# Tests of significance, correlation vs chi-squared [closed]

I am comparing two ordinal variables along a Likert scale. From previous help received on this forum, I have decided to use the Spearman Rho for my correlation statistic. My next question is how to test significance. I have used a Chi-squared, but there is also a significance test for the actual Spearman Rho statistic. I'm not quite clear on what the two should be telling me. I know the Chi-squared should tell me whether there is a significant relationship at all, whereas the Spearman Rho p-value will tell me if the correlation coefficient is significant, so please don't give me answers along that line. Is it possible to have one be significant but not the other?

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• If we have ordinal variables, how do measure them on Likert scale? Do not seem to be compatible ? How do you conduct chi square test here ? – Subhash C. Davar Jan 17 '18 at 15:28
• @subhashc.davar en.wikipedia.org/wiki/Pearson%27s_chi-squared_test – Carl Jan 18 '18 at 21:34
• @subhashc.davar - why do you think a Likert scale isn't ordinal in nature? An ordinal variable is a categorical variable for which the possible values are ordered (lifted from a website), and that seems to describe the Likert scale precisely. – jbowman Jan 26 '18 at 3:34
• To me ordinal variable is a variable that reflects the ranking of say; students in terms of I.Q. The Likert scale measurments do not indicate rank of (a subject. It simply measures the strength or magnitude of a phenomenon(say, it a variable). you seem to have a visited an inappropriate site. Moreover, likert type scale is a bipolar scale It can take a positive or negative value. – Subhash C. Davar Jan 26 '18 at 10:29
• @subhashc.davar Likert scales are one of the best examples of ordinal data: You know the order of the responses but you cannot tell that the intervals between the levels are equal. Some Likert scales are bipolar, some are not, but that's not really relevant. – Peter Flom Jan 26 '18 at 13:41

No two different tests can reasonably be expected to yield the same probability of significance, that is implicit to their being genuinely different. The Pearson Chi-squared test implies the four assumptions of

(1) A simple random sample, that is, the sample data is a random sampling
from a fixed distribution or population where every collection of members
of the population of the given sample size has an equal probability of
selection. Variants of the test have been developed for complex samples

(2) A sufficiently large sample size. If a chi squared test is conducted on
a sample with a smaller size, then the chi squared test will yield an
inaccurate inference, e.g., committing a Type II error.

(3) Sufficiently large expected cell counts. Some require 5 or more, and
others require 10 or more. A common rule is 5 or more in all cells of a
2-by-2 table, and 5 or more in 80% of cells in larger tables, but no cells
with zero expected count. When this assumption is not met, Yates's
correction is applied.

(4) Independence.  The observations are always assumed to be independent of
each other. This means chi-squared cannot be used to test correlated data
(like matched pairs or panel data).


Since a certain frequency of data in one category correlates by subtraction to the other categories for the question at hand, condition (4) above, independence, is not met exactly leading in the binary response case to the Cochran's Q test. For an ordinal response, the Friedman test and the Durbin test are considerations.

By way of contrast, the Spearman's rho significance tests the strength of monotonicity of ranked data against no monotonicity and in so doing it adjusts for monotonic nonlinearity in the data. It seems to be the more appropriate test than Chi-squared. It is not only possible for one of these tests to be significant and the other not significant, but which is which is no doubt dependent on the characteristics of the data themselves, e.g., appropriateness of assumptions being a common explanation, another consideration the power of each test, but noise is another factor. That is, which is more significant can be, but does not have to be, due to chance alone.

Power for the Spearman's rho is almost identical to the Mann-Kendall test for detecting monotonicity in time series, also see corrigendum. The power of the Mann-Kendall test, in turn, is competitive with t-testing of slope, i.e., slight worse for linear conditions (as expected) and more powerful for nonlinearity. Power for the Friedman's test is modest and similar to Cochran Q is equivalent to the sign test's power. Thus, overall, I would expect Spearman's rho to be a better discriminator of significance than Friedman's test, in those circumstances where there is actual monotonicity.

• I am comparing two ordinal variables along a Likert scale. What is data your answer has used ? – Subhash C. Davar Jan 19 '18 at 12:42
• @subhashc.davar All are at least ordinal except Cochran's Q which is binary in multiple panels. – Carl Jan 19 '18 at 22:42
• Please respond the querry . I am not interested in Q- statistic. – Subhash C. Davar Jan 20 '18 at 0:20
• what is "all" and " at least ordinal" ? – Subhash C. Davar Jan 20 '18 at 0:24
• @subhashc.davar All means Pearson, Spearman, Friedman, Mann-Kendall, Durbin tests above. At least means could be real numbers as well, although in some contexts it also includes the complex field. These are basic questions, would suggest this language is used in the links I included which links were provided for those who desire further reading. – Carl Jan 20 '18 at 1:06

You wrote

I know the Chi-squared should tell me whether there is a significant relationship at all, whereas the Spearman Rho p-value will tell me if the correlation coefficient is significant, so please don't give me answers along that line. Is it possible to have one be significant but not the other?

If you really knew the first part, you would not need to ask the question.

The chi-square statistic ignores the ordinality of the data. The rank correlation does not. Either might be what you want, and it is certainly possible for one to be significant and the other not. There are also other statistics that use ordinal data; I like the Jonckheere-Terpstra test which seems to me to look at "ordinality" most strictly.