(Pretty) large sample size, unequal variance, unequal sample sizes, non-normal distribution. T-test alternative?

So, I've never had all these problems at the same time before, but I have: - Likert data, from a 9 point scale - n = 317 v 177 respectively - SD = 2.08 v. .274; 1.9 v. 2.6; 2.5 v. 2.9 etc. (you get the picture: it's not terrible, but Levene's test is significant) - nonnormal distribution (positively skewed; quite flat)

If it's just unequal variances, but not unequal sample sizes, I'd be happy running a Student's. If it's just unequal variances and sample sizes, I'd run Welch's. But the normality violation is bothering me. I know that I shouldn't worry too much if the sample sizes are sufficiently large, but I've never really know how large is large enough not to worry.

Thoughts?

• I am puzzled about the remarks on "normality violation." It's impossible for data on a nine-point discrete scale to look even remotely normal except with very small data sizes. With 317 or even 177 data of course the responses will be demonstrably non-normal. But so what? – whuber Feb 11 '15 at 23:15
• I'm not sure what you mean. I've had plenty of data sets for Likert scale data that, while failing on the standard normality tests that are sample size sensitive, were fine if you inspect the histogram or run a significance test for skew and kurtosis. In this present case, the data is just uncontroversially non-normal, no matter how you look at it. – Jon Feb 12 '15 at 9:16
• What is the research question? If the two distributions also have clearly different shape, maybe a comparison only of means do not answer your questions. So maybe you need a method which aims at characterizing more fully the distributional differences? – kjetil b halvorsen Aug 17 '15 at 10:43