What proportion of missing data can be considered acceptable for inference with a mixed-effects model I am wondering what proportion of missing data can be considered acceptable for use of a mixed effects model?
I am analysing a clinical trial testing the efficacy of an agonist drug in reducing the number of days of illicit drug use. When the trial started there were 67 participants in the placebo group and 61 in the experimental group. 
By the end of the trial period 12 weeks later there were thirty in each group, which is 45% of the original number left in the placebo group and 49% left in the placebo group. 
By the time of the post-treatment follow-up, 12 weeks after the trial ended (i.e. 24 weeks from the commencement of the trial) there were 29 left in the placebo group (43% of original numbers) and 26 left in the placebo group (also 43% of original). These levels of attrition are quite normal in prospective Clinical Alcohol and Drug research.
Does this level of missingness catastrophically reduce the value of the inferences you would make from a longitudinal mixed effects model?
and 
If that level of missingness is catastrophic what other methods might be used? 
 A: 
Does this level of missingness catastrophically reduce the value of the inferences you would make from a longitudinal mixed effects model?

Not necessarily. A great deal depends on the reasons for dropout. If the data are missing at random (MAR), then a suitable multiple imputation approach can result in unbiased, or at least much less biased, estimates. Here, by "suitable" I mean, a multiple imputation scheme that handles the clustering of data within individuals. 
If the data are missing completely at random (MCAR), then unbiased point estimates may be obtained from a mixed model using complete case analysis, but standard errors will be biased upwards. Again, a suitable multiple imputation approach will reduce this.
On the other hand, if the data are missing not at random (MNAR), then you may have a very difficult time ahead.
A good review of approaches to this problem can be found here:
Huque, M.H., Carlin, J.B., Simpson, J.A. and Lee, K.J., 2018. A comparison of multiple imputation methods for missing data in longitudinal studies. BMC medical research methodology, 18(1), p.168.
https://www.ncbi.nlm.nih.gov/pubmed/30541455
