The general principle of bootstrap method is the resampling of a large number of times (n) a sample initially taken from a population [20]. Parameter estimation and statistical inference are then based on the results of the bootstrapped samples obtained.

The best known application of the bootstrap is to estimating the mean, µ say, of a population with distribution function F, from data drawn by sampling randomly from that population. The mean is:

μ = ∫ x d F ( x ) (1)

The sample mean is the same functional of the empirical distribution function, i.e. of

F ^ ( x ) = 1 n ∑ i = 1 n I ( X i ≤ x ) (2)

where X 1 , ⋯ , X n denote the data. Therefore the bootstrap estimator of the population mean, μ, is the sample mean, X ¯ :

X ¯ = ∫ x d F ^ ( x ) = 1 n ∑ i = 1 n X i (3)

Likewise, the bootstrap estimator of a population variance is the corresponding sample variance; the bootstrap estimator of a population correlation coefficient is the corresponding empirical correlation coefficient; and so on.

More generally, if θ 0 = θ ( F ) denotes the true value of a parameter, where θ is a functional, then

θ ^ = θ F ( θ ^ ) (4)

is the bootstrap estimator of θ 0 .

Parametric (simulation), semi-parametric (adding noise) and nonparametric bootstrap (resampling) are the three variants of bootstrap. These variants differ in the way the sample is drawn from the population data. In contrast to nonparametric and semi-parametric bootstraps, the parametric bootstrap assumes that the data comes from a known distribution of unknown parameters (for example, data may come from Poisson or negative binomial distributions for counts, or normal distribution for continuous characters). Three methods of bootstrap are used to draw sampling from population: Ordinary, Balanced, and Moving Block Bootstraps. In ordinary bootstrap, the bootstrap samples are drawn by a simple random sample, with replacement, from the values in the approximated population. The number of bootstrap for accurate estimation of the standard errors varies from 50 to 1000 while 1000 or more bootstrap samples are recommended to calculate the confidence intervals [15]. In this study, 1000 was considered appropriate to obtain desired results.

Many statistical problems can be represented as follows: given a functional f t from a class { f t : t ∈ τ } , we wish to determine the value of a parameter t that solves the population equation,

E { ( f t ( F 0 , F 1 ) | F 0 ) } = 0 (5)

where F 0 denotes the population distribution function and F 1 = F ^ is the empirical distribution function, computed from the sample X = X 1 , ⋯ , X n .

The basic steps in the bootstrap procedure are [15]:

· Step 1. Construct an empirical probability distribution, F n , from the sample by placing a probability of 1/n at each observed value x 1 , ⋯ , x n of the sample. This is the empirical distribution function of the sample, which is the nonparametric maximum likelihood estimate of the population distribution, F.

· Step 2. From the empirical distribution function, F n , draw a random sample of size n with replacement. This is a resample.

· Step 3. Calculate the statistic of interest, T n , for this resample, yielding T n ∗ .

· Step 4. Repeat steps 2 and 3 B times, where B is a large number, in order to create B resamples. The practical size of B depends on the tests to be run on the data. Typically, B is at least equal to 1000 when an estimate of confidence interval around T n is required as in this study.

· Step 5. Construct the relative frequency histogram from the B number of T n ∗ ′ by placing a probability of 1/B at each point, T n ∗ 1 , T n ∗ 2 , ⋯ , T n ∗ B . The distribution obtained is the bootstrapped estimate of the sampling distribution of T n . This distribution can now be used to make inferences about the parameter, which is to be estimated by T n .

According to Phillips, quantitative ethnobotany is the application of quantitative techniques to analyze plant use data [21]. These methods can be used to answer the question: how important is species, or plants to people? Quantitative ethnobotany gives a response on the Relative Cultural Importance (RCI) of plants and vegetation to people [22]. The most used technique by ethnobotanists to assess the RCI of plants is the calculation of an ethnobotanical index. In function of RCI, ethnobotanical indices can be grouped and divided into three categories [3]: Informant consensus, subjective allocation, and uses totaled. The Informant consensus approach is the most used in quantitative ethnobotany [23] [24]. The indices considered in this study are selected from the informant consensus approach (Table 1). They are defined to determine the number of interviewees who use a given species and the distribution of the use among the interviewees (ID), to measures the degree of homogeneity of the interviewee’s knowledge (IE), and to measures the importance of the use categories of the species and their contribution to the total use value (UE).

In total, 44 localities were surveyed at a rate of four per Department of Benin except the Department of Littoral where the agricultural production systems are almost non-existent. The sample size (704) for this study was calculated using the standard approximation of the binomial distribution [26], with a margin error of 3% and a proportion of individuals who had previously used the organs of J. curcas of 79.17% (obtained from an exploratory study). So, ethnobotanical surveys were conducted among 704 respondents (individual). The respondents were randomly selected from each locality and the data collected from each respondent concerned the use forms of J. curcas.

The sampling distribution of each ethnobotanical index (Table 1) was studied by simulation using Matlabv: 2011. The initial sample was a binary matrix of 704 lines representing the respondents and 10 columns representing the different use forms. The knowledge of a given use form was coded 1 and 0 otherwise. The algorithm for characterizing the distribution of the ethnobotanical indices considered is illustrated below:

Step 1. From the initial sample x of 704 interviewees, one bootstrapped sample x * , of size n, was obtained by simple random sample with replacement;

Step 2. Compute four ethnobotanical indices bootstrapped ID*, IE*, UD* et UE*;

Step 3. Step 1 and 2 are repeated 1000 times and the average values of the four ethnobotanical indices were computed;

Step 4. Step 1 to 3 were repeated for 200 different values of sample size n between 50 and 10,000 in increments of 50;

Step 5. Computed position (mean, median, 1^st quartile, 3^rd quartile, minimum, maximum) and dispersion (standard deviation) parameters for the 200 values of

each ethnobotanical index to characterize their empirical distribution; frequency histograms of index values were also constructed;

Step 6. For each index, a trend line was constructed by considering the 200 index values as a function of the n-sample sizes considered.

The 704 respondents were classified per sociolinguistic group (Bariba, Dendi, Fon, Haoussa, Lokpa, Nago, Otamari, Peulh, Wama), age group, and gender. Young people were those under 30 years, adults were those between 30 and 60 years, and old people were those over 60 years. For each category related to age, gender or sociolinguistic group, the sample was scaled 1000 times with the same size of 100 to eliminate the effect of sample size on the index values. The different categories were compared using permutation variance analysis [27].

The local people interviewed listed a total of ten uses for J. curcas, with medicinal and bioenergy uses predominating (Table 2). The characteristics of the sampling distribution for the ethnobotanical indices are presented in Table 3. Figure 1 presents the simulated population values of the four ethnobotanical indices considered. This figure shows that use diversity value (UD) (Figure 1(c)) and use equitability value (UE) (Figure 1(d)) varied very little in function of the number of respondents, with average values of 1.82 and 0.55, respectively. The interviewee’s diversity (ID) (Figure 1(a)) and equitability (IE) (Figure 1(b)) beliefs, on the other hand, varied significantly depending on the number of respondents considered. The interviewee’s diversity (ID) (Figure 1(a)) values ranged

*Total number of citations for each use divided by the total number of citations for all uses.

from 5 to 13 on a scale of 1 to 13. The largest values of these indices are also those with the highest frequencies in the simulated population (Figure 1). ID (Figure 1(a)) was more sensitive to the sample size used, enabling only comparisons of various cases with the same number of respondents, partly supporting Reyes-García et al. findings [11]. UD (Figure 1(c)) and UE (Figure 1(d)), on the other hand, are less affected by sample size than ID (Figure 1(a)). These findings back up previous findings in species ecology. In fact, Pielou’s fairness in measuring the homogeneity of the distribution of species in a population does not depend on the number of species [28]. Furthermore, UD (Figure 1(c)) discovered to be more sensitive to the number of uses considered than to the size of the respondents, which explains its relative low variance as the size of respondents increases.

Figure 2 presents the evolutionary trend of the four ethnobotanical indices considered in function of the samples size. The value of the interviewee’s diversity (ID) (Figure 2(a)) increases steadily with the sample size while the other three indices have a plateau starting from characteristic values. For interviewee equitability (IE) (Figure 2(c)), the maximum value (0.99) was noted around 1000 interviewees (greater than the study sample size) while for the use diversity (UD) (Figure 2(b)), the maximum value of 1.82 was obtained for a sample size of 750 (slightly higher than the study sample size of 704). The same trend was noted for the use equitability (UE) (Figure 2(d)).