Does calculus help in principles of economics courses? estimates using matching estimators.
Bosshardt, William ; Manage, Neela
I. Introduction
Researchers in economic education explore why some students perform
better than others in the learning of economic concepts. While intrinsic
student characteristics can play a role in determining the learning of
economics, the background of the student is also important. In the
principles of economics course, researchers have often focused on the
student's mathematical background or whether the student has had
prior economics training, such as high school economics or another
principles course (because principles courses are usually taught as two
courses, students often have had one before the other).
In order to measure the impact of prior economic or mathematical
coursework on learning in economics principles courses, the researcher
must observe the performance of principles students who have had a prior
course and those who have not had a prior course. The problem is that
the researcher rarely has control over which students have had the prior
course and which ones do not. Ideally, a researcher can run an
experiment and randomly assign students to take a mathematics course
before principles and have others take principles without the course.
One example of such an experiment is found in Fizel and Johnson (1986),
who found that the sequence of microeconomics--macroeconomics is better
than the reverse. However, the researcher today often has less control
to conduct such an experiment. Students demand control over their
schedules; institutional review boards prefer voluntary participation in
the experimental designs. (1)
In the absence of an experiment, the researcher is left to forces
outside his or her control to generate the data required. If the
researcher is fortunate, an event takes place--such as a change in
prerequisites or some other institutional change--that generates a
natural experiment. Unfortunately for most researchers, these
occurrences are rare and often serve to confound ongoing studies as
opposed to generate productive new ones.
In general, economic education researchers use data in which the
subjects themselves select the sequence of courses or the extent of
their mathematical background. With this data in hand, the researcher
estimates an educational production function using regression analysis.
Typically, an indicator variable measures whether the student has had a
certain type of background or not. Unfortunately, if the variable is
significant, the researcher is not sure whether the prior course itself
is responsible for the significance or the mere fact that the student
elected to take the course is responsible.
An alternative to regression analysis is to use matching methods to
analyze the impacts of having mathematical background such as a calculus
course. The idea behind matching methods is to pair individuals with
similar observable characteristics and backgrounds so that the only
difference between the two is that one individual, by pure chance, has
decided to take a calculus course while the other individual did not. By
pairing each individual in the sample with a calculus course with
another similar individual who did not, any resulting differences in the
student's performance can be attributed to the calculus course as
opposed to other characteristics.
This paper examines the impact of taking a calculus course on a
student's performance in microeconomic principles and macroeconomic
principles by utilizing two estimation procedures. Matching estimation
is utilized to obtain estimates of the impact of calculus on student
performance in principles of economics courses and the results are
compared with least squares estimates. Section II contains a brief
review of the literature on the effect of a student's background on
the student's performance in economic principles courses. Section
III outlines the role of matching estimation in addressing selection
issues that arise in non-experimental settings. The data and the model
are described in Section IV. The estimates are reported in Section V
along with a comparison of the matching estimates and those derived from
a regular OLS model.
II. Literature Review
Many studies have examined the determinants of student learning in
principles of economics courses. Various specifications of educational
production functions have been estimated to investigate the impact of
student characteristics or background, instructor characteristics, class
characteristics, or instructional techniques. In terms of student back
ground, studies have focused on high school economics (Brasfield,
Harrison, McCoy, 1993) or prior principles courses (Fizel, Johnson,
1986).
Another important determinant of success in principles courses is
mathematics ability. Generally, the impact of mathematical ability has
been measured by the estimation of an educational production function
with various measures of mathematical ability as independent variables.
Typical variables include the results of tests such as the SAT or ACT or
a test given to the students before the class (Douglas and Sullock,
1995, Elzinga and Melaugh, 2009). Other measures include prior courses
in mathematics, such as algebra or calculus (Anderson, Benjamin, Fuss,
1994 for one example).
Ballard and Johnson (2004) use a wide array of measures including
whether the student has had calculus or a remedial course. They also use
ACT scores and the results of a short math quiz as measures of
mathematical ability. All measures significantly impacted student
performance in a microeconomic principles course. One oddity in their
results was that taking a macroeconomics course before the
microeconomics course significantly reduced the student's score in
the course. While the authors speculated that students with prior
learning were lulled into complacency, another explanation is that
student's choice of the sequence of courses is the reason for the
"odd" sign. Students who are weaker at mathematics avoid the
microeconomics course by taking macroeconomics, the lesser of two evils,
first. The fact that they have taken the effort to avoid the
microeconomics course is yet another indicator for future poor
performance--beyond what is measured in the other variables. Another
similar explanation is that students that are off the usual track are
not necessarily the best students, period.
The "odd" sign found in Ballard and Johnson (2004)
suggests that it is important to examine the path by which the student
has reached the principles course in order to fully understand the
impacts of a prior mathematics course on performance in economic
principles.
III. OLS, Selection Bias, and Matching Estimation
Matching estimators provide an alternative to regression analysis
for estimating the causal effect of a treatment in a non-experimental
setting. Based upon the treatment-effects literature, for students who
take calculus prior to an economics principles course, the effect of
taking calculus is measured by the average treatment effect on the
treated (ATT) which is the difference between the mean outcome (in terms
of the course grade) observed for those students who actually took
calculus and the mean outcome that would have been observed for these
same students if they had not taken calculus.
The basic problem in estimating the average treatment effect is
that for students who have taken calculus, the counterfactual mean is
not observed. If it is substituted by the mean outcome for students who
did not take calculus, it is likely to result in a biased estimator of
the treatment effect. The bias results from the fact that the treated
and nontreated groups are likely to systematically differ in terms of
observed characteristics that also affect the outcome. (2) For example,
a student with a strong academic background is more likely to take
calculus and is also more likely to perform better in the economics
course.
Matching estimators avoid bias by matching treated units with
non-treated (control) units that have similar observable pretreatment
characteristics. By conditioning on these observable characteristics,
the difference in the outcomes is then attributed to the treatment. The
matching approach thus imitates a controlled or randomized experiment
and can provide unbiased estimates of the treatment effect. When there
are a large number of observable characteristics that affect the
outcome, practical applications of the matching approach are limited.
Rosenbaum and Rubin (1983) suggest matching the treated and control
units on their propensity scores in order to reduce the dimensionality
problem caused by the large number of covariates. (3) The propensity
score, which can be estimated with probit or logit models, is the
probability of receiving the treatment (taking calculus) conditional on
the observable characteristics. Biases due to observable characteristics
are thus eliminated by conditioning on the propensity score.
If the choice of taking calculus before economics is based only
upon observable factors, matching individuals based on their propensity
scores is similar to an experimental design and propensity score
matching provides an unbiased estimate of the treatment effect of taking
calculus. On the other hand, if the choice is also based on unobservable
factors, then estimates obtained via propensity score matching are
biased estimates of the treatment effect. Agodini and Dynarski (2004)
find that the size of the bias depends on the extent to which the
treatment group and the control group differ along unobservable
characteristics. The data utilized in this study contains information on
several observable student characteristics affecting the calculus
decision (mathematical ability, overall ability, prior coursework,
demographic factors) and institutional factors related to course
offerings and time-of-day constraints influencing student choices that
allow student matching. This study also uses students from the same
institution who all enter as freshmen--a homogeneous setting which might
minimize the unobservable factors. The fairly extensive data on each
student combined with the fact that the control group comes from the
same institution as the treatment group imply that, in this case,
unobservable factors may be relatively small. (4)
IV. The Data and Model
1. Data
Our initial data set consists of over 2,993 students who entered as
a freshman at a university of approximately 25,000 students from fall of
2000 to fall of 2003. Because the focus of our study is the group of
students who intend to take both calculus and a principles of economics
course, we requested data on anyone who has taken a principles course of
any major or anyone who is, or was at one time, a business major.
Business majors are required to take calculus, microeconomic principles
and macroeconomic principles, so our initial sample consists of anyone
who has taken or had intentions of taking (as best we can determine) an
economic principles course. Since none of the three courses (calculus,
micro, or macro) is a prerequisite of another, students may elect to
take the courses in any order they wish--and they do. Some in our sample
did not take any of the three courses as of the time the data was
collected in summer of 2007, leaving 2,295 in the sample as having one
of the three courses. Of those who had taken at least one, about 47%
take calculus as their first course of the three, 20% take
macroeconomics as their first course and 21% take microeconomics. The
rest take a combination of the courses in their first semester with one
of the three courses. Of the students who had completed all three, the
most popular sequence was calc/macro/micro. Almost as popular was
calc/micro/macro, followed by macro/micro/calc and then
calc-concurrent-with-macro/micro. Since the instructors in the
principles courses cannot expect their students to have had calculus
before the principles class, calculus is not explicitly used in the
courses. Nonetheless, the use of graphical analysis in the calculus
course should provide students with skills that are useful in completing
the principles course.
Data was collected from the registrar's office. The data
included:
1. Course information, including grades, semester taken and time of
day taken.
2. Standardized test information such as SAT scores and ACT scores.
3. Demographic information such as age, gender and race.
4. High school GPA.
Variables for our estimations are constructed from these data
sources. The variable of interest is the performance of the student in
principles courses. Because no standardized test is given at the end of
principles courses, we are left with the student's grade as an
indicator of their performance. While not a perfect measure of learning,
using the final course grade is appealing in two ways. First, it is a
result that is of interest to students wanting to do well in a course.
Second, departments have an interest in promoting the success of their
students as measured by their grades. The final course grade is
expressed as a grade point average with the typical 0 to 4 scale. The
institution uses + and--grading, so the variable has fairly fine
gradations. Two alterations to the grade are made. First, the average
grade for the section that the student attended is subtracted from the
grade in order to offset grading differences by individual professors.
Secondly, a grade of "W" (withdrawal) was turned into a grade
of "F." While not all "W" grades are due to poor
performance before the deadline (see Bosshardt 2004), in general, the
probability of success is lower.
In addition to the course grades in the principles courses, we also
have information on all courses taken including their grades and when
the course was taken. In terms of standardized test scores, we use
primarily the SAT verbal and math scores. When these were not available,
we used the predicted verbal and math scores based on the student's
ACT scores (Dorans, 1999). While not perfect substitutes, the
correlation between the ACT and SAT math is .89. The demographic data
and high school GPA data are self-explanatory.
Table 1 shows a brief description of the variables used and their
means for the two estimations: the effect of calculus on a
microeconomics grade and the effect of calculus on a macroeconomics
grade. The number of observations is lower for the subgroups because
some students did have one principles but not the other. It should be
noted that students who did not take all three courses because of a
change in major or because they left school have not been eliminated
from the sample.
Table la shows the raw differences in grades between those who have
had calculus and those who have not. Apparently, the raw difference is
approximately 1/2 a grade. This estimate, of course, does not account
for any selection issues at all.
2. Model
With the matching methodology, our goal is to estimate a propensity
score for a student's probability of choosing to take calculus
before their principles class, P(C). The general specification we chose:
P(C) = f(Overall ability, Mathematical ability, Courses taken,
Demographic Information, Time of Day)
The student's high school grade point average (HSGPA) and the
quantitative (QUANT) and verbal (VERBAL) scores on the SAT are included
as predictors of student performance in college. In addition, QUANT
might also reflect the mathematical background of the student. An
indicator variable for macroeconomics (microeconomics) principles taken
before microeconomics (macroeconomics) is assumed to capture the effect
of prior economics coursework (MACBMIC and MICBMAC). All these variables
are assumed to have a positive impact on the probability of taking
calculus before economics.
An indicator variable for a student having taken algebra sometime
in their college career (ALGEBRA) is included for two reasons. First,
taking algebra might delay taking calculus making it less likely that
the student will take calculus before economics. Second, if the student
is required to take algebra it might just be indicative of poor
mathematical skills.
The propensity score estimation also considers the role of course
scheduling and time of the day students prefer to take courses. For
example, students might have time preferences due to work related
constraints. This might be further complicated by certain courses being
offered only at certain times of the day. Such preferences are captured
by the percent of courses a student has taken in the morning during
their career (MORN) and by the percent of courses a student has taken in
the evening during their career (EVEN). The variable measuring the
percent of courses a student has taken in the afternoon during their
career is omitted from the regression to avoid perfect multicollinearity
because the sum of the three time-of-day variables equals one. In this
sample, students who take a larger percent of courses in the morning or
evening are more likely to take calculus before economics. (5)
FEMALE, BLACK and HISPANIC are dummy variables for female, black
and hispanic students. Finally, squared terms for the QUANT, and VERBAL
variables and an interaction term (which equals the product of ALGEBRA
and HSGPA) were included in the propensity score estimation to satisfy
the balancing property. (6) The balancing property is satisfied when,
within each interval of the propensity score, the means of each
covariate do not differ between the treated and control units. The test
of the balancing property is restricted to the common support. (7)
With the propensity score in hand, then the estimates, using the
various matching schemes described above, can be calculated.
V. Results
1. Matching Model
Matching estimates of the treatment effect of taking calculus on
performance in microeconomics principles and macroeconomics principles
are obtained in two parts. First, the propensity scores are estimated
and second, the average treatment effects on the treated are calculated
using matching methods.
The estimates of the propensity scores from the probit models are
reported in Table 2. The pseudo R-square in the estimated probit models
for taking calculus before economics is. 17 for the microeconomics model
and. 15 for the macroeconomics model.
The estimates of the propensity scores for microeconomics and
macroeconomics were similar. Stronger students, as measured by their
high school GPA or SAT math score, were more likely to take calculus
before their principles class. In terms of coursework, if students took
one of the principles courses before the other, this, probably by virtue
of having taken more time, increased the probability of the student also
having had calculus. The time-of-day variable, MORN, has a positive and
significant coefficient implying that student choices about taking
calculus early on might be influenced by the flexibility they have in
the scheduling of courses as well as the times these courses are
offered. Students with lower SAT verbal scores were more likely to take
calculus before their microeconomics principles course. Students who had
algebra were less likely to have had calculus before their
microeconomics principles course. These two variables did not have any
significant effects on the probability of students taking calculus
before macroeconomics. The HISPANIC variable had a significant impact in
both the models. The FEMALE variable was also significant in the
macroeconomics model, suggesting that race and gender played some role
in the calculus decision. Overall, the results suggest that students
with strong mathematical backgrounds and academic records are less
likely to postpone taking calculus and more likely to take calculus
before macroeconomics and microeconomics.
Based on the estimates of the propensity scores the treatment
effect on the treated was estimated for the performance in the economics
principles (microeconomics and macroeconomics) courses. Matching
estimates of the treatment effect of taking calculus are reported in
Table 3. Based on the magnitude of the estimated ATT's and their
statistical significance, the results from matching estimation suggest
that taking calculus helps in both the economics principles courses. (8)
For those who expected that mathematics preparation is more important
for microeconomics, the surprising result is that the effect is about
the same for both courses (the average treatment effect based on
stratification matching is .291 for microeconomics and .267 for
macroeconomics). (9)
2. Regression Analysis
The determinants of performance in principles of economics are also
estimated using regression analysis of Ballard and Johnson (2004). Our
results are reported in Table 4. Columns (1) and (2) present the same
model specification that was used in the matching estimation, and are
estimated with the same observations. In other words, it is assumed that
all variables that impact the decision to take the course also impact
performance in the course. The variable of interest--whether the student
has had calculus before the principles course--is also included.
The coefficients of the calculus dummy variables are positive and
significant for both microeconomics and macroeconomics but the magnitude
of these coefficients is slightly smaller than the mean of the average
treatment effects obtained using the four matching methods. The
regressions also suggest that the high school GPA, SAT math scores, (10)
and taking the other principles course are systematically related to
performance in both microeconomics and
macroeconomics. The grades for black students are lower in both
microeconomics and macroeconomics and one of the time-of-day variables
(EVEN) affects the performance in both courses. The grades for female
students are lower only in the microeconomics course whereas higher
verbal SAT scores improve performance in the macroeconomics course
suggesting that these variables do not reveal any consistent impacts on
performance.
3. Comparison
Overall, the estimates from the matching estimation are similar to
those found from the regression analysis. Both estimation procedures
suggest a small, significant impact of having calculus before a
principles course. A comparison of the propensity score estimates and
the regression estimates does reveal some differences. For example,
Hispanic students are more likely to have calculus before principles,
they do not, according to the regression results, do better. Black
students are no more or less likely to have had calculus, but tend to do
worse in principles.
Perhaps more is learned from examining the distribution of the
gains from a calculus course over different types of students. By
grouping students by their propensity to take calculus, we can estimate
which students (those with high propensity or low propensity) are likely
to gain the most from taking calculus. For example, our estimations show
that those who have a very low propensity to have taken calculus before
principles are not likely to be helped by doing so. In fact, for the
macroeconomics course, the 22 who did have calculus did worse than the
94 who did not. Second, the largest gains seemed to be for the groups
who were 20% to 60% likely to take calculus. For the group of students
whose propensity to take calculus was between 20% and 40%, the gain was
about a third of a grade for microeconomic students and almost
two-thirds of a grade for macroeconomics students For those students in
the 40% to 60% propensity range, the average gain was about one third of
a grade for microeconomic students and two-fifths of a grade for
macroeconomics students. In general, the majority of the students above
60% propensity did not gain as much. (11) In sum, the largest gains from
taking calculus are, apparently, for those who are not particularly
inclined to have taken it, but perhaps smart enough to learn it. These
students tend to have lower high school GPAs and have lower quantitative
SAT scores. Those students who have a good quantitative score and good
high school GPA tend not to benefit from the calculus course, probably
because the level of mathematics in a principles course is readily
understood by a student with reasonable mathematical and scholastic
abilities.
VI. Conclusions
The results of this study indicate that calculus taken before a
principles course generally helps a student's performance. For the
OLS regression and matching estimation, the overall impact is roughly
the same for a microeconomic course as a macroeconomic course. They gave
estimates lower than a raw difference in the means between those who had
calculus and those who did not.
An important advantage to the matching data technique is that more
information might be drawn from the estimation of the propensity scores
themselves as opposed to the calculation of the final estimate. In our
case, we note that those with lower probability of calculus - but not
the lowest - might benefit most from calculus. While matching techniques
may not provide a silver bullet in solving the problem of
nonexperimental designs faced by economic researchers, they nonetheless
provide economic educators with another tool with which to examine the
effects of classes or teaching techniques on student learning.
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The authors would like to thank an anonymous referee and Sheryl
Ball and other participants who attended a presentation of this paper in
a NAEE/NCEE session at the 2008 ASSA meetings for helpful comments. The
Office of Institutional Effectiveness at Florida Atlantic University
provided invaluable assistance in gathering the data for this project.
Notes
(1.) See Lopus, Grimes, Becker, and Pearson (2007) for a discussion
of institutional review boards in economic education.
(2.) See Heckman, Ichimura and Todd (1998) and Dehejia and Wahba
(2002) for more technical details about the selection bias problem.
(3.) Several propensity score matching methods have been employed
in the literature to obtain matching estimators: nearest neighbor
matching, radius matching, kernel matching, and stratification matching.
These methods differ in terms of how they construct the counterfactual
outcome for the estimation of the treatment effect. See Becker and
Ichino (2002).
(4.) Michalopoulos, Bloom and Hill (2004) found that having good
data and choosing a control group from the same local labor market and
with comparable measures from a common data source improved labor market
program impact estimates obtained via nonexperimental evaluation methods
such as propensity score matching.
(5.) Calculus courses are offered primarily during the day, and not
in the later evenings.
(6.) See Dehejia and Wahba (2002) for a discussion on use of square
and interaction terms in propensity score estimation.
(7.) Propensity score matching makes the assumption of common
support, which requires the availability of comparable control units for
each unit that has received the treatment. Becker and Ichino (2002)
explain that the test of the balancing property is performed only on
those observations whose propensity score belongs to the intersection of
the supports of the propensity scores of treated and control units.
(8.) The results based on stratification matching are reported in
Table 3. The stratification matching estimator partitions the common
support of the propensity score into blocks such that the propensity
score is balanced within each block. Treatment impacts are computed for
each block by taking the mean difference in outcomes between treated and
control units and the average treatment effect is computed as a weighted
average of the block-specific treatment effects, with the weights being
the percentage of the total number of the treated within each stratum.
(9.) Matching estimates of the treatment effect of taking calculus
are also obtained by using nearest neighbor matching, kernel matching
(bandwidth of 0.06), and radius matching (with radii of .1 and .005).
The estimates of the average treatment effect of calculus on the grade
in microeconomics and macroeconomics principles are statistically
significant for each matching method. Across the four matching methods
employed in this study, the average of the estimated ATT's is .31
for microeconomics and .32 for macroeconomics.
(10.) The SAT math score is significant in a linear specification
(when only QUANT is included), but the squared term, QUANT2, is included
in the model reported in Table 4 to provide a comparison to the results
from the matching estimation.
(11.) The highest propensity group had a large gain as well, but
the estimations were based on only two students without calculus.
William Bosshardt, Associate Professor of Economics, Florida
Atlantic University, 561-297-2908, wbosshar@fau.edu
Neela Manage, Associate Professor of Economics, Florida Atlantic
University, 561-297-3226, manage@fau.edu
TABLE 1.
Variable Descriptions and Means
Variable Description MICRO MACRO
Macrogr Grade in first macro course, -0.116
modified
Microgr Grade in first micro course, -0.092
modified
Calcmac Calculus before macro 0.499
Calcmic Calculus before micro 0.509
Algebra Had algebra at some time 0.804 0.796
Hsgpa High School GPA 3.317 3.323
Algebra*Hsgpa Interaction between above 2.654 2.628
Quant SAT Mathematics score (or 512.615 516.230
predicted based on ACT)
Verbal SAT Verbal score (or predicted 493.717 496.134
based on ACT)
Female Gender 0.476 0.451
Black Race indicator 0.180 0.148
Hispanic Race indicator 0.134 0.145
Morn % of courses in career taken 0.392 0.393
in morning
Even % of courses in career taken 0.146 0.148
in evening
Macbmic Had macro before micro 0.371
Micbmac Had micro before macro 0.372
N * 1541 1493
* Number of students who had course and grade for course
TABLE IA.
Unadjusted Differences in Grades by Whether the
Student Has Had Calculus
Microeconomics Macroeconomics
Grade Grade
Without Calculus -0.396 -0.390
With Calculus 0.201 0.160
Difference 0.597 0.550
TABLE 2.
Estimation of Propensity Scores of Having
Calculus Before a Principles Course
Variable CALCMIC CALCMAC
Algebra -1.3984 ** 0.3013
Quant 0.0177 *** 0.0137 **
[Quant.sup.2] -0.00001 ** -0.000008
Verbal -0.0097 ** -0.0037
[Verbal.sup.2] 0.000006 0.0000001
Hsgpa 1.0272 *** 0.8093 ***
Algebra*Hsgpa -0.5598 *** -0.2561
Female -0.1063 -0.1403 *
Black 0.0226 -0.0095
Hispanic 0.1905 * 0.2528 **
Morn 1.0306 *** 0.5806 **
Even 0.4306 0.3980
Macbmic 0.4813 ***
Micbmac 0.4916 *
Constant -6.5008 *** -5.8018 ***
N 1500 1456
N with common support 1482 1443
# control units 722 722
# treated units 760 721
* p < .1, ** p < .05, *** p < .01
TABLE 3.
Estimates of the Treatment Effect of Taking Calculus on Economics
Principles
Matching Estimates of the
Average Treatment Effect
of Calculus on Performance in
Economics Principles (a)
Microeconomics Macroeconomics
ATT (c) 0.291 *** 0.267 ***
Std. Error (0.090) (d) (0.067) (d)
#Treated 760 721
#Control 722 722
Total Obs 1482 1443
Regression Estimates
Microeconomics Macroeconomics
ATT (c) 0.219 *** 0.292 ***
Std. Error (0.062) (0.064)
#Treated
#Control
Total Obs 1482 1443
(a) The results are based on stratification matching and utilize
the pscore and atts commands in Stata.
(b) Reported figures for OLS are the coefficient estimates of the
calculus dummy from Table 4 and their corresponding standard errors.
(c) Average treatment effect on the treated
(d) Bootstrapped standard errors based on 200 replications of the
data.
*** P < .01
TABLE 4.
Regression Estimates of the Determinants of
Performance in Principles of Economics Courses
Variable MICROGR (1) MACROGR (2)
Calcmic 0.2189 ***
Calcmac 0.2922 ***
Algebra 0.7588 * 0.8851
Quant 0.0033 -0.0032
[Quant.sup.2] -0.0000006 0.000005
Verbal 0.0016 0.0033
[Verbal.sup.2] -0.000001 -0.000002
Hsgpa 0.6607 *** 0.6995 ***
Algebra*Hsgpa -0.2606 ** -0.2475 *
Female -0.1653 *** -0.0806
Black -0.1841 ** -0.1998 **
Hispanic -0.0371 -0.0283
Morn 0.7319 *** 0.2284
Even 1.4881 *** 0.7169 **
Macbmic 0.2554 ***
Micbmac 0.1752 ***
Constant -4.8509 *** -3.7932 ***
N 1482 1443
* p < .1, ** p < .05, *** p < .01
