paper_CR_usp
Database
The database consists of students who entered the University of São Paulo (USP) in 2010 in the Statistics, Physics, and Mathematics programs, totaling 458 students. The distribution among the programs was 82 students in Statistics, 163 in Physics, and 213 in Mathematics.
Among the enrolled students, approximately 28.17% were female and 71.83% were male. The average age at admission was approximately 24.3 years, with a standard deviation of 7.55 years. The first quartile (Q1) was 19 years and the third quartile (Q3) was 26 years, indicating that half of the students were between 19 and 26 years old. The minimum recorded age was 17 years, while the maximum was 73 years, showing considerable heterogeneity in the age distribution of the students.
Regarding the study shift, most students were enrolled in evening programs, with 246 students, representing approximately 72.9% of the sample.
Students’ Status in 2017
In 2017, the students’ academic status was classified as follows:
- Active: 99 students remained enrolled, representing approximately 21.6% of the total sample.
- Graduated: 106 students completed their programs, corresponding to approximately 23.1%.
- Dropped out: 253 students left the university without completing their programs, representing 55.2% of the total.
These results indicate that more than half of the students left their programs without graduation, while only around one-quarter completed their degrees. The proportion of active students in 2017 also suggests that a relevant fraction of the initial cohort had not yet completed their academic trajectory within the observed period.
The highest dropout percentages occurred during the first four years after admission. Graduation rates remained low or close to zero in the initial years and started to increase more noticeably around the fourth year, which is consistent with the expected duration of the programs, as students generally need to complete a sequence of required courses before becoming eligible for graduation. Exceptions may occur for students with previous academic experience, such as those entering through transfer or readmission processes.
## # A tibble: 3 × 3
## SITUACAO2017_1 count percentage
## <chr> <int> <dbl>
## 1 Active 99 21.6
## 2 Dropped out 253 55.2
## 3 Graduated 106 23.1
Cumulative Incidence Function (CIF)
The Cumulative Incidence Function (CIF) represents the cumulative probability of experiencing an event due to cause \(k\) up to time \(t\), in the presence of competing risks. It is defined as
\[ F_k(t) = P(T \leq t, C = k) \]
and can be estimated by
\[ \widehat{F}_k(t) = \sum_{u_i \leq t} \widehat{S}(u_i^-) \frac{dN_k(u_i)}{n(u_i)}. \]
Here, \(\widehat{S}(u_i^-)\) is the estimated overall survival probability immediately before time \(u_i\), \(dN_k(u_i)\) is the number of events due to cause \(k\) at time \(u_i\), and \(n(u_i)\) is the number of individuals at risk at that time.
The CIFs were estimated using the cmprsk package in R
and were used to describe the cumulative probabilities of Graduation and
Dropout over time, both overall and according to the covariates
considered in the study.
Gray’s Test
The Gray’s test is used to evaluate whether the cumulative incidence functions (CIFs) are statistically different among groups in the presence of competing risks. This test compares the incidence curves of a specific event across the categories of a covariate, considering that other competing events may occur. The null hypothesis assumes that the CIFs are equal among the groups, while the alternative hypothesis assumes that at least one group presents a different CIF.
For example, considering the variable race and the event Graduation:
Null hypothesis (\(H_0\)):
The cumulative incidence functions of graduation are equal between white
and non-white students:
\[ H_0: CIF_{White}(t)=CIF_{Non-white}(t) \]
Alternative hypothesis (\(H_1\)):
The cumulative incidence functions of graduation are different between
white and non-white students:
\[ H_1: CIF_{White}(t)\neq CIF_{Non-white}(t) \]
A significant p-value indicates evidence that the CIF curves differ among the groups, suggesting that the covariate is associated with different probabilities of experiencing the event over time.
Overall CIF
CIF by Gender
## stat pv df
## 1 8.274424 0.004020741 1
## 2 5.906479 0.015085292 1CIF by Race
## stat pv df
## 1 11.044326 0.0008895906 1
## 2 5.849172 0.0155843834 1CIF by Age
## stat pv df
## 1 9.351636 0.002227865 1
## 2 6.993058 0.008182645 1CIF by Shift
## stat pv df
## 1 6.093203 0.01357029 1
## 2 3.327201 0.06814276 1CIF by Program
## stat pv df
## 1 11.160124 0.003772332 2
## 2 4.302244 0.116353556 2The results presented above show that, for most covariates, the cumulative incidence curves differ at the 10% significance level (p-value < 0.10), indicating differences in the probability of experiencing graduation or dropout over time among the analyzed groups.
The only exception was the comparison of dropout curves among the programs, for which no statistically significant difference was observed (p-value = 0.1163). However, from a descriptive perspective, differences can still be observed. Throughout the follow-up period, the dropout incidence for the Statistics program remained lower than that observed for Physics and Mathematics. Additionally, after the fourth year, the dropout incidence for Mathematics remained below that of Physics.
Overall, the results indicate that female students (particularly after four years of follow-up), non-white students, younger students (age below the median, also mainly after four years of follow-up), students enrolled in non-evening programs, and students from the Statistics and Physics programs present higher graduation incidence (represented by the black curves). In contrast, male students (especially after two years of follow-up), white students, older students, students enrolled in evening programs, and students from the Physics and Mathematics programs show higher dropout incidence (represented by the red curves).
These interpretations provide a general overview of the differences observed among groups. More detailed analyses may be performed by examining specific periods of CIF, allowing the identification of particular patterns at the beginning, middle, or end of the follow-up period for each covariate of interest.