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I would like to test whether the fish abundance (response variable) differs among three levels (low, intermediate and high) of aquatic plant biomass. However, I have only three replicates of each level of plant biomass.

How possible is to use an ANOVA for three values per biomass level?

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  • $\begingroup$ Is abundance a count? $\endgroup$
    – Glen_b
    Commented Oct 12, 2020 at 3:51
  • $\begingroup$ Yes. This is one example, but actually, I would like to test the abundance (count), richness (count) and diversity (index) of fish among those levels of biomass plant. The assumptions of normality and homogeneity of variance were met for the three response variables, but I don't know if I really can use an ANOVA with just nine samples (three samples per level). $\endgroup$
    – Barbara Quirino
    Commented Oct 12, 2020 at 12:31
  • $\begingroup$ Pointless testing for normality when you know it cannot actually be normal. (Well it's not really a great idea even when it can be normal either.) I'd usually be inclined to use an analysis related to models for count data. Are these counts are necessarily >0, or are 0's potentially possible? $\endgroup$
    – Glen_b
    Commented Oct 13, 2020 at 6:00

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If I understand correctly your description of your design, you have nine observations in all. So the equation for the model is:

$$Y_{ij} = \mu + \beta_i + e_{ij},$$

for $i = 1,2,3$ levels of biomass and $j = 1,2,2$ replications per level, where $\mu$ is the grand mean, $\beta_i$ express possible differences (effects) due to to biomass, and $e_{ij} \stackrel{iid}{\sim}\mathsf{Norm}(0, \sigma).$

Then your ANOVA table will show $DF(Factor)=2,\,$ $DF(Resid) = 3(2)$ and $DF(Total) = 2+6 = 8.$

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  • $\begingroup$ Yes, you are correct. I have nine observations (three per level). Indeed, the ANOVA table showed Df (Factor)= 2 and Df(Resid)= 6. Does this mean that it is possible to obtain the p-value reliably? I mean, isn't it wrong, although I know that more power would be added to the analysis if I had more observations per level? $\endgroup$
    – Barbara Quirino
    Commented Oct 12, 2020 at 12:44
  • $\begingroup$ Seems to be a correct ANOVA. You would have more power if you had more replications per level of biomass. But what P-value do you get? If it's sufficiently small to reject the null hypothesis that level of biomass doesn't make a difference, then maybe you've got what you need. If so, a couple of ad hoc tests (even with only 3 replications per cell) might allow you to say whether medium biomass is significantly different from hi and lo. // I guess I never understood why you are suspicious something may be fundamentally wrong. $\endgroup$
    – BruceET
    Commented Oct 12, 2020 at 16:07
  • $\begingroup$ The p-value of ANOVA was extremely low (>0.01). Subsequently, I run the Tukey test to verify differences between specific pairs, which also showed p-values less than 0.01 for two out of three combinations. I also believe it is correct, but a reviewer of my paper asked me about this possibility. Thank you very much for your attention. $\endgroup$
    – Barbara Quirino
    Commented Oct 12, 2020 at 17:27

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