A bivariate response model for studying the marksobtained in two jointly-dependent modules in higher education

SORT 41 (2) July-December 2017, 255-276 DOI: 10.2436/20.8080.02.59 A bivariate response model for studying the marks obtained in two jointly-dependent modules in higher education Emilio G´omez-D´eniz1,NancyD´avila C´ardenes1and Mar´ıaD.Garc´ıa Artiles2 Abstract We study the factors which may affect students’ marks in two modules, mathematics and statistics, taught consecutively in the first year of a Business Administration Studies degree course. For this purpose, we introduce a suitable bivariate regression model in which the dependent variables have bounded support and the marginal means are functions of explanatory variables. The marginal probability density functions have a classical beta distribution. Simulation experiments were performed to observe the behaviour of the maximum likelihood estimators. Comparisons with univariate beta regression models show the proposed bivariate regression model to be superior. MSC: 62P25, 62E99, 97D60. Keywords: Beta distribution, bivariate beta distribution, conditional distributions, covariate, marginal distributions, regression, mathematics, statistics, business studies. 1. Introduction Event counts such as the number of claims for third-party liability, other claims under guarantee, medical consultations, the use of prescription drugs, and voluntary and/or involuntary job changes, among many others, are likely to be jointly dependent. In these cases, it is of interest to study how different covariates or factors may simultaneously affect the two random (dependent) variables involved. Bivariate Poisson regression models, bivariate negative binomial regression models (see Maher, 1990) and their extensions (see Gurmu and Elder, 2000), among other approaches, have been applied in these settings. Nevertheless, few such studies have been conducted when the dependent variables are continuous and bounded. 1Department of Quantitative Methods and TiDES Institute. University of Las Palmas de Gran Canaria, Spain. [email protected], [email protected] 2Department of Quantitative Methods. University of Las Palmas de Gran Canaria, Spain. [email protected] Received: March 2016 Accepted: March 2017 256 A bivariate response model for studying the marks obtained in two jointly-dependent... In the univariate case, Papke and Wooldridge (1996) examined potential econometric alternatives when the dependent variable is fractional, in a study of employee participation rates in 401(k) pension plans. More recently, Papke and Wooldridge (2008) analysed test pass rates and the portfolio choices of Australian households. Other research work related to the beta regression model includes Cepeda-Cuervo (2001), Paolino (2001), Ferrari and Cribari-Neto (2004) and Huang and Oosterlee (2011). The model proposed by G´omez-D´eniz, Sordo and Calder´ın-Ojeda (2013) provides an alternative to the beta regression model, and affords a better fit, at least in an actuarial setting. The model proposed by P´erez-Rodr´ıguez and G´omez-D´eniz (2015) also appears to be comparable to the beta regression approach in financial econometrics. Using Bayesian methodology, Bayes, Baz´an and Garc´ıa (2012) presented a variation of the beta regression model, while Cepeda-Cuervo and N´u˜nez-Ant´on (2013), used spatial regression in an analysis of the quality of education. In the bivariate case, Cepeda-Cuervo, Achcarb and Garrido (2014) proposed a bivariate beta regression model with joint modelling of the mean and dispersion parameters. As an extension of the works related above, we propose a flexible bivariate fractional response model in which the dependent variables are bounded and the marginal means are functions of explanatory variables. Although a bivariate regression model could be built by using, for instance, copulas from the Sarmanov family of distributions (see Lee (1996)), we chose the bivariate beta regression proposed by Olkin and Liu (2003) for this study because it is a simple model with which to compute marginal distributions, means and variances. In this model, the beta distribution has a straightforward formulation in which the Euler Gamma function is the only one considered. In this respect, Cepeda-Cuervo et al. (2014) used copulas to obtain a bivariate beta regression model in which, as in our own model, the marginal distributions are beta. These authors assumed weak dependence between the variables of interest and modelled the dependence using aFarlie-Gumbel-Morgenstern copula function. The model we propose is less complex than that presented in Cepeda-Cuervo et al. (2014) and therefore, by the Ockham’s razor principle, it might be preferable (Jaynes, 1994). In this paper, we study how some covariates may simultaneously affect the marks (ranging from 0 to 10) obtained by students in two first-degree subjects – Mathematics for Business and Basic Statistics in Business Administration Studies – taught at the University of Las Palmas de Gran Canaria (Spain) during two consecutive terms (first mathematics and then statistics). We assume that a good knowledge of mathematics will significantly influence the student’s understanding of statistics and therefore that there exists a positive correlation between these two variables. Accordingly, the model proposed would be suitable for studying this relationship. The importance of mathematical skills in other quantitative disciplines has been widely examined. In the fields of business and economics, many studies have analysed basic mathematical abilities as determinant factors of academic performance among first year university students: see, for example, Johnson and Kuennen (2006), Dolado and Morales (2009), Lunsford and Poplin (2011) and Arnold and Straten (2012). Emilio G´ omez-D´ eniz, Nancy D´ avila C´ ardenes and Mar´ıa D. Garc´ıa Artiles 257 Study plans in business and economics courses are organised in different ways, depending on the institution, but all have in common a requirement of basic mathematics to favour a better understanding of subjects that require this skill as a tool to develop more complex theories. In the present study, we focus on the above-mentioned mathematics and statistics modules, to determine whether certain common factors might explain the students’ marks obtained in each subject. The rest of this paper is structured as follows. Section 2 describes the bivariate model proposed by Olkin and Liu (2003), from which we derive the proposed bivariate regression model. This model and its parameters are studied in Section 3. The data are described in detail in Section 4. In Section 5 we fit the marginal beta regression models and the bivariate beta regression models, comparing the univariate and the bivariate models. Finally the results obtained and the main conclusions drawn are reported in Section 6. 2. Modelling bivariate marks Although mathematics and statistics are known to be logical and effective means of solving certain problems, most Business Administration students, especially those in the first and second years of their degree courses, are not interested in these course subjects. Indeed, numerous students in this area of study present some form of rejection of mathematics and statistics. Nevertheless, our empirical evidence shows that the marks obtained by students in statistics are positively related to those achieved in mathematics. We assume this is because mathematics is an instrumental subject that influences the results achieved in statistics. Let Y1and Y2be two random variables which represent the marks achieved in mathematics and statistics, respectively. To address the study goal presented in the introduction, and taking into account the above comments, we need a bivariate distribution that meets the following conditions: a)The support of the distribution should be bounded, since the marks are usually restricted to a given interval. b)The bivariate distribution should provide a dependence structure. c)The correlation between the two random variables should be positive. That is, ρ(Y1,Y2)>0. d)Preferably, Pr(Y2>y2|Y1>y1)should be a nondecreasing function in y1for all y2. Thus, the higher the mathematics mark, the greater the probability of obtaining higher marks in statistics. e)Because we wish to study the factors which may affect the marks obtained in the two courses, using a regression analysis, the marginal mean (the response variable) should be expressed as a function of the explanatory variables through a simple expression. 258 A bivariate response model for studying the marks obtained in two jointly-dependent... In this case, the standard beta distribution may be extended to the bivariate case. Many bivariate beta distributions have been derived from an application or as extensions to or generalisations of other well-known bivariate beta distributions. Since the latter are used in a wide variety of applications, the development and derivation of new bivariate beta distributions has been extensively studied. Nevertheless, few such distributions present these five features simultaneously. One, however, was proposed by Olkin and Liu (2003), with the following probability density function (pdf): f(y1,y2)=ya1−1 1ya2−1 2(1−y1)a2+a3−1(1−y2)a1+a3−1 B(a1,a2,a3)(1−y1y2)a1+a2+a3,(1) where 0 <yi<1(i=1,2), ai>0(i=1,2,3) and where B(a1,a2,a3)is given by B(a1,a2,a3)=∏3 i=1Γ(ai)/Γ(3 i=1ai),whereΓ(·)is the Euler Gamma function. Henceforth, we use the expression (Y1,Y2)∼BB(a1,a2,a3)when the two random variables (Y1,Y2)fit the pdf (1). The marginal distributions of Y1and Y2are beta distributions with parameters (a1,a3) and (a2,a3), respectively. Thus, the marginal means, the variances and the cross moment are given by E(Yi)= ai ai+a3 ,i=1,2,(2) var(Yi)= aia3 (ai+a3)2(ai+a3+1),i=1,2. E(Y1Y2)=a1a2Γ(a1+a3)Γ(a2+a3) mΓ(a3)Γ(m+1)3F2({m1,m2,m};{m+1,m+1};1),(3) where mi=ai+1(i=1,2),m=a1+a2+a3and 3F2is the generalised hypergeometric function. For details about this special function see, for instance, Gottschalk and Maslen (1988). This can be computed using the Mathematica package (see Wolfram (2003)). Using (2) and (3) we can obtain the covariance, cov(Y1,Y2), and the correlation between Y1and Y2,ρ(Y1,Y2). For reasons of space, these large expressions are not shown here. Olkin and Liu (2003) showed that the correlation is always positive, with values in the interval (0,1). The following result is obtained for the conditional distribution: f(y1|y2)= ya1−1 1(1−y1)a2+a3−1(1−y2)a1 B(a1,a2+a3)(1−y1y2)a1+a2+a3,(4) f(y2|y1)= ya2−1 2(1−y2)a1+a3−1(1−y1)a2 B(a2,a1+a3)(1−y1y2)a1+a2+a3.(5) Emilio G´ omez-D´ eniz, Nancy D´ avila C´ ardenes and Mar´ıa D. Garc´ıa Artiles 259 After some algebra, we derive the conditional mean obtained from (4) and (5). Thus E(Y1|Y2=y2)= a1 a1+a2+a3 2F1(1,a2+a3;a1+a2+a3;y2), E(Y2|Y1=y1)= a2 a1+a2+a3 2F1(1,a1+a3;a1+a2+a3;y1),(6) where 2F1represents the hypergeometric function (see Gradshteyn and Ryzhik, 1994). One of the advantages of using the pdf given in (1) is that for this distribution we have Pr(Y2>y2|Y1>y0 1)≤Pr(Y2>y2|Y1>y1 1),y0 1<y1 1, Pr(Y2≤y2|Y1≤y0 1)≥Pr(Y2≤y2|Y1≤y1 1),y0 1<y1 1, for all y2.Inotherwords,Pr(Y2>y2|Y1>y1)is a nondecreasing function in y1for all y2 and Pr(Y2≤y2|Y1≤y1)is a nonincreasing function in y1for all y2, because the pdf (1) is positively likelihood ratio dependent (see Tong (1980) and Olkin and Liu (2003) for details). This is corroborated by the fact that in our case the random variables Y1and Y2 are positively quadrant dependent, a concept introduced by Lehmann (1996). Thus, we have Pr(Y2>y2|Y1>y1)≥Pr(Y2>y2)Pr(Y1>y1), Pr(Y2≤y2|Y1≤y1)≥Pr(Y2≤y2)Pr(Y1≤y1). A possible interpretation of the parameters of the distribution in (1) is this. LetWbe a random variable measuring a student’s lack of mathematics skills for use in subjects such as mathematics, statistics and physics. Empirical evidence shows that when Business Administration students are asked about their skills in mathematics and statistics, most of them acknowledge inadequacy in this field. Let Ui(i=1,2)be the random variable representing the student’s willingness to study these subjects i(i=1,2). Assuming that Wand Uican take values in (0,∞), then the marks obtained in subject ican be represented by the random variables Yi=1 1+W/Ui =Ui Ui+W,i=1,2. The gamma distribution provides a flexible representation of a variety of distribution shapes, by varying the shape parameter. Let us now assume that the random variables U1,U2and Ware independent and follow a standard gamma distribution with shape parameters a1,a2and a3, respectively. Then, the random variable (Y1,Y2)follows the distribution given in (1). 260 A bivariate response model for studying the marks obtained in two jointly-dependent... In conclusion, the pdf given in (1) seems to be a suitable distribution to model the joint random variables corresponding to mathematics and statistics marks when the latter are influenced by the former. 3. Regression model and estimation Let us now consider a more realistic model, in which covariates are included. The linear regression model, which makes no distributional assumptions, is likely to be unsatisfactory because certain combinations of parameters and regressors could violate the nonnegative restriction and the upper limit on the mean. To avoid this situation we propose a parametric model based on using the distributional assumptions presented in the previous section. When a regression analysis is to be performed, it is often useful to model the mean of the response. By equating the mean given in (2) to μi(i=1,2), solving for ai(i=1,2), taking a3=θand replacing the resulting expression in the pdf of the bivariate beta distribution in (1), we obtain the following reparametrisation. f(y1,y2)= yφ1μ1−1 1yφ2μ2−1 2(1−y1)φ2−1(1−y2)φ1−1 B(φ1μ1,φ2μ2,θ)(1−y1y2)(1−μ1μ2)φ1φ2/θ,(7) where φi=θ/(1−μi),0<μi<1, i=1,2; with 0 <y1<1, 0 <y2<1andθ>0. Under this reparametrisation of the bivariate beta distribution, the marginal mean is E(Yi)=μi, for i=1,2. Now, let x x xT κi=(x1i,x2i,...,xpi)be a vector of the pcovariates associated with the ith observation. This is a vector of linearly independent regressors that are thought to determine (y1,y2).Fortheith observation, the model takes the form (Y1i,Y2i)∼BB(μ1i,μ2i,θ), μκi(x x xκi,β β βκ)≡μκi=exp(x x xT κiβ β βκ) 1+exp(x x xT κiβ β βκ),κ=1,2. Here, i=1,...,ndenotes the number of observations, x x xκidenotes a vector of p explanatory variables for the ith observation and β β βκ=(βκ1,...,βκp)T,κ=1,2, denotes the corresponding vectors of regression coefficients. It is clear that each variable Y1 and Y2may be influenced by different characteristics and variables. For this reason, the explanatory variables that are used to model each mean μκi, may not be the same. Furthermore, observe that the logit link assumed ensures that μκifalls within the interval (0,1). Emilio G´ omez-D´ eniz, Nancy D´ avila C´ ardenes and Mar´ıa D. Garc´ıa Artiles 261 Under this model the log-likelihood function takes the form given in the Appendix, which shows the equations used to provide the estimates of the parameters. The above model presents the advantage of simplicity; on the other hand, the normal equations require the use of the digamma function, ψ(z)= d dz log(Γ(z)),z>0, in order to estimate all the model parameters. However, this problem is overcome by means of Mathematica routines (see Wolfram, 2003) and RATS (see Brooks, 2009), which work well with this special function. Because the equations which provide the estimates of the parameters cannot be solved explicitly, they must be addressed either by numerical methods or by directly maximising the log-likelihood function; in this study, the latter approach is adopted. Since the global maximum of the log-likelihood surface is not guaranteed, different initial values of the parametric space can be considered as seed points. In this sense, we have used the FindMaximum function of the Mathematica software package v.11.0 (Wolfram, 2003). Moreover, other methods provided by Mathematica, such as Newton, PrincipalAxis and QuasiNewton (all of which are available in Mathematica) obtain the same result. Finally, the standard errors of the estimated parameter are approximated by inverting the Hessian matrix. This can also be done by approximating the Hessian matrix and recovering it from the Cholesky factors. These parameters were also computed by the RATS package, and the same values were obtained. 3.1. Marginal effects The marginal effect reflects the variation of the conditional mean produced by a oneunit change in the jth covariate (j=1,...,p). The marginal effect can be calculated as δj= ∂ μκi ∂ xji =βκjμκi(1−μκi),κ=1,2; i=1,...,n;j=1,...,p. Thus, the marginal effect indicates that a one-unit change in the jth regressor increases or decreases the expectation of marks for the jth covariate by δjunits, j=1,...,p. This expression is the one normally obtained under the logit marginal effect. For indicator variables which takes only the values 0 or 1 the marginal effect is δj=E(yκ|xji =1)/E(yκ|xji =0)≈ exp(βκj),κ=1,2; i=1,...,n;j=1,...,p. Therefore, the conditional mean is exp(βκj) times larger if the indicator variable is one rather than zero. 3.2. Simulation study We now present some simulation results, obtained by a bootstrap experiment, to study the behaviour of the maximum likelihood estimators. The Mathematica package was used to create random variables from the pdf (7). In this process, the first component of the vector was generated from a marginal, and then a second one from a conditional distribution. The estimated values of the parameters were then computed directly using the FindMaximum function of Mathematica v.11.0 (Wolfram (2003)). The following sets of model parameters were considered: 262 A bivariate response model for studying the marks obtained in two jointly-dependent... Table 1: Average estimates (first row), the square root of the mean squared errors (second row in parenthesis) and the correlation (ρ) between estimated parameters based on 1000 replications. nμ1=0.15 μ2=0.25 θ=0.85 ρ(μ1,μ2)ρ(μ1,θ)ρ(μ2,θ) 25 0.1402 0.2213 0.9827 0.7524 –0.7481 –0.6804 (0.0367) (0.0585) (0.4490) 50 0.1932 0.3004 0.7256 0.6425 –0.8068 –0.7392 (0.0303) (0.0438) (0.1364) 75 0.1795 0.2537 0.7326 0.6009 –0.7757 –0.7566 (0.0246) (0.0310) (0.1116) 100 0.1526 0.2820 0.7292 0.4851 –0.7637 –0.5173 (0.0167) (0.0269) (0.0842) μ1=0.25 μ2=0.75 θ=0.5ρ(μ1,μ2)ρ(μ1,θ)ρ(μ2,θ) 25 0.2308 0.7273 0.6525 0.4976 –0.7454 –0.5647 (0.0558) (0.0347) (0.1829) 50 0.2542 0.7529 0.4906 0.5719 –0.7883 –0.5647 (0.0375) (0.0323) (0.0998) 75 0.2395 0.7190 0.5886 0.4495 –0.7168 –0.3814 (0.0326) (0.0274) (0.0715) 100 0.2992 0.7943 0.4170 0.4116 –0.7040 –0.3814 (0.0298) (0.0180) (0.0438) μ1=0.50 μ2=0.15 θ=1.50 ρ(μ1,μ2)ρ(μ1,θ)ρ(μ2,θ) 25 0.5210 0.1273 1.9765 0.1815 0.0217 –0.7145 (0.0484) (0.0268) (0.4869) 50 0.5713 0.1603 1.4338 0.4822 –0.4643 –0.5710 (0.0332) (0.0249) (0.1888) 75 0.5289 0.1600 1.3432 0.1993 –0.4655 –0.5616 (0.0293) (0.0199) (0.1618) 100 0.5063 0.1851 1.5894 0.5823 –0.6167 –0.6597 (0.0240) (0.0198) (0.2168) (μ1,μ2,θ)=(0.15,0.25,0.85), (μ1,μ2,θ)=(0.25,0.75,0.50), (μ1,μ2,θ)=(0.50,0.15,1.50). In all three cases, we have simulated observations with a sample size given by n=25,50,75 and 100. We report the average estimates and the square root of the mean squared errors based on 1000 replications, i.e. the bootstrap sample is taken from the original by using sampling with replacement 1000 times. Additional replications are considered unnecessary, as the computational time needed would be prohibitive; nevertheless, we acknowledge that the use of fewer replications might reduce the statistical accuracy obtained. The results are shown in Table 1. In general, as the sample size increases the estimates approach the true values and the biases and the mean squared er- Emilio G´ omez-D´ eniz, Nancy D´ avila C´ ardenes and Mar´ıa D. Garc´ıa Artiles 263                              Figure 1: Box-and-whisker charts showing the differences between the true parameter values and the estimates based on the data in Table 1. rors decrease. These outcomes corroborate the consistency of the maximum likelihood estimates. From the standard errors obtained, it is evident that the errors are smaller as the sample size increases. Furthermore, the correlation between the parameters is always positive for μ1and μ2and negative for μ2and θ. Hence, the correlation between these two sets of parameters is not very high. Finally, Figure 1 shows that the parameters estimated have a slight negative bias, which is more apparent in the θparameter. 4. Factors affecting the mathematics and statistics marks obtained In order to make use of the bivariate regression model, we examined the relation between the marks achieved by the students in two course subjects: Mathematics and Statistics in Business Administration. Most of these students, before entering the university had studied subjects focused on statistics, more so than basic mathematics. In fact, many of them believed they did not need mathematics and did not consider the two courses to be related. During the first term, difficulties were encountered in mathematics, but with the start of the statistics class, in the second term, the students believed their performance would be better. Therefore, at the beginning of the mathematics course, the students were informed of the analysis that would be conducted, and were asked to complete a questionnaire on this subject. The following section describes how the data were compiled and how many students comprised the final study sample. 4.1. The sample The data for this study were collected from eight student groups in the Mathematics for Business and Statistics modules taught during the first year of the Business Administration degree course at the University of Las Palmas de Gran Canaria (Spain). The study population was initially composed of 725 students enrolled in these groups. On the first day of classes in 2013, a questionnaire was handed out to 456 students. The final sample was composed of the 213 students who completed both modules (mathematics and statistics) and answered the survey. The questionnaire was divided 270 A bivariate response model for studying the marks obtained in two jointly-dependent... are the same as in the univariate model, AGE and ADMSCORE (see Table 6). In both cases, the marginal effect is positive. Thus, the older the students and the better their admission score, the higher the marks obtained in the Mathematics for Business course. However, for the AGE-SQUARED variable, the marginal effect is negative. The age factor may have a positive effect on the students who are retaking the course, due to the knowledge acquired from the previous year, in the case of those whose age is close to that of the non-retakers (i.e. 18 years). On the other hand, when the AGE-SQUARED variable is considered, the students’ additional age has a negative effect. We believe this is because older students have much greater difficulty in understanding the course contents. The same effects of the covariates were observed with respect to the statistics course. In the latter case, however, a further variable, NEWCOMER, was significantly present in the bivariate model, with a negative marginal effect. It may be relevant that the new students, before starting university studies, took a course focused on statistics, although not on calculus; however, this background does not seem to have any positive impact on their later performance. Table 5: Details for univariate fitted models including covariates. MATHEMATICS Personal and academic information Variable Coeff Std Error |t|-Stat p-value AGE 0.47428300 0.24773300 1.91450000 0.05695610 AGE-SQUARED –0.00859712 0.00535907 1.60422000 0.11021300 ADMSCORE 0.47627700 0.12479800 3.81639000 0.00017952 Skill in Mathematics Variable Coeff Std Error |t|-Stat p-value SYSTEM 0.35490300 0.12840300 2.76398000 0.00623282 DERRATEXP 0.46429600 0.18433700 2.51874000 0.01254520 SIMPLIFYDER 0.58471300 0.24651500 2.37192000 0.01862630 STATISTICS Personal and academic information Variable Coeff Std Error |t|-Stat p-value AGE 0.31052900 0.23665300 1.31217000 0.19091700 AGE–SQUARED –0.00528830 0.00513981 1.02889000 0.30473200 ADMSCORE 0.34814600 0.12009600 2.89889000 0.00414850 Other parameters Constant for Mathematics –6.80259000 2.81041000 2.42050000 0.01637510 Constant for Statistics –4.65696000 2.66416000 1.74800000 0.08194620 θfor Mathematics 4.88422000 0.43697500 11.17730000 0.00000000 θfor Statistics 5.06500000 0.45354600 11.16760000 0.00000000 Value of the AIC for Mathematics: –94.917 Value of the AIC for Statistics: –101.721 Emilio G´ omez-D´ eniz, Nancy D´ avila C´ ardenes and Mar´ıa D. Garc´ıa Artiles 271 Table 6: Details for bivariate fitted model including covariates. MATHEMATICS Personal and academic information Variable Coeff Std Error |t|-Stat p-value δj AGE 0.228201842 0.183687046 1.24234 0.21411099 1.256 AGE-SQUARED –0.003363921 0.004001199 0.84073 0.40050008 0.996 ADMSCORE 0.434494686 0.090404187 4.80613 0.00000154 1.544 Skill in Mathematics Variable Coeff Std Error |t|-Stat p-value δj FRACTIONS 0.254534936 0.089643529 2.83941 0.00451967 1.289 SYSTEM 0.412448241 0.124542939 3.31170 0.00092733 1.510 EQ2 –0.365876393 0.124355071 2.94219 0.00325899 0.693 SIMPLIFYEXP 0.421013978 0.110824320 3.79893 0.00014532 1.523 DERRATEXP 0.299258577 0.130434786 2.29432 0.02177237 1.348 SIMPLIFYDER 0.434442994 0.159227590 2.72844 0.00636346 1.544 STATISTICS Personal and academic information Variable Coeff Std Error |t|-Stat p-value δj AGE 0.152238309 0.210965922 0.72163 0.47052499 1.164 AGE-SQUARED –0.001916838 0.004500367 –0.42593 0.67015944 0.998 ADMSCORE 0.375708496 0.099697582 3.76848 0.00016424 1.456 FRESHMEN –0.357923647 0.108415456 3.30141 0.00096201 0.700 Skill in Mathematics Variable Coeff Std Error |t|-Stat p-value δj DERINTEXP 0.205826943 0.089275677 2.30552 0.02113741 1.228 Other parameters θ3.164211826 0.199656264 15.84830 0.00000000 Constant for Mathematics –3.989088242 2.067120538 1.929780 0.05363408 Constant for Statistics –2.720631590 2.433508707 1.117990 0.26357246 Value of the AIC: –285.820 In the bivariate model, with respect to mathematics skills, some of the variables observed in the univariate model were again found to be relevant; in addition, the folowing new ones appeared: FRACTIONS, SYSTEM, EQ2, SIMPLIFYEXP, DERRATEXP and SIMPLIFYDER. In every case, the marginal effects were positive. Thus, when students are competent with the basic algebra of rational expressions, they are more likely to obtain higher marks in mathematics. The same is true when they can correctly apply a method for resolving a linear equations system to generate a quadratic equation to be solved. Another factor that appears to be significant is the ability to simplify algebraic expressions, to derive polynomial functions with rational exponents and to sim- 272 A bivariate response model for studying the marks obtained in two jointly-dependent... plify the expression of the derivative function obtained. For these covariates, the positive marginal effects mean that the students’ marks increase when they are able to correctly complete the exercises in question. However, the corresponding results for the statistics course show that the only significant factor was the covariate defining whether the students were capable of determining the derivative of a polynomial expression with integer exponents. Success in this task was also associated with higher marks in the subject, possibly because this type of expression appears in some elements of the statistics course. 6. Results and conclusions As part of the Business Administration degree offered by the University of Las Palmas de Gran Canaria (Spain), a Mathematics for Business course is taught in the first term of the first year; this is followed by a course focusing on applied statistics in social sciences. In view of the obvious connection between these two courses, we decided to analyse the relationship between the marks obtained in each course and to determine which covariates might affect these marks. Accordingly, we considered a flexible bivariate regression model to be applied when the dependent variables are bounded and the marginal means are functions of the explanatory variables. This model was applied to study the personal and academic factors relevant to the students in our study sample and the basic mathematical skills that may affect the marks obtained in the above-mentioned courses (mathematics and statistics). In our opinion, the model proposed is competitive with that presented by Cepeda-Cuervo et al. (2014), who generated a bivariate beta regression model from copulas evaluating it using a Bayesian methodology. As in our own case, the marginal distributions of the latter model were beta, but these authors assumed a weak dependence between the variables of interest, which was modelled by a Farlie-Gumbel-Morgenstern copula function. The model we propose has fewer parameters and therefore is simpler. The results obtained in the present analysis show that the mean value of the marks obtained increases with the age of the students, in both courses. Specifically, the students who were born before 1995 had higher marks both for mathematics and for statistics. We interpret this finding as follows: some of the students in the final sample had been enrolled in the same mathematics course the previous year, and so they were not newcomers to the subject. Indeed, some had taken remedial courses, or had transferred from other undergraduate studies. Thus, following an initial lack of success, these students subsequently acquired mathematics skills enabling them to achieve better marks in the subject. With regard to the admission score variable, this too was significant for both subjects, with a positive marginal effect. Thus, the higher the admission mark the better the marks obtained for mathematics and statistics. In this respect, obviously, the best students were most likely to achieve the best marks in mathematics and statistics. Emilio G´ omez-D´ eniz, Nancy D´ avila C´ ardenes and Mar´ıa D. Garc´ıa Artiles 273 Among the other variables related to personal information, another relevant factor was whether the students were newcomers, i.e. studying these subjects for the first time. Nevertheless, this variable was only significant for the statistics subject, which probably reflects the background acquired in this respect in the Social Sciences track studied at high school. Finally, with regard to the influence of mathematical skills on the marks obtained for statistics, only the variable related to obtaining the derivative of polynomial expressions with integer exponents was found to be significant. It is striking that no other mathematical ability affected the marks for statistics. This might be because the basic statistics course in question is mainly descriptive, merely introducing the main concepts; consequently, most of the students were already acquainted with these concepts having opted for the Social Science track at high school. Despite these considerations, however, the marks obtained for statistics and the success rate in this course were even worse than for the business mathematics course. In the light of the results obtained, we conclude that the bivariate beta regression model is more suitable than the univariate model for the analysis described in this paper. Acknowledgements The authors would like to express their gratitude to the Associate Editor and to an anonymous reviewer for their valuable comments and suggestions. EGD was partially funded by grant ECO2013-47092 (Ministry of Economy and Competitiveness, Spain). Appendix We present the equations needed to perform the estimation using the maximum likelihood method when covariates are introduced into the model. Consider a sample consisting of nobservations (˜y1,˜y2)={(y11,y21),...,(y1n,y2n)}, taken from the probability function (7). The log-likelihood is given by ≡(θ,β1,β2;(˜y1,˜y2)) = n  i=1 [(φ2i−1)log(1−y1i)+(φ1i−1)log(1−y2i) +(φ1iμ1i−1)logy1i+(φ2iμ2i−1)logy2i −φ1iφ2i θ(1−μ1iμ2i)log(1−y1iy2i) −logB(φ1iμ1i,φ2iμ2i,θ)],(8) where φκi=θ/(1−μκi),κ=1,2. 274 A bivariate response model for studying the marks obtained in two jointly-dependent... From straightforward computation, we have ∂ μκi ∂ βκj =μκixκj, ∂ φκi ∂ βκj =1 θ(φκiμκixκj)2, from which we obtain the first partial derivatives of the log-likelihood function (8) with respect to θand βκj(κ=1,2,j=1,...,p),givenby ∂  ∂ θ= n  i=1log(1−y1i) 1−μ2i +log(1−y2i) 1−μ2i +μ1ilogy1i 1−μ1i +μ2ilogy2i 1−μ2i +φ1iφ2i θ2(1−μ1iμ2i)log(1−y1iy2i)−ψ(θ)−ψ(θ+2 κ=1φκiμκi) B(φ1iμ1i,φ2iμ2i,θ), ∂  ∂ β1j = n  i=1 μ1ix1jμ2i−φ1iμ1ix1j θ(1−μ1iμ2i)φ1iφ2i θlog(1−y1iy2i) +1+φ1i θμ2 1ix1jφ1ilogy1i+φ2 1iμ1ix1j θlog(1−y2i) +μ1iφ1ix1j B(φ1iμ1i,φ2iμ2i,θ)ψ(μ1iφ1i)−ψ(θ+ 2  κ=1 φκiμκi), ∂  ∂ β2j = n  i=1 μ2ix2jμ1i−φ2iμ2ix2j θ(1−μ1iμ2i)φ1iφ2i θlog(1−y1iy2i) +1+φ2i θμ2 2ix2jφ2ilogy2i+φ2 2iμ2ix2j θlog(1−y1i) +μ2iφ2ix2j B(φ1iμ1i,φ2iμ2i,θ)ψ(μ2iφ2i)−ψ(θ+ 2  κ=1 φκiμκi), where j=1,...,p. By equating these 2p+1 equations to zero and then solving, we obtain the maximum likelihood estimates of the model parameters. Emilio G´ omez-D´ eniz, Nancy D´ avila C´ ardenes and Mar´ıa D. 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