The cost function for the neural network is derived from the natural logarithm of the likelihood of a multinomial distribution ( $\log L$). This function is represented as

$\log L=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\underset{j=1}{\overset{J}{{\displaystyle \sum }}}{y}_{ij}\times \log \left({\pi }_{ij}\right)$,(1)

where ${\pi }_{ij}=P\left(y_{ij}=1|x_i\right)$. Define the cumulative probabilities $p_{ij}$ as $p_{ij}=P\left(y_{ij}=1|x_i\right)+P\left(y_{ij-1}=1|x_i\right)+\cdots +P\left(y_{i1}=1|x_i\right)$. In addition, $p_{iJ}=1$ and $p_{i0}=0$. Replacing ${\pi }_{ij}$ with the cumulative probabilities $p_{ij}$ in $\log L$ allows us to rewrite Equation (1) as

$\log L=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\underset{j=1}{\overset{J}{{\displaystyle \sum }}}{y}_{ij}\times \log \left(p_{ij}-p_{ij-1}\right)$.(2)

We are primarily concerned with directly modeling cumulative probabilities, and to accomplish this, we employed cumulative link models (CLMs).

CLMs are statistical models designed to analyze data with ordinal outcomes. As such, we are concerned with modeling the cumulative probabilities $p_{ij}$. In CLMs, $p_{ij}$ is linked to a function of predictor variables through a link function. This approach allows for the incorporation of the covariate vector $x_i$ to explain the variation in the ordinal outcome while maintaining the inherent order of the response categories. The four-link functions considered in this study are as follows:

1) Logit (logistic Distribution)

2) Probit (normal distribution)

3) Complementary log-log (Gompertz distribution)

4) Log Log (Gumbel Distribution)

By employing different link functions within a neural network framework, we can leverage the flexibility and power of machine-learning techniques while maintaining the structured approach of CLMs.

By employing the logit link, the cumulative probabilities are modeled as:

$p_{ij}^{\left[1\right]}=\frac{1}{1+{\text{e}}^{-\left(f\left(x_i\right)+b_j\right)}}$(3)

where $b_j$ can be thought of as the intercept parameter for the j^th level and $f\left(x_i\right)$ is a non-linear function of $x_i$. Considering the probit link to model cumulative probabilities leads to:

$p_{ij}^{\left[2\right]}={\displaystyle {\int }_{-\infty }^{f\left(x_i\right)+b_j}\frac{1}{\sqrt{2\pi }}{\text{e}}^{-\frac{1}{2}{x}^2}\text{d}x}$.(4)

When the cloglog link is considered, we have

$p_{ij}^{\left[3\right]}=1-{\text{e}}^{-{\text{e}}^{f\left(x_i\right)+b_j}}$(5)

Finally, utilizing the loglog link leads to:

$p_{ij}^{\left[4\right]}={\text{e}}^{-{\text{e}}^{f\left(x_i\right)+b_j}}$(6)

These cumulative probabilities can be substituted into Equation (2) and solved accordingly.

A neural network was used in this investigation. The neural network comprised one hidden layer with six nodes. A neural network with a single hidden layer balances simplicity and capability, leveraging universal approximation to approximate functions while minimizing complexity and reducing overfitting. The architecture is subsequently delineated. For the first layer, we applied the function

$Z^{\left[1\right]}=W^{\left[1\right]}X+b^{\left[1\right]}$(7)

where $W^{\left[1\right]}$ is a 6 by p matrix and $b^{\left[1\right]}$ is a column vector of length 6. As, such $Z^{\left[1\right]}$ is of dimension 6 by n. After this, the first activation function was applied and is defined as

$A^{\left[1\right]}=g^{\left[1\right]}\left(Z^{\left[1\right]}\right)$(8)

where $g^{\left[1\right]}\left(z\right)$ is called the Leaky Rectified Linear Unit (RELU) [28] and is defined as $\max \left(0.01z,z\right)$. For the second layer, we applied the function

$Z^{\left[2\right]}=W^{\left[2\right]}{A}^{\left[1\right]}+b^{\left[2\right]}$(9)

where $W^{\left[2\right]}$ is a J-1 by 6 matrix and $b^{\left[2\right]}$ is a column vector of length J-1 with the j^th element defined as $b_j$ from the previous section. As such $f\left(x_i\right)$, from the previous section, is defined as the value from the i^th column of $W^{\left[2\right]}{A}^{\left[1\right]}$. After this, the four link functions from Equations (3) through (6) were applied, elementwise, to $Z^{\left[2\right]}$. This is presented as

$A^{\left[2\right]\left[k\right]}=p^{\left[k\right]}\left(Z^{\left[2\right]}\right)$,(10)

where the function $p^{\left[k\right]}$ is the element-wise application of Equations (3) through (6) to the matrix $Z^{\left[2\right]}$. After this, the cost function from Equation (2) was constructed, where $p_{ij}=A_{ji}^{\left[2\right]\left[k\right]}$.

To obtain the optimal values for $W^{\left[1\right]}$, $b^{\left[1\right]}$, $W^{\left[2\right]}$, and $b^{\left[2\right]}$, the Adam optimization algorithm was applied [11] [29] . Once these optimal parameters were obtained, they were used to evaluate the neural network’s predictive capabilities on validation data. Hyperparameter tuning was not performed.

The simulated data procedure has been previously introduced [11] . The data were simulated according to four covariate scenarios. The covariate scenarios are as follows:

1) Autoregressive (1)

2) Compound Symmetric

3) Toeplitz

4) Unstructured

For each scenario, a total of 100 datasets were simulated. Each dataset was divided into subsets comprising 80% for training and 20% for validation. Each dataset was simulated to yield an ordinal outcome comprising four categories (levels). Due to the nature of the simulation, the sample size for each category varies, resulting in an imbalance among the classes in most cases. The four neural networks, predicated on the four link functions delineated in Equations (3) through (6), were implemented on each training dataset. After optimizing these neural networks with respect to their parameters of interest, the parameters were subsequently applied to the validation data, and the percentage of correctly classified observations was documented. The medians and interquartile ranges are reported for each neural network employed within each covariate scenario. Assuming and verifying non-normality via a Kolmogorov-Smirnov test, the Kruskal-Wallis test was used to determine the equality of medians among the four neural networks applied within each scenario level. The Wilcoxon rank-sum test was implemented to facilitate pairwise comparisons of the neural networks within each data scenario. Furthermore, the percentage of observations correctly classified per forward propagation iteration of the Adam optimizer was reported for a subset of the training data. All analyses were conducted using R statistical software [30] .

Hepatocellular carcinoma (HCC) is the most common type of primary liver cancer, accounting for approximately 90% of all liver cancers. It is the fifth most common cancer in men and the seventh most common cancer in women globally, with over half a million new cases diagnosed annually [31] [32] . The incidence and mortality rates of hepatocellular carcinoma vary significantly across different geographical regions, with the highest rates observed in sub-Saharan Africa and Southeast Asia, where hepatitis B virus infection is endemic [31] [33] . In contrast, while increasing, the incidence and mortality rates in Europe and the United States remain relatively low compared with other regions [31] .

Several factors contribute to the development of hepatocellular carcinoma, including viral hepatitis (hepatitis B and C), cirrhosis, nonalcoholic fatty liver disease, and various genetic and environmental factors [34] [35] . Understanding the epidemiology, risk factors, and treatment strategies of hepatocellular carcinoma is crucial for improving patient outcomes and reducing the burden of this disease worldwide.

The real-world dataset employed in this study to predict an HCC-related ordinal outcome is a subset of the genomic dataset found in the Gene Expression Omnibus data repository labeled GSE18081 [27] . The data has a sample size of 56. The ordinal outcome of interest is the classification of tissue samples as normal (n = 20), pre-malignant (cirrhosis; n = 16), or malignant (hepatocellular carcinoma; n = 20) using a small set of 46 molecular features. The study utilized data from the BeadArray Cancer Panel I. All technical replicate samples and matched cirrhotic samples from subjects with HCC were removed from the dataset. As such, all the samples in the reduced dataset were independent. The dataset is called the hccframe and is located in the ordinalgmifs R package [27] . Data were obtained using the following R code:

>install.packages("ordinalgmifs")

>library(ordinalgmifs)

>data(hccframe).

The input features were standardized to be between zero and one. To evaluate the neural networks against the dataset, a 10-fold cross-validation approach was employed. The dataset was randomly partitioned into ten segments. Nine segments (approximately 90%) were utilized for training the neural network, and the remaining 10% served as validation data. Subsequently, the percentage of correctly classified instances across the entire dataset was computed. This procedure was executed 100 times for each neural network. The median and interquartile range of 100 percent of the correctly classified observations were subsequently calculated for each neural network. Upon assuming and confirming non-normality via the Kolmogorov-Smirnov test, the Kruskal-Wallis test was employed to assess the equality of medians among the four neural networks. The Wilcoxon rank-sum test was also conducted to facilitate pairwise comparisons among the neural networks.

The proposed methodology, comprising four cumulative link models, was applied to the simulated data. The objective was to compare the performance of the four distinct cumulative link models. The models were evaluated based on four criteria: optimization, median percent correctly classified, IQR, and nonparametric tests to determine statistically significant differences across groups.

Figure 1 visually presents the accuracy (percentage of observations correctly classified) achieved by different models when classifying observations across iterations of the applied Adam optimization algorithm, specifically for four distinct covariate scenarios: Autoregressive (1), Compound Symmetric, Toeplitz, and Unstructured. The results are presented for the training datasets. The performance oscillated for all models across all covariate scenarios and then consistently increased for the given datasets. Once the method maximizes the proportion it can correctly estimate, it oscillates slightly around that value. The results indicated that the four models were suitable for optimizing the test datasets. The logit link model outperformed the other models regarding prediction accuracy across all covariate scenarios. However, it requires the most iterations to reach its maximum accuracy level, highlighting a trade-off between accuracy and computational complexity.

Focusing on the Autoregressive (1) covariate scenario, it is evident that the logit model, represented by the green line, exhibits the highest classification accuracy, attaining a maximum of 90% of observations correctly classified. The other models exhibited a gradual upward trend in their performance metrics. Notably, the probit and loglog models, illustrated in gray and orange, respectively, closely followed the logit model in terms of accuracy at a maximum of 86%. The cloglog (blue line) model had a maximum accuracy of 84%. It is also worth mentioning that the probit model required the most significant number of iterations to achieve optimization at 8861.

Similarly, the logit model exhibits superior performance for the Compound Symmetric covariate scenario, maintaining a high percentage of correct classifications, with a maximum of 89% correctly classified. The other models demonstrated comparable performance, with a maximum correctly classified rate of 86% for the probit, 84% for the cloglog, and 83% for the loglog. The cloglog model requires the most iterations to be optimized at 8357 iterations.

The logit model outperforms the others for the Toeplitz covariate scenario, with a maximum correctly classified percentage of 92%. Loglog, cloglog, and probit maximize accuracy at 85%, 83%, and 82%, respectively. The Logit requires the most iterations (15,158) to reach an optimal value of the percentage of observations correctly classified.

The logit model demonstrated the highest accuracy (90%) for the unstructured covariate scenario, while the cloglog, probit, and loglog derived models exhibited slower but steady improvements with maximum accuracies of 82%, 83%, and 81%, respectively. The logit model required the most iterations to be optimized at 8725.

The logit model consistently achieved the highest percentage of correctly classified data across all the scenarios. In addition, on average, the logit model requires the most iterations (9750) to attain its maximal value. In contrast, the probit model required the lowest number of iterations (6725) to achieve its maximum value.

Table 1 presents the performance of each method as applied to the validation datasets. The medians and interquartile ranges for all methods are reported. The validation data constituted 20% of the original dataset and were not used to develop the final models. The Kruskal-Wallis rank sum test, with a significance level of 0.05, was used to compare the methods’ median accuracy. The Wilcoxon rank-sum test was used for all pairwise comparisons within each covariate scenario.

Figure 2 graphically shows the same information as Table 1 . It displays the percentage of correctly classified data across iterations for four covariate scenarios.

For the Autoregressive (1) covariate scenario, the logit link model (green boxplot) demonstrated the highest accuracy, rapidly attaining a median accuracy of 87%. Other link models also achieved over 80% accuracy (81% - 82%) but exhibited significantly lower accuracy than the logit model. Furthermore, the logit model displayed fewer outlying values than the other three cumulative link models. For

¹Kruskal-Wallis rank sum test. For each covariate scenario, 100 data sets were created. Each dataset was divided into 80% for training and 20% for validation. After training the four Neural Networks, the models were applied to the validation datasets. The median proportion and interquartile range of correctly classified observations for the 100 datasets are reported. The results of the Kruskal-Wallis rank-sum test are reported for the three models.

the Compound Symmetric covariate scenario, among the models evaluated, the logit approach exhibited the highest accuracy regarding correct classifications. The cloglog, probit, and loglog models yielded comparable results, with a median of 82% correctly classified. For the Toeplitz covariate scenario, the logit model performs superiorly to other models, with the cloglog, probit, and loglog cumulative link models exhibiting moderate improvement. For the Unstructured covariate scenario, the logit model leads in accuracy, seconded by the probit model with a median accuracy of 83%. Cloglog and loglog have a median correctly classified accuracy of 82%. The logit model consistently achieved the highest percentage of correctly classified data across all scenarios, with the fewest outliers.

Table 1 compares the performances of four cumulative link models—cloglog, logit, loglog, and probit—to correctly classify the ordinal outcome from the validation data across different covariate scenarios. The logit model consistently demonstrated the highest median accuracy across all covariate scenarios and exhibited the smallest interquartile ranges. The median of correctly classified observations for the logit model was 86% - 87%, whereas the median correct for the other cumulative link models was 81% - 83%. For all covariate scenarios, the p-value for all Kruskal-Wallis rank sum tests was less than 0.001, indicating statistically significant differences in performance among the models. Pairwise comparisons were performed to elucidate these differences. Significant pairwise differences for the Autoregressive 1 covariate scenario include: logit-cloglog at <0.001, logit-loglog at <0.001, logit-probit at <0.001, and probit-cloglog at 0.01. Significant pairwise differences for the Compound Symmetric covariate scenario included logit-cloglog at <0.001, logit-loglog at <0.001, and logit-probit at <0.001. Significant pairwise differences for the Unstructured covariate scenario included logit-cloglog at <0.001, logit-loglog at <0.001, and logit-probit at <0.001. Significant pairwise differences for the Toeplitz Covariate Scenario included logit-cloglog at <0.001, logit-loglog at <0.001, and logit-probit at <0.001. The logit model’s performance is significantly different from that of the other three models across all pairwise comparisons.

The proposed methodology and the four cumulative link models were applied to the hccframe dataset. The goal was to compare the performance of the four different cumulative link models. Table 2 presents the performance of each method as measured by the median and interquartile range for all methods; 10-fold cross-validation was applied 100 times, yielding 100 estimates of percent correctly classified for the dataset. Kruskal-Wallis Rank sum test, with a significance level of 0.05, was used to compare median accuracy for the methods. In addition, the Wilcoxon rank-sum test was used to compare all the pairwise comparisons. Figure 3 visually presents the same information as Table 2 . Table 2 compares the performance of four cumulative link models, cloglog, logit, loglog, and probit, in correctly classifying ordinal outcomes. The logit model showed the highest median accuracy across all scenarios and the smallest interquartile range. The median number of correctly classified observations for the logit model was 86%. The loglog model comes in second with a median percentage correctly classified at

¹Kruskal-Wallis rank sum test. The percent correctly classified for the four models (Cloglog, Logit, Loglog, and Probit) applied to the hccframe dataset. The results are based on 10-fold cross-validation, repeated 100 times.

71%, with the cloglog and probit having values of 66% and 66%, respectively. The interquartile ranges were similar across all four models. The p-value for all Kruskal-Wallis rank-sum tests was less than 0.001, indicating significant differences in performance among the models. Pairwise comparisons were performed to determine these differences. All pairwise comparisons were considered statistically significant at a significance level of 0.05.