To ensure the data structure conformed to the machine learning formats, a preprocessing analysis was carried out. In order to ensure that there was no vital information unnecessarily got lost, mean imputation was performed for numeric features to address the missing data values while mode imputation was used for categorical features. Capping off the most extreme data values, categorical encoding, erasing all the duplicates was done before scaling the continuous variables into a zero to one range using normalization techniques so as to accommodate most of the ML algorithms.

Logistic regression method for classification was used Such that the dichotomous response variable of whether the applicant’s status is non-defaulter (0) or defaulter (1) such that; by letting the independent design matrix be X, then the response variable y with the binary values was:

$Y=\{\begin{array}{ll}0\hfill & \text{if non-defaulter}\hfill \\ 1\hfill & \text{if a defaulter}\hfill \end{array}$

Then the $\log \left(\text{Odds}\left(\text{defaulter}\right)\right)=\text{logit}\pi$ such that:

$\log \frac{\pi }{1-\pi }={\beta }_0+{\beta }_1{x}_1+\cdots +{\beta }_p{x}_p$ (5.1)

where $\pi$ is the likelihood of the credit risk score status of a loan applicant. The sigmoid function is most appropriate for performing logistic regression because it maps the linear combination of input features to a probability. The sigmoid function is given by Equation (5.2).

$S\left(x\right)=\frac{1}{1+{\text{e}}^{-x}}$ (5.2)

where $x$ is the input variable such that; As $x\to \infty$, the function approaches 1 and as $x\to -\infty$, the function approaches 0 resulting in the sigmoid shape illustrated in Figure 2 . The sigmoid function takes any real input log-odds and gives output probability. Such that $\sigma :ℝ\to \left(0,1\right)$. where x is a linear function of multiple explanatory variables such that

$S\left(x\right)=p\left(x\right)=\frac{1}{1+{\text{e}}^{{\beta }_0+{\beta }_i{x}_i}}$

In the logistic model, $p\left(x\right)$ is interpreted as the probability of a binary response variable $y$ whereby, the normal threshhold for the event being successful ranges between 0.5 to 1, otherwise it is deemed not successful. That is in this case, the probability of failing to pay will range between 0.5 to 1. Next, the log odds inverse is explained as

$\text{logit}p\left(x\right)=\ln \frac{p\left(x\right)}{1-p\left(x\right)}={\beta }_0+{\beta }_i{x}_i$ (5.3)

Then equivalently after exponentiation in both sides, we get the odds such that:

$\frac{p\left(x\right)}{1-p\left(x\right)}={\text{e}}^{{\beta }_0+{\beta }_i{x}_i}$ (5.4)

where $p\left(x\right)$ is the probability that the loan is not paid given some linear combination of the features. ${\beta }_0$ is the intercept from linear regression when the numeric feature is equal to zero and the categorical feature is equal to the referenced feature. The study used both the regression coefficients and the odds ratio to establish the feature contribution and importance in the model.

Decision tree which is a simple model that divides data into different categories based on certain characteristics, is the foundation of the random forest. Decision trees mostly use entropy and information gain to decide where to split the data. Such that letting

$\text{entropy}=H,\text{then}H\left(S\right)={\displaystyle {\sum }_i^n{p}_i{\log }_2{p}_i}$.

where $p_i$ is the probability of each class in the dataset and n represents the number of classes. Main idea is to select the feature that leads to the greatest reduction in entropy (information gain). According to [10] , “A Random Forest model is a robust supervised ML algorithm composed of a tree-like structure that uses a set of decision trees to make predictions. Whereby each tree is trained on a random subset (with replacement) and a random subset of features”. This concept is well illustrated in Figure 3 .

The final prediction for classification tasks is decided upon by a majority vote, while for the regression ones, it is the arithmetic mean of the predictions from all the trees.

The Extreme Gradient Boosting, also referred to as XGBoost, is a ML model used for supervised learning challenges. It uses training data to predict a target variable. Its main strength is that it is a scalable and optimized algorithm that does improve the speed and prediction performance of Gradient Boosting Machines (GMB). It does either classification or regression based on the explanatory variables which could either be quantitative and, or qualitative. The XGBoost relies on the intuition that the best possible succeeding model, when combined with preceding models, minimizes the overall prediction error as illustrated in Figure 4 . The errors for the classification tasks are minimized through the random forest process illustrated in Figure 4 . Each succeeding model majors on misclassifications or errors of the preceding model and tries to reduce it in the succeeding iterations.

The following classification metrics were used to evaluate the performance of different machine learning algorithms for this study. The selected metrics were preferred under the assumptions that, in a binary classification task, the instances of data are typically predicted to be either positive or negative such that, where the instance is present (loan default) is labeled positive while a negative implies that the instance does not differ from the baseline in this respect, no default. Consequently, each predicted binary label has four possible outcomes: True Positive (TP) which is a correctly predicted positive outcome, True Negative (TN) which is a correctly predicted negative outcome, False Positive (FP) which is a negative instance predicted to be positive, and a False Negative (FN) which is a positive instance predicted to be negative. Therefore, the metrics included; A confusion matrix, which is a 2 by 2 matrix containing the counts of TP, TN, FP, and FN observations. A confusion matrix shows how well the classification model is performing, and it was an output of Python and R codes used to analyze the data for this study. The confusion matrix is expressed as:

$\text{BS}=\frac{1}{N}\underset{i=1}{\overset{N}{{\displaystyle \sum }}}{\left(f_i-o_i\right)}^2$ (5.5)

where

1) $N$ = total number of predictions.

2) $f_i$ = predicted probability that default will occur.

3) $o_i$ = actual outcome (1 if default occurred, 0 otherwise).

The study used learning curve to choose a sample size appropriate for model training. Typically, the ideal sample was the saturation point where the curve no longer rises but plateau, indicating that there was no further significant performance improvement by adding more data. Learning models may stabilize at slightly different levels of sample size as shown in Figure 5 and the other models used in this study, plateau started at 3000 training sample size.

Therefore, based on the learning curves from the three models, the ideal sample size for training data ranged between 3000 to 6000. The study used 6770 individual and small business loan accounts from the 21 registered microfinance institutions operating within Machakos County. Each microfinance was stratified to preserve representation, and then a unique identifier was assigned for each account. The accounts were randomly sampled within each stratum proportional to size without replacement.

The study features included demographic factors such as marital status, gender, age of the applicant, loan amount, credit history of the applicant, operating area and the date of disbursement between 2020 to 2024. Each loan was identified by the unique loan ID with labeled information on the status of the default payments. Table 1 gives the features and label summary.

The study explored both the target and the features to get an overview of the target and features distribution and their general relationships. The findings are discussed in the following sections.

The distribution of the response variable among the sampled loan accounts was performed. About a third of the loan applicants defaulted on payment, as shown in Figure 6 . Based on the findings displayed in Figure 6 , non-defaulters outnumbered

the defaulters implying an imbalanced dataset. Since no resampling techniques such as SMOTE or under sampling were applied, the use of class weighting mitigated bias. The residual imbalance is acknowledged as a limitation. The dataset was partitioned into an 80/20 stratified train/test split. Further visualization was performed as displayed in the following sections.

This section related the target to the features of the study such as sex, marital status, credit history of the applicants and loan purpose as summarized in Figure 7 .

Proportionately the higher default was by loan applicants with secondary school certificates, more default was by business loan applicants 18% compared to loan applicants for personal reasons 15.1%. The least default was by those without any credit history followed by those with more than three years’ experience. Male applicants were more likely to default compared to their female counterpart applicants.

Several model performance evaluations were carried out, including confusion matrix, recall, and F1-score. A confusion matrix also known as error matrix was used to visualize the breakdown of the predicted class of loan default status based on the historical loan default from loan applicant accounts in Machakos County. The confusion matrix summarized the true positives, false positives, true negatives and false negatives as shown in Table 2 . Based on the Confusion matrix values, the model posted 935 true negatives but it also misclassified 45 accounts as positives. There were 237 false negatives alongside 137 true positives, the computed accuracy level was 0.791728212. The study also used ROC (Receiver Operating Characteristic) curve to plot the true positive rate (TPR) against the false positive rate (FPR) at various threshold settings as shown in Figure 8 . The Area Under the ROC Curve (AUC) was 0.785 implying a strong degree of separability between the binary classification of defaulters and non-defaulters. Other evaluation scores which included recall and F1-Score are summarized in Table 3 .

The necessary libraries were imported prior to data preprocessing and data analysis. Data were split into training and testing sets (80% train, 20% test) prior to training random forest model. The random forest algorithm performed better than the logistic classification algorithm in contrary to [11] whose logistic regression models through stepwise selection criteria outperformed the other models with an overall accuracy of 93.33% in his research on identifying the optimal model for financial distress prediction. However, the predictions in the test data by using the random forest model were in accordance with [9] . whose research on empirical evidence from small and medium businesses observed that one of the benefits of random forests over logistic regressions was their capacity to handle unbalanced data and variables with missing values. The predictions on the test data results of the random forest model are summarized in Table 4 . The model performed better than logistic classification model in precision, F1-score and accuracy.

The XG Boost findings are summarized in Table 5 . The model had a good accuracy level (83%) with a strong detection to non-defaulters which posted very high recall and high F1-score. Its detection to actual positives was comparatively low but good enough to detect two thirds. The weighted average was very good across different metric evaluation methods. The findings were in agreement with the study by [12] , that machine learning has come out as a strong apparatus in overcoming the limitations of traditional business analysis statistics.

For each model, the predicted probabilities of loan default were compared with the actual outcomes of the defaults in the test set. Calibration curves were constructed using 10 equally spaced probability bins, plotting the observed default rate within each bin against the mean predicted probability. The calibration curves generated to visualize the agreement between predicted probabilities and observed default rates is shown in Figure 9 . Among the three models, XG Boost showed the best calibration (lowest Brier score), followed by Logistic Regression, while Random Forest exhibited comparatively weaker probability calibration. This meant that Logistic Regression and XG Boost predictions aligned more closely with the diagonal line of perfect calibration.