆ
β
1
,
ˆ
β
2
,
ˆ
β3,
ˆ
β3, and
ˆ
β5).
In order to assess how the precision of the estimated parameters is
affected by unbalanced data, we computed the empirical variance
(V
MC
) of each estimated parameter
ˆ
θas follows:
∑
=−
=
θSθθ
V
[ˆ]1(ˆE[ˆ])
,
j
S
jMC
1
2
(4)
where
ˆ
θis the MLE of θfor the jth simulation. Finally, we compute the
empirical mean square error (ECM
MC
) of each
ˆ
θby:
=+
ˆˆ ˆ
θθ θ
E
CM [ ] V ( ) [B ( ) ]
MC MC MC 2(5)
We represented the variance and mean square error in the same units of
the estimated parameters by taking their square root, thus obtaining the
standard error (SE
ˆ
θ
[
]
) and their root mean squared error (RMSE
ˆ
θ
[
]
).
(b) Prediction capabilities. For each simulation and unbalanced data
scenario we computed prediction indexes of the logistic regression
model. In order to do so, we calculated the predicted probability of
mortality occurrence for the ith observation ( =
ˆ
πyX1
i
i
), as follows:
=+
=−′
ˆ
ˆ
π
e
1
1
,
yβ
XX
1
iii(6)
where X
i
is the matrix of predictor variables for the ith case and
ˆ
β
is the
vector of estimated parameters. We use a probability threshold of 0.5
for occurrence, that is to say, if ≥
=
ˆ
π0.5
yX1
ii
we assume that the event
occurs, and non-occurrence otherwise (Jones et al., 2010). Based on
these predicted probabilities, we computed the following eight pre-
diction indexes: commission error (proportion of ncases in which the
model erroneously predicts occurrence); commission accuracy (pro-
portion of ncases in which the model correctly predicts occurrence);
omission error (proportion of ncases in which the model erroneously
predicts non-occurrence); omission accuracy (proportion of ncases in
which the model correctly predicts non-occurrence); sensitivity (pro-
portion of the total cases of occurrence where the model correctly
predicts occurrence); and specificity (proportion of the total cases of
non-occurrence where the model correctly predicts non-occurrence).
We have also carried out all the above analyses (i.e., simulation and
statistical properties assessment) for a different dataset. We used data of
forest fire occurrence in central-Chile, as a way of representing how our
findings could change in a forest fire model, and the main results are
shown in Supplementary Material.
3. Results
The proportion of 0/1 in the data used for fitting a logistic regres-
sion model affects the distribution of the estimated parameters. The
variability of the estimated parameters tends to increase with an ex-
treme proportion of zero (or ones) in the data (Fig. 1).
Unbalanced data affects on the bias of the estimated parameters. All
the parameters estimates were nearly unbiased for the proportion of
zeros data assessed that is closer to the proportion of zeros in the entire
population (First row panel of Fig. 1). However, for the other un-
balanced data scenarios, all parameters are biasedly estimated (Fig. 1).
The bias increases as the proportion of zeros in the data decreases both
in nominal units (Fig. 1), as well as in percentage (Fig. 2a). The bias is
larger for the estimated intercept-parameter than for the other para-
meters, regardless the unbalanced data scenario. The only exception to
this trend is the estimate of the parameter β
2
, being also heavily biased,
which could be a result of its higher variability compared to the other
parameter estimates (Fig. 2b). More importantly, the greatest precision
of all estimated parameters occurs with balanced data (Fig. 2b), as well
as the lowest root mean squared error (Fig. 2c). The reported greatest
precision of the estimated parameter for the balanced-data scenario was
even more pronounced for the forest fire model (Fig. 4). This can be a
result of a stronger relationship among the response and the predictor
variables, than we found in the forest mortality model. Besides, the
forest fire model (Eq. (7) in Supplementary Material) has a lower
number of parameters, therefore multicollinearity should be a minor
problem than in a model with two more parameters (Eq. (1)). In fact, in
the mortality model there are two parameters representing function of
variables already present in the model (i.e.,
B
Ai
2
and Defnorm
i
×BA
i
),
therefore the model is affected by multicollinearity.
The prediction capabilities of the logit model are greatly affected by
the different proportions of zeros and ones. Both overall error (i.e., sum
of omission and commission errors) and overall accuracy (i.e., sum of
omission and commission accuracy) tend to be better, with a decreasing
and increasing trend, respectively, when extreme proportions of zeros
(or ones) are used for fitting the model (Table 1). Moreover, the larger
is the proportion of zeros in the data, the better is the prediction of non-
occurrence (i.e., higher values of omission accuracy). A similar trend,
but not completely linear, is found when the omission errors are used as
reference. On the contrary, the larger is the proportion of ones in the
data, the better is the prediction of occurrence (i.e., higher values of
commission accuracy). A similar trend is found, when the commission
errors are used as reference (Table 1). A clear pattern is observed if
C. Salas-Eljatib et al. (FRORJLFDO,QGLFDWRUV²
specificity or sensitivity are used as reference. Hence, specificity in-
creases with higher number of zeros, but sensitivity decreases as
number of zeros increases (Table 1).
4. Discussion
In this paper we demonstrated that the common unbalanced pro-
portion of zeros and ones found in ecological data affects the statistical
properties of logistic regression models being fitted. Because the var-
iances of the estimated parameters are affected by the proportion of 0/1
data, all the statistical inference (e.g., hypothesis testing) of the fitted
model will be affected. Thus, if we are investigating the driver variables
of a ecological phenomenon, such as species distribution across a geo-
graphic area, we could be erroneously determining them, because the
statistical significance of each parameter of the model is based upon
their respective variance estimator. Therefore, the practice of balancing
data must be carried out with caution, as well as fully considering its
implications for model performance. Some authors have argued that
there is no major effect in having unbalanced binary data (except for
the bias in the intercept parameter, Maddala, 1992), but our results
indicate that all statistical properties of the MLE parameters are af-
fected. Although all the parameters estimates are biased, the magnitud
of bias will diminish as soon as our sample mimic the proportion of
zeros that are found in the population (see the crossed lines in Fig. 2a).
Notice that it has previously been stated that all the parameters would
be biased for small samples sizes (Schaefer, 1983; King and Zeng,
2001), but that was not necessarily the case in the present study (where
n= 1000).
We also claim that the prediction capabilities of the logistic re-
gression model are affected, as also was found by McPherson et al.
(2004) and Maggini et al. (2006), but using slightly different statistical
models. Thus, a given ecological binary phenomenon could be erro-
neously predicted to occur (or not) if the fitted model suffers from
statistical issues derived from using unbalanced data. This is especially
critical for predicting habitat suitability for endangered species (and its
conservation) or for predicting the distribution range of exotic invasive
species and their subsequent control plans. In either case it can result in
allocating efforts and resources in an inefficient manner. In this study
we encourage researchers to carefully examine the nature of the data
they have available and the 0/1 proportion of it before fitting the
Fig. 1. Empirical distribution of estimated parameters for the forest mortality model (Eq. (1)) given different scenarios of zeros in the data. The vertical solid line and the vertical dashed
line, within each histogram, represents the parameter value and the Monte Carlo expected value of the estimated parameter, respectively.
C. Salas-Eljatib et al. (FRORJLFDO,QGLFDWRUV²
logistic regression model, as this can greatly improve both the statistical
properties of the estimated parameters of the model and the prediction
capabilities applied to ecological phenomena.
We did not focus on analyzing alternatives for overcoming the ef-
fects of unbalanced data nor on finding the best fitted model. Some
studies dealing with models in ecology have shown the necessity of
effectively correct biased analyses for better interpretation and pre-
diction capabilities (Lajeunesse, 2015; Ruffault et al., 2014), but we
focused on pointing out the effects of unbalanced data when fitting
logistic regression models. The bias in the intercept of a logistic model
could be diminished when using the correction given by Manski and
Lerman (1977), but this type of correction is better suited for disciplines
where higher proportion of ones in the population is more common to
find or sample (e.g., social, economy, and political sciences), than for
ecological populations.
The statistical inference of fitted logistic regression models is af-
fected by the unbalanced nature of ecological data. Our results show
that the largest standard error and root mean squared error of the es-
timated parameters are found when having extreme proportion of zeros
(or ones) in the data. More importantly, for the first time in the lit-
erature, as far as we are aware of, we described that the variability of
the maximum likelihood estimated (MLE) parameters decreases when
having a balanced sample. This finding may suggest that balancing data
is an appropriate practice, if statistical inference (e.g., hypothesis
testing), is what the researcher is concerned about. Hence, by using
unbalanced data, we might conclude that a predictor variable is sta-
tistically significant when in fact it is not, or otherwise. Furthermore,
this finding refutes what King and Zeng (2001) had claimed regarding
that the addition of ones into the data, would decrease the variance of
the MLE parameters.
We also found that unbalanced data heavily affects the prediction
capabilities of a logistic regression model. Our study reflects that the
occurrence of the event is better predicted when having larger pro-
portions of ones in the data. On the other hand, non-occurrence of the
event is better predicted when having larger proportions of zeros in the
data. This trend is expected, because the model is fitted by ML, where
the parameters estimates are those that maximize the likelihood of the
data at hand, therefore we should predict them concordantly
(Schabenberger and Pierce, 2002). Also, if we take into account the
trade-offof building a model that predicts occurrence and non-occur-
rence as best as possible, the balanced data scenario with a 50% of zeros
and ones offers a suitable way to proceed (Table 1). Overall, balancing
the data seems to be an appropriate practice to improve some statistical
properties and prediction capabilities of the fitted model. Regardless of
balancing or not balancing the data before fitting a logistic regression
model, we recommend to use the remaining sample (i.e., not used for
fitting the model) for validation purposes and behavior analyses.
5. Recommendations
Given that the proportion of 0/1 data affects the variance of the
estimated parameters of the fitted logistic regression model, the selec-
tion of the statistically significant predictor variables to conduct the
analyses may also being influenced, ultimately leading to a wrong
Fig. 2. Statistical properties of the estimated parameters for the forest
mortality model (Eq. (1)), given different scenarios of zeros in the data. (a)
Bias, (b) standard error, and (c) root mean squared error are shown as a
percentage of the real parameter value.
Table 1
Prediction indexes of the logistic regression model depending upon unbalanced data
scenarios. Each value is the empirical expected value of the respective index.
Proportions of zeros in the sample
10% 30% 50% 70% 90%
Commission
Error (%) 0.41 4.44 13.09 25.12 9.99
Accuracy (%) 89.59 65.56 36.91 4.88 0.01
Omission
Error (%) 9.54 21.33 21.52 4.54 0.01
Accuracy (%) 0.46 8.67 28.48 65.46 89.99
Sensitivity (%) 99.54 93.65 73.81 16.28 0.02
Specificity (%) 4.58 28.91 56.95 93.52 99.91
C. Salas-Eljatib et al. (FRORJLFDO,QGLFDWRUV²
conclusion. From our study, we have provided evidence that using a
balanced data scenario (i.e., 50% of zeros and ones) will yield smaller
variances for the maximum likelihood estimates of parameters, there-
fore offering less uncertainty in the estimation process, and ultimately
in identifying the driver variables for modelling presence/absence re-
sponse variables. This finding is extremely relevant in ecological ap-
plications, as an important amount of studies are currently dealing with
niche modeling and species distribution based on presence/absence
data, especially within the climate change context. Thus, we re-
commend that when modelling binary response variables, researchers
can safely use balanced datasets for fitting candidate models, in order to
choose the best model given the variables used for the analysis. Giving
our results, by performing these procedure, the analysis itself will gain
more certainty because the researcher could better distinguish between
the effects of the predictor variables being included in the model or
whether is the ecological phenomenon really important (McPherson
et al., 2004). Another issue to take into account is the drastic reduction
of the sample for balancing purposes. We recommend to use the re-
maining data (i.e., the one not used for fitting purposes) for assessing
the prediction capabilities of the models (using the indices and plots
recommend by Jones et al., 2010), and assessing the models behavior
by plotting the prediction of the response variable as a function of the
predictor variable(s).
Regarding interpreting the predicted outcomes, we recommend not
extrapolating the model results into areas where predictor variables
were not measured. This implies that the presence/absence of a given
organisms could be altered within certain ranges. In the case when
extrapolation is indeed necessary, researchers should differentiate their
predictions from those areas where no data were collected using, for
instance, color-codded results to distinguish them from prediction re-
sults by the model using real data. This situation will be specially ad-
vantageous for modelling any ecological phenomena that is a function
of spatially-recorded predictor variable(s).
6. Concluding remarks
The proportion of zeros and ones in a dataset affects the statistical
inference and prediction capabilities of a fitted logistic regression
model. Not only the accuracy of the estimated parameters is affected by
unbalanced data, but also their precision. More importantly, the sta-
tistical inference (e.g., hypothesis testing) is influenced by the propor-
tion of zeros and ones in the data. In addition, the prediction cap-
abilities of the fitted logistic regression model are affected as well,
therefore the model performance would greatly depend on the pro-
portion of 0/1 data. Overall, the 0/1 proportion might affect the con-
clusions that can arise from the fitted model and its further application.
Since unbalanced data in ecology are fairly common, this can have
great implications in model building of several ecological phenomena
being modelled by scientists.
Acknowledgements
This study was supported by the Chilean research grant Fondecyt
No. 1151495. AFR is supported by a Postdoctoral Scholarship from
Vicerrectoría de Investigación y Postgrado, Universidad de La Frontera,
Temuco, Chile.
Appendix A. Supplementary data
Supplementary data associated with this article can be found, in the
online version, at http://dx.doi.org/10.1016/j.ecolind.2017.10.030.
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