Omitted variable bias occurs in most statistical models. Whenever a confounding variable that is correlated with both dependent and independent variables is omitted from a statistical model, estimated effects of included variables are likely to be biased due to omitted variables. This issue is particularly problematic in physics education research where many research studies are quasiexperimental or observational in nature due to ethical and logistical limitations. In this paper, we illustrate the mechanisms behind omitted variable bias in explanatory modeling using authentic data and analytical solutions. We demonstrate that omitting confounding variables that are strongly correlated with included variables and have large effects on the dependent variable can significantly bias estimated effects for included variables. We also find that controlling for variables that are uncorrelated with other variables or have no effect on the dependent variable does not appreciably bias estimated effects and may or may not affect the precision of those estimates. These results suggest that removing from explanatory models variables that are not "statistically significant" can have unintended consequences on model and variable interpretations. Our results underscore the importance of carefully considering why or why not to include a variable in a model, informed by both data and theory.