Multiscale sample entropy (MSE) of human electroencephalogram (EEG) data from patients under different pathological conditions of Alzheimer's disease (AD) was evaluated to measure the complexity of the signal. Quantifying the complexity level with respect to various temporal scales, MSE analysis provides a dynamical description of AD development. When compared to EEG data from normal subjects, EEG data from subjects with mild cognitive impairment (MCI) showed nearly the same complexity profile, but a scale discrepancy which may occur from a spectral abnormality. EEG data from severe AD patients showed a loss of complexity over the wide range of time scales, indicating a destruction of nonlinear structures in brain dynamics. We compare the MSE method and spectral analysis to propose that nonlinear dynamical approach combining a multiscale method is crucial for revealing AD mechanisms.