In some problems using Analysis of Variance (ANOVA) arising in the Design of Experiments, the computed value of F-ratio like F_C = MS_C / MS_E turns out to be less than one, and it is a trivial calculation that this computed F-value is less than the table F-value at 5% or 1% level of significance and one concludes that the effect of this factor under investigation is insignificant.
I like to know the statistical significance or some more insight into this! Thanks!
This may help to get some insight
http://stats.stackexchange.com/questions/9242/what-is-the-meaning-of-an-f-value-less-than-1-in-one-way-anova
Assuming that there is no effect, the F-ratio has the F distribution. The precise
probability that a random variable with the F distribution is less than 1 depends
on the parameters, but it is at least 1/3.
Assuming the model is correct, the probability under H0 that a test statistic is less than the value actually observed will be uniformly distributed between 0 and 1. In the case of ANOVA, if H0 is false it will be more likely to be closer to 1. Despite this, in the past some writers would reject H0 in either tail. In the lower tail the fit was said to be "too good". This is not a sensible reason to reject H0, but it does suggest that something might be wrong, either with the model, or with the data (or it could be just a chance event).
Here are a couple of ways that you might get small F vaues.
1. Fisher discussed the problem of the fit being too good in paper on Mendel's data on genetics of peas. Later work suggests that this might have occurred due to rejecting samples that contradicted the expected results. At the time this might have seemed sensible. How do you determine the difference between a mistake and sampling variation? This was in the 1860s--I hope we know better today.
2. Over 30 years ago, I helped someone calculate an F-test on an old TI programmable calculator. The result was in the lower tail and I suggested that the data might not be genuine. When approached quietly (we didn't tell the researchers), the technician who did the chemical analysis admitted making up some figures to reduce the backlog. He said it wouldn't happen again and redid the analysis. Don't blame just the technician. It was really a management failure. The technicians worked for a number of researchers who independently gave them samples to analyse. There was no control of workload and this caused the backlog.
In your case, if you trust the data and the model (and the F-test is fairly robust to model mispecification), you can just pass off small F values as rare events that do not cast suspicion on H0. But you should not dismiss such events lightly--if it happens too often you have a problem. How often depends on whether you believe there is a genuine effect. Results below the 5% threshold should occur in much less than 5% of samples. It will be 5% only if H0 is true. And in many experiments it isn't true.
On re-reading the question, I realise you didn't mean F values in the bottom 1% or 5%. As, Hermann said, values a little less than 1 are not unlikely under H0. My reply was about values much less than 1.
PS. My confusion was due to the layout of F tables. You should have said "it is trivial that the F value is greater than the 5% or 1% level". The tables are arranged that way because departures from H0 lead to larger values of F.
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