Feedforward neural networks (FNNs) are typically viewed as pure prediction algorithms, and their strong predictive performance has led to their use in many machine-learning applications. However, their flexibility comes with an interpretability trade-off; thus, FNNs have been historically less popular among statisticians. Nevertheless, for suitably parsimonious shallow FNNs, classical statistical theory, such as significance testing and uncertainty quantification, may still provide useful regression-style summaries. Supplementing FNNs with methods of statistical inference, and covariate-effect visualisations, can shift the focus away from black-box prediction and move FNNs towards traditional statistical models. This can allow for more inferential analysis, and, hence, make FNNs more accessible within the statistical-modelling context. We investigate covariate-level Wald testing in the context of penalised FNNs, and also propose covariate-effect plots that emulate regression coefficients. Simulation studies are used to extensively investigate the performance of Wald-based inference, with particular emphasis on when this approach performs well. This statistical-based approach to neural networks is demonstrated through an application to insurance data.

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