（1）这是在 as 之后省略了 is ：

Studies have found obesity is associated with impairments in cognitive functioning, as is assigned by a range of learning and memory tests.

（2）这样，as is assigned by a range of learning and memory tests 是针对前文宾语从句的比较状语从句。也就是说，宾语从句所说的事实，正如由一系列学习和记忆测试所确定的结论，形成比较关系。

（3）传统语法把 as is assigned by... 视为“非限制性定语从句”。理由是：as 是关系代词，指代主句的内容，且在从句中作 assigned 的宾语。

（4）assign 的意思是“确定（价值、功能、时间、地点、结论）”

Studies have found obesity is associated with impairments in cognitive functioning, as is assigned by a range of learning and memory tests.

【翻译】研究发现，肥胖与一个人认知功能的损害有关，（此结论）是由一系列的学习和记忆测试所确定的。