Introduction
Germination is the most frequently described trait in seed science, and methods of statistical analyses for such studies, including analysis of variance (ANOVA), Student's ttest, Tukey, ScottKnott, KruskalWallis, and MannWhitney tests, appear to be validated by their continued use. However, the perspective of researchers that perform seed biology research is not shared by statisticians, primarily because of the nature of variables involved in germination. For seed scientists, the number of germinated seeds is converted into a percentage and an assumed normal distribution is attributed to this variable, for which ANOVA is the preferred statistical method (^{Sileshi 2012}). Developed by Fisher between 1920 and 1935, ANOVA models emphasize the importance of repetition, randomization and local control on experimental efficiency (^{Fisher 1925}; ^{1934}). Based on a normal linear model, ANOVA requires homoscedasticity, independent residuals and additivity effects between treatments and blocks (^{Steel & Torrie 1980}; ^{Sokal & Rohlf 1995}). Therefore, some researchers have opted to apply angular transformation, which is the widely known
Violations of ANOVA’s assumptions, the low statistical power of nonparametric tests and criticism to data transformation (^{Warton & Hui 2011}) have led to the development of statistical models with fewer requirements other than the normal distribution. The Generalized Linear Models (GLM) are the conjunction of statistical models formed by more flexible probability distributions, being ANOVA a particularization of these models (^{Nelder & Wedderburn 1972}; ^{Wilson & Hardy 2002}; ^{Warton & Hui 2011}). The GLMs include several error distributions including the binomial, Poisson, negative binomial, and beta binomial, bringing more alternatives for the variable distribution in seed studies due to the binary nature of germination (i.e. seeds either germinate or do not germinate), fixed number of seeds, independence of germination (the chances for a seed to germinate is not affected by the other seeds) and variation in germination occurrence.
In scientific research, the statistical advances involving counting variables are recognized, but some peculiarities of the process of seed formation from some Brazilian species may limit the analysis of the true seed germination potential. For species distributed on quartzitic campos rupestres, the combination of shallow, extremelyimpoverished soils (^{Benites et al. 2003}; ^{Oliveira et al. 2015}) with pollen limitation, and genetic load may result in a tradeoff between sexual vs. asexual reproduction. These characteristics can affect the seed development process and increase the frequency of embryoless seeds. Empty seeds are phylogenetically skewed and occur in several families (^{Dayrell et al. 2016}). As a result, it is common to attribute, albeit incorrectly, low percentages of germination to certain campos rupestres species.
The percentage of seeds with embryos can be increased through physical separation methods like an aspirator seed cleaning machine, sieves or Xray techniques. By identifying seeds without an embryo, such techniques have a direct impact on germination levels (^{Tonetti et al. 2006}). However, the absence of an embryo is not easily perceived or quantified for most campos rupestres seeds, and it is sometimes difficult to separate seeds with or without an embryo from the samples.
An alternative for most species is to quantify the number of seeds with embryos regardless of whether the embryo is dead or not. In this case, the germination percentage will not be divided by the sample size (e.g., 25, 50 or 100 seeds) but by the number of seeds with an embryo (sum of germinated, dormant and dead seeds). The relative germination minimizes underestimates of the germination potential for species when embryoless seeds are present in the sample. Researchers may question the need to calculate the relative germination percentage, because embryoless seeds are considered an impure fraction of the sample. This impure fraction approach is common among seed technologists, but not always shared by botanists interested in determining causes and reasons for embryoless seeds. Relative germination does not exclude the importance of the germination percentage as it is usually calculated; only the selection criteria are different, standardized for seed with embryo.
The evolution of statistical analysis for seed germination analysis, starting from the nonparametric tests and getting to the Generalized Linear Models, is the subject of this paper. It was also aimed to describe a relative measurement of germination for L. ericoides seeds, as an example of other species from campos rupestres with embryo development problems.
Materials and methods
To develop a chronology of statistical advancements in germination, an experiment was performed with seeds from Lychnophora ericoides Mart. (Asteraceae), a species in a genus that is endemic in campos rupestres from Brazilian states of Bahia, Goiás and Minas Gerais (^{Semir 1991}). Lychnophora ericoides capitula were collected in 2007 from individuals distributed at altitudes between 1,102 and 1,245 m from a population distributed in Serra da Bocaina, Brazil, which is formed by quartzite mountains at a maximum altitude of 1,350 m. The average maximum and minimum temperatures are 26.5 and 15.7 °C, respectively, and the pluviometric index is 1,574.7 mm per year.
The germination experiment was conducted in a germination chamber (Seedburo Equipment Company), set to a mean day temperature of 26.3 °C and a mean night temperature of 23.8 °C with a mean irradiance of 114.82 ± 8.36 µmol m^{2} s^{1}. Seeds of 20 individuals of L. ericoides were distributed in a completely randomized design with four repetitions, formed by 80 plots containing 50 seeds each (200 seeds per individual). Germination was scored daily until 70 days after sowing, and the evaluation criterion was radicle protrusion. After germination ceased, the nongerminated seeds were cut to determine the number of seeds with and without an embryo per individual. Seed mortality was scored only at the end of the experiment and, if containing an embryo, added to the number of viable seeds with an embryo, indicated by tetrazolium test, to calculate the relative germination percentage. The characteristics analyzed were germination percentage over 50 seeds and relative germination percentage over the number of seeds with an embryo.
The results were analyzed in detail using normal and binomial distributions (GLM) and nonparametric tests. Since the species aggregates the main characteristics and particular populations distributed in campos rupestres, there is no obstruction to implementing these statistics routines to other species.
Relative germination percentage was obtained by the quotient between the number of germinated seeds and the number of seeds with an embryo (germinated, not germinated or dead):
The usual expression is divided by the sample size:
Initially, the experimental results were statically analyzed by converting the number of germinated seeds into a percentage. Adjustment of residuals to the normal distribution was checked according to the KolmogorovSmirnov (KS) test and variance homogeneity was checked with the Levene (F) test. The assumptions were not met and
For GLM application, the random component for the germination experiment of L. ericoides seeds was expressed in two forms: as a percentage that assumed normal distribution and as a number of germinated seeds that assumed binomial distribution. Both distributions belong to the parametric exponential family and present probability density functions defined as follows.
Normal:
where y is the germination percentage (G or RG); μ is the mean; 𝜎 2 is the variance; and 𝜋 is the mathematical constant approximated by 3.14;
and
Binomial:
where y is the number of germinated seeds; n is the number of seeds per repetition (total seeds or only those with an embryo); and 𝜋 is the proportion of germinated seeds.
The explanatory variable represented by L. ericoides individuals and the effect of their respective germination corresponded to the systematic component of the model. For binomial distribution, the link function adopted was logit, and for normal distribution it was identity. Normal plots performed a graphical comparison between the studentized deviation component and the observed quartiles of the sample. Graphs were generated with intervals of 95 % of confidence to guarantee a better inference in the visual analysis. In all tests, the established significance was 0.05.
Results
A brief comparison between the means of the absolute percentage (G) and the relative percentage (RG) of germination showed evidences of the underestimation of germination potential of some individuals when germination was based on the number of sown seeds of the sample and not on the number of seeds with an embryo (Tab. 1). Individual 7 was erroneously quantified as not efficient by absolute germination (12 %), while this individual showed a high relative germination capacity (RG = 78.1 %). The high relative seed germination potential could also be observed for individuals 11, 13 and 20. The relative percentages did not underestimate the potential of seed germination of L. ericoides and provided an improved discrimination among individuals. However, some individuals presented both absolute (G) and relative (RG) low germination (e.g., individuals 1, 5 and 14).
Ind.  G (%)  Seeds without embryo  RG (%)  Ind.  G (%)  Seeds without embryo  RG (%) 

1  0.0  81.5  0.0  11  15.0  79.5  72.4 
2  2.5  88.0  15.7  12  0.0  71.5  0.0 
3  1.5  68.0  6.3  13  14.5  73.0  52.4 
4  1.5  82.0  9.4  14  1.0  65.5  3.6 
5  1.5  74.0  4.7  15  0.0  81.0  0.0 
6  9.5  77.0  41.8  16  6.5  83.5  45.3 
7  12.0  85.5  78.1  17  2.0  77.5  7.7 
8  4.0  80.5  18.8  18  6.0  78.0  30.8 
9  2.0  79.0  8.3  19  0.0  90.0  0.0 
10  1.5  80.5  13.6  20  9.0  85.0  66.9 
The first statistical check of assumptions of the model indicated low probabilities of KolmogorovSmirnov and Levene tests (P < 0.05), demonstrating nonNormal residuals and heterogeneous variances for the germination percentage (G) and relative germination (RG) of L. ericoides seeds, both on the original and transformed scale (Tab. 2). The GLM/ANOVA model was applied to G and RG data even with the violation of the assumptions. The nonparametric KruskalWallis (KW) test was also applied. In both procedures, probabilities lower than 0.05, associated with Snedecor's F statistic and KruskalWallis' H statistic, indicated one or more differences in the germination (G and RG) of L. ericoides individuals 1 and 20 (Tab. 3). An important detail of GLM/ANOVA was the high values of the coefficient of variation for both germination percentages (80.5 and 71.4 %), sensu ^{PimentelGomes (2000}).
KolmogorovSmirnov test  Levene test  

KS (Prob.)  Normal Residuals  F (Prob.)  Homogeneous variance  
G  
Original scale  0.269 (0.000)  No  2.856 (0.001)  No 

0.312 (0.000)  No  2.895 (0.001)  No 
RG  
Original scale  0.276 (0.000)  No  5.468 (0.000)  No 

0.295 (0.000)  No  4.841 (0.000)  No 
Prob. Probability; G germination percentage; RG relative germination
ANOVA  

Germination (G)  
Source  df  SS  MS  F statistic  Pvalue 
Individual  19  1856.0  97.684  7.438  0.000 
Residual  60  788.0  13.133  
CV=80.5%  
Relative germination (RG)  
Source  df  SS  MS  F statistic  Pvalue 
Individual  19  52253.157  2750.166  9.534  0.000 
Residual  60  17307.878  288.465  
CV=71.4%  
KruskalWallis test  
Df  H statistic  Pvalue  
Germination (G)  19  55.340  0.000  
Relative germination (RG)  19  58.773  0.000 
df degrees of freedom, SS sum of squares, MS mean square, F Snecedor statistics, P Probability, H KrukalWallis statistics, CV coefficient of variation
It was expected that tests for multiple comparisons would identify at least one difference between means or medians because the hypothesis of equal germination capacities of L. ericoides individuals was rejected. In fact, the ScottKnott test identified differences among individuals of L. ericoides, although these differences were not detected by Dunn's nonparametric test (Tab. 4). The ScottKnott test separated individuals of the species into two groups according to the germination percentage: seeds with germination means equal or less than 6.5 % and seeds with germination between 9 and 15 % (Tab. 4). With regards to the relative percentage, three groups were formed: group I, seeds with percentages higher than 66.9 %; group II, seeds with percentages between 41.8 and 52.4 %; and group III, seeds with percentages lower than 30.8 %. The results pointed to ScottKnott’s results are reliable with the lack of residuals adjustment to normal distribution and in the presence of heterogeneous variances.
ScottKnott test  Dunn test  

Individual  G (%)  Individual  RG (%)  Individual  G (%)  RG (%) 
6, 7, 11, 13, 20  915 a  7, 11, 20  66.978.1 a  120  015 a  081.3 a 
15, 810, 12, 1419  ≤ 6.5 b  6, 13, 16  41.8 52.4 b  
1 5, 810, 12, 14, 15, 17 19  ≤ 30.8 c 
Dunn's nonparametric test was not able to detect differences from 0 to 15 % in the absolute percentage of germination and from 0 to 81.3 % in the medians of the relative percentage of L. ericoides seeds (Tab. 4). It should be noted that the test that preceded it, KW, indicated that there was at least one difference between the medians for the two characteristics, G and RG (Tab. 3). The question arises whether the nonparametric tests form an alternative when GLM/ANOVA’s assumptions are not met.
A simple comparison between the parametric (GLM/ANOVA, ScottKnott) and nonparametric tests (KW and Dunn) showed a greater coherence of parametric tests, even though model assumptions were not met. The results question the efficiency of nonparametric tests. In this case, data can be analyzed by means of GLM for probability distributions other than the normal distribution.
It must be considered that the GLM  normal distribution with identity link function (Tab. 5) is the conventional ANOVA (Tab. 3). In this analysis, as shown previously, the P values were less than 0.05. For the GLM binomial distribution with logit link function, the G and RG variables also had P values lower than 0.05 (Tab. 5).
Generalized Linear Model  normal distribution  identity  

G (%)  
Source  Difference of df  Difference of deviance  F  Pvalue 
Individual  19  1856.0  7.438  < 0.001 
Residual  60  788.0  
D²  70.2%  
RG (%)  
Individual  19  52253.16  9.534  < 0.001 
Residual  60  17307.88  
D²  75.1%  
Generalized Linear Model  binomial Distribution  logit  
G (%)  
Source  Difference of df  Difference of deviance  Pvalue  
Individual  19  213.77  < 0.001  
Residual  60  99.74  
D²  64.5%  
RG (%)  
Individual  19  300.31  < 0.001  
Residual  60  104.27  
D²  74.2% 
The inferential similarity of the models did not imply a similarity of goodness of fit. Binomial distribution was considered to have a better fit with the quantilequantile graphic analysis (Fig. 1). It was possible to detect poor fitting of the observations of the simulated envelope for the normal distribution, mainly at the ends of the envelope. This leakage was only moderately observed in the binomial model.
Discussion
The correction of the germination percentage for the number of seeds with embryos and not the number of seeds of the sample is necessary for botanic families that have problematic seed formation, such as Asteraceae, Cyperaceae, Melastomataceae and Poaceae (^{Dayrell et al. 2016}). We show here that Lychnophora ericoides is a species that represents this issue. It cannot be inferred that the seeds of these species have low germination potential. What can be inferred is that some individuals produce large amounts of seeds without embryos, but when the embryo is present, seeds of some individuals show high germinability. This seed development problem is important for understanding the reproduction potential of a species, and it should be considered in statistical analysis for species that produce large amounts of embryoless seeds.
The detection of these empty seeds in the planning phase is not so simple and only with the imbibition due to the germination process, the absence of the embryo became evident. Moreover, the production of empty seeds has high variability and unpredictability, both at the individual tree and the population level (^{Perea et al. 2013}). One of the consequences of the unquantified presence of empty seeds is the recurrent and ineffective planning of experiments based on germination methods, especially those to overcome dormancy in order to increase the germination percentages.
Labeled as inappropriate, aberrant, and outdated, the most impetuous criticisms fell upon data transformation (^{Sakia 1992}; ^{Wilcox 1998}; ^{Sileshi 2007}; ^{2012}; ^{Osborne 2010}). The question is why ANOVA models (with or without transformations) still are the most applied statistical tools, even with the restrictions and the availability of other GLM models (^{Nelder & Wedderburn 1972}; Sileshi 2012)? Two factors seem to motivate the use of outdated techniques. One is the many statistical programs with procedures anchored in the normal model. The second is a historic labeling that some science variables naturally have a lack of fit to normal distribution and the historical investment in nonparametric techniques.
Normality is an important assumption in the theory of linear models and its deviation can lead to losses of efficiency in the analysis of variance. This loss can be recovered if the true distribution is known and used instead of normal, which is the competence of Generalized Linear Models (^{McCulloch et al. 2008}). However, there are some researchers that promote the robustness of GLM/ANOVA for small deviations from normality and even indicate the nonverification of this assumption (^{Sharpe 1970}; ^{Harwell et al. 1992}; ^{Driscoll 1996}; ^{Faraway 2006}). The problem with this approach is the impossibility to define the limits for small normality deviations. The low probabilities of the KS test are indicative of the large deviation of normal distribution from our data, which makes its robustness questionable for species such as L. ericoides.
The lack of fit of the germination percentages (G and RG) to a normal distribution could be explained because L. ericoides is an endemic, nondomesticated species from campos rupestres. There are several records in the literature which discard the possibility of adjustment of ecological data to a normal distribution (^{Hampel et al. 1986}; ^{Austin 1987}; ^{Biondini et al. 1988}). However, the origin of the data is not sufficient to judge adjustment to normal distribution, and therefore specific tests need to be performed. Germination articles published between 2000 and 2011 revealed that from the experiments that used ANOVA as a statistical tool, only 19.5 % tested the assumption of normality residuals (^{Sileshi 2012}). Based on this reference, in about 80 % of the publications, the fit to a normal distribution is unknown, which makes it impossible to generalize that native or even cultivated species have nonnormal germination data.
Our results also pointed to the inefficiency of angular transformations to approximate the residuals of both variables to normal distribution and to stabilize variances. The criticisms regarding this statistical resource are severe, not only related to its application, but also to the interpretation of the results on a different scale (^{Ahrens et al. 1990}; ^{Fernandez 1992}; ^{Sakia 1992}; ^{Sileshi 2007}; ^{2012}; ^{Jaeger 2008}; ^{Warton & Hui 2011}; ^{Valcu & Valcu 2011}). Although widely criticized, data transformation is the most widely statistical feature used in germination articles in an attempt to correct deviations from normal distribution. Failure to meet the assumptions with the transformed scale for L. ericoides seed germination is evidence that this attempt may not be effective.
Until the introduction and use of GLMs, nonparametric tests such as KruskalWallis (KW), Mann Whitney, Friedman and Dunn, were the only statistical tools available for analyzes of seed germination of species whose data failed to meet ANOVA’s assumptions, such as L. ericoides. In fact, failing to meet the assumptions may lead to loss of test reliability, problems with Type I or II errors (^{Glass et al. 1972}; ^{Bradley 1978}; ^{Levine & Dunlap 1983}; ^{Rasmussen 1985}) and with the level of significance of the test (^{Kempthorne 1952}; ^{Little & Hills 1978}; ^{Gomez & Gomez 1984}). However, these problems are amplified in the nonparametric tests. Differences not detected by the Dunn test in the order of 80% of RG between individuals are part of this problem. Although this result might seem circumstantial and particular for the species, it provided a numerical proof for simulations that have indicated the inefficiency and lower power of nonparametric tests. The observations of these problems related to nonparametric tests are not recent. ^{Box (1953}), stated that: "I do not think that we need necessarily go to the extreme of using nonparametric tests when it may well be that more powerful robust parametric tests can be found".
The problem of statistical analysis in seed germination is not in the denomination parametric or nonparametric, but in the nature of the variable involved and in the probability distribution associated to this variable. Historically analyzed as a continuous variable expressed as a percentage and associated with normal distribution, the nature of the germination data of L. ericoides (G or RG) is discrete. Scoring germination with a fixed n, originated from the total number of seeds or the number of seeds with an embryo, follow a binomial distribution. Many authors warn that there is no indication of the use of GLM/ANOVA for data with binomial nature (^{Zhao et al. 2001}; ^{Agresti 2002}; ^{Warton & Hui 2011}).
With the extension of statistical analyses to other probability distributions, such as binomial achieved with the GLMs, the models and methods used in germination could be revised. The qqplot pointed that binomial distribution was the most suitable model for the absolute and relative germination of L. ericoides. This result will probably be obtained when other species, regardless of the presence or absence of the embryo, are statistically analyzed by this model. ^{Montgomery (2000}) reports the superiority of the GLM approach compared to transformation and nonparametric statistics. Specifically, Jatropha curcas and orchid seedling germination data were better fitted to a binomial logistic model (^{Mora et al. 2008}; ^{Araújo 2012}).
The high values of the experimental coefficients of variation (CV = 80.5 and 71.4 %) for L. ericoides germination are not a consequence of the nonnormality of the residuals and the presence of heterogeneous variances. However, it does not exclude the fact that the problems that affected the CV also affected the assumptions negatively. For L. ericoides seeds, the greatest factor that increased the CV was the absence of germination in one or more repetitions. In fact, the presence of zeros inflates variability, as an immediate consequence in CV (^{Ahmad et al. 2006}), but it is not the only factor. The instability of germination between repetitions of the same individual also contributed for that CV increase.
The increased accuracy of the GLMs for variables with a binomial distribution indicates the future of statistical analysis for seed germination experiments and the need for new procedures that can be used as alternatives for ANOVAs. The germination experiment with L. ericoides seeds presented here is only one among a number of studies that has shown the low efficiency of ANOVA and data transformation. Regardless of the field of study, however, nonparametric tests and data altered by transformations should be avoided.
The divergence between seed scientists and statisticians leads to the question of whether scientific studies of germination that use GLM/ANOVA models, data transformation and nonparametric tests are incorrect. The answer is no. Even in the GLMs context, different distributions and link functions for germination data generate distinct efficiencies, and less efficient does not mean incorrect. This is the relationship between GLM/ANOVA and GLMs. It is possible to lose information when more contemporary techniques are ignored (^{Wilcox 1998}), but the use of GLMs to analyze germination experiments is not a guarantee of maximum efficiency.