We provide insights into new methodology for the analysis of multilevel binary data observed longitudinally, when the repeated longitudinal measurements are correlated. The proposed model is logistic functional regression conditioned on three latent processes describing the within- and between-variability, and describing the cross-dependence of the repeated longitudinal measurements. We estimate the model components without employing mixed-effects modeling but assuming an approximation to the logistic link function. The primary objectives of this article are to highlight the challenges in the estimation of the model components, to compare two approximations to the logistic regression function, linear and exponential, and to discuss their advantages and limitations. The linear approximation is computationally efficient whereas the exponential approximation applies for rare events functional data. Our methods are inspired by and applied to a scientific experiment on spectral backscatter from long range infrared light detection and ranging (LIDAR) data. The models are general and relevant to many new binary functional data sets, with or without dependence between repeated functional measurements.