Let's say we have cells from two genotypes (WT and KO), each tested with three drug concentrations on each of three separate days. Assume sphericity over the responses of 100 cells tested, 40 WT and 60 KO. Obviously the repeated measures do not apply to the factor genotype, only to the drug level and day. What are the degrees of freedom for the F(df1,df2) statistic?
df1= (2-1)*(2-1)*(3-1)=2 - correct?
df2= 600-5-99=496 or is it 600-7-99=494 or perhaps something else?
A two-way repeated measures ANOVA has three degrees of freedom (dfs). The first df is for the interaction between the two factors, the second df is for the first factor, and the third df is for the second factor.
Perhaps my question was not clear. To check the F statistic for significance in an F table, one needs the degrees of freedom for columns and another for rows. These are indicated as F(df1,df2), for example
F(2,10)=4.1028, p=0.05 see http://www.socr.ucla.edu/Applets.dir/F_Table.html
In the example I gave, df1 should be 2 and df2 is somewhere near 496 if my calculation is correct, which would give an F(2,496) between 3.0718 and 2.9957 using the same F table at p=0.05. A lower value of F would be significant at the 0.05 probability level. I am reviewing a paper where the authors are citing their F values in a repeated measures ANOVA design and I would like to cross-check these against an F-table using the correct degrees of freedom.
Hello Nikolaos C Aggelopoulos. It sounds to me as if you have a 3-factor (mixed design) ANOVA model. To simplify things, let's use the following notation:
- A = genotype (Between-Ss factor, 2 levels--i.e., 2 groups)
- B = drug concentration (Within-Ss factor, 3 levels)
- C = day (Within-Ss factor, 3 levels)
Does this sound correct to you? If so, there are F-tests for:
- A main effect
- B main effect
- C main effect
- A*B interaction
- A*C interaction
- B*C interaction
- A*B*C interaction
Which of these F-tests are you asking about? And what statistical software are you using? Thanks for clarifying.
Thank you Bruce. Your understanding is correct. First, before checking for the individual main effects and interactions, one should check if the overall ANOVA is significant, for which one should need an F(df1,df2) where df1 is the overall degrees of freedom for the factors and df2 is the overall df for the between groups comparison. At this point I only want the concept. I have SPSS and can probably run it with some hypothetical observations. However, if I remember correctly, SPSS does not state the resulting F with the associated degrees of freedom in the same way one would tend to report it in a publication.
Hi Nikolaos C Aggelopoulos. Thank you for clarifying. I guess there are different conventions in different fields, because I have never been taught (in psychology stats courses) that one must have a significant F for the overall model before proceeding to the F-tests for main effects and interactions. Can you point to resources recommending that approach? Thanks.
Second, if you are estimating your mixed design ANOVA using GLM REPEATED MEASURES, then I don't think there is any easy way to get one overall test for the model, because the tests of within-subjects effects and between-subjects effects appear in separate tables. If I really wanted an overall test for the model, I think I would restructure the data from wide to long and use MIXED. You can find examples here:
- https://stats.oarc.ucla.edu/spss/seminars/repeated-measures/
- https://stats.oarc.ucla.edu/spss/library/comp_repeat_spss/
HTH.
By contrast, I have always been taught that individual pairwise comparisons can lead to type 1 errors and the more you do, the greater the chance of a type 1 error. So ANOVA must be used as a statistical method to estimate the overall probability of rejecting the null-hypothesis before proceeding to examining the effects of each main factor and their interactions in the subgroups, ultimately carrying out pair-wise comparisons to identify what was significantly higher or lower to something else. Out of every 20 individual comparisons, 1 may by chance exceed the 0.05 probability level and in a large ANOVA design there will be many more than 20 such possible comparisons. It follows from this that one must first determine the overall F. Checking the F value using the appropriate degrees of freedom is more important when reading a publication when the exact design or numbers of observations may not be clear. As long as the F statistic with the correct degrees of freedom is stated, one can check against an F table for themselves the overall significance level.
Examples of why overall F must be tested first before testing other comparisons: http://menzerath.phonetik.uni-frankfurt.de/teaching/R/bw_anova_general_HR.pdf https://files.eric.ed.gov/fulltext/EJ1139744.pdf - almost certainly the argument is in Fisher's book but I do not have a copy to check.
NB: Having received no responses and having searched the literature, it appears that outside a one-way ANOVA and a repeated-measures ANOVA, in all other kinds of an ANOVA or ANOVA-type design (e.g. LMM), there is no overall F calculated but only F-statistics for each main effect and each specific interaction. I find this awkward but that is how it is. Please, correct me if my understanding is not correct.
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