What is a meaning of "p-value F" from Friedman test? I'm a new R user and had just tried running friedman on non-normal and heteroscedastic data on seagrass.  I am testing whether biomass is significantly different between sites across years.  R (friedman function from agricolae package) returned result like this:
Friedman's Test
===============
Adjusted for ties
Value: 0.01333333
Pvalue chisq : 0.9080726
F value : 0.01316678
Pvalue F: 0.9086942 0.9087002

Alpha     : 0.05
t-Student : 1.990847
LSD       : 17.34995

Means with the same letter are not significantly different.
GroupTreatment and Sum of the ranks
a   1   119 
a   2   118 

I know this means no significant difference given the p-value chisq.  But what does p-value F mean?  
 A: It seems the output is from the agricolae package using the method friedman.  The relevant lines for computing the two statistics in that function are:
T1.aj <- (m[2] - 1) * (t(s) %*% s - m[1] * C1)/(A1 - C1)
T2.aj <- (m[1] - 1) * T1.aj/(m[1] * (m[2] - 1) - T1.aj)

Comparing this with the formula in chl's answer, you'll notice that T2.adj ("F value") corresponds to $F_{obs}$ and T1.adj ("Value") to $F_r$.
A: I generally used friedman.test() which doesn't return any F statistic. If you consider that you have $b$ blocks, for which you assigned ranks to observations belonging to each of them, and that you sum these ranks for each of your $a$ groups (let denote them sum $R_i$), then the Friedman statistic is defined as 
$$
F_r=\frac{12}{ba(a+1)}\sum_{i=1}^aR_i^2-3b(a+1)
$$
and follows a $\chi^2(a-1)$, for $a$ and $b$ sufficiently large. Quoting Zar (Biostatistical Analysis, 4th ed., pp. 263-264), this approximation is conservative (hence, test has low power) and we can use an F-test, with
$$
F_{\text{obs}}=\frac{(b-1)F_r}{b(a-1)-F_r}
$$ 
which is to be compared to an F distribution with $a-1$ and $(a-1)(b-1$) degrees of freedom.
A: Probably $p_F$ refers to the F-statistic developed by Iman and Davenport? They showed that Friedman’s $\chi^2$ is undesirably conservative and derived a "better" statistic 
$F_F=\frac{(N-1)\chi^2_F}{N(k-1)-\chi^2_F}$
which is distributed according to the F-distributionwith k−1 and (k−1)(N−1) degrees of freedom. 
References: 
Demsar, J. (2006). Statistical Comparisons of Classifiers over Multiple Data Sets. Journal of Machine Learning Research, 7, p. 11.
Iman, R. L.  and Davenport,J. M. Approximations of the critical region of the Friedman statistic. Communications in Statistics, pages 571–595, 1980.
