Choosing Kruskal–Wallis test for group comparison with unequal sizes

[btomatoe]

I'm doing a survey that first seeks to identify the motivation types of university ESL students. My advisory committee added a comparison among university level groups as well (Freshmen, Sophomore, Junior, and Senior). The survey has 20 items which can be categorized into 7 motivation types.

The sample sizes are as follows: Freshmen - 103
Sophomore - 47
Junior - 61
Senior - 36

The sample sizes are unequal due to the dropout rate in the department. When I carried out the tests for normality, the Shapiro-Wilk test showed the data's distribution is not equal to a normal distribution in all areas except for the Senior group in two motivation types. Levene's Test of Homogeneity of Variances showed statistically significant differences for one motivation type. So the basic assumptions for ANOVA are not met which I'm not really sure is important because I think I probably need to be using a nonparametric test anyway. Also my second research question will be very similar but dealing with self-identity changes. Do I even need to consider ANOVA or just move straight to the nonparametric test because of the study design?

After reading and trying to understand it, I think I may need to choose the Kruskal–Wallis test for group comparisons. My advisor says she doesn't know about statistics, and what can I say? I'm an English major.

Any help would be greatly appreciated! I'm trying to learn.

[Karabiner]

Krukal-Wallis is a well-founded choice here. If you find a statistically
significant overall effect for the factor "university level" and want to
perform pairwise comparisons between the groups thereafter, you may
use Mann-Whitney U-tests as post-hoc tests (with significance level
corrected for multiple testing, e.g. by using the => Bonferroni method).

With kind regards

K.

[btomatoe]

Thank you very much!