Hello everybody. I am dealing with the analysis of an experiment where I want to compare a variable after measuring it in several cells of three plants per treatment.
My hypothesis is that the treatment influences the accumulation of a carbohydrate in plant cells. So, I take three plants for the mock group and three for the treatment group. I stain the carbohydrate in the leaves and image them in confocal microscope. As the accumulation takes place in citoplasmic spots, I run a macro with ImageJ to identify these spots in the confocal images and measure the pixel mean intensity relative to the area of the spots (IntDen). Then, for each plant, I calculate the mean of IntDen, using each spot as a replicate.
I consulted with colleagues who advised me to analyze this experiment with an ANOVA block design, with the treatment as my FACTOR and each plant as a BLOCK. Two way ANOVA was another advise, taking the plants as the second FACTOR.
However, residuals don't pass the normality and the variance homogeneity tests, even with variable transformation.
I know that for simple ANOVA the Kruskall Wallis test is the nonparametric option. However, I am looking for a nonparametric equivalent to ANOVA in block desing and its implementation in R.
Also, I would like to know if the use of ANOVA in blocks or two way ANOVA is appropriate for my experiment.
Thanks in advance
Nicolas
How do the residual diagnostic plots for log(IntDen) (given plant and treatment) look like?
I strongly believe what you need is not a Non-parametric equivalent to ANOVA block design but to a single ANOVA design! Three issues gave me concern: the size of your sample (s(3) for each treatment and placebo group); non normality of the errors (residuals) which is more because of the sample size, sample from sorting, and possibly values close to zero with outlier at least one; and lastly heteroscedasticity!
Please be informed that a lack of independence of cases has been stated as the most serious assumption to fail and not non normality. Now: "The one-way ANOVA is considered a robust test against the normality assumption. This means that it tolerates violations to its normality assumption rather well. As regards the normality of group data, the one-way ANOVA can tolerate data that is non-normal (skewed or kurtotic distributions) with only a small effect on the Type I error rate. However, platykurtosis can have a profound effect when your GROUP SIZES ARE SMALL. This leaves you with two options: (1) transform your data using various algorithms so that the shape of your distributions become normally distributed or (2) choose the nonparametric Kruskal-Wallis H Test or Mood's Median Test which does not require the assumption of normality." ☑️
NOTE some argument:
1. Small Sample, sorted. The Mood's median test is used to compare the medians for two samples to find out if they are different. While not as powerful as the Kruskal-Wallis Test, it is useful for smaller sample sizes, when there are a few outliers or errors in the data as it focuses only on the median value. https://www.google.com/url?sa=t&source=web&rct=j&url=https://sixsigmastudyguide.com/moods-median-non-parametric-hypothesis-test/&ved=2ahUKEwiE7Jr0grnrAhXKT8AKHYaFAIkQFjABegQIAhAB&usg=AOvVaw0PPQH6IMxSI6iLNqGXrn9S. "Mood's median test is a special case of Pearson's chi-squared test. It is a nonparametric test that tests the null hypothesis that the medians of the populations from which two or more samples are drawn are identical. {https://en.m.wikipedia.org/Median test }
2. Failed Homogeneity Test? There are two tests that you can run that are applicable when the assumption of homogeneity of variances has been violated: (a) Welch T Test or (b) Brown and Forsythe Test. Alternatively, you could run a Kruskal-Wallis H Test. For most situations it has been shown that the Welch test is best only when the two populations are normally distributed, an approximated t distribution is obtained. However, the approximation can be affected by non-normal distributions, especially by the third central moments at smaller sample sizes! See the discussion here 👉https://www.researchgate.net/publication/301292970
For larger samples , suggested was a Weighted Regression which minimizes the sum of the weighted squared residuals. "When you use the correct weights, heteroscedasticity is replaced by homoscedasticity."
https://statisticsbyjim.com.
Conclusion: You are left with Kruskal-Wallis , Cox transformation or log transform won't work, neither is it ok to treat the three units as blocks.(2Way?)
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