My question is because one reviewer asked me to perform Tukey or Student test to perform statistical comparison between levels of inclusion of Passion fruit peel, as a substitute of Tifton-85 hay. I had the following levels: 0% (control); 20%; 40%; and 60%. As far as I know, the proper approach is to utilize regression analyses instead of Tukey or Student t test. What arguments can I use to counter-argue the reviewer?
Because your treatment variable is a continuous, quantitative variable, a regression based approach is preferred for characterizing the response. Tukey or Student t would be better used for non-quantitative variables like forage species or breed of cattle. Generally, when I conduct an experiment with a quantitative factor, I use polynomial contrasts to test the treatment effect. In this case you have four levels of the treatment factor, so you could look at single degree of freedom contrasts for linear, quadratic, and cubic effects of passion fruit peel. This is a very strong test because all of the other sources of variation remain in the model (unlike a straight regression model where the variation from those sources flows into the error term), and it also does not depend on a significant effect of the overall F test for "level of passion fruit peel" with three degrees of freedom. That test is effectively the average of the linear, quadratic, and cubic effects, whereas the single degree of freedom contrasts approach allows you to look exclusively at each of those patterns of response. Since your levels of the treatment factor are equally spaced it is very easy to get the coefficients for the polynomial contrasts directly from a statistics book.
Because your treatment variable is a continuous, quantitative variable, a regression based approach is preferred for characterizing the response. Tukey or Student t would be better used for non-quantitative variables like forage species or breed of cattle. Generally, when I conduct an experiment with a quantitative factor, I use polynomial contrasts to test the treatment effect. In this case you have four levels of the treatment factor, so you could look at single degree of freedom contrasts for linear, quadratic, and cubic effects of passion fruit peel. This is a very strong test because all of the other sources of variation remain in the model (unlike a straight regression model where the variation from those sources flows into the error term), and it also does not depend on a significant effect of the overall F test for "level of passion fruit peel" with three degrees of freedom. That test is effectively the average of the linear, quadratic, and cubic effects, whereas the single degree of freedom contrasts approach allows you to look exclusively at each of those patterns of response. Since your levels of the treatment factor are equally spaced it is very easy to get the coefficients for the polynomial contrasts directly from a statistics book.
Thank you very much, Dr. Sollenberger. I appreciate!
The single-degree-of-freedom contrasts are probably a good idea, especially since the four levels of passion fruit peel are equally spaced. I would also consider whether the effect of that variable is well described by a polynomial. Some patterns of effect would be more compatible with a linear spline.
You didn't mention the sample sizes in the four groups. How many degrees of freedom do you have for "error" after fitting a separate effect in each group?
Sorry, should have limited that comment to Tukey. Yes, qualitative variables is what I was referring to.
I will add here the summary of this study, to facilitate comprehension:
"Two experiments were performed to evaluate the substitution of Tifton 85 (Cynodon spp.) hay by dehydrated passion fruit (Passiflora edulis f. flavicarpa) peel on intake, digestibility, nitrogen balance, and carcass traits in Santa Inês x Dorper rams. In Experiment 1, 12 rams were penned in individual metabolic crates for 21 days to evaluate DMI, nitrogen balance, and digestibility. Four treatments were evaluated: (1) control, with no inclusion of passion fruit peel; (2) 20% substitution of Tifton 85 hay by passion fruit peel; (3) 40% substitution of Tifton 85 hay by passion fruit peel; and (4) 60% substitution of Tifton 85 hay by passion fruit peel (n = 3 rams per treatment). In Experiment 2, 20 rams were randomly assigned into individual dry-lot pens for 63 days, and the same treatments from Experiment 1 were randomly assigned to pens (n = 5 pens per treatment). Body weight was weekly determined, and at the end of the experimental period, rams were humanely slaughtered, for assessment of carcass traits. No effect (P>0.33) of passion fruit peel inclusion was observed for DM and CP apparent digestibility. However, NDF apparent digestibility had a quadratic effect (P=0.01). Quadratic regression (P=0.02) was observed for nitrogen intake, but no effect (P>0.16) was observed for fecal and urine nitrogen excretion, and nitrogen balance. Daily DMI, DMI as % of BW, and NDF and CP intake had quadratic effect (P
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