297
included are
parent dis
ples (Si, S2)
Lbution a
Lfference be
Dtained as
(2)
amber of
an estimate
but ion given
(3)
of S , S2 .
(2) , has
The second is an accurate procedure, by testing the
normality of sample distribution, equality of
variances and finally significance of differences in
mean values. The samples Sj do not necessarily belong
to a normal parent distribution and, if so, the
variances of their parent distribution are not neces
sarily equal. The goodness of fit of Si, S2 to a
normal distribution can be assessed (Steel and Torrie
1960) by calculating the value of the criterion:
(4)
where o^, e^ are the observed respectively expected
occurrences in class i. To calculate e^, the normally
distributed variable:
(X  Xj)
(5)
is applied, with xj, Oj being the mean respectively
the variance of sample Sj. Then the expected number
of sample elements e^ in the ith class, i.e. between
zi and z^ + i are calculated on the basis of a normal
distribution and of the total number of occurrences
l i0i . Finally, the thus obtained x 2 ~value is compared
with threshold x 2 values, corresponding with preset
probabilities, to assess whether the observed dis
tribution significantly deviates from a normal dis
tribution.
Assuming that the observed distributions belong to
normal parent distributions, we have now to test the
equality of variances. The following criterion can be
applied (Kenney and Keeping 1959):
F =
(6)
2 . 2
where U is the larger and V the smaller estimated
variance between:
N 1  1
and
2 2
s 2 = nT^T
(7)
The Fvalue thus obtained is compared with the select
ed threshold F, e.g.^with a probability P = 0.01 of
F being larger than F.
After having established that both samples belong
to the same normal distribution, the Student's ttest
can be applied according to the first procedure. If
not, the following procedure has to be applied (Steel
and Torrie 1960) :
t =
and
(0) 2 _ V? + V2
N
1 + N 2
(8)
(9)
The threshold tvalue is obtained as:
t* =
Vi + V 2
(10)
with Wl = J and w 2 = Л
threshold tvalues at the
while t^ , t^ are the
same level of probability
Table 4. Goodness of fit of observed distributions
against the normal distribution; observed values of
TVI in field plots in the Grande Bonifica Ferrarese
and East Sesia irrigation districts (Po valley, Italy);
reflectance measurements obtained with the LANDSAT
TM, band 4 and 3; x 2 values having a probability
P = 0.05 respectively 0.01 to be exceeded are in
dicated (s = significant, ns = not significant)
Variable
Plots
ВАЗ
CA4
CA2
IA2
0B2
Total frequency
4
25
49
25
77
Skewness
0.1 7
0.05
2.4
0.05
1.41
Kurtosis
317.1
664
2289.9
44.7
3.81
X 2
0.24
7.4
32
6.9
18.4
X 2 (P = 0.05)
3.84
11.1
7.8
6.0
7.8
X 2 (P = 0.01)
Deviation from
6.63
15.1
11.1
9.2
1 1.3
normal distr.
ns
ns
s
ns
s
Table 5. Significance of differences in TVIvalues,
as obtained with LANDSAT TM 4 and TM 3 measurements
(2 May 1985), Grande Bonifica Ferrarese, Povalley,
Italy; significance assessed by applying the accurate
procedure respectively the simplified procedure
(within brackets); ns = not significant, s = signif
icant at probability P = 0.05, hs = highly signif
icant at P = 0.01; for plot coding see text; largest
observed differences in seeding dates: BB1, 30 April;
BB2, 16 May; 0B1, 5 April; 0B2, 12 May; IB1, 10 April;
IB2, 7 May
Plots
BB1
BB2
0B1
0B2
IB1
IB2
BB1
hs (hs)
hs(hs)
hs (hs)
hs(hs)
hs(hs)
BB2

hs(ns)
s (ns)
hs (hs)
s( s)
0B1

s( s)
hs(hs)
hs( s)
0B2

hs (hs)
hs( s)
IB1

hs(hs)
IB2

Table 6. Significance of differences in TVIvalues,
as obtained with LANDSAT TM 4 and TM 3 measurements
(30 April 1985); East Sesia, Povalley, Italy; for
explanation of plot coding see text: ns = not sig
nificant, s  significant at P  0.05, hs = highly
significant at P = 0.01; samples include between 20
and 77 pixels; largest observed phenological dif
ferences: CA1, full cover = 20 March; CA2, full cover
= 30 April; BA1, seeding date = 15 April; BA2, seed
ing date  24 April; LA1, seeding date = 1 April;
IA2, seeding date = 5 May
Plots
CA1 CA2
BA1
BA2
IA1
IA2
CA1
 s
hs
hs
hs
hs
CA2

hs
hs
hs
hs
BA1

ns
hs
hs
BA2

hs
hs
IA1

hs
IA2
