Model Overview and Parameter Estimation

The forward selection procedure results in a multinomial logistic regression model fitted using the MNLogit() function. Although the model matrix $X$ contains eight columns, the fitted model reports only five parameter estimates. This discrepancy arises because three parameters are aliased due to linear dependence among categorical indicators.

The estimated parameters returned by the model are:

  • Intercept: $-2.3409513$
  • Scholarship holder_No: $1.3392169$
  • Debtor_No: $-1.3240921$
  • Age at enrollment: $0.0610271$
  • Attendance_Daytime: $0.2436942$

The missing parameters correspond to Scholarship holder_Yes, Debtor_Yes, and Attendance_Evening. These parameters are aliased and are therefore assigned values of zero by convention. While the presence of aliased parameters implies that coefficient estimates are not unique, an important theoretical result guarantees that predicted probabilities remain unchanged and reliable.

The complete parameter set is therefore:

  • Intercept: $-2.3409513$
  • Scholarship holder_No: $1.3392169$
  • Scholarship holder_Yes: $0$ (aliased)
  • Debtor_No: $-1.3240921$
  • Debtor_Yes: $0$ (aliased)
  • Age at enrollment: $0.0610271$
  • Attendance_Daytime: $0.2436942$
  • Attendance_Evening: $0$ (aliased)

Variable Definitions and Indicator Coding

For interpretation, the response variable $Y$ denotes Dropout. The predictors are defined as follows:

  • $A$: Attendance
  • $D$: Debtor status
  • $S$: Scholarship holder status
  • $G$: Age at enrollment

Indicator variables are defined in a standard dummy-variable framework:

  • Attendance:
    $A=\text{Daytime}=1$ if attendance is daytime, otherwise $0$
    $A=\text{Evening}=1$ if attendance is evening, otherwise $0$
  • Debtor:
    $D=\text{No}=1$ if not a debtor, otherwise $0$
    $D=\text{Yes}=1$ if a debtor, otherwise $0$
  • Scholarship holder:
    $S=\text{No}=1$ if not a scholarship holder, otherwise $0$
    $S=\text{Yes}=1$ if a scholarship holder, otherwise $0$

Logistic Regression Model Formulation

The logistic regression model expresses the log-odds of dropout as a linear combination of predictors:

$\log\left(\frac{p_1}{1-p_1}\right)=\beta_0+\beta_{S=\text{No}}S_{\text{No}}+\beta_{S=\text{Yes}}S_{\text{Yes}}+\beta_{D=\text{No}}D_{\text{No}}+\beta_{D=\text{Yes}}D_{\text{Yes}}+\beta_G G+\beta_{A=\text{Daytime}}A_{\text{Daytime}}+\beta_{A=\text{Evening}}A_{\text{Evening}}$

Because aliased parameters are set to zero, the effective contributions arise only from the non-aliased terms.

Attendance Mode Comparison via Odds Ratios

Two scenarios are considered to isolate the effect of attendance mode:

Scenario I:
Attendance is Daytime, Debtor = No, Scholarship holder = Yes, Age = 18

Scenario II:
Attendance is Evening, Debtor = No, Scholarship holder = Yes, Age = 18

The difference in log-odds between these two scenarios is:

$\log\left(\frac{p_1^{(I)}/(1-p_1^{(I)})}{p_1^{(II)}/(1-p_1^{(II)})}\right)=\beta_{A=\text{Daytime}}-\beta_{A=\text{Evening}}$

Using the estimated coefficients, this becomes:

$0.2436942-0=0.2436942$

Exponentiating yields the odds ratio:

$\exp(0.2436942)=1.28$

This indicates that, holding all other factors constant, attending daytime classes increases the odds of dropping out by a factor of 1.28 relative to evening attendance. From an advising perspective, evening attendance is therefore preferable for reducing dropout risk in this profile.

Age at Enrollment Comparison via Odds Ratios

Next, the effect of enrollment age is examined:

Scenario I: Age = 18
Scenario II: Age = 25

All other characteristics remain identical. The difference in log-odds is:

$\log\left(\frac{p_1^{(II)}/(1-p_1^{(II)})}{p_1^{(I)}/(1-p_1^{(I)})}\right)=7\beta_G$

With $\beta_G=0.0610271$, the change in log-odds is:

$7\times0.0610271=0.427190$

The corresponding odds ratio is:

$\exp(0.427190)=1.53$

Thus, deferring enrollment by seven years increases the odds of dropout by a factor of 1.53, suggesting that earlier enrollment is advisable for retention.

Model Assessment: Measures of Association

To evaluate model performance, several pseudo–$R^2$ measures are used. Let $L_M$ and $l_M=\log L_M$ denote the likelihood and log-likelihood of the fitted model, and let $L_0$ and $l_0$ denote those of the intercept-only model, both trained on the same $n$ observations.

McFadden’s $R^2$ is defined as:

$R_{MF}^2=1-\frac{l_M}{l_0}$

Cox–Snell $R^2$ is defined as:

$R_{CS}^2=1-\left(\frac{L_0}{L_M}\right)^{2/n}$

Nagelkerke $R^2$ rescales Cox–Snell’s measure:

$R_N^2=\frac{R_{CS}^2}{1-\left(L_0\right)^{2/n}}$

Tjur’s $D$ (Coefficient of Discrimination) is defined as the difference between the mean predicted event probability among event observations and non-event observations:

$D=\frac{1}{n_1}\sum_{i\in S_1}\hat{p}i^{(1)}-\frac{1}{n_0}\sum{i\in S_0}\hat{p}_i^{(1)}$

where $S_1$ and $S_0$ index event and non-event observations respectively.

Empirical Results

The computed values for the college student retention model are:

  • McFadden: $0.1199$
  • Cox–Snell: $0.1397$
  • Nagelkerke: $0.1954$
  • Tjur: $0.1452$

All measures indicate that the model provides a solid initial fit, capturing meaningful associations between predictors and dropout risk. At the same time, the magnitude of these values suggests substantial room for further refinement through additional predictors, feature engineering, or alternative model specifications.

Concluding Remarks

This forward-selection logistic regression model demonstrates how categorical and continuous predictors jointly influence dropout behavior. Aliased parameters do not compromise predictive validity, odds ratios provide clear interpretability, and multiple association measures offer complementary perspectives on model performance. Together, these elements form a coherent framework for both explanation and assessment in applied retention analytics.