Can I look at reported standard errors (SE) and decide if means differ?

No guarantees, but roughly if means differ by 3*SE then they are statistically significant. This is based on the Least Significant Difference, which is 2*sqrt(2)*SE. Often people use non-overlapping confidence intervals as a decision rule, but this is equivalent to 4*SE, which is a bit conservative.

Things that make 3*SE fail:

1) Actually statistical differences depend on the standard error of difference, SED, not SE. Anything in the model that makes these differ will make the rule fail, such as covariates and blocking factors.

2) In general, mixed models with random effects will make the rule fail, because random variance is included in SE, but not in SED. But this will make 3*SE rule conservative, 3*SED will be even smaller. If 3*SE suggests a statistical difference, difference most likely exists.

Also take a look at Error Bars paper.

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