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I want to test if the means of different groups (tissue types) are different using one-way ANOVA. Most of the groups have replicates but there is one group that only has one sample. Like this:

heart, heart, heart, skin, skin, skin,skin, bone, bone, bone, brain

In the above example, only one sample/data point for brain and the other three tissues have replicates.

So can ANOVA work for this data or it has to require all groups have replicates?

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It does not require that all groups have replicates. It is just a special kind of regression model, and the usual underlying assumption is that all observations have the same error SD. Under that assumption, ANOVA pools the information from groups that do have replicates to create an estimate of the error SD. Having no replicates in one or more groups makes it hard to do diagnostics to check on that equal-SD assumption. So you have to hope that the assumption actually holds, else the analysis is invalid.

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