Is it ok to use a random intercept model without testing for random slopes? In university we calculated a random intercept model for two-level nested data. We compared the random intercept model with all level-1 variable with the random intercept model with all level-1 and all level-2 variables. Like this we got main effects for level-1 and also main effects for level-2 variables. But we did not run a model to see if there are random slopes too. (In another step we even included an interaction term between a level-1 and a level-2 variable in the model, which was significant. But this was still without testing for random slopes before).
Now I am wondering if one shouldn`t always test for random slopes too? What happens if actually the random slope model would better fit the data than the random intercept model? Would that mean that the coefficients in the random intercept model are not estimated right?
 A: Models with mixed effects have 2 stochastic ingredients:


*

*Residual errors

*Random effects
If you use a random slope and notice that the model performs better, it means that there is variability that was not properly captured by the residual errors, the random intercept, and fixed effects. Another direction that you can take is to be flexible on the residual errors (e.g. by using a Student-t instead of a normal distribution for the residual errors). Now the question is: which model performs better? The one with normal errors or with Student-t errors? The one with random slope and normal errors? The one with random slope and Student-t errors? ... And so on. If you can implement all of them and compare them, you can gain understanding on the features of the data. For example, if your model selection favours the model with Student-t errors and no random slope, it tells you that a single slope fits the data better (no variability among subjects in terms of the slope) but there might be some outliers that require a distribution with heavier tails than normal.
A: I would echo @East's advocacy for model exploration, but it was already stated very well.  Per Barr et all (2013; full citation below), a failure to fit random slopes when present could result in an inflated Type-I error rate (incorrect rejections of the null hypothesis when the null hypothesis actually is true).  So, (as I understand it) if the random slope model would better fit the data than the random intercept model, there will be a tendency for the fixed effect error to be under estimated.  In practice (again, as I understand it), the fixed effects coefficients will be estimated reasonably well, but the errors will be estimated incorrectly (with bias towards lower magnitude).  
Barr, D. J., Levy, R., Scheepers, C., & Tily, H. J. (2013). Random effects structure for confirmatory hypothesis testing: Keep it maximal. Journal of Memory and Language, 68(3), 255-278.
