  Regression with random data objects is becoming increasingly common in modern
data analysis. Unfortunately, this novel regression method is not immune to the
trouble caused by unusual observations. A metric Cook's distance extending the
original Cook's distances of Cook (1977) to regression between metric-valued
response objects and Euclidean predictors is proposed. The performance of the
metric Cook's distance is demonstrated in regression across four different
response spaces in an extensive experimental study. Two real data applications
involving the analyses of distributions of COVID-19 transmission in the State
of Texas and the analyses of the structural brain connectivity networks are
provided to illustrate the utility of the proposed method in practice.
