SB phenotype of MMC and non-MMC (meningocele, fatty filum with tethered cord, lipomyelomeningocele, split cord malformation, or terminal myelocystocele) (these are the only subphenotypes eligible for the registry).

Children aged <5 years were excluded because obtaining accurate manual muscle testing is relatively unreliable until the age of 5 years.²⁴ In addition, patients with only one recorded visit were excluded to ensure that we were assessing future ambulation capabilities.

All statistical analyses were conducted using IBM SPSS Statistics version 28.0.^a A multilayer perceptron (MLP) RNN was used to predict ambulation ability. The MLP RNN was run twice, using 2 different training datasets. The first training dataset was randomly selected from the full dataset, making up 70% of valid cases. Because there were large differences in frequencies between ambulation categories, this is considered an unbalanced training dataset. A second, balanced training dataset was created where the frequencies of ambulation ability values were all equal. The frequency itself was determined by taking 70% of the least frequent variable, and cases were randomly selected from ambulation group. Using a validation dataset is one way to address potential overfitting. Once the training datasets were designated, all cases were run through the MLP, and prediction accuracy, recall, and precision were assessed. Accuracy measures how well the model predicts the outcome variable and is measured by dividing all correct predictions by the total number of predictions. Recall measures the ability of the model to identify true positives and is calculated by dividing the number of correctly predicted positives by the number of actual positives. Precision measures how often the model correctly predicts a class and is calculated by dividing the number of correct predictions from a class by the number of total predictions of that class. RNN models have the advantage of identifying hidden trends or relationships in large datasets with many independent variables that are nonlinear. They consist of several layers: input, output, and one or multiple hidden layers, with weights attached to each connection.²⁵ Data are offered to the network via the input layer, which communicates to one or more hidden layers where the actual processing is completed by a system of weighted connections. The hidden layers then link to an output layer, where the final solution can be found. The nodes are connected from the input layer outward, and the error between the predicted output value and the actual value is fed back through the network to update the initial values assigned to the weights (see RNNs with 1 or 2 hidden layers below). The goal of this process is to minimize the error between anticipated and expected outputs. The ability to recognize patterns was trained in the hidden layers, and when new relationships are presented, the model is able to identify them. The RNN models provide measures of features’ importance in accounting for outcome variance, as illustrated in fig 1.²⁶ The RNN models can use many activation functions. The choice of activation function in the hidden layer will control how well the network model learns the training dataset. The choice of activation function in the output layer will define the type of predictions the model can make. In this analysis, a single-hidden layer MLP transformation function was used. MLPs are good for problems where inputs are assigned a class or label, or where a real-valued quantity is predicted given a set of inputs. MLPs can learn complex patterns and relationships in structured data because they can handle input data with fixed dimensions and independent features. They can also approximate any continuous function, not just linear functions, by combining several neurons in at least 3 layers. MLPs are scalable, easy to implement, and can handle large amounts of input data well. They can make quick predictions after training and achieve the same accuracy ratio even with smaller samples. Traditionally, an MLP can be viewed as a logistic regression classifier where the input is first transformed using a learned nonlinear transformation Φ. This transformation projects the input data into a space where it becomes linearly separable. A single-hidden layer is sufficient to make MLPs a universal approximator.

Formally, a one hidden layer MLP is a function f:RD→RL, where R is a set of real vectors, D is the size of input vector x, and L is the size of the output vector f(x), such that in matrix notation f(x)=G(b(2)+W(2)(s(b(1)+W(1)x))), with bias vectors b(1) and b(2); weight matrices W(1) and W(2), and activation functions G and s. The bias vector is a set of values (one for each neuron) added to the weighted input before the activation function is applied.

The vector h(x)=Φ(x)=s(b(1)+W(1)x) constitutes the hidden layer of the nonlinear transformation, Φ. The hidden layer vector is composed of the activation function, s, which pulls in both bias vectors, b(1); and weight matrices, W(1). To avoid poor initial values, such as a saddle point where gradients are zero but the value is suboptimal, weights are initialized randomly. This random initialization is key to escaping local optima and allows gradient descent to update the weights effectively. The weights are initialized as random matrices drawn from a Gaussian distribution. The dimensions of the weight matrices are determined by the dimensions of the input data and the number of classes. Finally, the output vector is then obtained as o(x)=G(b(2)+W(2)h(x)), which is a prediction (in our case, category) based on the probability of class membership, derived from the inputs transformed by the hidden layer h(x), activation function, (G), bias vector b(1), and weight matrices, W(2).

To train an MLP, all of the parameters must be obtained. The set of parameters to learn is the set: θ={W(2),b(2),W(1),b(1)}, which included collection of the weight matrices and the bias vector.