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I have a data set with two independent variables (species and life-stage) and one dependent variable (concentration of a protein). There are 5 replicates within each level (please see table below). I want to perform two tasks with this data: (1) visualize data as bar graph where mean value of independent variable (concentration of protein) will be on y axis and one independent variable will be on x axis (life-stage) and the data will be faceted by second independent variable (species). Please see the sample figure below. (2) Apply two-way ANOVA on this data. The original data did not meet equal variance assumption; hence, I had to square root transform the data before applying two-way ANOVA. Is it acceptable if I use the original data for creating the graphs but the transformed data for statistical analysis?

Sample Table:

  Species | Life-stage | protein concentration (nmol) | square root transformed |  |  
 ---------|------------|------------------------------|------------------------|--|-- 
  rt      | early      |                            5 |                2.24    |  |  
  rt      | early      |                            7 |                   2.65 |  |  
  rt      | early      |                            6 |                   2.45 |  |  
  rt      | early      |                            4 |                      2 |  |  
  rt      | early      |                            5 |                   2.24 |  |  
  rt      | late       |                            8 |                   2.83 |  |  
  rt      | late       |                           10 |                   3.16 |  |  
  rt      | late       |                            6 |                   2.45 |  |  
  rt      | late       |                            8 |                   2.83 |  |  
  rt      | late       |                           10 |                   3.16 |  |  
  ws      | early      |                            7 |                   2.65 |  |  
  ws      | early      |                            8 |                   2.83 |  |  
  ws      | early      |                            5 |                   2.24 |  |  
  ws      | early      |                            5 |                   2.24 |  |  
  ws      | early      |                            8 |                   2.83 |  |  
  ws      | late       |                           14 |                   3.74 |  |  
  ws      | late       |                            7 |                   2.65 |  |  
  ws      | late       |                           17 |                   4.12 |  |  
  ws      | late       |                           10 |                   3.16 |  |  
  ws      | late       |                           14 |                   3.74 |  |  

Figure: bar graphs represent mean +- SEM (non transformed values) Figure: bar graphs represent mean +- SEM (non transformed values)

Below is two way ANOVA on square root transformed data:

enter image description here

As you can see here that I have created graph with original data (non-transformed) but I have applied two-way ANOVA on square root transformed data (because equal variance test was failing on original data). Is this approach correct? Thank you!

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  • $\begingroup$ What is the goal of your visualization ? $\endgroup$
    – deemel
    Jun 3 '19 at 7:02
  • $\begingroup$ Just to show the trend of data. I want to create bar graphs with mean values of the dependent variable on y axis against independent variables on x axis. $\endgroup$
    – kash91
    Jun 3 '19 at 7:40
  • $\begingroup$ You should specify which variables you are transforming and which ones you want to plot on this graph and, ideally, add some actual plots to the question so we understand what you are referring to. Right now, we don't know anything about your variables or about any trends that you say exist, so the question is pretty vague. $\endgroup$
    – AlexK
    Jun 3 '19 at 7:52
  • $\begingroup$ Thanks! I have modified the question and added some details. $\endgroup$
    – kash91
    Jun 3 '19 at 9:02
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You have such a small sample size that you can and should show all the values without risk of obscuring any overall pattern.

My exploration indicates that you would be much better off on log scale. In the graph below, log scale is used and the reference lines show geometric means.

The habit of reducing data to bars for means and error bars indicating variability -- in my experience most common in certain biological sciences -- is often denounced with positive suggestions for much better graphs.

See for example Weissgerber, T.L., N.M. Milic, S.J. Winham and V.D. Garovic. 2015. Beyond bar and line graphs: time for a new data presentation paradigm. PLoS Biology 13(4): e1002128. link here

If you work on logarithmic scale, the interaction term proves dispensable, a further indication that such a scale makes scientific sense.

enter image description here

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