I have a question regarding a Split Plot ANOVA I had to calculate in STATISTICA:
In total, I analysed four factors: Two fixed factors: OAW and Nutrients, and two random factors: Mesocosms and Sibling group.
The factors OAW and N have two levels each and are fully crossed. Each OAWx N combination has three replicates, that are three mesocosms. So, for four treatment combinations I have 12 mesocosms in total.
I nested the factor mesocosm within the treatment combination OAWxN. This was what the reviewers asked me to do. Sibling group was just treated as a random factor.
So, the design in statistica was: oaw*N (mesocosm) + oaw + N + oaw*N + sib + n*sib + oaw*sib
The random factor mesocosm is statistically significant, so: OAW x N (mesocosm): p-value < 0.05
Does this mean, that the difference between mesocosm over all 12 mesocosms (and therefore over all 4 treatment combinations) are statistically significant? Or does this mean, that the difference of three mesocosms within each treatment combination is statistically different?
How should this be interpreted in the article? Is the OAW effect, which I found, still reliable?
Thanks in advance,
"Den. Syn. Error df " is the Denominator degrees of freedom for corresponding synthesized error terms constructed from random variations, and is computed using Satterthwaite's method.
"To test the significance of effects in mixed or random models, error terms must be constructed that contain all the same sources of random variation except for the variation of the respective effect of interest. This is done using Satterthwaite's method of denominator synthesis (Satterthwaite, 1946), which finds the linear combinations of sources of random variation that serve as appropriate error terms for testing the significance of the respective effect of interest. To perform the tests of significance of effects, ratios of appropriate Mean squares are then formed to compute F statistics and p-values for each effect. Denominator degrees of freedom for corresponding synthesized error terms are computed using Satterthwaite's method. The resulting F tests generally are approximate, rather than exact, and can be based on fractional degrees of freedom, reflecting fractional sources of random variation from which the error terms are synthesized.".
Above paragraph is copied from our online help page on variance computation. You can take a look of how these variance components are computed for mixed model Anova.
If we consider the general calculation of MS = SS / df, similarly, Den.Syn. Error MS should be calculated as synthesized random variation error term (for example, name it as Den.Syn Error SS) divided by "Den.Syn Error df", in which, Den.Syn Error SS is however not shown in the result table. That is, "Den.Syn Error df" should have been used to calculate Den.Syn. Error MS.
Hope it explains.
I am sorry that I am not able to make any comments on your data analysis result or give suggestions for your article to be published, due to our company legal policy. We are good to provide technical help if you encounter any problems using Statistica.
One thing I noticed that, you mentioned "I nested the factor mesocosm within the treatment combination OAWxN". If mesocosm is nested within the treatment combination OAWxN, the nested term should be expressed as mesocosm(OAW*N). oaw*N (mesocosm) means oaw*N is nested within mesocosm.
When factor B is nested within levels of factor A, the nested term in Statistica will be termed as B(A). In nested ANOVA analysis, this expression for nested term is generally used as a thumb of rule.
So you may want to double check which variable is nested within another and if you have specified the nested terms correctly in Statistica?
If the nested term B(A) is significant, generally it means there is significant variability among the levels/groups of B nested in A. Comprehensive interpretation of nested ANOVA results is a bit complex which involves like variability, mean squared error, residual error term analysis and should consider other factors in the model as well.
If you need statistical help, we have a professional team who can provide statistical analysis services to customers at reasonable charges. They are contactable at email address: DSG.Sales.Statistica.NAM.PSO@software.dell.com.
Thank you very much.
thanks a lot for the reply, that was helpful.
Are you sure that the term in STATISTICA, if mesocosms are nested within the treatment combination OAWxN, should be: mesocosms (OAWxN)? I am not so familiar with the commands in this software.
I repeated the tests accordingly and found that there were only minor differences of the statistical result, when using the right command now. I will keep the calculation that you suggested.
"If the nested term B(A) is significant, generally it means there is significant variability among the levels/groups of B nested in A."
So, if I have in total four treatment combinations OAWxN (because each of both factors has two factor levels), and mesocosms are nested within each treatment combination (there are always three mesocosms per treatment combination), then it means that within at least one of this treatment combinations, the three mesocosms are different from each other, or?
Thank you, best regards,
Yes. If B is nested with A, it will be B(A) in Statistica. You can refer to this Statistica Help webpage on nested Anova for evidence.
For your 2nd question, If B(C*D) is significant, B has 3 groups in very combinations of C*D, I would not say "the three B groups are different from each other", I would say there is significant difference among the 3 groups of B (i.e. at least two groups of B are different from each other). There is not enough evidence that all 3 groups of B are different from each other, just based on the ANOVA test. One can also say the effect of other model factors on the dependent variable significantly depends on the value of B.
Hope it helps. Thanks.
thanks a lot for the reply, that was helpful. I have another question:
Within the Split Plot ANOVA, I got a result table including the following parameters (columns)
Source of variation; df, SS, MS, Den. Syn. Error df, Den. Syn. Error MS, F-value, p-value
According to the help page, the Den. Syn. Error df, and Den. Syn. Error MS, were generated according to Satterthwaite 1946. I also saw that the "Den. Syn. Error MS." is used to calculate the F-value and p-value for each of the factors. This is done by dividing MS/ Den.Syn. Error MS.
Then, what is the "Den. Syn. Error df" used for? I would have say in the article what it is used for in order to show these values in the result table.
ok, I saw the description of the online help page. However, now I understand that there is also a calculated Den. Syn. Error SS which has not been shown but which was used in the table to calculate the F value. The important thing is to know now that the den.syn. error df was used for calculating the f-value.
Thanks a lot,
You are most welcome :)
All the best!