Does DiD work when not all individuals have pre-treatment observations? I would like to make causal statements on the effect of a treatment on a group of individuals. I have panel data on these individuals but the treatments do not occur at the same time in their life. Some individuals get the treatment as adults, some get it pre-birth, and some don't get it at all. 
I thought I would do a difference-in-difference analysis, but for those who receive it pre-birth there isn't a pre-treatment period. What can I do to make causal statements in this case?
 A: 
I have panel data on these individuals but the treatments do not occur at the same time in their life.

Difference-in-differences (DiD) approaches can handle variation in treatment timing. 

Some individuals get the treatment as adults, some get it pre-birth, and some don't get it at all.

I would be interested to know how or why some individuals receive treatment pre-birth, while others receive it later in adulthood. Are these qualitatively different treatments depending upon which phase (epoch) of life individuals (entities) enter into? Obviously, this requires intimate knowledge of your treatment. 

I thought I would do a difference-in-difference analysis, but for those who receive it pre-birth there isn't a pre-treatment period.

In the absence of pre-event data, you cannot possibly defend the validity of the DiD design to a reader/audience. Moreover, you cannot credibly demonstrate, visually (or statistically), the presence of common trends prior to treatment exposure, which is crucial. The DiD approach performs a double difference across groups and across times. The pre-birth cohort of individuals (entities) have no pretreatment data, and therefore there is no pretreatment differences to assess. Moreover, you cannot make any statements about the stability (parallelism) of the group trends before treatment goes into effect. With post-treatment (posttest) only observations, you may be further reinforcing issues concerning selection of individuals/units into treatment. See the top answer here for a hypothetical example of why it would be unwise to make causal claims in this setting. This post also addresses questions regarding a panel of units that begin in the treatment condition.

What can I do to make causal statements in this case?

It will be difficult to make causal statements about the impact of your policy/treatment/intervention without sufficient pre-even data. If you're opting to go with a DiD estimation strategy, then I would limit your analysis to individuals/units with serial observations pretreatment. According to your post, you already have a group(s) of individuals (adults) never subjected to treatment. From there, you can start by inspecting the group trends and investigate if DiD is worthwhile. 
In my opinion, I would subject the pre-birthers to a separate analysis. 
