INTRODUCTION:

In applied sciences, such as agricultural sciences, experimentation is particularly important (^{ANDRIAZZI, 2007}). Using it, one can determine the best dose of fertilizer to be applied to the soil for a given culture to have maximum productivity and to know the ideal period to harvest a fruit, in order to provide the most appreciated flavor, or the best type of pruning for a perennial culture for a larger yield (ARAÚJO et al., 2007; ^{BRUNETTO et al., 2008}).

An example of area where experimentation is very useful is the fruit culture, particularly for growing grapevines, for table grapes or grapes for winemaking. In Brazil, the South, Southeast and Northwest regions are the regions where the grape cultivation is more effective, where each cultivar is produced for a particular purpose. In the South, grape production is destined for the production of wines and juices, while the southwestern region consumes grape *in natura. *In the Northeast, the most of the production is for table grapes (^{ELIAS, 2008}).

There are over 60,000 varieties of grapes around the world, among them there are Merlot, Sayra, Cabernet Sauvignon and Niagara.In particular, the Niagara grape is the most widely produced in Brazil, mainly in the northwest of the state of São Paulo.It has low production cost to be somewhat susceptible to diseases caused by fungi (^{BOTELHO et al., 2004}).The Niagara is the main Brazilian table grape and it has compact medium bunch delicate berries (^{GOMES & FERRAZ, 2011}).

On the other hand, the grape has high perishability, which causes economic losses to growers, since the postharvest loss is estimated around 20-95% (^{ELIAS, 2008}). Therefore, combining preservative processes in pre or post-harvest might be important. In these cases, the use of substances to prolong the post-harvest life has become an alternative.

In fruit culture and other areas of agricultural sciences, there are experiments whose designs and schemes can lead to tricky analysis. For instance, one can study the effect that cultivars, soil fertilization rates, foliar fertilization and harvest seasons have on the characteristics of the fruit, such as sugar content, pH, content of total solids, soluble solids, among others.

For these and other reasons, it is not uncommon the designs, statistical models and hence the analysis to become complex, what demands constant research. An example of a relatively complex experimental situation is the split double factorial experiment with a single additional treatment in the plot (BEZZERRA NETO et al., 2005; ^{VIDAL NETO et al., 2005}; ^{YASSIN et al., 2002}). This additional treatment is an extra or control treatment and is conducted within the experiment, but it is analyzed separately by contrast against the test treatments (HEALY, 1956).

In these situations, the analysis requires the researcher to perform several runs using popular software such as Sisvar (^{FERREIRA, 2000}), Assistat (^{SILVA & AZEVEDO, 2006}) and R (^{R CORE TEAM, 2014}), and complement these analyses on his/her own, making the whole procedure more difficult and uncertain. Namely, it is required to combine results of regular double factorials, triple factorials, regular split-plot design and single factor one-way ANOVA for providing the necessary source of variation.

The objectives of this research are: (i) discuss and explain how to deal with the analysis of experiments conducted in completely randomized design (CRD) and split double factorial design with additional treatment in the plot; (ii) program a function able to perform such analysis in a single run, using R language (R CORE TEAM, 2014); and (iii) illustrate the discussion analyzing data from an experiment on post-harvest of Niagara grapes.

MATERIAL AND METHODS:

Data from the experiment with Niagara (**Vitislabrusca**) grapes were kindly provided by Dr. Heloisa Helena de Siqueira, pos-doc fellow at the Federal University of Lavras, Brazil, in order to demonstrate the analysis of experiments conducted in completely randomized design (CRD) and split double factorial design with additional treatment in the plot and also the function set in R code (^{R CORE TEAM, 2014}).

Grapes were treated in the pre-harvest with three preservatives: calcium chloride, calcium nitrate and calcium lactate at 0%, 0.5%, 1% and 2%, and stored at 0, 10, 20 and 30 days. All the preservatives evaluated at 0% represented the control (additional) treatment. Thus, the experiment features a split double factorial with additional treatment in the plot (3x3+1) x 4, under 3 replications.

RESULTS AND DISCUSSIONS:

Let a triple factorial experiment conducted in CRD with factors a, b and g considered in *a, b* and *c* levels, respectively, replicated *J *times. Also, consider that these three factors are arranged in split-plot scheme with the double factorial formed by a and b, allocated in the plot, and the factor g, in the subplot. Finally, consider additional treatment allocated in the plot, which contains the *c* levels of g and is also replicated *J *times.

As a first result of this research, it was sketched the layout of the randomization of the experiment (Figure 1) which has a split double factorial in CRD with an additional factor. In the plot one can see the combination of factors a and b levels or the additional treatment. On the other hand, in the subplot, the *c* levels of factor g can be found.

From this arrangement, one can note that sources of variation involved come, at a first sign, either from the plots or the subplots. Regarding the plots, one can subdivide it into several sources variation: factor a, factor b, interaction aband contrast between the treatment and additional treatment allocated in the plot *Ad _{p} *. In turn, the variation in the subplots,can be divided in: factor g, double interactions ga and gb, triple interaction abg and contrast between additional treatment and subplot treatments. This way of designing the experiment leads to the following statistical model:

with *i*=1,2,...,*a*; *j*=1,2,...,*b*; *k*=1,2,...,*c* and *l*=1,2,...,*J*; where *Y _{ijkl} * is the

*l*-th repetition of the plot subject to the level

*i*of factor a, to the level

*j*of factor b and the level

*k*of factor g; m is a common constant to all observations; a

_{i}the effect of the

*i*-th level of factor a; b

_{j}is the effect of the

*j*-th level of factor b; (ab)

_{ij}is the interaction of a

_{i }and b

_{j};

*Ad*is the contrast between the additional treatment and double factorial;

_{p }*e*is the error associated to the plot; g

_{A}_{k}is the effect of

*k*-th level of the factor g; (ag)

_{ik}is the interaction of a

_{i }and g

_{k}; (bg)

_{jk }is the interaction of b

_{j}and g

_{k}; (abg)

_{ijk}is the triple interaction between a

_{i}, b

_{j}and g

_{k};

*Ad*is the contrast between the additional treatments and the subplots;

_{s }*e*is the random error associated with subplots. All terms but the errors are considered fixed, leading to a fixed model; and and .

_{B}Given the statistical model is easy to get how the analysis of variance table should be built. Its sources of variation and respective degrees of freedom, sums of squares, mean squares and computed F's are given in table 1.

It is worth noting that the interpretation of the contrast between the additional treatment and subplot treatments (Ads) is not straightforward established, since the additional treatment contains the c level of g within itself. Therefore, it is advised that this effect can be confounded with the error B, inflating the latter and increasing a unit in degrees of freedom. Let e*B= eB+ Ads be the new error. Given this confounding, both statistical model and the framework of analysis of variance are reduced, to:

where all the effects are as previously described and the new table of variance is in table 2.

Regarding the plots, it is easy to estimate the contrast between the factorial average and the additional treatment, and get all conclusions usually provided by a split double factorial.

Under this framework, one derives the sums of squares of the sources of variation, inspired by ^{BANZATTO and KRONKA (2006}). The corrected overall sum of squares (*SQ _{T} *) in this case is given by: , where is the quadratic mean, also called

*correction*.

The a and b sum of squares are given respectively by and , where Ai and Bj are the totals of the levels i and j of a and b, respectively, and the other correction Co is given by:

For g sum of squares it is necessary to consider the additional treatment (sum of squares that makes up the subplot): , where CFatk is the total of the k-th level of the factor g within the factorial and CAdk is total of the k-th level of the factor g within the additional treatment.

The sum of squares of the interaction ab is obtained from the difference between the sum of squares of the treatments from the double factorial and the sums of squares of such factors, as: , where ABij is the total of the plots receiving ai and bj.

Similarly, the sums of squares of the interactions ag and bg can be obtained, respectively, by:

and

, where *AC _{ik} * and

*BC*are, respectively, the total of the treatments generated by the factorial combination of a and g, and b and g.

_{jk}The triple interaction abg can also be obtained by difference. It is the difference between the sum of squares of treatments obtained by the factorial combination of a, b and g, and simple effects and double interactions, as:

where *ABC _{ijk} * is the total of the

*ijk*-th treatment (combining levels of a, b and g).

In turn, the sum of squares of the contrast between the additional treatment and the factorial average is given by:

where is the contrast of total additional treatment *Ad _{p } *combining the totals of the plot treatments

*A*and

_{i }*B*, and

_{j}*Z*are the coefficients of the combination of the total of

_{i}*A*and

_{i}*B*.

_{j }The plots sum of squares can be described as follows:

where AB_{ijl} is the total of each replication of AB_{ij}.

Finally, the sum of the squares of the first error (*e _{A} *) may also be obtained from the difference between the plots sum of squares and other sources of variation present in the plot, namely:

Similarly, the sum of squares of the second error (*e _{B} *) is obtained by difference between the overall sum of squares (

*SQ*),plots sum of square (

_{T}*SQ*), g sum of squares and its interactions:

_{P}

The whole analysis of variance table is straightforward, given the sum of squares, degrees of freedom and mean squares.

The function sub.fat.ad() allows to analyze a split double factorial in CRD with one additional treatment, through analysis of variance. It receives ten arguments, as follows: sub.fat.ad(factor1, factor2, factor3, addit, rep, resp, quali=c(T,T,T), fac.names=c('F1','F2','F3'), sigF=0.05where factor1, factor2 and factor3 are vectors containing the levels of the factors1, 2 and 3 (which, in the experiment with Niagara grapes are, respectively, the factors a b and g); addit contains the additional treatment; rep contains the replications indices; resp contains the response variable (for the three factors); quali=c(T,T,T) is a logical variable, if TRUE (default), where respective factor is qualitative; otherwise, quantitative; fac.names are labels for factors 1, 2 and 3; sig F is the significance for F test (the default is 5%).

To illustrate the behavior of Niagara grapes during storage (factor g), under several combinations of kinds of preservatives (factor a) and doses (factor b), four physicochemical variables were picked up, namely pH, firmness (Table 3), total sugar and phenolic acids content, whose results will be discussed below. All analysis were performed using function sub.fat.ad(), in R software (R CORE TEAM, 2014). Assumptions of normality, homoscedasticity and independence were satisfied.

Those physicochemical variables were significantly affected by the factor time. There was a reduction in firmness, total sugar and phenolic acids content along time (P=0.0192, P<10^{-4}, P<10^{-4}, respectively), while pH slightly increased (P<10^{-4)}. Regarding the preservatives, they were considered statistically similar for pH (P=0.4249). For firmness, calcium nitrate and calcium chloride provided the firmest berries (P=0.0203). On the other hand, the highest level of phenolic acids and total sugar was provided by calcium chloride and calcium lactate (both P<10^{-4)}. Note that calcium chloride had the best overall behavior.

Finally, the doses of preservatives were considered statistically similar for pH and firmness (P = 0.1002 and P = 0.9928, respectively). For total sugar and phenolic acids, the dose 1% provided the highest content (both p < 10^{-4)}. Therefore, 1% seems to provide good results for all variables.

CONCLUSION:

Split double factorial designs with an additional treatment in the plot raise when researchers plan an experiment along time (for instance) and, at the same time, want to study two other factors, which absence of one factor or its null level leads to the same treatment. The way this design is approached in this paper allows the researcher to analyze such experiments in a single run, particularly, using the function sub.fat.ad(). In the experiments with Niagara grapes presented here, the time of storage affected the fruits significantly in different ways, as expected. As preservative, calcium chloride is recommended to be used at dose 1%.