Jonckheere-Terpstra test: two-sided and one-sided, directed and undirected hypotheses

Question

I need to perform a Jonckheere-Terpstra test, but I am not sure how to do it and how to interpret the results.

I have following data: Three groups doctors1, doctors2, doctors3 And how often they ask their patients about their emotional state (0 never - 4 always).

I predict such order doctors1 < doctors2 < doctors3

In SPSS I chose the non-parametric tests JT and than-"from smallest to largest" (groups 1-2-3) and pairwise comparison. My results are p value 0.077 (two-sided)

My questions are: I dont really understand what exactly "from smallest to largest" means. Does it mean just 1-2-3 order of the groups or is it my hypothesis, that the first group has the smallest value?

Second question is - should I divide p value by two? I have a directional hypothesis... if i do it, than i would get statistical sign results. I am really confused.

Answer 1

Jonckheere-Terpstra test is appropriate for ordered samples only. If your three groups of doctors are knowingly ordered upon some characteristic which may be important in treating their patient's emotional state - then you may use the test. For example, group1 could be students, group2 interns and group3 experienced doctors. You might expect that group3>group2>group1 in skills in treatment of psychiatric patients. But if you can't order your groups of doctors meaningfully before the statistical test - don't use Jonckheere-Terpstra, use Kruskal-Wallis instead.

I recommend you to read Jonckheere-Terpstra interpretation with a number of good answers related to your question.

In the present answer I want to focus on the interpretation of p-value of the test. What statistical hypotheses does the singificance pertain to?

According to SPSS Statistics Algorithms document, Jonckheere-Terpstra outputs 2-sided sig. $p$$^1$. This is significance of the two sided test (let S1, S2,...,Sk denote the location parameters of the corresponding ordered samples which codes ascend like 1, 2,...,k):

H0: S1=S2=...Sk vs H1: {S1<=S2<=...Sk with at least one strict inequality (<) or S1>=S2>=...Sk with at least one strict inequality (>)}.

So the alternative hypothesis H1 of this test is double. If you take in consideration the sign of the standardized test statistic T you can say in favour of which part of H1 the null is being rejected; namely in favour of the S1<=S2<=...Sk - when T>0, or in favour of the S1>=S2>=...Sk - if T<0. Consequently, $p/2$ is one-sided significance for case "T>0 and H1: S1<=S2<=...Sk", as well as for case T<0 and H1: S1>=S2>=...Sk".

In full (for any sign of T) the two one-sided tests are formulated as:

H0: S1=S2=...Sk vs H1asc: S1<=S2<=...Sk with at least one strict inequality (<). The one-sided p-value of it is: $p/2$, if T>0, and $1-p/2$ otherwise.

H0: S1=S2=...Sk vs H1desc: S1>=S2>=...Sk with at least one strict inequality (>). The one-sided p-value of it is: $p/2$, if T<0, and $1-p/2$ otherwise.

When you request Jonckheere-Terpstra in SPSS, it wants you to specify "Hypothesis order": largest to smallest or smallest to largest. You will notice that whichever you choose does not influence the output two-sided $p$ (and the value or sign of T). The specification is important not for the omnibus result but for the post-hoc multiple comparisons of the samples (if you request to do it after the omnibus test). The omnibus significance that the multiple comparisons will be based on is the significance of the described above one-sided test: smallest to largest corresponds to H1asc while largest to smallest to H1desc.

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$^1$ Which is, by the way, equal to the significance of Kendall' tau-b correlation between those dependent and independent variables.

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Example.

   group response

       1      1.0
       1      1.0
       1      1.0
       1      1.0
       1      1.0
       1      2.0
       1      2.0
       1      2.0
       1      2.0
       1      3.0
       1      3.0
       2      1.0
       2      1.0
       2      1.0
       2      1.0
       2      2.0
       2      2.0
       2      2.0
       2      2.0
       2      2.0
       2      3.0
       3      1.0
       3      1.0
       3      2.0
       3      2.0
       3      2.0
       3      2.0
       3      2.0
       3      2.0
       3      2.0
       3      2.0
       3      2.0
       3      2.0
       3      2.0
       3      2.0
       3      2.0
       3      2.0
       3      3.0
       3      3.0
       3      3.0
       3      3.0
       3      3.0
       3      3.0
       3      3.0
       3      3.0

Like Kruskal-Wallis, Jonckheere-Terpstra compares stochastic dominance (not means or medians) or nonparametric locations, and the most appropriate statistic to display now for the 3 groups would be Hodges-Lehmann pseudomedian. I'm showing, however, mean and median, because they still relate the fact that the 3rd group is the highest response while the 1st and the second are similar:

Request Jonckheere-Terpstra test with "Hypothesis order": smallest to largest (ascending):

NPTESTS 
  /INDEPENDENT TEST (response) GROUP (group) JONCKHEERE_TERPSTRA(ORDER=ASCENDING COMPARE=PAIRWISE) 
  /MISSING SCOPE=ANALYSIS USERMISSING=EXCLUDE
  /CRITERIA ALPHA=0.05  CILEVEL=95.

The 2-sided $p=.016$ is significant. If you want a one-sided, directed alternative hypothesis, which one is it? If it is ascending, S1<=S2<=...Sk one, then its p-value is $p/2=.008$ because the standardized statistic T is positive. If it is descending, S1>=S2>=...Sk one, its p-value is $1-.008$.

Pairwise multiple comparisons (based on Mann-Whitney U):

reveal significant differences group3>group2 and group3>group1 (though Bonferroni-corrected remains only the first difference).

If you requested "Hypothesis order": largest to smallest you'd have every statistic the same, yet there would be no significant post-hoc differences:

which is understandable since the omnibus test significance for the descending-order one-sided alternative is $1-.008$, as mentioned.