Research（主语）//（修饰主语后置定语；由过过去分词充当） carried out at the University of North Carolina //（谓语动词）indicates //（that宾语从句）that the speed //（定语从句：介词at+关系代词which引导）at which human nails are growing has increased by close to 25 per cent over the last 70 years.

参考译文：北卡罗来纳大学的研究表明，在过去的70年的时间里里，人类指甲的生长速度提高了将近近25%。

定语从句语法结构分析如下

the speed（主句主语）// at which（定语从句状语）// human nails（定语从句主语） are growing（定语从句谓语）// has increased（主句谓语）//（主句状语） by close to 25 per cent //（主句时间状语）over the last 70 years.