This type of estimation requires one way ANOVA if the data is normally distributed (the parametric test). In case your data is non normal, then you should use KW test to identify whether there are some differences between the groups. This test would yield the significance test to establish presence of differences between or absence of differences. To identify further, which group is radically different (high, or low), Dunn's test (1), or the more powerful but less well known Conover-Iman (2) test would help. 

1. Dunn, Olive Jean (1964). "Multiple comparisons using rank sums". Technometrics. 6 (3): 241–252. doi:10.2307/1266041

2. Conover, W. Jay; Iman, Ronald L. (1979). "On multiple-comparisons procedures" (PDF) (Report). Los Alamos Scientific Laboratory. Retrieved 2016-10-28

tldr: You can use Kruskal-Wallis, but it may not be the best approach.

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Your question seems to be, Can you use Kruskal-Wallis for unbalanced data.  The answer is, as far as I know, yes.

Some authors don't like using K-W or other traditional nonparametric tests with Likert data, because 1) the tests were derived for continuous data, and 2) there are likely to be many ties in the dependent variable.  Even though these tests can handle ties, they may not be designed to handle _many_ ties correctly.

I tend to think using traditional nonparametric tests with Likert data is an acceptable practice.

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More important problems are having only 2 observations in one group, and having many groups.  It seems that treating Years-of-experience as a continuous variable rather than a categorical variable might be a better approach, depending on what your data look like.

Dear Manjula

For ordinal data and more than two groups, your best choice is Kruskal-Walis test.

Good Luck

I'm not sure why this confusion (the number of responses in each category i.e., each years of experience are not equal). It is natural to have varying number for varying years of experience. As long as you have a DV and a IV it is possible to apply KW. In your case the DV is rating of 'peer-reviewed' (an ordinal type) and IV is years of experience (a ratio scale). So you have to have two or more groups of an independent variable. I think data set of 129 is good enough to run KW. 

Yes, the best choice for your study is KW if you have more than two categories.

But if you have categories with only two values (2 authors with 6 years of experience) your results will be less powerful and less reliable. To solve this problem you can merge two or three categories to increase the number of values. For example, 1-2 years of experience, 3-4 years of experience, 5-6 years of experience, ..... or 1-3 years of experience, 4-6 years of experience, 7-9 years of experience, 10-12 years of experience, ..... and so on.

Dear All,

Thanks for giving me some valuable clues. Almost all the answers imply that I still have the chance to apply KW test.

@ Tahir-

Yes, I already tested the way you have proposed after realizing the power of the test could too little due to the presence of very small groups. 

@Salvatore-

What do you mean by "traditional nonparametric tests with Likert data"? Could you please suggest me some examples?

Thanks.

By "traditional nonparametric tests" I mean Mann-Whitney, Kruskal-Wallis, Friedman, and others.