The only true statistical test for homogeneity is the Chi-Square test of Q.

Tau-squared, the between-study variance, is derived from this Q and subsequently used to conduct random effects analysis. Under an inverse-variance model, there is no need to switch to a fixed effect model in the absence of heterogeneity, as the two results will be exactly identical. That is, the weights for studies will be based on the inverse of a study's variance, which only under a random effects model will include tau-squared. Hence, if that term is zero, the random effects and fixed effect model are the same.

I-Squared above 50% can typically be interpreted as more than half of the total heterogeneity stems from between-study variance that cannot be explained by sampling error (alone).

Dear Robert, thank you for your very informative answer. So if it is initially assumed that fixed effects is chosen (and thus tau squared is not taken into account), is it acceptable if Q test is not significant (p>.5)? Or should I just choose random effects?

I suggest that you read the attached paper by Borenstein et al (2016) which discusses the heterogeneity issue. Selection of the fixed-effect model or random-effects model should not be done on the basis of whether you conclude there is heterogeneity or not. You select the random-effects model when the research designs among the studies are not the same (which in my opinion is most of the time). If the Chi-Square test of Q yields a nonsignificant p-value, that can be due to two things, i.e., 1) there really is no heterogeneity or 2) you do not have the statistical power to detect it (e.g., not enough studies in your meta-analysis).

Thank you very much, the attached paper was very helpful. So a random effects model makes sense almost always but is more conservative.