When examining relationships between HEI scores as exposures and other variables as outcomes, interest is on quantifying the relationship between the HEI and the outcome.
The recommended approaches for estimating means and distributions of scores for a population, subpopulation, or group are intended to minimize the effects of measurement error in dietary intake data such that the results better reflect HEI scores for usual intake.
For analyses that require estimating scores for each person rather than means or distributions for a group, such as examining relationships between diet and health or other variables, there currently are few options for correcting for error in dietary intake data. Further, little is known about the impact of measurement error on the results of analyses that make use of the HEI.
Biomarkerbased validation studies focusing on energy and protein have shown that the observed effects of diet on health are biased (typically toward the null, or attenuated) when diet is measured with error (see Freedman et al 2014; Freedman et al 2015). Energy adjustment appears to lessen, though not eliminate, this problem. The extent to which these findings apply to models using the HEI is not yet understood. Further, techniques for dealing with measurement error in such analyses have not yet been developed. Until more is known about the effects of measurement error on analyses using HEI total or component scores as exposures in regression models, researchers should consider the potential for bias due to error in the interpretation of their results.
Below is a table that summarizes what methods are possible and which are recommended given various types of dietary intake data available.
Examining Association between Diet and Another Variable (e.g. health status)
Purpose  Method  Data  Considerations & Caveats 

To Calculate Total and Component HEI Scores and Estimate Regression Coefficients  Simple HEI Scoring Algorithm + Regression  Single 24HR on all  Yields HEI scores for a single day, contrary to the recommendation that HEI be based on usual intake. Will be biased to the extent that the 24HRreport of the intake of the component is biased. 
Single 24HR on all, multiple 24HRs on at least a subset  Yields HEI scores for a single day or the mean intake over a limited number of days, contrary to the recommendation that HEI be based on usual intake Will be biased to the extent that the 24HRreport of the intake of the component is biased. 

FFQ on all  Can be done, but is biased because FFQreported intakes are usually biased.  
Bivariate Approach + Regression  Single 24HR on all, multiple 24HRs on at least a subset  Can only be used to put a single component (or other ratio) into a regression model, unlike the MCMC method which allows for the multivariate HEI total score to be included in the model Doesn’t account for relationship between HEI components Gives estimates of single component (or other ratio) consistent with the MCMC method More userfriendly than MCMC method Including the FFQ data would only be recommended when the subset with 24HRs is very small compared to total sample and/or the subset with the 24HRs is not a random subsample of the total sample. (Outside of these cases the recommendation would be to just use the information from the 24HRs as the FFQ would not add much new information.) 

FFQ on all, multiple 24HRs on at least a subset  
Multivariate Approach (MCMC) + Regression  Single 24HR on all, multiple 24HRs on at least a subset  This method is under development, but is not currently available Allows the multivariate total HEI score to be included in a regression model, accounting for the relationship between the HEI components. The MCMC method is computationally intensive and has a steep learning curve. 

FFQ on all, multiple 24HRs on at least a subset  
Multiple 24HRs and FFQ on all participants 