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ABSTRACT The purpose of this article is to provide
researchers, editors, and readers with a set of guidelines for
what to expect in an article using logistic regression tech-
niques. Tables, figures, and charts that should be included to
comprehensively assess the results and assumptions to be ver-
ified are discussed. This article demonstrates the preferred
pattern for the application of logistic methods with an illustra-
tion of logistic regression applied to a data set in testing a
research hypothesis. Recommendations are also offered for
appropriate reporting formats of logistic regression results
and the minimum observation-to-predictor ratio. The authors
evaluated the use and interpretation of logistic regression pre-
sented in 8 articles published in The Journal of Educational
Research between 1990 and 2000. They found that all 8 studies
met or exceeded recommended criteria.
Key words: binary data analysis, categorical variables,
dichotomous outcome, logistic modeling, logistic regression
any educational research problems call for the
analysis and prediction of a dichotomous outcome:
whether a student will succeed in college, whether a child
should be classified as learning disabled (LD), whether a
teenager is prone to engage in risky behaviors, and so on.
Traditionally, these research questions were addressed by
either ordinary least squares (OLS) regression or linear dis-
criminant function analysis. Both techniques were subse-
quently found to be less than ideal for handling dichoto-
mous outcomes due to their strict statistical assumptions,
i.e., linearity, normality, and continuity for OLS regression
and multivariate normality with equal variances and covari-
ances for discriminant analysis (Cabrera, 1994; Cleary &
Angel, 1984; Cox & Snell, 1989; Efron, 1975; Lei &
Koehly, 2000; Press & Wilson, 1978; Tabachnick & Fidell,
2001, p. 521). Logistic regression was proposed as an alter-
native in the late 1960s and early 1970s (Cabrera, 1994),
and it became routinely available in statistical packages in
the early 1980s.
Since that time, the use of logistic regression has
increased in the social sciences (e.g., Chuang, 1997; Janik
& Kravitz, 1994; Tolman & Weisz, 1995) and in education-
al research—especially in higher education (Austin,Yaffee,
& Hinkle, 1992; Cabrera, 1994; Peng & So, 2002a; Peng,
So, Stage, & St. John, 2002. With the wide availability of
sophisticated statistical software for high-speed computers,
the use of logistic regression is increasing. This expanded
use demands that researchers, editors, and readers be
attuned to what to expect in an article that uses logistic
regression techniques. What tables, figures, or charts should
be included to comprehensibly assess the results? What
assumptions should be verified? In this article, we address
these questions with an illustration of logistic regression
applied to a data set in testing a research hypothesis. Rec-
ommendations are also offered for appropriate reporting
formats of logistic regression results and the minimum
observation-to-predictor ratio. The remainder of this article
is divided into five sections: (1) Logistic Regression Mod-
els, (2) Illustration of Logistic Regression Analysis and
Reporting, (3) Guidelines and Recommendations, (4) Eval-
uations of Eight Articles Using Logistic Regression, and (5)
Summary.
Logistic Regression Models
The central mathematical concept that underlies logistic
regression is the logit—the natural logarithm of an odds
ratio. The simplest example of a logit derives from a 2 × 2
contingency table. Consider an instance in which the distri-
bution of a dichotomous outcome variable (a child from an
inner city school who is recommended for remedial reading
classes) is paired with a dichotomous predictor variable
(gender). Example data are included in Table 1. A test of
independence using chi-square could be applied. The results
yield χ
2
(1) = 3.43. Alternatively, one might prefer to assess
An Introduction to Logistic Regression
Analysis and Reporting
CHAO-YING JOANNE PENG
KUK LIDA LEE
GARY M. INGERSOLL
Indiana University-Bloomington
Address correspondence to Chao-Ying Joanne Peng, Depart-
ment of Counseling and Educational Psychology, School of Edu-
cation, Room 4050, 201 N. Rose Ave., Indiana University, Bloom-
ington, IN 47405–1006. (E-mail: peng@indiana.edu)
3
M
a boy’s odds of being recommended for remedial reading
instruction relative to a girl’s odds. The result is an odds ratio
of 2.33, which suggests that boys are 2.33 times more like-
ly, than not, to be recommended for remedial reading class-
es compared with girls. The odds ratio is derived from two
odds (73/23 for boys and 15/11 for girls); its natural loga-
rithm [i.e., ln(2.33)] is a logit, which equals 0.85. The value
of 0.85 would be the regression coefficient of the gender pre-
dictor if logistic regression were used to model the two out-
comes of a remedial recommendation as it relates to gender.
Generally, logistic regression is well suited for describing
and testing hypotheses about relationships between a cate-
gorical outcome variable and one or more categorical or con-
tinuous predictor variables. In the simplest case of linear
regression for one continuous predictor X (a child’s reading
score on a standardized test) and one dichotomous outcome
variable Y (the child being recommended for remedial read-
ing classes), the plot of such data results in two parallel lines,
each corresponding to a value of the dichotomous outcome
(Figure 1). Because the two parallel lines are difficult to be
described with an ordinary least squares regression equation
due to the dichotomy of outcomes, one may instead create
categories for the predictor and compute the mean of the out-
come variable for the respective categories. The resultant plot
of categories’ means will appear linear in the middle, much
like what one would expect to see on an ordinary scatter plot,
but curved at the ends (Figure 1, the S-shaped curve). Such a
shape, often referred to as sigmoidal or S-shaped, is difficult
to describe with a linear equation for two reasons. First, the
extremes do not follow a linear trend. Second, the errors are
neither normally distributed nor constant across the entire
range of data (Peng, Manz, & Keck, 2001). Logistic regres-
sion solves these problems by applying the logit transforma-
tion to the dependent variable. In essence, the logistic model
predicts the logit of Y from X.As stated earlier, the logit is the
natural logarithm (ln) of odds of Y, and odds are ratios of
probabilities (π) of Y happening (i.e., a student is recom-
mended for remedial reading instruction) to probabilities (1 –
π) of Y not happening (i.e., a student is not recommended for
remedial reading instruction). Although logistic regression
can accommodate categorical outcomes that are polytomous,
in this article we focus on dichotomous outcomes only. The
illustration presented in this article can be extended easily to
polytomous variables with ordered (i.e., ordinal-scaled) or
unordered (i.e., nominal-scaled) outcomes.
The simple logistic model has the form
(1)
For the data in Table 1, the regression coefficient (β) is the
logit (0.85) previously explained. Taking the antilog of
Equation 1 on both sides, one derives an equation to predict
the probability of the occurrence of the outcome of interest
as follows:
π = Probability(Y = outcome of interest | X = x,
a specific value of X) = (2)
where π is the probability of the outcome of interest or
“event,” such as a child’s referral for remedial reading class-
es, α is the Y intercept, β is the regression coefficient, and
e = 2.71828 is the base of the system of natural logarithms.
X can be categorical or continuous, but Y is always categor-
ical. According to Equation 1, the relationship between
logit (Y) and X is linear. Yet, according to Equation 2, the
relationship between the probability of Y and X is nonlinear.
For this reason, the natural log transformation of the odds in
Equation 1 is necessary to make the relationship between a
categorical outcome variable and its predictor(s) linear.
The value of the coefficient β determines the direction of
the relationship between X and the logit of Y. When β is
greater than zero, larger (or smaller) X values are associated
with larger (or smaller) logits of Y. Conversely, if β is less
than zero, larger (or smaller) X values are associated with
smaller (or larger) logits of Y. Within the framework of infer-
ential statistics, the null hypothesis states that β equals zero,
or there is no linear relationship in the population. Rejecting
such a null hypothesis implies that a linear relationship exists
between X and the logit of Y. If a predictor is binary, as in the
Table 1 example, then the odds ratio is equal to e, the natural
logarithm base, raised to the exponent of the slope β (e
β
).
4 The Journal of Educational Research
Table 1.—Sample Data for Gender and Recommendation for
Remedial Reading Instruction
Gender
Remedial reading instruction Boys Girls Total
Recommended (coded as 1) 73 15 88
Not recommended (coded as 0) 23 11 34
Total 96 26 122
Figure 1. Relationship of a Dichotomous Outcome Variable,
Y (1 = Remedial Reading Recommended, 0 = Remedial Read-
ing Not Recommended) With a Continuous Predictor, Reading
Scores
Reading Score
1.0–
–
–
–
–
0.0–
|| |||||
40 60 80 100 120 140 160
logit( ) log( ) ln .Y natural odds X==
−
=+
π
π
αβ
1
e
e
x
x
αβ
αβ
+
+
+1
,
Extending the logic of the simple logistic regression to
multiple predictors (say X
1
= reading score and X
2
= gender),
one can construct a complex logistic regression for Y (rec-
ommendation for remedial reading programs) as follows:
(3)
Therefore,
π = Probability (Y = outcome of interest | X
1
= x
1
, X
2
= x
2
(4)
where π is once again the probability of the event, α is the
Y intercept, βs are regression coefficients, and Xs are a set
of predictors. α and βs are typically estimated by the max-
imum likelihood (ML) method, which is preferred over the
weighted least squares approach by several authors, such as
Haberman (1978) and Schlesselman (1982). The ML
method is designed to maximize the likelihood of reproduc-
ing the data given the parameter estimates. Data are entered
into the analysis as 0 or 1 coding for the dichotomous out-
come, continuous values for continuous predictors, and
dummy codings (e.g., 0 or 1) for categorical predictors.
The null hypothesis underlying the overall model states
that all βs equal zero. A rejection of this null hypothesis
implies that at least one β does not equal zero in the popu-
lation, which means that the logistic regression equation
predicts the probability of the outcome better than the mean
of the dependent variable Y. The interpretation of results is
rendered using the odds ratio for both categorical and con-
tinuous predictors.
Illustration of Logistic Regression Analysis
and Reporting
For the sake of illustration, we constructed a hypothetical
data set to which logistic regression was applied, and we
interpreted its results. The hypothetical data consisted of
reading scores and genders of 189 inner city school children
(Appendix A). Of these children, 59 (31.22%) were recom-
mended for remedial reading classes and 130 (68.78%)
were not. A legitimate research hypothesis posed to the data
was that “the likelihood that an inner city school child is
recommended for remedial reading instruction is related to
both his/her reading score and gender.” Thus, the outcome
variable, remedial, was students being recommended for
remedial reading instruction (1 = yes, 0 = no), and the two
predictors were students’ reading score on a standardized
test (X
1
= the reading variable) and gender (X
2
= gender). The
reading scores ranged from 40 to 125 points, with a mean of
64.91 points and standard deviation of 15.29 points (Table
2). The gender predictor was coded as 1 = boy and 0 = girl.
The gender distribution was nearly even with 49.21% (n =
93) boys and 50.79% (n = 96) girls.
Logistic Regression Analysis
A two-predictor logistic model was fitted to the data to
test the research hypothesis regarding the relationship
between the likelihood that an inner city child is recom-
mended for remedial reading instruction and his or her read-
ing score and gender. The logistic regression analysis was
carried out by the Logistic procedure in SAS version 8
(SAS Institute Inc., 1999) in the Windows 2000 environ-
ment (SAS programming codes are found in Table 3). The
result showed that
Predicted logit of (REMEDIAL) = 0.5340
+ (−0.0261)*READING + (0.6477)*GENDER. (5)
According to the model, the log of the odds of a child
being recommended for remedial reading instruction was
negatively related to reading scores (p < .05) and positively
related to gender (p < .05; Table 3). In other words, the high-
er the reading score, the less likely it is that a child would be
recommended for remedial reading classes. Given the same
reading score, boys were more likely to be recommended
for remedial reading classes than girls because boys were
coded to be 1 and girls 0. In fact, the odds of a boy being
recommended for remedial reading programs were 1.9111
(= e
0.6477
; Table 3) times greater than the odds for a girl.
The differences between boys and girls are depicted in
Figure 2, in which predicted probabilities of recommenda-
tions are plotted for each gender group against various read-
ing scores. From this figure, it may be inferred that for a
given score on the reading test (e.g., 60 points), the proba-
bility of a boy being recommended for remedial reading
programs is higher than that of a girl. This statement is also
confirmed by the positive coefficient (0.6477) associated
with the gender predictor.
Evaluations of the Logistic Regression Model
How effective is the model expressed in Equation 5?
How can an educational researcher assess the soundness of
a logistic regression model? To answer these questions, one
must attend to (a) overall model evaluation, (b) statistical
tests of individual predictors, (c) goodness-of-fit statistics,
and (d) validations of predicted probabilities. These evalua-
tions are illustrated below for the model based on Equation
5, also referred to as Model 5.
Overall model evaluation. A logistic model is said to pro-
vide a better fit to the data if it demonstrates an improvement
over the intercept-only model (also called the null model). An
September/October 2002 [Vol. 96(No. 1)] 5
Table 2.—Description of a Hypothetical Data Set for Logistic
Regression
Remedial Total
reading sample Boys Girls
recommended? (N)(n
1
)(n
2
) MSD
Yes 59 36 23 61.07 13.28
No 130 57 73 66.65 15.86
Summary 189 93 96 64.91 15.29
Reading score
logit( ) ln .YXX=
−
=+ +
π
π
αβ β
1
11 2 2
=
+
++
++
e
e
XX
XX
αβ β
αβ β
11 22
11 22
1
,
intercept-only model serves as a good baseline because it con-
tains no predictors. Consequently, according to this model, all
observations would be predicted to belong in the largest out-
come category. An improvement over this baseline is exam-
ined by using three inferential statistical tests: the likelihood
ratio, score, and Wald tests. All three tests yield similar con-
clusions for the present data (Table 3), namely, that the logis-
tic Model 5 was more effective than the null model. For other
data sets, these three tests may not lead to similar conclusions.
When this happens, readers are advised to rely on the likeli-
hood ratio and score tests only (Menard, 1995).
Statistical tests of individual predictors. The statistical
significance of individual regression coefficients (i.e., βs) is
tested using the Wald chi-square statistic (Table 3). Accord-
ing to Table 3, both reading score and gender were signifi-
cant predictors of inner city school children’s referrals for
remedial reading programs (p < .05). The test of the intercept
(i.e., the constant in Table 3) merely suggests whether an
intercept should be included in the model. For the present
data set, the test result (p > .05) suggested that an alternative
model without the intercept might be applied to the data.
Goodness-of-fit statistics. Goodness-of-fit statistics
assess the fit of a logistic model against actual outcomes
(i.e., whether a referral is made for remedial reading pro-
grams). One inferential test and two descriptive measures
are presented in Table 3. The inferential goodness-of-fit test
is the Hosmer–Lemeshow (H–L) test that yielded a χ
2
(8) of
7.7646 and was insignificant (p > .05), suggesting that the
model was fit to the data well. In other words, the null
hypothesis of a good model fit to data was tenable.
The H–L statistic is a Pearson chi-square statistic, calcu-
lated from a 2 × g table of observed and estimated expected
frequencies, where g is the number of groups formed from
the estimated probabilities. Ideally, each group should have
an equal number of observations, the number of groups
should exceed 5, and expected frequencies should be at least
5. For the present data, the number of observations in each
group was mostly 19 (3 groups) or 20 (5 groups); 1 group
had 21 observations and another had 11 observations. The
number of groups was 10, and the expected frequencies were
at or exceeded 5 in 90% of cells. Thus, it was concluded that
the conditions were met for reporting the HL test statistic.
Two additional descriptive measures of goodness-of-fit
presented in Table 3 are R
2
indices, defined by Cox and
Snell (1989) and Nagelkerke (1991), respectively. These
indices are variations of the R
2
concept defined for the OLS
regression model. In linear regression, R
2
has a clear defin-
ition: It is the proportion of the variation in the dependent
variable that can be explained by predictors in the model.
Attempts have been devised to yield an equivalent of this
concept for the logistic model. None, however, renders the
meaning of variance explained (Long, 1997, pp. 104–109;
Menard, 2000). Furthermore, none corresponds to predic-
tive efficiency or can be tested in an inferential framework
(Menard). For these reasons, a researcher can treat these
two R
2
indices as supplementary to other, more useful eval-
uative indices, such as the overall evaluation of the model,
tests of individual regression coefficients, and the good-
ness-of-fit test statistic.
Validations of predicted probabilities. As we explained
earlier, logistic regression predicts the logit of an event out-
come from a set of predictors. Because the logit is the nat-
ural log of the odds (or probability/[1–probability]), it can
be transformed back to the probability scale. The resultant
predicted probabilities can then be revalidated with the
actual outcome to determine if high probabilities are indeed
associated with events and low probabilities with non-
events. The degree to which predicted probabilities agree
with actual outcomes is expressed as either a measure of
association or a classification table. There are four measures
6 The Journal of Educational Research
Table 3.—Logistic Regression Analysis of 189 Children’s Referrals for Remedial Reading Programs by
SAS PROC LOGISTIC (Version 8)
Wald’s e
β
Predictor β SE βχ
2
df p (odds ratio)
Constant 0.5340 0.8109 0.4337 1 .5102 NA
Reading –0.0261 0.0122 4.5648 1 .0326 0.9742
Gender (1 = boys, 0 = girls) 0.6477 0.3248 3.9759 1 .0462 1.9111
Test χ
2
df p
Overall model evaluation
Likelihood ratio test 10.0195 2 .0067
Score test 9.5177 2 .0086
Wald test 9.0626 2 .0108
Goodness-of-fit test
Hosmer & Lemeshow 7.7646 8 .4568
Note. SAS programming codes: [PROC LOGISTIC; MODEL REMEDIAL=READING GENDER/CTABLE PPROB=(0.1 TO
1.0 BY 0.1) LACKFIT RSQ;]. Cox and Snell R
2
= .0516. Nagelkerke R
2
(Max rescaled R
2
) = .0726. Kendall’s Tau-a = .1180.
Goodman-Kruskal Gamma = .2760. Somers’s D
xy
= .2730. c-statistic = 63.60%. All statistics reported herein use 4 decimal
places in order to maintain statistical precision. NA = not applicable.
of association and one classification table that are provided
by SAS (Version 8).
The four measures of association are Kendall’s Tau-a,
Goodman-Kruskal’s Gamma, Somers’s D statistic, and the
c statistic (Table 3). The Tau-a statistic is Kendall’s rank-
order correlation coefficient without adjustments for ties.
The Gamma statistic is based on Kendall’s coefficient but
adjusts for ties. Gamma is more useful and appropriate than
Tau-a when there are ties on both outcomes and predicted
probabilities, as was the case with the present data (see
Appendix A). The Gamma statistic for Model 5 is 0.2760
(Table 3). It is interpreted as 27.60% fewer errors made in
predicting which of two children would be recommended
for remedial reading programs by using the estimated prob-
abilities than by chance alone (Demaris, 1992). Some cau-
tion is advised in using the Gamma statistic because (a) it
September/October 2002 [Vol. 96(No. 1)] 7
Figure 2. Predicted Probability of Being Referred for Remedial Reading Instructions Versus Reading
Scores
Note. Plotting symbols A = 1 observation, B = 2 observations, C = 3 observations, and so forth.
A
B
A
FA
CC
EA
AI
E
PC
BC
HB
D
AC A
CB AC
BB ABA
AIB A
CJ BA
E
AKE B
AFA B
BB A
CA
AAA
ACA
AB A
B A A A
A
A
Reading Score
Estimated Probability
0.6–
0.5–
0.4–
0.3–
0.2–
0.1–
0.0–
|| | |||
40 60 80 100 120 140
boys
girls
has a tendency to overstate the strength of association
between estimated probabilities and outcomes (Demaris),
and (b) a value of zero does not necessarily imply indepen-
dence when the data structure exceeds a 2 × 2 format
(Siegel & Castellan, 1988).
Somers’s D is a preferred extension of Gamma whereby
one variable is designated as the dependent variable and the
other the independent variable (Siegel & Castellan, 1988).
There are two asymmetric forms of Somers’s D statistic: D
xy
and D
yx
. Only D
yx
correctly represents the degree of associa-
tion between the outcome (y), designated as the dependent
variable, and the estimated probability (x), designated as the
independent variable (Demaris, 1992). Unfortunately, SAS
computes only D
xy
(Table 3), although this index can be cor-
rected to D
yx
in SAS (Peng & So, 1998).
The c statistic represents the proportion of student pairs
with different observed outcomes for which the model cor-
rectly predicts a higher probability for observations with the
event outcome than the probability for nonevent observations.
For the present model, the c statistic is 0.6360 (Table 3). This
means that for 63.60% of all possible pairs of children—one
recommended for remedial reading programs and the other
not—the model correctly assigned a higher probability to
those who were recommended. The c statistic ranges from 0.5
to 1. A 0.5 value means that the model is no better than assign-
ing observations randomly into outcome categories. A value
of 1 means that the model assigns higher probabilities to all
observations with the event outcome, compared with non-
event observations. If several models were fitted to the same
data set, the model chosen as the best model should be asso-
ciated with the highest c statistic. Thus, the c statistic provides
a basis for comparing different models fitted to the same data
or the same model fitted to different data sets.
In addition to these measures of association, SAS output
includes a classification table that documents the validity of
predicted probabilities (Table 4). The first two rows in Table
4 represent the two possible outcomes, and the two columns
under the heading “Predicted” are for high and low proba-
bilities, based on a cutoff point. The cutoff point may be
specified by researchers or set at 0.5 by SAS. According to
Table 4, with the cutoff set at 0.5, the prediction for children
who were not recommended for remedial reading programs
was more accurate than that for those who were. This obser-
vation was also supported by the magnitude of sensitivity
(3.39%) compared to that of specificity (99.23%). Sensitiv-
ity measures the proportion of correctly classified events
(i.e., those recommended for remedial reading programs),
whereas specificity measures the proportion of correctly
classified nonevents (those not recommended for remedial
reading programs). Both false positive and false negative
rates were a little more than 30%. The false positive rate
measures the proportion of observations misclassified as
events over all of those classified as events. The false nega-
tive therefore measures the proportion of observations mis-
classified as nonevents over all of those classified as non-
events. The overall correction prediction was 69.31%, an
improvement over the chance level. In the opinion of Hos-
mer and Lemeshow (2000, p. 160), “the classification table
is most appropriate when classification is a stated goal of
the analysis; otherwise it should only supplement more rig-
orous methods of assessment of fit.”
Table 4 was prepared with SAS using a reduced-bias
algorithm. The algorithm minimizes the bias of using the
same observations both for model fitting and for predicting
probabilities (SAS Institute Inc., 1999). According to a
recent comparative study of six statistical packages that can
be used for logistic regression (Peng & So, 2002b), SAS is
the only package that uses this algorithm. Thus, entries in
Table 4 would be slightly different if other software (such
as SPSS) was used to prepare it.
Reporting and Interpreting Logistic Regression Results
In addition to the data presented in Tables 3 and 4 and
Figure 2, it is helpful to demonstrate the relationship
between the predicted outcome and certain characteristics
found in observations. For the present data, this relationship
is demonstrated in Table 5 for four cases (1–4) extracted
from Appendix A, as well as for four observations (5–8) for
whom reading scores were hypothesized at two levels for
both genders. For the first four cases, the predicted proba-
bilities of referrals for remedial reading programs were cal-
culated using Equation 5. Even though these four cases
were not perfectly predicted, the correct prediction rate was
better than chance.
The last four hypothetical cases show the descending pre-
dicted probabilities of referrals for remedial reading programs
as the reading scores increase for children of both genders.
For each point increase on the reading score, the odds of
being recommended for remedial reading programs decrease
from 1.0 to 0.9742 (= e
–0.0261
; Table 3). If the increase on the
reading score was 10 points, the odds decreased from 1.0 to
0.7703 (= e
10*[–0.0261]
). However, when the reading score was
held as a constant, boys were predicted to be referred for
remedial reading instructions with greater probability than
girls. The differences between boys and girls are graphically
shown in Figure 2 and confirmed previously by the positive
coefficient (0.6477) of the gender predictor in Equation 5.
8 The Journal of Educational Research
Table 4.—The Observed and the Predicted Frequencies for
Remedial Reading Instructions by Logistic Regression With
the Cutoff of 0.50
Predicted
Observed Yes No % Correct
Yes 2 57 3.39
No 1 129 99.23
Overall % correct 69.31
Note. Sensitivity = 2/(2+57)% = 3.39%. Specificity = 129/(1+129)% =
99.23%. False positive = 1/(1+2)% = 33.33%. False negative =
57/(57+129)% = 30.65%.
The odds of a boy being recommended for remedial reading
programs were 1.9111 (= e
0.6477
; Table 3) times greater than
the odds for a girl.
In terms of the research hypothesis posed earlier to the
hypothetical data—“the likelihood that an inner city school
child is recommended for remedial reading instruction is
related to both his/her reading score and gender”—logistic
regression results supported this proposition. Specifically,
the likelihood of a child being recommended for remedial
reading instruction was negatively related to his or her read-
ing scores. However, given the same reading score, boys
were more likely to be recommended for remedial reading
classes than girls. We reached this conclusion with multiple
evidences: the significant test result of the logistic model,
statistically significant test results of both predictors,
insignificant HL test of goodness-of-fit, and several
descriptive measures of associations between predicted
probabilities and data.
Guidelines and Recommendations
What Tables, Figures, or Charts Should Be Included to
Comprehensively Assess the Result?
In presenting the assessment of logistic regression
results, researchers should include sufficient information to
address the following:
• an overall evaluation of the logistic model
• statistical tests of individual predictors
• goodness-of-fit statistics
• an assessment of the predicted probabilities
Table 3 illustrates the presentation of the first three types
of information and Table 4 the fourth. To illustrate the
impact of a statistically significant categorical predictor
(e.g., gender in our example) on the dichotomous dependent
variable (e.g., recommendation for remedial reading pro-
grams), it is helpful to include a figure such as Figure 2. It
is our recommendation that logistic regression results be
reported, similar to those in Tables 3 and 4 and Figure 2, to
help communicate findings to readers.
A model’s adequacy should be justified by multiple indi-
cators, including an overall test of all parameters, a statistical
significance test of each predictor, the goodness-of-fit statis-
tics, the predictive power of the model, and the interpretabil-
ity of the model. Furthermore, researchers should pay atten-
tion to mathematical definitions of statistics (such as D
xy
)
generated by the statistical package of choice. Among the
packages that perform logistic regression, none was found
to be error free (Peng & So, 2002b). A reference to the
software should inform readers of programming mistakes
and limitations, and help researchers verify results with
another statistical package. A recent review of six statistical
software programs, conducted by Peng and So (2002b, pp.
55–56) for performing logistic regression, concluded that
The versatile SAS logisitic and BMDP LR [were recom-
mended] for researchers experienced with logistic regression
techniques and programming. . . . Several unique goodness-
of-fit indices and selection methods are provided in SAS. Its
ability to fit a broad class of binary response models, plus its
provision to correct for over-sampling, over-dispersion, and
bias introduced into predicted probabilities, sets it apart from
the other five. . . . If either SPSS LOGISTIC REGRESSION
or SYSTAT LOGIT is the only package available,
researchers must be aware that both compute the goodness-
of-fit and diagnostic statistics from individual observations.
Consequently, these statistics are inappropriate for statistical
tests. With dazzling graphic interfaces, both packages are
user-friendly.
MINITAB BLOGISTIC is the simplest to use. It adopts the
hierarchical modeling restriction in direct modeling. . . . A
substantial number of goodness-of-fit indices are available
including the unique Brown statistic. However, the absence
of predictor selection methods may make it less appealing to
some researchers. . . . STATA LOGISTIC provides the most
detailed information on parameter estimates, yet its good-
ness-of-fit indices are limited. We recommend MINITAB
and STATA for beginners, although experienced researchers
may also employ them for logistic regression.
What Assumptions Should Be Verified?
Unlike discriminant function analysis, logistic regression
does not assume that predictor variables are distributed as a
multivariate normal distribution with equal covariance
matrix. Instead, it assumes that the binomial distribution
describes the distribution of the errors that equal the actual
Y minus the predicted Y. The binomial distribution is also
the assumed distribution for the conditional mean of the
dichotomous outcome. This assumption implies that the
same probability is maintained across the range of predictor
September/October 2002 [Vol. 96(No. 1)] 9
Table 5.—Predicted Probability of Being Referred for Remedial Reading Instructions for 8 Children
Predicted probability of
Case Reading score Gender Intercept being referred for Actual outcome
number β = –0.0261 β = 0.6477 = 0.5340 remedial reading program 1 = Yes, 0 = No
1 52.5 Boy 0.5340 0.4530 1
2 85 Boy 0.5340 0.2618 0
3 75 Girl 0.5340 0.1941 1
4 92 Girl 0.5340 0.1250 0
5 60 Boy 0.5340 0.4051 —
6 60 Girl 0.5340 0.2627 —
7 100 Boy 0.5340 0.1934 —
8 100.5 Girl 0.5340 0.1115 —
values. The binomial assumption may be tested by the nor-
mal z test (Siegel & Castellan, 1988) or may be taken to be
robust as long as the sample is random; thus, observations
are independent from each other.
Recommended Reporting Formats of Logistic Regression
In terms of reporting logistic regression results, we rec-
ommend presenting the complete logistic regression model
including the Y-intercept (similar to Equation 5), odds
ratios, and a table such as Table 5 to illustrate the relation-
ship between outcomes and observations with profiles of
certain characteristics. Odds ratios are directly derived from
regression coefficients in a logistic model. If β
j
represents
the regression coefficient for predictor X
j
, then exponentiat-
ing β
j
yields the odds ratio. When all other predictors are
held at a constant, the odds ratio means the change in the
odds of Y given a unit change in X
j
. It is one of three epi-
demiological measures of effect that have been recently rec-
ommended by psychologists for informing public policy
makers (Scott, Mason, & Chapman, 1999). Three condi-
tions must be met before odds ratios can be interpreted sen-
sibly: (a) the predictor X
j
must not interact with another pre-
dictor; (b) the predictor X
j
must be represented by a single
term in the model; and (c) a one-unit change in the predic-
tor X
j
must be meaningful and relevant. It is worth noting
that odds ratios and odds are two different concepts. They
are related but not in a linear fashion. Likewise, the rela-
tionship between the predicted probability and odds, though
positive, is not linear either.
Recommended Minimum Observation-to-Predictor Ratio
In terms of the adequacy of sample sizes, the literature
has not offered specific rules applicable to logistic regres-
sion (Peng et al., 2002). However, several authors on multi-
variate statistics (Lawley & Maxwell, 1971; Marascuilo &
Levin, 1983; Tabachnick & Fidell, 1996, 2001) have rec-
ommended a minimum ratio of 10 to 1, with a minimum
sample size of 100 or 50, plus a variable number that is a
function of the number of predictors.
Evaluations of Eight Articles Using Logistic Regression
To help understand how logistic regression has been
applied by authors of articles published in The Journal of
Educational Research (JER), we reviewed articles that used
this technique between 1990 and 2000. During this period,
eight articles were found to have used logistic regression.
The criterion used in selecting articles was simple: at least
one empirical analysis in the article must have been con-
ducted to derive the logistic model and its regression coeffi-
cients. This criterion excluded any article that relied on oth-
ers’ work to derive the model or merely performed a
logarithm or logit transformation of the dependent or the
independent variable. A complete list of these eight articles
is found in Appendix B.
A breakdown of the articles by year showed that, prior to
1993, there was no article that used logistic regression. In
1993, 1994, 1996, and 1997, one article per year applied
logistic regression; in 1998 and 2000, there were two per
year. This trend mirrors the pattern that was found in high-
er education journals (Peng et al., 2002, except that the rise
of logistic regression began a year earlier, in 1992, in high-
er education journals.
The research questions addressed in the eight articles
included American Indian adolescents’ educational com-
mitment (Trusty, 2000), school performance and activities
(Alexander, Dauber, & Entwisle, 1996; McNeal, 1998;
Smith, 1997), students at-risk (Meisels & Liaw, 1993; Rush
& Vitale, 1994), family connectedness (Machamer & Gru-
ber, 1998), and parents’ conceptions of kindergarten readi-
ness (Diamond, Reagan, & Bandyk, 2000). One central
theme shared by all was education-related adjustment and
performance. The dependent variable was dichotomous,
whether it was retention in school, dropping-out from high
school, or readiness for kindergarten. The predictors typi-
cally included a combination of demographic characteris-
tics (such as age, gender, and ethnicity) and cognitive,
affective, or personality-related measures. The objective of
each study was to predict or to distinguish the outcome cat-
egories on the basis of predictors.
To test pertinent research hypotheses, the authors of these
eight articles used three modeling approaches: direct,
sequential, and stepwise modeling. Of these three, only
direct and sequential models were controlled and imple-
mented by researchers (Peng & So, 2002a). Three studies
investigated interactions among predictors (Alexander,
Dauber, & Entwisle, 1996; Meisels & Liaw, 1993; Trusty,
2000); the others did not. Though not all prior studies have
always followed the guidelines and recommendations out-
lined in the previous section, all authors are credited for
making substantive contributions as well as for introducing
logistic regression into the field of educational research.
The Assessment of Logistic Regression Results
Four groups of authors (Alexander, Dauber, & Entwisle,
1996; Diamond, Reagan, & Bandyk, 2000; McNeal, 1998;
Rush & Vitale, 1994) evaluated the overall logistic model;
all reported tests of individual predictors, such as those
shown in Table 3. Evidence of the goodness-of-fit of logis-
tic models was provided by the R
2
index for either the entire
model or for each predictor (Alexander, Dauber, &
Entwisle, 1996; Diamond, Reagan, & Bandyk, 2000; Rush
& Vitale, 1994; Trusty, 2000). None reported the HL test.
Only one study (Rush & Vitale, 1994) validated predicted
probabilities against data in the Table 4 format. Our review,
however, uncovered two minor discrepancies in Rush and
Vitale’s (1994) classification table (Table 5, p. 331). In
Table 5, the hit rate was reported to be 90.6%, and misclas-
sifications were 223 for at-risk children and 112 for non-at-
risk children. The text on page 332 reported a hit rate of
10 The Journal of Educational Research
90.71%, and the misclassifications were 223 versus 115,
written on page 329. None reported measures of association
such as Kendall’s Tau-a, Goodman-Kruskal’s Gamma,
Somers’s D statistic, or the c statistic. None mentioned the
statistical package that performed the logistic analysis,
although Rush and Vitale (1994) used SPSS-X to perform
factor analysis, and those results were subsequently incor-
porated into logistic regression.
Verification of the Binomial Assumption
As stated earlier, logistic regression has only one
assumption: The binomial distribution is the assumed dis-
tribution for the conditional mean of the dichotomous out-
come. This assumption implies that the same probability is
maintained across the range of predictor values. Though
none of the eight studies verified or tested this assumption,
the binomial assumption is known to be robust as long as
the sample is random; thus, observations are independent
from each other. Samples used in the eight studies did not
appear to be nonrandom, nor did they have inherent depen-
dence among observations. Thus, the binomial assumption
appeared to be robust underlying all logistic analyses con-
ducted by these eight studies.
Reporting Formats of Logistic Regression Results
Five of the articles (Alexander, Dauber, & Entwisle,
1996; Diamond, Reagan, & Bandyk, 2000; Machamer &
Gruber, 1998) did present the logistic model. Of those five,
three (Meisels & Liaw, 1993; Smith, 1997; Trusty, 2000)
did not include intercepts in the logistic model. Odds ratios
were reported in three studies (McNeal, 1998; Meisels &
Liaw, 1993; Rush & Vitale, 1994), and odds were reported
in one (Trusty, 2000).
One study presented results in terms of marginal proba-
bilities (McNeal, 1998). The use of marginal probabilities
has been criticized by Long (1997, pp. 74–75) and Peng et
al. (2002) because marginal probabilities do not correspond
to a fixed change in the predicted probabilities that will
occur if there is a discrete change in one predictor (e.g.,
reading), while other predictors are realized at a constant. In
other words, the marginal probability corresponding to a
change in reading from 50 points to 60 points is different
from that associated with another 10-point change from,
say, 60 to 70 points. Furthermore, if other predictors (e.g.,
age) are held at their respective means, the corresponding
marginal probability for reading is different from that com-
puted at other values (e.g., the mode). One study did not
explain how a categorical predictor was coded in the data
(Diamond, Reagan, & Bandyk, 2000). These reporting for-
mats create difficulties for readers to verify results with
another sample or at another time or place.
One study (Trusty, 2000) coded a dichotomous predictor
as 1 (do not have a computer in the home) and 2 (do have a
computer), instead of the recommended 0 and 1, or –1/2 and
+1/2 (Peng & So, 2002b). This practice is not necessarily
incorrect; it simply makes the interpretation of the regres-
sion coefficient awkward and less direct.
Observation to Predictor Ratio
As stated earlier, the literature has not offered specific
rules that are applicable to logistic regression (Peng et al.,
2002). On the basis of the general rule of a minimum ratio of
10 to 1, with a minimum sample size of 100, all eight studies
met and even exceeded our recommendation. Therefore, the
results reported in these studies were considered stable.
Summary
In this paper, we demonstrate that logistic regression can
be a powerful analytical technique for use when the out-
come variable is dichotomous. The effectiveness of the
logistic model was shown to be supported by (a) signifi-
cance tests of the model against the null model, (b) the sig-
nificance test of each predictor, (c) descriptive and inferen-
tial goodness-of-fit indices, (d) and predicted probabilities.
During the last decade, logistic regression has been gain-
ing popularity. The trend is evident in the JER and higher
education journals. Such popularity can be attributed to
researchers’easy access to sophisticated statistical software
that performs comprehensive analyses of this technique. It
is anticipated that the application of the logistic regression
technique is likely to increase. This potential expanded
usage demands that researchers, editors, and readers be
coached in what to expect from an article that uses the
logistic regression technique. What tables, charts, or figures
should be included? What assumptions should be verified?
And how comprehensive should the presentation of logistic
regression results be? It is hoped that this article has
answered these questions with an illustration of logistic
regression applied to a data set and with guidelines and rec-
ommendations offered on a preferred pattern of application
of logistic methods.
ACKNOWLEDGMENTS
We wish to thank James D. Raths and one anonymous consulting editor
for their very helpful comments on earlier drafts of this article.
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12 The Journal of Educational Research
APPENDIX A
Hypothetical Data for Logistic Regression
Remedial
Reading reading
ID Gender score recommended?
1
2
No
No
91.0
77.5
Boy
Boy
Remedial
Reading reading
ID Gender score recommended?
Girl
Girl
Girl
Boy
Girl
Boy
Girl
Girl
Boy
Boy
Boy
Boy
Girl
Girl
Boy
Girl
Boy
Girl
Boy
Girl
Girl
Boy
Boy
Girl
Girl
Girl
Boy
Boy
Boy
Girl
Boy
Girl
Boy
Girl
Girl
Girl
Girl
Girl
Boy
Girl
Boy
Girl
Girl
Girl
Girl
Boy
Girl
Boy
Girl
Boy
Girl
Girl
Boy
Boy
Boy
Boy
Boy
Boy
Boy
Boy
52.5
54.0
53.5
62.0
59.0
51.5
61.5
56.5
47.5
75.0
47.5
53.5
50.0
50.0
49.0
59.0
60.0
60.0
60.5
50.0
101.0
60.0
60.0
83.5
61.0
75.0
84.0
56.5
56.5
45.0
60.5
77.5
62.5
70.0
69.0
62.0
107.5
54.5
92.5
94.5
65.0
80.0
45.0
45.0
66.0
66.0
57.5
42.5
60.0
64.0
65.0
47.5
57.5
55.0
55.0
76.5
51.5
59.5
59.5
59.5
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
(Appendix continues)
September/October 2002 [Vol. 96(No. 1)] 13
APPENDIX A—continued
Remedial
Reading reading
ID Gender score recommended?
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
Boy
Girl
Boy
Boy
Boy
Boy
Girl
Boy
Girl
Boy
Boy
Boy
Girl
Boy
Girl
Girl
Boy
Girl
Girl
Girl
Boy
Boy
Boy
Boy
Boy
Girl
Girl
Girl
Girl
Boy
Girl
Girl
Girl
Girl
Girl
Girl
Girl
Girl
Girl
Girl
Boy
Girl
Boy
Boy
Girl
Girl
Girl
Boy
Girl
Boy
Girl
Girl
Girl
Boy
Girl
Boy
Girl
Boy
Girl
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
55.0
70.0
66.5
84.5
57.5
125.0
70.5
79.0
56.0
75.0
57.5
56.0
67.5
114.5
70.0
67.0
60.5
95.0
65.5
85.0
55.0
63.5
61.5
60.0
52.5
65.0
87.5
62.5
66.5
67.0
117.5
47.5
67.5
67.5
77.0
73.5
73.5
68.5
55.0
92.0
55.0
55.0
60.0
120.5
56.0
84.5
60.0
85.0
93.0
60.0
65.0
58.5
85.0
67.0
67.5
65.0
60.0
47.5
79.0
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
Boy
Girl
Girl
Girl
Girl
Girl
Girl
Girl
Girl
Boy
Girl
Boy
Boy
Boy
Boy
Boy
Boy
Boy
Girl
Girl
Girl
Boy
Boy
Girl
Girl
Boy
Girl
Boy
Boy
Boy
Girl
Girl
Girl
Girl
Boy
Boy
Girl
Boy
Boy
Girl
Boy
Boy
Boy
Boy
Girl
Boy
Boy
Girl
Girl
Boy
Boy
Boy
Boy
Boy
Girl
Girl
Girl
Girl
Boy
No
No
No
No
No
No
No
No
No
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
80.0
57.5
64.5
65.0
60.0
85.0
60.0
58.0
61.5
60.0
65.0
93.5
52.5
42.5
75.0
48.5
64.0
66.0
82.5
52.5
45.5
57.5
65.0
46.0
75.0
100.0
77.5
51.5
62.5
44.5
51.0
56.0
58.5
69.0
65.0
60.0
65.0
65.0
40.0
55.0
52.5
54.5
74.0
55.0
60.5
50.0
48.0
51.0
55.0
93.5
61.0
52.5
57.5
60.0
71.0
65.0
60.0
55.0
60.0
Remedial
Reading reading
ID Gender score recommended?
(Appendix continues)
14 The Journal of Educational Research
APPENDIX A—continued
Remedial
Reading reading
ID Gender score recommended?
181
182
183
184
185
186
187
188
189
Boy
Boy
Girl
Boy
Girl
Boy
Boy
Boy
Girl
77.0
52.5
95.0
50.0
47.5
50.0
47.0
71.0
65.0
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
APPENDIX B
List of JER Articles Reviewed
1. Alexander, K. L., Dauber, S. L., & Entwisle, D. R. (1996).
Children in motion: School transfers and elementary school per-
formance. The Journal of Educational Research, 90(1), 3–11.
2. Diamond, K. E., Reagan, A J., & Bandyk, J. E. (2000). Par-
ents’ conceptions of kindergarten readiness: Relationships with
race, ethnicity, and development. The Journal of Educational
Research, 94(2), 93–100.
3. Machamer, A. M., & Gruber, E. (1998). Secondary school,
family, and educational risk: Comparing American Indian adoles-
cents and their peers. The Journal of Educational Research, 91(6),
357–369.
4. McNeal, R. B., Jr. (1998). High school extracurricular activi-
ties: Closed structures and stratifying patterns of participation.
The Journal of Educational Research, 91(3), 183–191.
5. Meisels, S. J., & Liaw, F.-R. (1993). Failure in grade: Do
retained students catch up? The Journal of Educational Research,
87(2), 69–77.
6. Rush, S., & Vitale, P. A. (1994). Analysis for determining fac-
tors that place elementary students at risk. The Journal of Educa-
tional Research, 87(6), 325–333.
7. Smith, J. B. (1997). Effects of eighth-grade transition pro-
grams on high school retention and experiences. The Journal of
Educational Research, 90(3), 144–152.
8. Trusty, J. (2000). High educational expectations and low
achievement: Stability of educational goals across adolescence.
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