\section{Results} \label{aplic}

The data considered here were collected on a citrus field 
with presence of \MSC{} within the farm Vale Verde, municipality of
Comendador Gomes, Minas Gerais State, Brazil.
The trees are arranged in 20 rows of 48 plants with spacing of
$7,5~m$ between rows and $4~m$ within rows. Data were collected at 11 time points
between  $05/11/2001$ and  $07/10/2002$.
The incidence ranged from  $14,9\%$ at the first visit to
$45,73\%$ on the final data. The response variable used here is the presence/absence
of \MSC{} on each tree.

\begin{table}[h]
\begin{center}
\caption{Incidence, parameter estimates and p-values
for models $m1$, $m2$, $m3$}
\label{tab:mod}
  \vspace{0.2cm}
\small
\begin{tabular}{rr|rr|rr|rr|rr}
\hline
& & \multicolumn{2}{|c|}{Model $m1$} & \multicolumn{2}{c|} {Model $m2$} 
& \multicolumn{4}{c} {Model $m3$}\\
\hline
& & & & & & \multicolumn{2}{|c|}{Present time} & 
\multicolumn{2}{|c}{Previous time} \\
\hline
 Av. & Incidence & $\hat{\gamma_1}$ &  p-value  &
$\hat{\gamma_1}$ &  p-value & $\hat{\gamma_1}$ &  p-value  & 
$\hat{\gamma_1}$ &  p-value  \\
\hline
1  & 0.14895 & -2.02052 & 0.27365 & & & & & \\
2  & 0.17293 & -1.97306 & 0.16735 & 0.35758 & 0.0542  & 0.4166  & 0.0462 & $-$0.0342 & 0.4350\\
3  &  0.21875 & -1.84436 & 0.00221 & 0.44081 & 0.0054 & 1.0271  & 0.0000 & $-$0.5060 & 0.0041\\
4  & 0.23840 & -1.78096 & 0.00036 & 0.63954 & 0.0000 & 0.9160  & 0.0000 & $-$0.2393 & 0.0600\\
5  & 0.26354 & -1.68169 & 0.00126 & 0.61800 & 0.0000 & 0.3901  & 0.0163 &    0.2437 & 0.0246\\
6  & 0.27812 & -1.63307 & 0.00025 & 0.59488 & 0.0000 & 0.8874  & 0.0000 & $-$0.2445 & 0.0305\\
7  & 0.32292 & -1.45117 & 0.00018 & 0.58248 & 0.0000 & 0.5437  & 0.0006 &    0.0972 & 0.1955\\
8  & 0.33125 & -1.39161 & 0.00014 & 0.61067 & 0.0000 & 0.5732  & 0.0002 &    0.0703 & 0.2597\\
9  & 0.34167 & -1.28953 & 0.00005 & 0.60794 & 0.0000 & 0.1542  & 0.1672 &    0.4721 & 0.0000\\
10 & 0.37500 & -0.90676 & 0.00190 & 0.50256 & 0.0000 & 0.0641 & 0.3341 &    0.4443 & 0.0000\\ 
11 & 0.45729 & -0.90008 & 0.00028 & 0.43635 & 0.0000 & 0.6371 & 0.0000 & $-$0.1196 & 0.1179\\
\hline
\end{tabular}
\end{center}
\end{table}

The models presented in Section~\ref{met} were fitted to the data.
Table~\ref{tab:mod} shows 
 significant effects only for
the covariate number of neighbours within row  for models $m1$ e $m2$
and the spatial covariate was not significant for the first and second
data collections. Overall similar results were found for model~$m2$.

Model~$m3$ includes two spatial covariates: $S_1$ is 
number of within rows neighbours at present time and $S_2$ is
number at previous time. 
Estimated coefficients and $p$-values are also 
shown in Table~\ref{tab:mod}.
Some combinations of relevant results are as follows.
Both spatial covariates are significant 
at  5\% significance level for times 3, 5 e 6; 
for times 2, 4, 7, 8 e 11, just $S_1$ was significant; 
and only $S_2$ for times 9 e 10. It is important to notice 
a potential (nearly) collinearity effect 
since the values of the two covariates can be similar,
specially when the incidence is nearly the same between two consecutive observations in time.

Table~\ref{aic} shows the Akaike Information Criteria (AIC) 
which is used to assess the fitted models. This criteria points that
model $m1$ is the preferable one for most of the observation periods
($2$,$4$,$5$,$6$,$7$,$8$ e $11$),
Model $m3$ is better supported for time~$3$ and $m2$ for times
 $9$ and $10$.

\begin{table}[ht]
\begin{center}
\caption{AIC values for the tree fitted models}
\label{aic}
\begin{tabular}{rrrr}
\hline
 &  Model $m1$ & Model $m2$ & Model $m3$ \\
\hline
2 & 725.55 & 726.76 & 727.54 \\
3 & 813.25 & 824.66 & 812.33 \\
4 & 851.58 & 858.66 & 853.08 \\
5 & 908.32 & 909.09 & 909.81 \\
6 & 932.52 & 936.61 & 934.17 \\
7 & 992.94 & 997.26 & 994.80 \\
8 & 1003.70 & 1004.79 & 1005.68 \\
9 & 1019.30 & 1018.58 & 1020.50 \\
10 & 1067.11 & 1064.87 & 1066.82 \\
11 & 1109.49 & 1121.87 & 1111.08 \\
\hline
\end{tabular}
\end{center}
\end{table}

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