How to consider replicated measurements over time as a random effect?

[vitormacedo]

Dear all.

I have a data set as follows:

data test;
input Person Day Drug$ Response;
cards;
1 1 A .
1 2 A .
1 3 A .
2 1 A .
2 2 A .
2 3 A .
3 1 B .
3 2 B .
3 3 B .
4 1 B .
4 2 B .
4 3 B .
5 1 C .
5 2 C .
5 3 C .
6 1 C .
6 2 C .
6 3 C .
;

My goal is to test the difference between drugs. To increase the amount of data, assessment was made over three days (Day) on each Person. Initially, I thought of the following command to analyse the data:

proc mixed data=test;
class Droug Day;
model Response=Droug/htype=3;
random Day/type=covariance_structure;
lsmeans Droug;
run;

"random Day" because Day is a random effect factor as block, and "type = covariance_structure" because days are correlated, because they are measured in the same person.

However, I read the following paper: https://support.sas.com/resources/papers/proceedings/proceedings/sugi29/198-29.pdf . And maybe the correct command is as follows:

proc mixed data=test;
class Droug Day Person;
model Response=Droug/htype=3;
random Droug/type=covariance_structure subject=Person;
repeated Droug/type=covariance_structure subject=Day(Person);
lsmeans Droug;
run;

I'm in doubt!!!

For my goal. Can I use the first command? Or do I have to use the second command?

I appreciate all the help. Thanks.

[sld · Accepted solution]

If you assume that PERSON is a random effect and that DAY is a random effect, then PERSON and DAY are crossed random effects: each PERSON is observed for each DAY (each PERSON is a block for DAYs), and each DAY is observed for each PERSON (each DAY is a block for PERSONs). More commonly, random effects are nested, but crossed is possible. Generally, you would consider PERSONs to be independent, but DAYs might possibly be correlated.

In that scenario, I would consider this code

proc mixed data=test;
    class person drug day;
    model response = drug;
    random person(drug);
    random day / subject=intercept type=<some covariance structure> g;
    lsmeans drug;
    run;

The paper that you link to does not address this type of model structure, so I do not think it provides an appropriate template for analysis.

I hope this helps.

[vitormacedo]

Thank you @sld for your help.