Multiple pairwise comparison tests of treatment means are of great interest in applied research. Two modifications for the Tukey test were proposed. The power of unilateral and bilateral Student, Waller-Duncan, Duncan, SNK, REGWF, REGWQ, Tukey, Bonferroni, Sidak, unilateral Dunnet statistical tests and the modified tests, Sidak, Bonferroni 1 and 2, Tukey 1 and 2, has been compared using the Monte Carlo method. Data were generated for 600 experiments with eight treatments in a randomized block design, of which 400 had four and 200 eight blocks. The differences between the treatment means in relation to the control were 30%, 20%, 15%, 10%, 5%. Two extra treatments did not differ from the control. A coefficient of variation of 10% and a probability Type I error of a = 0.05 were adopted. The power of all the tests decreased when the differences to the control, decreased. The unilateral and bilateral Student t, Waller-Duncan and Duncan tests showed greater number of significative differences, followed by unilateral Dunnett, modified Sidak, modified Bonferroni 1 and 2, modified Tukey 1, SNK, REGWF, REGWQ, modified Tukey 2, Tukey, Sidak and Bonferroni. There is great loss of efficiency for all tests in relation to the unilateral Student t test for each difference of the treatment to the control, when the differences between means decrease. The modified tests were always more efficient than their original ones.
RESUMO -Os delineamentos sistemáticos se destacam pela sua compacidade e abrangência e por permitir testar maior número de espaçamentos possíveis. No entanto, não é utilizado devido ao arranjo sistemático (não casualizado) das plantas e à alta sensibilidade para valores perdidos. O objetivo deste trabalho foi descrever o modelo geoestatístico e métodos associados de inferência no contexto de análise de experimentos não aleatorizados, reportando resultados aplicados para identificar a dependência espacial em um particular experimento em delineamento sistemático tipo leque de Eucalyptus dunnii. Também foram propostas, analisadas e comparadas diferentes alternativas para tratar dados faltantes que pudessem advir de falhas e, ou, mortalidade de plantas. Os dados foram analisados seguindo-se três modelos que diferiram, com co-variáveis, na forma de tratar os dados faltantes. Para cada um destes foi construído um semivariograma, com o ajuste de três modelos de função de correlação, sendo os parâmetros estimados pelo método de máxima verossimilhança e selecionados pelo critério de Akaike. Esses modelos, com e sem o componente espacial, foram comparados pelo teste da razão de verossimilhança. De acordo com os resultados, verificou-se que: (1) as co-variáveis interagiram positivamente com a variável de resposta, evitando que dados coletados sejam desperdiçados; (2) a comparação dos modelos, com e sem o componente espacial, não confirmou a existência de dependência; (3) a incorporação da estrutura de dependência espacial aos modelos observacionais recuperou a capacidade de fazer inferências válidas na ausência de aleatorização, permitindo contornar problemas operacionais e, assim, garantindo que os dados possam ser submetidos a uma análise clássica.Palavras-chave: Eucalyptus dunnii, semivariograma, máxima verossimilhança, critério de informação de Akaike e teste de razão de verossimilhança. APPLICATION OF GEOSTATISTICAL METHODS TO IDENTIFY SPATIAL DEPENDENCE IN THE DATA ANALYSIS OF A FOREST SPACING EXPERIMENT WITH A FAN SYSTEMATIC DESIGN
The efficiency of segregation, by particle size, in reducing the aflatoxin content of corn lots under the Brazilian conditions was studied. The aflatoxin content, in the corn fractions obtained after sieving thirty contaminated samples, with approximately 3.0 kg each, was determined. The samples were passed through 4.5 mm round-hole sieves and the fractions with particles ³4.5 mm (fraction A) and <4.5 mm (fraction B) and then analyzed for aflatoxins by TLC. The results showed that the distribution of the aflatoxins among the fractions A and B varied from sample to sample. However, the fraction B presented higher content (6 to 1,422 µg/kg) of aflatoxins (P<5%), than samples of fraction A (4 to 389 µg/kg). In spite of that the segregation of this fraction did not reduce significantly the contamination of the samples. This probably occurred because the contribution of the fraction B for the estimated total of the concentration of aflatoxin of the sample was smaller than that of the fraction A, due to the smallest amount of the fraction B in the sample weight total. In spite of being significant for the studied samples the correlation among the percentages of the groups of grains with defects a) burning grains, sprouted, burned and moldy and b) total injured, with the estimated level of contamination for the samples, more studies will be necessary so that better conclusions can be drawn for this correlation.
Results of practical importance had been discarded testing formulated hypothesis with the aid of statistical analysis of experimental data because of the power of the utilized test. This study compares the power of two Bonferroni's Modified and one Sidak's Modified tests with known tests analyzing 1200 simulated experiments. All differences of means were obtained in relation to the mean of the adopted control to guarantee parametrical magnitude of mean differences. Student's test (type I comparisonwise error) and Waller-Duncan's (Bayesian error) showed the highest percentage of significative differences, followed by Duncan's, BM 2 , S i M, BM 1 , DunnettU's, S i N, BN, Dunnettu's, SNK's, REGWF's, REGWQ's, Tukey's, Sidak's and Bonferroni's tests. For differences equal to zero, Student's and Waller-Duncan's test exhibit 5% frequency of rejection of the null hypothesis, in accordance the nominal error I adopted (α = 0.05). All other tests had values below 0.05, generally ranging on 0.01 to 0.02 or less. Depending of the number of zero differences and considering the type I experimentwise error I, Student's, Waller-Duncan's and Duncan's tests showed crescent values of errors (> 0.05), proportional to the number of null differences included in the experiment; all other tests exhibit showed of type I experimentwise error < 0.05, most nearing 0.01-0.02 or less. Efficiency of the three "Modified Tests" was close to DunnettU's test, but higher than the other testes of type I experimentwise error nature (MEER). Key words: statistical tests, efficiency of tests, multiples comparisons TESTE MODIFICADO DE BONFERRONI E SIDAKRESUMO: Para a comprovação de hipóteses, experimentos são conduzidos e os resultados obtidos analisados estatisticamente. Entende-se que, em função dos testes utilizados, muito material de importância prática tem sido descartado. Neste estudo, dois testes de Bonferroni Modificados e um teste de Sidak Modificado foram desenvolvidos, o poder desses testes avaliados através da simulação de 1.200 experimentos e sua eficiência comparada às de testes de significância mais conhecidos. As diferenças das médias foram, todas elas, obtidas em relação à média do tratamento controle de forma a garantir, parametricamente, a magnitude das diferenças. O teste de Student (do tipo erro I, por comparação), WallerDuncan teste (erro de natureza bayesiana) mostraram mais alta porcentagem de diferenças significativas seguida pelos testes de Duncan, BM 2 , S i M, BM 1 , Dunnett, SNK, REGWF, REGWQ, Tukey, Sidak e Bonferroni. Para diferenças iguais a zero, os testes de Student, Waller-Duncan exibem frequência de rejeição da hipótese nula próxima a 0,05, de acordo com o erro tipo I adotado (α = 0,05). Os outros testes exibem valores < 0,05, quase todos com valores entre 0,01 a 0,02 ou menos. Considerando o erro experimental tipo I, por experimento, os testes de Student, Waller-Duncan e Duncan exibem, para diferenças nulas, valores crescentes de erro I, proporcional ao número de diferenças nulas incluidas nos experimentos; ...
Leaf area measurements are required in several agronomical studies. Usually, there is an interest for measurement methods that are simple, quick and that will not destroy the leaf. The objectives of this work were to evaluate leaf area (y), length (l) and width (w) of 20 half-sibling progenies of custard apple tree (Annona squamosa L.), and to fit regression equations of the type y = a + bx, where x = l.w, that will allow y to be estimated based on l and w. The experiment was conducted as random blocks with five replicates and four plants per plot. Five mature leaves were randomly collected from each plant. Leaf area was measured with an automatic measuring device and leaf dimensions were determined with a ruler. All values of b were different from zero. Differences occurred only in 11% of the 190 possible comparison pairs between progenies, with regard to the estimates of b. No differences were observed between progenies with respect to leaf length, width and area. In view of this fact, the equation y = 0.72 x (R² = 0.77) was fitted for all progenies.
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