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 = 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.
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:
C results in higher scores for
However reading the course Material from Prof. Harrell, I have the "idea" that the hypothesis should be:
Treatment interacts with
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
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
Let us assume the interactions would have been significant (
TreatmentC:TimeNoon) and $H_0$ is rejected:
My conclusion would be that the best scores are reach for Treatment
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?