Repeated ordinal dependent variable with ordinal independent

[Jorge Ruiz Moreno]

I have a long format data set with a repeated by year ordinal dependent variable Y (0=mild, 1=moderate, 2=severe), the repeated factor year (0=2014, 1=2016) and centre (1, 2, 3, 4). I would like to test the marginal homogeneity in the variable centre. What I’m doing is:
use dataset_long, clear
xtset id year
ologit Y i.centre##year b4.i.centre

Obtaining (cut points omitted)

Y Coef. Std. Err. z P>z [95% Conf. Interval]
centre
1 -.1290855 .3169848 -0.41 0.684 -.7503643 .4921933
2 -.5908961 .2714226 -2.18 0.029 -1.122875 -.0589175
3 -.1446371 .3012552 -0.48 0.631 -.7350864 .4458122
1.year -.760835 .2815766 -2.70 0.007 -1.312715 -.2089551
centre#year
1 1 | 1.094596 .4460686 2.45 0.014 .2203171 1.968874
2 1 | .8680099 .3843708 2.26 0.024 .1146569 1.621363
3 1 | 1.053251 .4283335 2.46 0.014 .2137328 1.892769

Could someone tell me if this code is correct? If so, I would appreciate if you may indicate me some reference for interpreting centre#year.
Thank you very much.

[Jorge Ruiz Moreno]

Thank you very much Joseph

[Joseph Coveney]

How to tell Stata that this is a panel (repeated measures) dataset?

Maybe something like that below. Like Clyde, I'm not sure what you mean "to test the marginal homogeneity in the variable centre". I gave it my best guess. Begin at the "Begin here" comment.

.ÿversionÿ15.1

.ÿ
.ÿclearÿ*

.ÿ
.ÿprogramÿdefineÿgrologit
ÿÿ1.ÿÿÿÿÿÿÿÿÿversionÿ15.1
ÿÿ2.ÿÿÿÿÿÿÿÿÿsyntaxÿvarname(numeric)ÿ,ÿGenerate(name)ÿCuts(numlist)ÿ[Start(integerÿ1)]
ÿÿ3.ÿ
.ÿÿÿÿÿÿÿÿÿlocalÿindexÿ1
ÿÿ4.ÿÿÿÿÿÿÿÿÿforeachÿcutÿofÿnumlistÿ`cuts'ÿ{
ÿÿ5.ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿtempvarÿcut`index'
ÿÿ6.ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿgenerateÿdoubleÿ`cut`index''ÿ=ÿinvlogit(`varlist'ÿ+ÿ`cut')
ÿÿ7.ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿlocalÿcut_listÿ`cut_list'ÿ`cut`index''
ÿÿ8.ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿlocalÿ++index
ÿÿ9.ÿÿÿÿÿÿÿÿÿ}
ÿ10.ÿ
.ÿÿÿÿÿÿÿÿÿtempvarÿrandu
ÿ11.ÿÿÿÿÿÿÿÿÿgenerateÿdoubleÿ`randu'ÿ=ÿruniform()
ÿ12.ÿ
.ÿÿÿÿÿÿÿÿÿgenerateÿbyteÿ`generate'ÿ=ÿ`start'
ÿ13.ÿÿÿÿÿÿÿÿÿforeachÿcutÿofÿlocalÿcut_listÿ{
ÿ14.ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿlocalÿ++start
ÿ15.ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿquietlyÿreplaceÿ`generate'ÿ=ÿ`start'ÿifÿ`randu'ÿ>ÿ`cut'
ÿ16.ÿÿÿÿÿÿÿÿÿ}
ÿ17.ÿend

.ÿ
.ÿsetÿseedÿ`=strreverse("1462188")'

.ÿ
.ÿ//ÿcentres
.ÿquietlyÿsetÿobsÿ4

.ÿgenerateÿbyteÿcidÿ=ÿ_n

.ÿ
.ÿ//ÿpersonsÿwithÿanÿunfavorableÿevent
.ÿquietlyÿexpandÿ250

.ÿgenerateÿintÿpidÿ=ÿ_n

.ÿgenerateÿdoubleÿpid_uÿ=ÿrnormal()

.ÿ
.ÿ//ÿtheÿrepeatedÿfactorÿyear
.ÿquietlyÿexpandÿ2

.ÿbysortÿpid:ÿgenerateÿbyteÿtimÿ=ÿ_n

.ÿlabelÿdefineÿYearsÿ1ÿ2014ÿ2ÿ2016

.ÿlabelÿvaluesÿtimÿYears

.ÿ
.ÿ//ÿordinalÿdependentÿvariableÿY
.ÿgenerateÿdoubleÿxbÿ=ÿ///
>ÿÿÿÿÿÿÿÿÿ0ÿ*ÿcidÿ+ÿ///
>ÿÿÿÿÿÿÿÿÿpid_uÿ+ÿ///
>ÿÿÿÿÿÿÿÿÿ0ÿ*ÿtim

.ÿlocalÿthresholdÿ=ÿ_piÿ/ÿsqrt(3)

.ÿgrologitÿxb,ÿg(out)ÿc(-`threshold'ÿ`threshold')ÿs(0)

.ÿlabelÿdefineÿSeveritiesÿ0ÿMildÿ1ÿModerateÿ2ÿSevere

.ÿlabelÿvaluesÿoutÿSeverities

.ÿ
.ÿ*
.ÿ*ÿBeginÿhere
.ÿ*
.ÿmeologitÿoutÿi.cidÿi.timÿ||ÿpid:ÿ,ÿnolrtestÿnolog

Mixed-effectsÿologitÿregressionÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿNumberÿofÿobsÿÿÿÿÿ=ÿÿÿÿÿÿ2,000
Groupÿvariable:ÿÿÿÿÿÿÿÿÿÿÿÿÿpidÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿNumberÿofÿgroupsÿÿ=ÿÿÿÿÿÿ1,000

ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿObsÿperÿgroup:
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿminÿ=ÿÿÿÿÿÿÿÿÿÿ2
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿavgÿ=ÿÿÿÿÿÿÿÿ2.0
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿmaxÿ=ÿÿÿÿÿÿÿÿÿÿ2

Integrationÿmethod:ÿmvaghermiteÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿIntegrationÿpts.ÿÿ=ÿÿÿÿÿÿÿÿÿÿ7

ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿWaldÿchi2(4)ÿÿÿÿÿÿ=ÿÿÿÿÿÿÿ2.34
Logÿlikelihoodÿ=ÿ-1783.7521ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿProbÿ>ÿchi2ÿÿÿÿÿÿÿ=ÿÿÿÿÿ0.6738
------------------------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿoutÿ|ÿÿÿÿÿÿCoef.ÿÿÿStd.ÿErr.ÿÿÿÿÿÿzÿÿÿÿP>|z|ÿÿÿÿÿ[95%ÿConf.ÿInterval]
-------------+----------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿcidÿ|
ÿÿÿÿÿÿÿÿÿÿ2ÿÿ|ÿÿÿ.1301819ÿÿÿ.1603452ÿÿÿÿÿ0.81ÿÿÿ0.417ÿÿÿÿ-.1840889ÿÿÿÿ.4444528
ÿÿÿÿÿÿÿÿÿÿ3ÿÿ|ÿÿ-.0985216ÿÿÿ.1602031ÿÿÿÿ-0.61ÿÿÿ0.539ÿÿÿÿ-.4125139ÿÿÿÿ.2154708
ÿÿÿÿÿÿÿÿÿÿ4ÿÿ|ÿÿÿ.0647923ÿÿÿ.1605154ÿÿÿÿÿ0.40ÿÿÿ0.686ÿÿÿÿ-.2498122ÿÿÿÿ.3793968
ÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿtimÿ|
ÿÿÿÿÿÿÿ2016ÿÿ|ÿÿ-.0325286ÿÿÿ.0941207ÿÿÿÿ-0.35ÿÿÿ0.730ÿÿÿÿ-.2170017ÿÿÿÿ.1519445
-------------+----------------------------------------------------------------
ÿÿÿÿÿÿÿ/cut1ÿ|ÿÿ-1.716974ÿÿÿ.1381387ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ-1.987721ÿÿÿ-1.446227
ÿÿÿÿÿÿÿ/cut2ÿ|ÿÿÿ1.893418ÿÿÿ.1411132ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ1.616841ÿÿÿÿ2.169994
-------------+----------------------------------------------------------------
pidÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿvar(_cons)|ÿÿÿ.9954363ÿÿÿ.2174425ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ.6487502ÿÿÿÿ1.527388
------------------------------------------------------------------------------

.ÿ
.ÿ/*ÿRejectÿnullÿhypothesisÿofÿheterogeneityÿatÿαÿ=ÿ5%ÿinÿfavorÿofÿalternative
>ÿÿÿÿhypothesisÿofÿhomogeneityÿ→ÿeachÿandÿeveryÿ90%ÿCIÿcontainedÿwithin,ÿsay,ÿ½ÿandÿ2ÿ*/
.ÿforvaluesÿiÿ=ÿ1/3ÿ{
ÿÿ2.ÿÿÿÿÿÿÿÿÿquietlyÿlincomÿ_b[4.cid]ÿ-ÿ_b[`i'.cid],ÿlevel(90)ÿor
ÿÿ3.ÿÿÿÿÿÿÿÿÿdisplayÿinÿsmclÿasÿtextÿ"Centreÿ`i':ÿ"ÿasÿresultÿ%04.2fÿr(lb),ÿ%04.2fÿr(ub)
ÿÿ4.ÿ}
Centreÿ1:ÿ0.82ÿ1.39
Centreÿ2:ÿ0.72ÿ1.22
Centreÿ3:ÿ0.90ÿ1.53

.ÿ
.ÿ/*ÿTestÿforÿhomogeneityÿif,ÿsay,ÿtheÿcentersÿwereÿinterventionsÿwhereÿequality
>ÿÿÿÿisÿtenableÿandÿinterestingÿ*/
.ÿtestparmÿi.cid

ÿ(ÿ1)ÿÿ[out]2.cidÿ=ÿ0
ÿ(ÿ2)ÿÿ[out]3.cidÿ=ÿ0
ÿ(ÿ3)ÿÿ[out]4.cidÿ=ÿ0

ÿÿÿÿÿÿÿÿÿÿÿchi2(ÿÿ3)ÿ=ÿÿÿÿ2.22
ÿÿÿÿÿÿÿÿÿProbÿ>ÿchi2ÿ=ÿÿÿÿ0.5281

.ÿ//ÿor
.ÿestimatesÿstoreÿFull

.ÿquietlyÿmeologitÿoutÿi.timÿ||ÿpid:

.ÿlrtestÿFull

Likelihood-ratioÿtestÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿLRÿchi2(3)ÿÿ=ÿÿÿÿÿÿ2.22
(Assumption:ÿ.ÿnestedÿinÿFull)ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿProbÿ>ÿchi2ÿ=ÿÿÿÿ0.5275

.ÿ
.ÿexit

endÿofÿdo-file


.

[Jorge Ruiz Moreno]

Dear Clyde,
Regarding “clogit” I did not mean Stata clogit command. I meant the cumulative logits of cumulative probabilities log[P(moderate, severe)/P(mild)] and log[P(severe)/P(mild, moderate)] following (Y>m)/P(Y≤m).
Thank you!

[Clyde Schechter]

1. Regarding clogits, when you say "odds of being above vs. at or below any of the the levels of Y", just to be sure, do you mean P(Y>m)/P(Y≤m).

Yes, that is what I mean.

If so, in this case we would have two clogits with the unfavorable event in the numerator

I have no idea what this means. What does clogit have to do with this?

log[P(mod, sev)/P(mild)]
and
log[P(sev)/P(mild, mod)]

In the -ologit- model, there is a constraint (assumption) that these two are equal, so there is only one such probability ratio.

My interpretation of 1 1: the effects of year 2016 (year = 1) over distribution of Y in centre 1 is 2.98 times higher than in center 4. Is this interpretation correct?

It is correct if by effect you mean ratio of odds ratios.

My interpretation of 0 1: for year 2014 in centre 1, we expect to find 0.88 persons with an unfaforable event for every person without it
My interpretation of 1 1: for 2016 in centre 1 we expect to find 1.22 persons with an unfavorable event for every person without it
Unfavorable event refers to P(Y>m)/P(Y≤m)
Are these interpretations correct?

Correct.

[Jorge Ruiz Moreno]

Clyde and Joseph, thank you very much for the answers.

After reflecting:

1. Regarding clogits, when you say "odds of being above vs. at or below any of the the levels of Y", just to be sure, do you mean P(Y>m)/P(Y≤m). If so, in this case we would have two clogits with the unfavorable event in the numerator
log[P(mod, sev)/P(mild)]
and
log[P(sev)/P(mild, mod)]

2. After exponentiate coefficients I have obtained odds ratio (centre=4 as reference):

ologit Y i.centre##year b4.i.centre, or

(excerp)
Y Odds Ratio Std. Err. z P>z [95% Conf. Interval]
centre#year
1 1 2.987974 1.332842 2.45 0.014 1.246472 7.162607
2 1 2.382165 .9156349 2.26 0.024 1.121489 5.059982
3 1 2.866956 1.228013 2.46 0.014 1.238292 6.637724

My interpretation of 1 1: the effects of year 2016 (year = 1) over distribution of Y in centre 1 is 2.98 times higher than in center 4. Is this interpretation correct?

3. On the other hand I used the margins command for obtaining odds for combinations of year and centre (following Maarten L.Buis, STATA tip 87: Interpretation of interactions in nonlinear models, The Stata Journal (2010) 10, Number 2, pp. 305-308)

margins, over(year centre) expression(exp(xb())) post

Delta-method
Margin Std. Err. z P>z [95% Conf. Interval]
year#centre
0 1 .8788988 .2785976 3.15 0.002 .3328576 1.42494
0 2 .5538308 .1503222 3.68 0.000 .2592047 .8484569
0 3 .8653362 .260687 3.32 0.001 .3543991 1.376273
0 4 1 . . . . .
1 1 1.227126 .3833175 3.20 0.001 .4758378 1.978415
1 2 .6164851 .1679434 3.67 0.000 .287322 .9456481
1 3 1.159256 .3513713 3.30 0.001 .4705814 1.847931
1 4 .4672761 .131574 3.55 0.000 .2093958 .7251564

My interpretation of 0 1: for year 2014 in centre 1, we expect to find 0.88 persons with an unfaforable event for every person without it
My interpretation of 1 1: for 2016 in centre 1 we expect to find 1.22 persons with an unfavorable event for every person without it
Unfavorable event refers to P(Y>m)/P(Y≤m)
Are these interpretations correct?

4. As pointed out by Joseph it is not necessary to use xtset before ologit command. Then, How to tell Stata that this is a panel (repeated measures) dataset?

Again thank you for your valuable help.

[Joseph Coveney]

Just be aware that

xtset id year

doesn't do much with ologit

[Clyde Schechter]

Your code looks like a correct implementation of an ordinal logistic regression of outcome Y with predictors center and year and their interaction.

I am not sure what you mean by "test the marginal homogeneity in the variable centre." If you wish to test the hypothesis that the marginal distribution of Y is the same in all four centers, you can do that by following your -ologit- command with:

margins center, contrast

The overall test of homogeneity will be in the output row that is called "Joint"

The interaction terms tell you whether the effect of center on the distribution of Y differs between the two years. (Or, equivalently, they tell you whether the effect of year on the distribution Y differs among the centers.) The direct interpretation of the individual interaction term coefficients is somewhat complicated because the -ologit- model is, itself, somewhat complicated to explain. If you exponentiate each coefficient you will get ratios (2016 vs 2014) of odds ratios for each center. The term "odds ratio" in an ordinal logistic model refers to the odds of being above vs. at or below any of the the levels of Y in association with a unit difference in the predictor. In this case, each predictor is an indicator for a center, so it refers to the odds of being above vs. at or below any of the levels of Y in association with being in that center vs being in center 4 (the reference center).