Originally, I intended to conduct independent samples t-test and one-way ANOVA in SPSS for analysing my data. However, when I examined the normality of the dependent variable (DV) for checking their assumptions, it showed that my DV is highly skewed as the attached photo.
My DV is measured through an open-ended question and it is a continuous variable about participants’ predictions of the duration of an emotion, ranging from 0 to almost 300 unit. Sample size is around 1000.
Since the normality assumption is violated, I am wondering:
Thank you in advance for your advice.
IYH Dear Pcl Lee
tL/dr: given the skewness of your DV, a GLM might be a better choice if you want to examine interactions btw variables, but requires careful consideration of the assumption of homoscedasticity.
Kindly allow me to justify this recommendation:
a) Nonparametric Tests vs. GLM Given the skewness of your DV, like you concluded a nonparametric tests such as the Mann-Whitney U Test for independent samples t-tests and the Kruskal-Wallis H Test for one-way ANOVA is more suited. Both require the assumption of identical distributions across groups, which can be checked visually using QQ plots or formally tested using the Shapiro-Wilk test.
However, a GLM w a gamma distribution and log link function allows for the modeling of the mean & variance of the response variable simultaneously, whereas nonparametric tests only provide information about differences in location (median). Additionally, GLMs can handle covariates and interactions, allowing for the examination of potential moderators.
Hence, given your expressed interest in examining the moderating effect of a categorical variable on the relationship btw a continuous IV and a DV, a GLM w a gamma distribution and log link function might be a better choice since it allows for the examination of interactions btw variables.
You can still examine the interaction between a continuous IV and a categorical moderator in a GLM, even if the DV is not normally distributed. The key assumption to check in this case is homoscedasticity, meaning equal variances across levels of the moderator. Violations of this assumption can lead to biased estimates of the interaction term. One way to check for heteroscedasticity is to plot residuals against predicted values separately for each level of the moderator
b) Reporting Results: If you decide to conduct both nonparametric and parametric tests, best honest research practice is to report both sets of results to ensure transparency and completeness. Then discuss the implications of any discrepancies btw the results obtained from the two approaches. Should the results be similar, it is acceptable to report only the parametric results (no harm in reporting the other tests, though).
Good answer from Daniel! Be careful with nonparametric tests that are based on ranks (alternatively you may bootstrap the null distribution of the test statistic - this also non-parametric but not rank-based, so it allows to address non-additive effects). I am sure that the distributions in your groups are not comparable (except a location shift) - there will be a correlation of mean and variance, making the interpretation of rank-based tests difficult.
Something he forgot to mention: linear models (from which t-tests and ANOVAs are derived) do not require that the DV is normal distributed but that the residuals are normal distributed. So when you check the distribution, check the residuals, not the raw data. There are a couple of useful "residual diagnostic plots" that help you judging if the assumptions about the residual distribution are reasonable (including variance homogeneity).
I`d first check how the (residual!) distribution looks like when you use the log-transformed values of the DV. If the residual plots look inconspicuous, you can continue with that data (the log-transformation is monotone, so if the mean log in group A is larger than that in group B, and if the variances of the log data are similar in both groups, then then mean of A must also be larger than the mean of B).
Log-transformation or log-link (in a GLM) will change the interpretation of interactions: the non-additivity is then analyzed on the log-scale, what means non-multiplicativity on the original scale. Note that the decision what makes sense is a subject-matter decision, not a statistical decision.
If you have a (a lot of) zero values you may consider a quasi-Poisson model or a Tweedie-model or a hurdle model.
Thanks to Jos for reminding that it's about reaction times. There is a lot of literature about the distributions and the analysis of reaction times, e.g.
Article What Are the Shapes of Response Time Distributions in Visual Search?
https://dominiquemakowski.github.io/post/2020-05-18-analyze_rt/
https://minhoshin.net/blog/2021-08-17-rt-analysis/
The DV can be log transformed which will reduce the skewness and bring the data close to normality. Then you can carry out the usual analysis.
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