Analysis for ordinal categorical outcome

Question

I am not sure how to evaluate my data correctly, since the outcome (dependent variable) is on an ordinal categorical scale and there are two independent variables.

To simplify my experiment:

Each subject is assigned to one of three different Times (morning, noon or afternoon) and receives one Treatment (A,B or C). Afterwards a test is performed and the outcome is an ordinal categorical Score (1, 2 or 3).

The data can be summarized in three tables (one for each Time):

Time = Morning

            Score
 Treatment  "1"     "2"     "3"
   A        19      10      20
   B        6       7       5
   C        17      16      11

Time = Noon

            Score
 Treatment  "1"     "2"     "3"
   A        23      14      12
   B        15      33      15
   C        14      15      19


Time = Afternoon

        Score
 Treatment  "1"     "2"     "3"
   A        11      15      12
   B        17      18      15
   C        9       15      19

So the research question is: which treatment dependent on the time results in the best scores?

If the outcome would be some continuous normal distributed data, I would apply pair-wise ANOVAs and post hoc t-tests.

However in the given situation I have no experience and googling did not help me, that is why I am asking here for help.

Here it is stated that for tables larger than 2x2, in the case that the chi-squared test is significant, post hoc chi-squared tests of the format 2x2 can be applied.

My question is the following: Do I summarize the data in one big table, something like:

Morning/A
Morning/B
Morning/C
Noon/A
Noon/B
Noon/C
...

applying the chi-squared test and then following by pair-wise tests (like Morning/A vs Morning/B)? Or which statistical approach is the appropriate with post hoc tests to answer the research question?

P.S. The analysis is done with R, so if anyone has a tutorial or a link with an example that would be really helpful.

Edit 1)

I decide to edit my question and append my current approach, as it is still a part of the current question, if I should open a new question please let me know.

Taking the answer of @(Frank Harrell) into account I tried using "Ordinal Logistic Regression" (OLR).

My hypothesis is that:

Treatment C results in higher scores for Noon and Afternoon.

However reading the course Material from Prof. Harrell, I have the "idea" that the hypothesis should be:

Treatment interacts with Time.

Question E1) Am I correct about the adjustment of the hypothesis for (OLR)?

So this is what I tried(I should say that my data has two more score values than in my example above, it was designed exemplary, in my case there are the following scores 0,1,2,3 and 4 ):

polr(formula = Outcome ~ Treatment + Time , data = Data, Hess = T)

getting the following results

                          Value       Std. Error   t value    p value
TreatmentB               -0.26373775  1.0059962 -0.26216576   0.793
TreatmentC               -0.62342836  0.8070730 -0.77245595   0.440
TimeNoon                  0.08236084  0.8257454  0.09974121   0.921
TimeAfterNoon             0.74163895  0.9282499  0.79896475   0.424
TreatmentB:TimeNoon      -0.53242839  1.1327578 -0.47002844   0.638
TreatmentC:TimeNoon       0.19885488  0.9406062  0.21141141   0.833
TreatmentB:TimeAfterNoon -0.56413612  1.2321034 -0.45786428   0.647
TreatmentC:TimeAfterNoon -1.32190703  1.0447585 -1.26527520   0.206
0|1                      -3.96290826  0.8499135 -4.66271949   0.000
1|2                      -2.60173952  0.7747232 -3.35828277   0.001
2|3                      -0.56198118  0.7460030 -0.75332292   0.451
3|4                       2.46049479  0.7878248  3.12314968   0.002

Questions E2)

As the interaction are all not significant, $H_0$ can not be rejected? So the conclusion here is: The outcome does not depend on the interaction between Treatmentand Time?

Let us assume the interactions would have been significant (TreatmentB:TimeNoon and TreatmentC:TimeNoon) and $H_0$ is rejected:

My conclusion would be that the best scores are reach for Treatment B and C during Noon, is this assumption correct? In the second step I would like to make a statement which of the two treatments is the better one. Is it enough to check whether on of both is significant or which approach is the correct one here? Do I need further post hoc test to make a statement like this?

Answer 1

For ordinary response variables Y, use a method that takes advantage of the ordinality, to increase power by reducing the number of parameters in the model. Ordinary $\chi^2$ tests for contingency (frequency) tables involve a lot of parameters because they consider Y as polytomous (multinomial; categorical).

To handle the repeated-within-subject nature of the data, you can incorporate random subject effects. So combining a proportional odds ordinal logistic model with random effects is advised, e.g., R package ordinal. Even better (fewer approximations) would be a Bayesian mixed effects ordinal model.