I have a set of 35 independent variables. I do not have response variable for my data set. I used density plots to identify multi-modal distribution in my independent variables. Hence, I used Gaussian mixture clustering technique to group the data. Upon clustering, I obtained 6 clusters.
I designed hypothesis to test my results as follows
Hypothesis 1: H0: there is no significant difference in means in the clusters formed.
Before proceeding to ANOVA, I did Shapiro - Wilk normality test (rejected null hypothesis W = 0.99132, p - value = 1.623e-12) and outlier test (found that there are outliers in the data)
Next, I did Levene 's Test for Homogeneity of Variance and found that variances for groups are unequal. Results were same for Bartlett test, and Fligner - Killeen test.
From this, I failed to meet the ANOVA assumption. Now, that ANOVA is out of the picture what non-parametric test should I use to test that the clusters I have are unique or distinct. Or should I use Test for Homogeneity of Variance to redefine my hypothesis and conclude my finding?
Just out of curiosity, I did do ANOVA and MANOVA analysis and from the results, I could reject null hypothesis.
I have looked into different validation indices such as silhouette width (using this to confirm the number of clusters formed are optimal), Dunn, pearsongamma etc.
I have over 100 similar data sets I need to validate. Any help regards to this is very much appreciated.
PS: my data is normalized data with mean = 0 and SD=1
Cluster comparison is incorporated in cluster analysis. It may not be necessary to further investigate the cluster effects.
You have more than 100 sample sets and not clusters. By the use of their dendrogram you can group them into clusters. The process involves distinguishing them accordingly. The distinction between them was the basis of their clustering in the first place.
I am sorry but I don't understand what you are trying to do. First, are you clustering observations or variables?.Second, what is the point of hypothesis 1? Having different means doesn't guarantee the clusters are different perhaps there is outlier contamination. In fact, Third. The S-W test rejected Ho:popn normal and found outliers with a test which is not too surprising. This doesn't say you can't use ANOVA. What about a Box-Cox transform or something if the ANOVA model RESIDUALS turn out to be non-normal. Note this is not the same as finding the cluster distributions are non-normal. Fourth If the clusters have different variances, the cluster distributions must be different except for the presence of those outliers which again muddy the water. It looks to me like you have used a lot of computer time without a plan to achieve your goal which you have not mentioned. HELP. DEB BTW As Dr Etuk points out if the cluster analysis algorithm makes sense and was properly applied the clusters should be different. Now you might want to consider the outliers depending on your goal in doing the analysis.
Dr. Ette Etuk Thank you for your feedback and clarification.
Dr. Booth, I am actually clustering based on observations. I was trying to show statistical significance among cluster groups. In a way my assumption testing ended up overtesting.
I am building a real-time application that clusters the data (mostly signal data in real time as well) into groups. I will be using these clusters to study the signals and eventually come up with a classification model. Before jumping into studying the clusters, I wanted to make sure the clusters were statistically significant.
Update: I did some research and found non-parametric approach called PERMANOVA for testing the data for statistical significance. This technique is mostly used in ecological data. I still need to do more research on this.
The clusters are as significantly different as you are going to get given the data and the method of clustering. I would now concentrate on the original research question using the clusters that you found. Hope this helps alittle. Best, David
It is important to note that after you have clustered data objects into groups, statistical testing for differences between groups is biased (because the determination of groups has already taken the data into account).
I think Kruskal-Wallis would work as a non-parametric alternative to ANOVA :)
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