Introduction Assessing cognitive and functional changes at the early stage of Alzheimer's disease (AD) and detecting treatment effects in clinical trials for early AD are challenging. Methods Under the assumption that transformed versions of the Mini–Mental State Examination, the Clinical Dementia Rating Scale–Sum of Boxes, and the Alzheimer's Disease Assessment Scale–Cognitive Subscale tests'/components' scores are from a multivariate linear mixed-effects model, we calculated the sample sizes required to detect treatment effects on the annual rates of change in these three components in clinical trials for participants with mild cognitive impairment. Results Our results suggest that a large number of participants would be required to detect a clinically meaningful treatment effect in a population with preclinical or prodromal Alzheimer's disease. We found that the transformed Mini–Mental State Examination is more sensitive for detecting treatment effects in early AD than the transformed Clinical Dementia Rating Scale–Sum of Boxes and Alzheimer's Disease Assessment Scale–Cognitive Subscale. The use of optimal weights to construct powerful test statistics or sensitive composite scores/endpoints can reduce the required sample sizes needed for clinical trials. Conclusion Consideration of the multivariate/joint distribution of components' scores rather than the distribution of a single composite score when designing clinical trials can lead to an increase in power and reduced sample sizes for detecting treatment effects in clinical trials for early AD.