Análise estatística da distribuição de Poisson

The general properties of POISSON distributions and their relations to the binomial distribuitions are discussed. Two methods of statistical analysis are dealt with in detail: X2-test. In order to carry out the X2-test, the mean frequency and the theoretical frequencies for all classes are calculated. Than the observed and the calculated frequencies are compared, using the well nown formula: f(obs) - f(esp) 2; i(esp). When the expected frequencies are small, one must not forget that the value of X2 may only be calculated, if the expected frequencies are biger than 5. If smaller values should occur, the frequencies of neighboroughing classes must ge pooled. As a second test reintroduced by BRIEGER, consists in comparing the observed and expected error standard of the series. The observed error is calculated by the general formula: δ + Σ f . VK n-1 where n represents the number of cases. The theoretical error of a POISSON series with mean frequency m is always ± Vm. These two values may be compared either by dividing the observed by the theoretical error and using BRIEGER's tables for # or by dividing the respective variances and using SNEDECOR's tables for F. The degree of freedom for the observed error is one less the number of cases studied, and that of the theoretical error is always infinite. In carrying out these tests, one important point must never be overlloked. The values for the first class, even if no concrete cases of the type were observed, must always be zero, an dthe value of the subsequent classes must be 1, 2, 3, etc.. This is easily seen in some of the classical experiments. For instance in BORKEWITZ example of accidents in Prussian armee corps, the classes are: no, one, two, etc., accidents. When counting the frequency of bacteria, these values are: no, one, two, etc., bacteria or cultures of bacteria. Ins studies of plant diseases equally the frequencies are : no, one, two, etc., plants deseased. Howewer more complicated cases may occur. For instance, when analising the degree of polyembriony, frequently the case of "no polyembryony" corresponds to the occurrence of one embryo per each seed. Thus the classes are not: no, one, etc., embryo per seed, but they are: no additional embryo, one additional embryo, etc., per seed with at least one embryo. Another interestin case was found by BRIEGER in genetic studies on the number os rows in maize. Here the minimum number is of course not: no rows, but: no additional beyond eight rows. The next class is not: nine rows, but: 10 rows, since the row number varies always in pairs of rows. Thus the value of successive classes are: no additional pair of rows beyond 8, one additional pair (or 10 rows), two additional pairs (or 12 rows) etc.. The application of the methods is finally shown on the hand of three examples : the number of seeds per fruit in the oranges M Natal" and "Coco" and in "Calamondin". As shown in the text and the tables, the agreement with a POISSON series is very satisfactory in the first two cases. In the third case BRIEGER's error test indicated a significant reduction of variability, and the X2 test showed that there were two many fruits with 4 or 5 seeds and too few with more or with less seeds. Howewer the fact that no fruit was found without seed, may be taken to indicate that in Calamondin fruits are not fully parthenocarpic and may develop only with one seed at the least. Thus a new analysis was carried out, on another class basis. As value for the first class the following value was accepted: no additional seed beyond the indispensable minimum number of one seed, and for the later classes the values were: one, two, etc., additional seeds. Using this new basis for all calculations, a complete agreement of the observed and expected frequencies, of the correspondig POISSON series was obtained, thus proving that our hypothesis of the impossibility of obtaining fruits without any seed was correct for Calamondin while the other two oranges were completely parthenocarpic and fruits without seeds did occur.

A determinação dos números de indivíduos mínimos necessários na experimentação genética

The main object of the present paper consists in giving formulas and methods which enable us to determine the minimum number of repetitions or of individuals necessary to garantee some extent the success of an experiment. The theoretical basis of all processes consists essentially in the following. Knowing the frequency of the desired p and of the non desired ovents q we may calculate the frequency of all possi- ble combinations, to be expected in n repetitions, by expanding the binomium (p-+q)n. Determining which of these combinations we want to avoid we calculate their total frequency, selecting the value of the exponent n of the binomium in such a way that this total frequency is equal or smaller than the accepted limit of precision n/pª{ 1/n1 (q/p)n + 1/(n-1)| (q/p)n-1 + 1/ 2!(n-2)| (q/p)n-2 + 1/3(n-3) (q/p)n-3... < Plim - -(1b) There does not exist an absolute limit of precision since its value depends not only upon psychological factors in our judgement, but is at the same sime a function of the number of repetitions For this reasen y have proposed (1,56) two relative values, one equal to 1-5n as the lowest value of probability and the other equal to 1-10n as the highest value of improbability, leaving between them what may be called the "region of doubt However these formulas cannot be applied in our case since this number n is just the unknown quantity. Thus we have to use, instead of the more exact values of these two formulas, the conventional limits of P.lim equal to 0,05 (Precision 5%), equal to 0,01 (Precision 1%, and to 0,001 (Precision P, 1%). The binominal formula as explained above (cf. formula 1, pg. 85), however is of rather limited applicability owing to the excessive calculus necessary, and we have thus to procure approximations as substitutes. We may use, without loss of precision, the following approximations: a) The normal or Gaussean distribution when the expected frequency p has any value between 0,1 and 0,9, and when n is at least superior to ten. b) The Poisson distribution when the expected frequecy p is smaller than 0,1. Tables V to VII show for some special cases that these approximations are very satisfactory. The praticai solution of the following problems, stated in the introduction can now be given: A) What is the minimum number of repititions necessary in order to avoid that any one of a treatments, varieties etc. may be accidentally always the best, on the best and second best, or the first, second, and third best or finally one of the n beat treatments, varieties etc. Using the first term of the binomium, we have the following equation for n: n = log Riim / log (m:) = log Riim / log.m - log a --------------(5) B) What is the minimun number of individuals necessary in 01der that a ceratin type, expected with the frequency p, may appaer at least in one, two, three or a=m+1 individuals. 1) For p between 0,1 and 0,9 and using the Gaussean approximation we have: on - ó. p (1-p) n - a -1.m b= δ. 1-p /p e c = m/p } -------------------(7) n = b + b² + 4 c/ 2 n´ = 1/p n cor = n + n' ---------- (8) We have to use the correction n' when p has a value between 0,25 and 0,75. The greek letters delta represents in the present esse the unilateral limits of the Gaussean distribution for the three conventional limits of precision : 1,64; 2,33; and 3,09 respectively. h we are only interested in having at least one individual, and m becomes equal to zero, the formula reduces to : c= m/p o para a = 1 a = { b + b²}² = b² = δ2 1- p /p }-----------------(9) n = 1/p n (cor) = n + n´ 2) If p is smaller than 0,1 we may use table 1 in order to find the mean m of a Poisson distribution and determine. n = m: p C) Which is the minimun number of individuals necessary for distinguishing two frequencies p1 and p2? 1) When pl and p2 are values between 0,1 and 0,9 we have: n = { δ p1 ( 1-pi) + p2) / p2 (1 - p2) n= 1/p1-p2 }------------ (13) n (cor) We have again to use the unilateral limits of the Gaussean distribution. The correction n' should be used if at least one of the valors pl or p2 has a value between 0,25 and 0,75. A more complicated formula may be used in cases where whe want to increase the precision : n (p1 - p2) δ { p1 (1- p2 ) / n= m δ = δ p1 ( 1 - p1) + p2 ( 1 - p2) c= m / p1 - p2 n = { b2 + 4 4 c }2 }--------- (14) n = 1/ p1 - p2 2) When both pl and p2 are smaller than 0,1 we determine the quocient (pl-r-p2) and procure the corresponding number m2 of a Poisson distribution in table 2. The value n is found by the equation : n = mg /p2 ------------- (15) D) What is the minimun number necessary for distinguishing three or more frequencies, p2 p1 p3. If the frequecies pl p2 p3 are values between 0,1 e 0,9 we have to solve the individual equations and sue the higest value of n thus determined : n 1.2 = {δ p1 (1 - p1) / p1 - p2 }² = Fiim n 1.2 = { δ p1 ( 1 - p1) + p1 ( 1 - p1) }² } -- (16) Delta represents now the bilateral limits of the : Gaussean distrioution : 1,96-2,58-3,29. 2) No table was prepared for the relatively rare cases of a comparison of threes or more frequencies below 0,1 and in such cases extremely high numbers would be required. E) A process is given which serves to solve two problemr of informatory nature : a) if a special type appears in n individuals with a frequency p(obs), what may be the corresponding ideal value of p(esp), or; b) if we study samples of n in diviuals and expect a certain type with a frequency p(esp) what may be the extreme limits of p(obs) in individual farmlies ? I.) If we are dealing with values between 0,1 and 0,9 we may use table 3. To solve the first question we select the respective horizontal line for p(obs) and determine which column corresponds to our value of n and find the respective value of p(esp) by interpolating between columns. In order to solve the second problem we start with the respective column for p(esp) and find the horizontal line for the given value of n either diretly or by approximation and by interpolation. 2) For frequencies smaller than 0,1 we have to use table 4 and transform the fractions p(esp) and p(obs) in numbers of Poisson series by multiplication with n. Tn order to solve the first broblem, we verify in which line the lower Poisson limit is equal to m(obs) and transform the corresponding value of m into frequecy p(esp) by dividing through n. The observed frequency may thus be a chance deviate of any value between 0,0... and the values given by dividing the value of m in the table by n. In the second case we transform first the expectation p(esp) into a value of m and procure in the horizontal line, corresponding to m(esp) the extreme values om m which than must be transformed, by dividing through n into values of p(obs). F) Partial and progressive tests may be recomended in all cases where there is lack of material or where the loss of time is less importent than the cost of large scale experiments since in many cases the minimun number necessary to garantee the results within the limits of precision is rather large. One should not forget that the minimun number really represents at the same time a maximun number, necessary only if one takes into consideration essentially the disfavorable variations, but smaller numbers may frequently already satisfactory results. For instance, by definition, we know that a frequecy of p means that we expect one individual in every total o(f1-p). If there were no chance variations, this number (1- p) will be suficient. and if there were favorable variations a smaller number still may yield one individual of the desired type. r.nus trusting to luck, one may start the experiment with numbers, smaller than the minimun calculated according to the formulas given above, and increase the total untill the desired result is obtained and this may well b ebefore the "minimum number" is reached. Some concrete examples of this partial or progressive procedure are given from our genetical experiments with maize.

Análise da variação qualitativa em amostras pequenas

Na aplicação do X2-teste devemos distinguir dois casos : Á) Quando as classes de variáveis são caracterizadas por freqüências esperadas entre p = 0,1 e p = 0,9, podemos aplicar o X2-teste praticamente sem restrição. É talvez aconselhável, mas não absolutamente necessário limitar o teste aos casos nos quais a freqüência esperada é pelo menos igual a 5. e porisso incluimos na Táboa II os limites da variação de dois binômios ( 1/2 + 1/2)n ( 1/4 + 3/4)n para valo r es pequenos de N e nos três limites convencionais de precisão : ,5%, 1% e 0,1%. Neste caso, os valores dos X2 Índividuais têm apenas valor limitado e devemos sempre tomar em consideração principalmente o X2 total. O valor para cada X2 individual pode ser calculado porqualquer das expressôe seguintes: x2 = (f obs - f esp)²> f. esp = ( f obs - pn)2 pn = ( f obs% - p)2.N p% (100 - p%) O delta-teste dá o mesmo resultado estatístico como o X2-teste com duas classes, sendo o valor do X2-total algébricamente igual ao quadrado do valor de delta. Assim pode ser mais fácil às vezes calcular o X2 total como quadrado do desvio relativo da. variação alternativa : x² = ( f obs -pn)² p. (1-p)N = ( f obs - p %)2.N p% (100 - p%) B) Quando há classes com freqüência esperada menor do que p = 0,1, podemos analisar os seus valores individuais de X2, e desprezar o valor X2 para as classes com p maior do que 0,9. O X2-teste, todavia, pode agora ser aplicado apenas, quando a freqüência esperada for pelo menos igual ou maior do que 5 ou melhor ainda, igual ou maior do que 10. Quando a freqüência esperada for menor do que 5, a variação das freqüências observadas segue uma distribuição de Poisson, não sendo possível a sua substituição pela aproximação Gausseana. A táboa I dá os limites da variação da série de Poisson para freqüências esperadas (em números) desde 0,001 até 15. A vantagem do emprego da nova táboa I para a comparação, classe por classe, entre distribuições esperadas e observadas é explicada num exemplo concreto. Por meio desta táboa obtemos informações muito mais detablhadas do que pelo X2-teste devido ao fato que neste último temos que reunir as classes nas extremidades das distribuições até que a freqüência esperada atinja pelo menos o valor 5. Incluimos como complemento uma táboa dos limites X2, pára 1 até 30 graus de liberdade, tirada de um outro trabalho recente (BRIEGER, 1946). Para valores maiores de graus da liberdade, podemos calcular os limites por dois processos: Podemos usar uma solução dada por Fischer: √ 2 X² -√ 2 nf = delta Devem ser aplicados os limites unilaterais da distribuição de Gauss : 5%:1, 64; 1%:2,32; 0,1%:3,09: Uma outra solução podemos obter segundo BRIEGER (1946) calculando o valor: √ x² / nf = teta X nf = teta e procurando os limites nas táboas para limites unilaterais de distribuições de Fischer, com nl = nf(X2); n2 = inf; (BRIEGER, 1946).

Formas de cultura de Trypanosoma rotatorium Mayer, 1843: isolado da rã Leptodactylus ocellatus do Brasil

Os autores estudaram o comportamento "in vitro" do Trypanosoma encontrado nas rãs brasileiras, visando critérios adicionais na caracterização específica deste grupo. Utilizaram diferentes meios de cultura (NNN, Novy e Mac Neal, SNB 9 de Diamond 1954, Boné & Steinert, 1956 Boné & Parent 1963 e Halevy & Gisry 1964) no isolamento do Trypanosoma rotatorium encontrado com certa freqüência na rã Leptodactylus com larga distribuição na região Neotropical. Observamso que o comportamento do T. rotatorium das rãs desta região em meios de cultura mostra características bem diferentes daquelas observadas com tripanosomas de outras regiões, quer seja pela dificuldade de manutenção em subcultura, quer pelas formas de divisão desenvolvidas. Empregamos os mesmos meios de cultura utilizados nos isolamentos dos tripanosomas de rã da Europa e como pode ser visto no Quadro I os resultados obtidos com material da região Neotropical são concordantes, surgerindo, pelo menos uma variação dentro da espécie.

Contribuição para o conhecimento dos parasitos de peixes do litoral do Estado da Guanabara - parte II

Neste presente trabalho os autores criam uma nova espécie para o subgênero Plagioporus (Plagioporus) Stafford, 1904, P. (P.) dollfusi sp. n. fica no grupo "b" da distribuição de travassos & cols. (1966), mais se aproximando de P. (P.) multilobatus Travassos & cols. 1966, distinguindo-se principalmente por possuir o limite anterior dos vitelinos na zona acetabular, poro genital bifurcal, bolsa do cirro alcançando a zona acetabular e ovos menores. Apresentam como novas ocorrências Enenterum pimelopteri Nagaty 1942 e Pseudopecoelus priacanthi (Mac Callum, 1921) manter, 1947. Apresentam ainda Garrupa sp. como novo hospedeiro de Pseudopecoelus priacanthi. Referem a presença de Hysterolecitha elongata manter, 1931, Bucephalus varicus manter, 1940, Metadena spectanda travassos, Freitas & Bührnheim, 1967 e uma fêmea jovem de Echinorhynchideae Southwell & Macfie, 1925 (Acanthocephala).

Técnica para a observação do comportamento do Triatoma infestans (Klug, 1834) em uma miniatura de casa de pau-a-pique e sapê

Com o objetivo de observar e registrar o comportamento do Triatoma infestans quanto a movimentos, posturas e estados fisiológicos, como preconizam os etólogos, foi construída uma réplica de uma casa de paua-pique e sapê com uma proteção externa de acrílico transparente. Para o registro das atividades empregou-se a cinematografia com lapso de tempo, através de uma filmadora super-8 sincronizada a um flash eletrônico e programada para disparos simultâneos de 1 fotograma a cada 30 segundos. A análise dos dados foi feita com um projetor super-8 e um editor, que permitiu observar cada fotograma. Com um período de registros durante 6 dias ininterruptos, os resultados permitiram concluir que: a) na ausência de estímulo alimentar, não ocorre atividade locomotora no T. infestans, independente de ser dia ou noite, mesmo com o inseto privado de alimentos; b) em presençaa do estímulo alimentar a atividade locomotora ocorre durante as 24 horas do dia, embora em proporção significantemente maior no período de obscuridade.

Marcação individual de triatomíneos para estudos comportamentais e ecologicos

Idealizou-se uma técnica de marcação de insetos adultos, visando principalmente à identificação individual de triatomineos, que consiste na elaboração de códigos correspondentes a números, através de cinco cores basicas (vermelho, branco, azul, verde e amarelo) representadas por pintas coloridas feitas com tinta esmalte e depositadas do pronoto ao escutelo do inseto manualmente, com um fino pincel de seda. As pintas não devem se estender às asas sobrepostas, porque estas mudam constantemente de posição, encobrindo assim a marcação. A tinta é indelével e, por não apresentar toxicidade, não afeta a longevidade e o comportamento dos insetos. A técnica pode ser utilizada tanto para insetos no laboratório quanto no campo principalmente em trabalhos relacionados à Ecologia e ao Comportamento.

Virological study of a dengue type 1 epidemic at Rio de Janeiro

A dengue outbreak started in March, 1986 in Rio de Janeiro and spread very rapidly to other parts of the country. The great majority of cases presented classical dengue fever but there was one fatal case, confirmed by virus isolation. Dengue type 1 strains were isolated from patients and vectors (Aedes aegypti) in the area by cultivation in A. albopictus C6/36 cell line. The cytopathic effect (CPE) was studied by electron microscopy. An IgM capture test (MAC-ELISA) was applied with clear and reproducible results for diagnosis and evaluation of virus circulation; IgM antibodies appeared soon after start of clinical disease, and persisted for about 90 days in most patients. The test was type-specific in about 50% of the patients but high levels of heterologous response for type 3 were observed. An overall isolation rate of 46,8% (813 virus strains out of 1734 specimens) was recorded. The IgM test increased the number of confirmed cases to 58,2% (1479 out of 2451 suspected cases). The importance of laboratory diagnosis in all regions where the vectors are present is emphasized.

Implementació i simulació funcional en MATLAB del protocol IEEE 802.15.4 de WSN

El principal objectiu que es pretén assolir és dissenyar un simulador de la capa MAC, definida a l’estàndard IEEE 802.15.4. Convé matisar que l’objectiu no és implementar un simulador, sinó utilitzar una plataforma genèrica existent, MATLAB, i definir sobre ella, una metodologia que, permeti utilitzar-la més com a plataforma específica de desenvolupament i simulació de protocols per a xarxes de sensors, que no pas com a simple simulador.

Study of two different enzyme immunoassays for the detection of Mayaro virus antibodies

This paper presents the evaluation of an enzyme immunoassay in which Mayaro virus-infected cultured cells ara used as antigen (EIA-ICC) and an IgM antibody capture ELISA (MAC-ELISA) for Mayaro serologic diagnosis using 114 human sera obtained during a Mayaro outbreak occurred in Bolivia, in 1987. Results were compared with those obtained by haemagglutination-inhibition test (HAI). MAC-ELISA was the most sensitive technique for anti-Mayaro IgM detection. MAC-ELISA was twice sensitive as IgM EIA-ICC. The data shows that MAC-ELISA is a practical and valid technique for diagnosis of recent mayaro infection. IgG-ICC showed hight sensitivity and high specificity compared to HAI. The combination of anti-Mayaro IgG and IgM EIA-ICC results presented the highest sensitivity of the study. Anti-Mayaro IgG and IgM simultaneous detection by ELISA-ICC can be used for recent infection diagnosis (in spite of a less sensitive IgM detection than by MAC-ELISA), for surveillance and sero-epidemiologic studies, and for studies of IgG and IgM responses to Mayaro infection.

Cumbre de Viena: energía, polarización y búsqueda de liderazgos

La IV Cumbre Unión Europea – América Latina y el Caribe (UE-ALC), celebrada en Viena los días 11, 12 y 13 de mayo, estuvo marcada por una creciente polarización interna latinoamericana, por los problemas europeos para definir un liderazgo sobre el futuro del proyecto de integración y por la crisis global de energía.

Serological diagnosis of dengue by an Elisa inhibition method (EIM)

An ELISA Inhibition Method (EIM) was proposed for the serologic diagnosis of dengue, comparing its results with the Hemagglutination Inhibition (HI) and the IgM capture-ELISA (MAC-ELISA). Advantages and disadvantages of both methods are discussed according to sensitivity, specificity, performance and usefulness. As a conclusion we recommend the complementary inclusion of the EIM and MAC-ELISA substituting the HI for laboratories engaged in the diagnosis and surveillance of dengue.

Disseminated M. avium complex infection in the Swiss HIV Cohort Study: declining incidence, improved prognosis and discontinuation of maintenance therapy.

Introduction of potent antiretroviral combination therapy (ART) has reduced overall morbidity and mortality amongst HIV-infected adults. Some prophylactic regimes against opportunistic infections can be discontinued in patients under successful ART. (1) The influence of the availability of ART on incidence and mortality of disseminated M. avium Complex infection (MAC). (2) The safety of discontinuation of maintenance therapy against MAC in patients on ART. The Swiss HIV-Cohort Study, a prospective multicentre study of HIV-infected adults. Patients with a nadir CD4 count below 50 cells/mm3 were considered at risk for MAC and contributed to total follow-up time for calculating the incidence. Survival analysis was performed by using Kaplan Meier and Cox proportional hazards methods. Safety of discontinuation of maintenance therapy was evaluated by review of the medical notes. 398 patients were diagnosed with MAC from 1990 to 1999. 350 had a previous CD4 count below 50 cells/mm3. A total of 3208 patients had a nadir CD4 count of less than 50 cells/mm3 during the study period and contributed to a total follow-up of 6004 person-years. The incidence over the whole study period was 5.8 events per 100 person-years. In the time period of available ART the incidence of MAC was significantly reduced (1.4 versus 8.8 events per 100 person-years, p < 0.001). Being diagnosed after 1995 was the most powerful predictor of better survival (adjusted hazard ratio for death: 0.27; p < 0.001). None of 24 patients discontinuing maintenance therapy while on ART experienced recurrence of MAC during a total follow-up of 56.6 person-years (upper 95% confidence limit 5.3 per 100 person-years). Introducing ART has markedly reduced the risk of MAC for HIV-infected individuals with a history of very low CD4 counts. Survival after diagnosis of MAC has improved after ART became available. In patients responding to ART, discontinuation of maintenance therapy against M. avium may be safe.

Possible absence of attraction to odor in Panstrongylus megistus (Hemiptera: Reduviidae) under laboratory conditions

We tested the attraction of Panstrongylus megistus odor under laboratory conditons, between males and females of this species and by individuals of each sex on recently fed virgin couples. We employed a system of choice boxes both with or without aeration over the stimuli in the tested situations. We also observed a clear trend among the insects to remain in the central box where they had been placed in the beginning of the tests.

Estudio de los mecanismos terapéuticos en la inducción de tolerancia immunológica en un modelo animal de esclerosis múltiple

En una serie de estudios previos demostramos que la infusión de células de médula ósea (MO) modificadas genéticamente para la expresión del autoantígeno MOG40-55 en ausencia de mieloablación inducía tolerancia antígenoespecífica en un modelo murino de esclerosis múltiple. También observamos que este efecto terapéutico no requería injerto hematopoyético. Nos propusimos estudiar si el efecto tolerogénico está inducido por una subpoblación de células generadas durante la transducción de la MO y el papel de las células T reguladoras en la inducción de la tolerancia. Las células de MO fueron cultivadas y transducidas usando medio complementado con 20% FCS y medios condicionados como fuente de stem cell factor (SCF) e IL-3 murinos. Las diferentes poblaciones celulares se separaron por citometría de flujo y se analizó la capacidad supresora de las poblaciones candidatas. Por otro lado se analizó la presencia de células T reguladoras en bazo y SNC de los ratones recuperados después de la infusión de células de MO transducidas. A los cinco días de cultivo, la mayoría de células presentaban fenotipo mieloide (Mac-1+Gr-1low/-:31,9+-10,2%; Mac-1+Gr-1high:26,0+-3,3%). Ambos fenotipos se corresponden con dos subpoblaciones de células mieloides supresoras (MDSC, tipo monocí¬tico y granulocítico respectivamente) descritas recientemente. Se estudió la capacidad de ambas poblaciones para suprimir la respuesta proliferativa específica de esplenocitos frente a MOG40-55 in vitro, observando una mayor capacidad de supresión de las MDSC monocíticas, que se correspondí¬a con niveles significativamente superiores de actividad de las enzimas arginasa-1 y sintasa de óxido nítrico (ambos mecanismos supresores característicos de las MDSC). A los 7 días del tratamiento no se observaron diferencias significativas en el porcentaje de células T reguladoras (Treg y Tr1) entre el grupo tratado (liM) y los grupos de control.