Thank you for your quick reply. Embarrassingly, it seems that the analysis wasn't the same... 
 
However, regarding using ANOVA to generate lsmeans: when I use ANOVA I get very different results from when I use REML. The trait ("ADI") and the effect ("UniqueID") are the same in both analyses. Any insight as to why that might be the case? (Apologies if this should be a different discussion topic--first time posting here--and also apologies if it ends up being another silly error on my part!)
REML:
Fit Model(
	Y( :ADI ),
	Effects,
	Random Effects( :UniqueID ),
	NoBounds( 1 ),
	Personality( "Standard Least Squares" ),
	Method( "REML" ),
	Emphasis( "Minimal Report" ),
	Run(
		:ADI << {Summary of Fit( 1 ), Analysis of Variance( 0 ),
		Parameter Estimates( 1 ), Scaled Estimates( 0 ),
		Plot Actual by Predicted( 0 ), Plot Regression( 0 ),
		Plot Residual by Predicted( 0 ), Plot Studentized Residuals( 0 ),
		Plot Effect Leverage( 0 ), Plot Residual by Normal Quantiles( 0 )}
	),
	SendToReport(
		Dispatch( {"Response ADI"}, "Effect Details", OutlineBox, {Close( 0 )} ),
		Dispatch(
			{"Response ADI", "Effect Details", "UniqueID"},
			"Least Squares Means Table",
			OutlineBox,
			{Close( 0 )}
		)
	)
);
ANOVA: 
Oneway( Y( :ADI ), X( :UniqueID ), Means( 1 ), Mean Diamonds( 1 ) )