Sample size importantly affect the significance of the test,

while smaller, suggesting that the difference between the averages should be very relevant, "may not really be" advised calculate effect size.

Anova is not a tool to analyse difference between groups, so nothing here wil help you in interpretation of difference between groups. Full stop.

Anova is about analyzing the effect of an entire predictor (or of a whole set of such predictors) in a more complex model, and this is done by calculation of the F statistic. Under the assumption of normal distribution of the residuals (then and only then! as Fausto pointed out), the F statistic has a known distribution under the hypothesis that the considered predictors have no systematic influence on the response ("F-distribution"). The exact shape of the F-distribution depends on the number of predictors, their and their structure and on the sample size; this information is contained in the "degrees of freedom", DF1 and DF2. So, the DF values tell you how the F-distribution looks like under the null hypothesis, and the F-value is the F obtained from the observed data. Now one can asses how likely an F value as large as the observed or larger is expected under the null hypothesis. If this is very unlikely, this is usually taken as "evidence against" the null hypothesis.

Fausto,

for survival times, Anova is not appropriate. You know this. So I do not really get your question or why you have chosen this example. If we forget that these are survival times and only consider that fact that there are data of k=3 groups with n1=6, n2=7 and n3=5, then

df1 = k-1 = 3-1 = 2

df2 = Sum(n1, n2,,, nk) - k = (6+7+5) - 3 = 15

The 0.9-quantile of F[2,15] is 2.7 (this is the critical value to reject H0 at 90% confidence).

Fausto, sorry, I don't understand what you want.