In many applications, statistical models for real data often have natural constraints or restrictions on some model parameters. For example, the growth rate of a child is expected to be positive, and patients receiving anti-HIV treatments are expected to exhibit a decline in their viral loads. Hypothesis testing for certain model parameters incorporating the natural constraints is expected to be more powerful than testing ignoring the constraints. Although constrained statistical inference, especially multi-parameter order-restricted hypothesis testing, has been studied in the literature for several decades, methods for models for complex longitudinal data are still very limited. We develop innovative multi-parameter order-restricted (or one-sided) hypothesis testing methods for modelling the following complex data: (1) multivariate normal data with non-ignorable missing values; (2) semi-continuous longitudinal data; and (3) left censored or truncated longitudinal data due to detection limits. We focus on testing mean parameters in the models, and the approaches are based on the likelihood methods. Some asymptotic results are obtained, and some computational challenges are discussed. Simulation studies are conducted to evaluate the proposed methods. Several real datasets are analyzed to illustrate the power advantages of the proposed new tests.