Occupational and environmental epidemiological studies often involve ordinal data, including antibody titer data, indicators of health perceptions, and certain psychometrics. Ideally, such data should be analyzed using approaches that exploit the ordinal nature of the scale, while making a minimum of assumptions. In this work, we first review and illustrate the analytical technique of ordinal logistic regression called the "proportional odds model". This model, which is based on a constrained ordinal model, is considered the most popular ordinal model. We use hypothetical data to illustrate a situation where the proportional odds model holds exactly, and we demonstrate through derivations and simulations how using this model has better statistical power than simple logistic regression. The section concludes with an example illustrating the use of the model in avian and swine influenza research. In the middle section of this work, we show how the proportional model assumption can be relaxed to a less restrictive model called the "trend odds model". We demonstrate how this model is related to latent logistic, normal, and exponential distributions. In particular, scale changes in these potential latent distributions are found to be consistent with the trend odds assumption, with the logistic and exponential distributions having odds that increase in a linear or nearly linear fashion. Actual data of antibody titer against avian and swine influenza among occupationally- exposed participants and non-exposed controls illustrate the fit and interpretation of the proportional odds model and the trend odds model. Finally, we show how to perform a multivariable analysis in which some of the variables meet the proportional model assumption and some meet the trend odds assumption. Likert-scaled data pertaining to violence among middle school students illustrate the fit and interpretation of the multivariable proportional-trend odds model. In conclusion, the proportional odds model provides superior power compared to models that employ arbitrary dichotomization of ordinal data. In addition, the added complexity of the trend odds model provides improved power over the proportional odds model when there are moderate to severe departures from proportionality. The increase in power is of great public health relevance in a time of increasingly scarce resources for occupational and environmental health research. The trend odds model indicates and tests the presence of a trend in odds, providing a new dimension to risk factors and disease etiology analyses. In addition to applications demonstrated in this work, other research areas in occupational and environmental health can benefit from the use of these methods. For example, worker fatigue is often self-reported using ordinal scales, and traumatic brain injury recovery is measured using recovery scores such as the Glasgow Outcome Scale (GOS).