Factors that Influence Final Math Grades

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1 Introduction

This data set was obtained from [Data World] https://data.world.com. The data consists of information on students gathered from two different schools in Portugal about students habits and lives outside of school to see what impact these external factors might have on their final grade in mathematics. The data was collected through school surveys and questionnaires.

1.1 Variable Description

• school(x1) - student’s school (binary: ‘GP’ - Gabriel Pereira or ‘MS’ - Mousinho da Silveira)
• sex(x2) - student’s sex (binary: ‘F’ - female or ‘M’ - male)
• age(x3) - student’s age
• address(x4) - student’s home address type (binary: ‘U’ - urban or ‘R’ - rural)
• famsize(x5) - family size
• Pstatus(x6) - parent’s cohabitation status (binary: ‘T’ - living together or ‘A’ - apart)
• Medu(x7) - mother’s education (numeric: 0 - none, 1 - primary education (4th grade), 2 – 5th to 9th grade, 3 – secondary education or 4 – higher education)
• Fedu(x8) - father’s education
• Mjob(x9) - mother’s job (nominal: ‘teacher’, ‘health’ care related, civil ‘services’ (e.g. administrative or police), ‘at_home’ or ‘other’)
• Fjob(x10) - father’s job
• reason - reason to choose this school (nominal: close to ‘home’, school ‘reputation’, ‘course’ preference or ‘other’)
• guardian(x11) - student’s guardian (nominal: ‘mother’, ‘father’ or ‘other’)
• traveltime(x12) - home to school travel time (numeric: 1 - <15 min., 2 - 15 to 30 min., 3 - 30 min. to 1 hour, or 4 - >1 hour)
• studytime(x13) - weekly study time (numeric: 1 - <2 hours, 2 - 2 to 5 hours, 3 - 5 to 10 hours, or 4 - >10 hours)
• failures(x14) - number of past class failures (numeric: n if 1<=n<3, else 4)
• schoolsup(x15) - extra educational support
• famsup(x16) - family educational support
• paid - extra paid classes within the course subject
• activities(x17) - extra-curricular activities
• nursery(x18) - attended nursery school
• higher(x19) - wants to take higher education
• internet(x20) - Internet access at home
• romantic(x21) - with a romantic relationship
• famrel(x22) - quality of family relationships (numeric: from 1 - very bad to 5 - excellent)
• freetime(x23) - free time after school (numeric: from 1 - very low to 5 - very high)
• goout(x24) - going out with friends (numeric: from 1 - very low to 5 - very high)
• Dalc(x25) - workday alcohol consumption (numeric: from 1 - very low to 5 - very high)
• Walc(x26) - weekend alcohol consumption (numeric: from 1 - very low to 5 - very high)
• health(x27) - current health status (numeric: from 1 - very bad to 5 - very good)
• absences(x28) - number of school absences
• G1 -(x29) first period grade (numeric: from 0 to 20)
• G2 -(x30) second period grade (numeric: from 0 to 20)
• G3(y) - final grade (numeric: from 0 to 20, output target)

1.2 Practical Question

The question trying to be answered throughout this data analysis is what the association is between students study habits and lives outside of school, and their final math grade.

2 Exploratory Data Analysis

First, the data is uploaded.

students0 <- read.csv("https://raw.githubusercontent.com/AvaDeSt/STA-321/refs/heads/main/student-mat.csv", header = TRUE)
students=read.table("https://raw.githubusercontent.com/AvaDeSt/STA-321/refs/heads/main/student-mat.csv",sep=";",header=TRUE)

model = lm(G3 ~ Medu + Fedu + traveltime +  failures + freetime +  famsize + goout + Walc + Dalc + famrel + absences + health + studytime , data = students)
kable(summary(model)$coef, caption ="Statistics of Regression Coefficients")

Statistics of Regression Coefficients

	Estimate	Std. Error	t value	Pr(>|t|)
(Intercept)	9.0797575	1.6801061	5.4042762	0.0000001
Medu	0.6241196	0.2549488	2.4480195	0.0148147
Fedu	-0.0559475	0.2543342	-0.2199761	0.8260075
traveltime	-0.4317401	0.3140260	-1.3748548	0.1699845
failures	-1.8833355	0.3065013	-6.1446257	0.0000000
freetime	0.3498435	0.2304954	1.5177890	0.1298969
famsizeLE3	0.8561716	0.4729041	1.8104548	0.0710129
goout	-0.6475661	0.2207275	-2.9337814	0.0035512
Walc	0.3367182	0.2381571	1.4138492	0.1582227
Dalc	-0.1982684	0.3192112	-0.6211197	0.5348923
famrel	0.2536250	0.2441219	1.0389275	0.2994973
absences	0.0240459	0.0271578	0.8854119	0.3764930
health	-0.1444087	0.1554869	-0.9287521	0.3536056
studytime	0.2688117	0.2685105	1.0011220	0.3174032

stepwise_model <- step(model, direction = "both")

Start:  AIC=1148.3
G3 ~ Medu + Fedu + traveltime + failures + freetime + famsize + 
    goout + Walc + Dalc + famrel + absences + health + studytime

             Df Sum of Sq    RSS    AIC
- Fedu        1      0.86 6736.0 1146.3
- Dalc        1      6.82 6742.0 1146.7
- absences    1     13.86 6749.0 1147.1
- health      1     15.25 6750.4 1147.2
- studytime   1     17.72 6752.9 1147.3
- famrel      1     19.08 6754.3 1147.4
- traveltime  1     33.41 6768.6 1148.3
<none>                    6735.2 1148.3
- Walc        1     35.34 6770.5 1148.4
- freetime    1     40.72 6775.9 1148.7
- famsize     1     57.94 6793.1 1149.7
- Medu        1    105.94 6841.1 1152.5
- goout       1    152.15 6887.3 1155.1
- failures    1    667.44 7402.6 1183.6

Step:  AIC=1146.35
G3 ~ Medu + traveltime + failures + freetime + famsize + goout + 
    Walc + Dalc + famrel + absences + health + studytime

             Df Sum of Sq    RSS    AIC
- Dalc        1      6.76 6742.8 1144.8
- absences    1     14.22 6750.2 1145.2
- health      1     15.72 6751.8 1145.3
- studytime   1     18.61 6754.6 1145.4
- famrel      1     19.09 6755.1 1145.5
- traveltime  1     32.84 6768.9 1146.3
<none>                    6736.0 1146.3
- Walc        1     35.06 6771.1 1146.4
- freetime    1     41.49 6777.5 1146.8
- famsize     1     58.73 6794.8 1147.8
+ Fedu        1      0.86 6735.2 1148.3
- Medu        1    146.08 6882.1 1152.8
- goout       1    152.85 6888.9 1153.2
- failures    1    674.80 7410.8 1182.1

Step:  AIC=1144.75
G3 ~ Medu + traveltime + failures + freetime + famsize + goout + 
    Walc + famrel + absences + health + studytime

             Df Sum of Sq    RSS    AIC
- absences    1     13.60 6756.4 1143.5
- health      1     16.06 6758.9 1143.7
- studytime   1     18.80 6761.6 1143.8
- famrel      1     19.65 6762.4 1143.9
- Walc        1     29.32 6772.1 1144.5
<none>                    6742.8 1144.8
- traveltime  1     35.55 6778.3 1144.8
- freetime    1     37.35 6780.1 1144.9
- famsize     1     57.05 6799.9 1146.1
+ Dalc        1      6.76 6736.0 1146.3
+ Fedu        1      0.80 6742.0 1146.7
- Medu        1    141.66 6884.5 1151.0
- goout       1    149.96 6892.8 1151.4
- failures    1    685.66 7428.5 1181.0

Step:  AIC=1143.55
G3 ~ Medu + traveltime + failures + freetime + famsize + goout + 
    Walc + famrel + health + studytime

             Df Sum of Sq    RSS    AIC
- health      1     17.20 6773.6 1142.5
- studytime   1     17.68 6774.1 1142.6
- famrel      1     19.50 6775.9 1142.7
- freetime    1     33.81 6790.2 1143.5
<none>                    6756.4 1143.5
- Walc        1     35.00 6791.4 1143.6
- traveltime  1     36.64 6793.0 1143.7
+ absences    1     13.60 6742.8 1144.8
- famsize     1     58.65 6815.0 1145.0
+ Dalc        1      6.15 6750.2 1145.2
+ Fedu        1      1.14 6755.3 1145.5
- goout       1    150.81 6907.2 1150.3
- Medu        1    155.07 6911.5 1150.5
- failures    1    674.71 7431.1 1179.2

Step:  AIC=1142.55
G3 ~ Medu + traveltime + failures + freetime + famsize + goout + 
    Walc + famrel + studytime

             Df Sum of Sq    RSS    AIC
- famrel      1     16.00 6789.6 1141.5
- studytime   1     19.12 6792.7 1141.7
- Walc        1     30.35 6804.0 1142.3
- freetime    1     31.28 6804.9 1142.4
<none>                    6773.6 1142.5
- traveltime  1     35.96 6809.6 1142.6
+ health      1     17.20 6756.4 1143.5
+ absences    1     14.75 6758.9 1143.7
- famsize     1     61.38 6835.0 1144.1
+ Dalc        1      6.46 6767.1 1144.2
+ Fedu        1      1.72 6771.9 1144.5
- goout       1    143.80 6917.4 1148.8
- Medu        1    157.93 6931.5 1149.7
- failures    1    685.65 7459.3 1178.6

Step:  AIC=1141.48
G3 ~ Medu + traveltime + failures + freetime + famsize + goout + 
    Walc + studytime

             Df Sum of Sq    RSS    AIC
- studytime   1     19.77 6809.4 1140.6
- Walc        1     24.72 6814.3 1140.9
<none>                    6789.6 1141.5
- traveltime  1     35.63 6825.2 1141.5
- freetime    1     39.24 6828.8 1141.8
+ famrel      1     16.00 6773.6 1142.5
+ absences    1     14.49 6775.1 1142.6
+ health      1     13.70 6775.9 1142.7
- famsize     1     60.63 6850.2 1143.0
+ Dalc        1      6.92 6782.7 1143.1
+ Fedu        1      1.66 6787.9 1143.4
- goout       1    136.40 6926.0 1147.3
- Medu        1    154.57 6944.2 1148.4
- failures    1    698.49 7488.1 1178.2

Step:  AIC=1140.63
G3 ~ Medu + traveltime + failures + freetime + famsize + goout + 
    Walc

             Df Sum of Sq    RSS    AIC
- Walc        1     16.68 6826.1 1139.6
- freetime    1     33.24 6842.6 1140.6
<none>                    6809.4 1140.6
- traveltime  1     38.91 6848.3 1140.9
+ studytime   1     19.77 6789.6 1141.5
+ famrel      1     16.65 6792.7 1141.7
+ health      1     14.94 6794.4 1141.8
+ absences    1     13.32 6796.1 1141.9
- famsize     1     57.44 6866.8 1142.0
+ Dalc        1      7.17 6802.2 1142.2
+ Fedu        1      2.88 6806.5 1142.5
- goout       1    128.72 6938.1 1146.0
- Medu        1    155.50 6964.9 1147.5
- failures    1    743.24 7552.6 1179.5

Step:  AIC=1139.6
G3 ~ Medu + traveltime + failures + freetime + famsize + goout

             Df Sum of Sq    RSS    AIC
- traveltime  1     33.72 6859.8 1139.5
<none>                    6826.1 1139.6
- freetime    1     34.72 6860.8 1139.6
+ absences    1     17.32 6808.7 1140.6
+ Walc        1     16.68 6809.4 1140.6
+ health      1     11.78 6814.3 1140.9
+ studytime   1     11.73 6814.3 1140.9
+ famrel      1     11.52 6814.5 1140.9
- famsize     1     64.17 6890.2 1141.3
+ Fedu        1      2.27 6823.8 1141.5
+ Dalc        1      0.11 6825.9 1141.6
- goout       1    112.20 6938.3 1144.0
- Medu        1    151.95 6978.0 1146.3
- failures    1    730.38 7556.4 1177.8

Step:  AIC=1139.54
G3 ~ Medu + failures + freetime + famsize + goout

             Df Sum of Sq    RSS    AIC
<none>                    6859.8 1139.5
+ traveltime  1     33.72 6826.1 1139.6
- freetime    1     36.72 6896.5 1139.7
+ absences    1     17.62 6842.2 1140.5
+ studytime   1     15.15 6844.6 1140.7
+ famrel      1     12.01 6847.8 1140.8
+ health      1     11.81 6848.0 1140.9
+ Walc        1     11.49 6848.3 1140.9
- famsize     1     59.08 6918.9 1140.9
+ Fedu        1      1.34 6858.4 1141.5
+ Dalc        1      0.21 6859.6 1141.5
- goout       1    117.06 6976.8 1144.2
- Medu        1    178.98 7038.8 1147.7
- failures    1    748.93 7608.7 1178.5

summary(stepwise_model)

Call:
lm(formula = G3 ~ Medu + failures + freetime + famsize + goout, 
    data = students)

Residuals:
     Min       1Q   Median       3Q      Max 
-12.5232  -2.0881   0.4697   2.7259   9.1755 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   9.6298     0.9771   9.855  < 2e-16 ***
Medu          0.6378     0.2002   3.186  0.00156 ** 
failures     -1.9333     0.2967  -6.517 2.23e-10 ***
freetime      0.3196     0.2215   1.443  0.14983    
famsizeLE3    0.8551     0.4672   1.830  0.06796 .  
goout        -0.5156     0.2001  -2.576  0.01035 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 4.199 on 389 degrees of freedom
Multiple R-squared:  0.1705,    Adjusted R-squared:  0.1599 
F-statistic: 15.99 on 5 and 389 DF,  p-value: 2.432e-14

Students.num <- select(students,"Medu", "failures", "freetime", "goout", "G3", "famsize")

I got rid of the variable “school” because which school the students go to is not really being taken into account since the data is being collected from only two schools that are in a similar location. I also got rid of the separate period grades because in this study, I am mostly looking at the final grade. I also got rid of age because that is just a demographic. I also did stepwise selection since some of the p-values for some of the variables were pretty high. This left me with the variables “Medu”, “failures”, “freetime”, “goout”, and “famsize”.

2.1 Full Model

full.model = lm(G3 ~ Medu +  failures + freetime +  famsize + goout, data = Students.num)
kable(summary(full.model)$coef, caption ="Statistics of Regression Coefficients")

Statistics of Regression Coefficients

	Estimate	Std. Error	t value	Pr(>|t|)
(Intercept)	9.6298289	0.9771261	9.855258	0.0000000
Medu	0.6377844	0.2001929	3.185849	0.0015598
failures	-1.9333273	0.2966648	-6.516875	0.0000000
freetime	0.3195798	0.2214706	1.442989	0.1498282
famsizeLE3	0.8551003	0.4671671	1.830395	0.0679558
goout	-0.5155712	0.2001114	-2.576420	0.0103508

par(mfrow=c(2,2))
plot(full.model)

The QQ plot indicates that the data is not quite a normal distribution. The variance of the data is also not constant and the data is clumped more together at the right side of the graph. There also appears to be one outlier to the far right in the lower right graph.

vif(full.model)

    Medu failures freetime  famsize    goout 
1.073128 1.087442 1.093401 1.003687 1.108887 

barplot(vif(full.model), main = "VIF Values", horiz = FALSE, col = "steelblue")

Since all of the VIF values are close to 1 and do not exceed 4, multicollinearity is not an issue.

2.2 Transformations

To help correct the non constant variance of the data, I am going to perform a boxcox transformation.

par(pty = "s", mfrow = c(2, 2), oma=c(.1,.1,.1,.1), mar=c(4, 0, 2, 0))
Students.num$G3_adjusted <- Students.num$G3 + 3

boxcox(G3_adjusted ~ Medu + freetime + famsize + goout + log(failures +1)  
       , data = Students.num, lambda = seq(0, 1, length = 10), 
       xlab=expression(paste(lambda, ": log-failures")))

boxcox(G3_adjusted ~ Medu + freetime + famsize + goout + failures  
       , data = Students.num, lambda = seq(0, 1, length = 10), 
       xlab=expression(paste(lambda, ": failures")))

boxcox(G3_adjusted ~ Medu + log(freetime) + famsize + goout + failures
       , data = Students.num, lambda = seq(0, 1, length = 10), 
       xlab=expression(paste(lambda, ": log-freetime")))

boxcox(G3_adjusted ~ Medu + freetime + famsize + goout + failures   
       , data = Students.num, lambda = seq(0, 1, length = 10), 
       xlab=expression(paste(lambda, ": freetime")))

In order to do the boxcox transformation, I had to change my response variable “G3” to be positive. Also, when taking the log of “failures” i also had to change that to be positive by adding 1.

2.3 Square Root Transformation

sqrt.G3.log.fa = lm((G3_adjusted)^0.5 ~ Medu + log(failures +1) + freetime + famsize + goout, data = Students.num)
kable(summary(sqrt.G3.log.fa)$coef, caption = "log-transformed model")

log-transformed model

	Estimate	Std. Error	t value	Pr(>|t|)
(Intercept)	3.4895320	0.1562355	22.335082	0.0000000
Medu	0.0884132	0.0319745	2.765116	0.0059612
log(failures + 1)	-0.6048948	0.0895496	-6.754855	0.0000000
freetime	0.0492631	0.0353184	1.394829	0.1638632
famsizeLE3	0.1408791	0.0745153	1.890607	0.0594198
goout	-0.0732050	0.0319050	-2.294469	0.0222956

par(mfrow = c(2,2))
plot(sqrt.G3.log.fa)

Similar to the boxcox model, I had to adjust G3 and “failures” to be positive.

2.4 Log Transformation

log.G3 = lm(log(G3_adjusted) ~ Medu + failures + freetime + famsize + goout , data = Students.num)
kable(summary(log.G3)$coef, caption = "log-transformed model")

log-transformed model

	Estimate	Std. Error	t value	Pr(>|t|)
(Intercept)	2.4138563	0.1097202	22.000113	0.0000000
Medu	0.0556047	0.0224794	2.473584	0.0138027
failures	-0.2110588	0.0333121	-6.335803	0.0000000
freetime	0.0338659	0.0248686	1.361792	0.1740516
famsizeLE3	0.0962690	0.0524576	1.835179	0.0672425
goout	-0.0426323	0.0224702	-1.897276	0.0585318

par(mfrow = c(2,2))
plot(log.G3)

## Goodness of Fit measures

select=function(m){ 
 e = m$resid                           
 n0 = length(e)                        
 SSE=(m$df)*(summary(m)$sigma)^2       
 R.sq=summary(m)$r.squared             
 R.adj=summary(m)$adj.r                
 MSE=(summary(m)$sigma)^2              
 Cp=(SSE/MSE)-(n0-2*(n0-m$df))         
 AIC=n0*log(SSE)-n0*log(n0)+2*(n0-m$df)          
 SBC=n0*log(SSE)-n0*log(n0)+(log(n0))*(n0-m$df)  
 X=model.matrix(m)                     
 H=X%*%solve(t(X)%*%X)%*%t(X)          
 d=e/(1-diag(H))                       
 PRESS=t(d)%*%d   
 tbl = as.data.frame(cbind(SSE=SSE, R.sq=R.sq, R.adj = R.adj, Cp = Cp, AIC = AIC, SBC = SBC, PRD = PRESS))
 names(tbl)=c("SSE", "R.sq", "R.adj", "Cp", "AIC", "SBC", "PRESS")
 tbl
}

output.sum = rbind(select(full.model), select(sqrt.G3.log.fa), select(log.G3))
row.names(output.sum) = c("full.model", "sqrt.G3.log.fa", "log.G3")
kable(output.sum, caption = "Goodness-of-fit Measures of Candidate Models")

Goodness-of-fit Measures of Candidate Models

	SSE	R.sq	R.adj	Cp	AIC	SBC	PRESS
full.model	6859.77432	0.1705139	0.1598521	6	1139.5449	1163.4182	7080.63373
sqrt.G3.log.fa	174.57636	0.1679182	0.1572231	6	-310.5268	-286.6535	180.35971
log.G3	86.49313	0.1470004	0.1360364	6	-587.9342	-564.0609	89.50556

The best model to use regarding this data set is the full model because it has the highest R^2 value. It therefore is best able to explain the variance for the final math grade of the students based on the explanatory variables.

2.5 Final Model

kable(summary(full.model)$coef, caption = "Inferential Statistics of Final Model")

Inferential Statistics of Final Model

	Estimate	Std. Error	t value	Pr(>|t|)
(Intercept)	9.6298289	0.9771261	9.855258	0.0000000
Medu	0.6377844	0.2001929	3.185849	0.0015598
failures	-1.9333273	0.2966648	-6.516875	0.0000000
freetime	0.3195798	0.2214706	1.442989	0.1498282
famsizeLE3	0.8551003	0.4671671	1.830395	0.0679558
goout	-0.5155712	0.2001114	-2.576420	0.0103508

3 Summary of the Model

The model can be written as the following:

G3 = 9.63 + (0.638)Medu - (1.933)failures + (0.320)freetime + (0.855)famsize - (0.516)goout

From this model, we can see that the variables mother’s education, amount of free time, and the size of the family positively impact the final grade a student receives in math. The number of class failures, and the amount of time spent going out typically cause the final grade in math to decrease. We can see from the model that when the amount of time going out increases by one point, the final math grade decreases by 0.51.

4 Conclusion and Discussion

To conclude, I only took into account the explanatory variables that were numeric. I also did stepwise regression to determine which variables were the most significant. This left five numeric variables. Each model contains the same number of variables. Even though it ended up being the best model, the full model had several violations. The variance of the model was not constant, and the box cox transformation was done in order to try to correct this. The data was also not normal, this violation remains uncorrected. There also appears to be an outlier in the data. Despite the full model having the highest R-squared value, it is still fairly low at only 0.17. An explanation for this can be that there is a model better suited for this data that was not included in this project. There also may have been better explanatory variables selected that would better explain why a student might have a certain math grade.