Using a Normal Approximation to Test for the Binomial Distribution

The commonly used method to test for the binomial distribution is the x²-test. In this paper, we introduce an alternative method to test for the binomial distribution. Suppose N is the number of sample groups with n individuals each, x_ij is the jth sample in ith group, a Bernoulli variable with parameter and VVI=s²/[m(1 - m)/n]. Then it is well know that the asymptotic distribution of the statistic (N - 1) VVI is x²(N - 1) under the hypothesis p₁ = p₂ = … = p_N. Here we find that VVI has an asymptotic normal distribution N(1, 2(1 - 1/n)/(N - 1)). Unlike the x²-statistic, the variance of the normal test statistic is a function of n. This method is convenient in detecting spatial patterns and dispersion in the study of diseased organisms (e.g., plants) in field samples.     

A theoretical and experimental analysis of the qP and qN coefficients of chlorophyll fluorescence quenching and their relation to photochemical and nonphotochemical events

The initial (F₀), maximal (F_M) and steady-state (F_S) levels of chlorophyll fluorescence emitted by intact pea leaves exposed to various light intensities and environmental conditions, were measured with a modulated fluorescence technique and were analysed in the context of a theory for the energy fluxes within the photochemical apparatus of photosynthesis. The theoretically derived expressions of the fluorescence signals contain only three terms, X=J₂p_2F/(1–G), Y=T/(1–G) and V, where V is the relative variable fluorescence, J₂ is the light absorption flux in PS II, p_2F is the probability of fluorescence from PS II, G and T are, respectively, the probabilities for energy transfer between PS II units and for energy cycling between the reaction center and the chlorophyll pool: F₀=X, F_M=X/(1–Y) and F_S=X(1+(YV/(1–Y))). It is demonstrated that the amplitudes of the previously defined coefficients of chlorophyll fluorescence quenching, q_P and q_N, reflect, not just photochemical (q_P) or nonphotochemical (q_N) events as implied in the definitions, but both photochemical and nonphotochemical processes of PS II deactivation. The coefficient q_P is a measure of the ratio between the actual macroscopic quantum yield of photochemistry in PS II (41-1) in a given light state and its maximal value measured when all PS II traps are open (41-2) in that state, with 41-3 and 41-4. When the partial connection between PS II units is taken into consideration, 1-q_P is nonlinearily related to the fraction of closed reaction centers and is dependent on the rate constants of all (photochemical as well as nonphotochemical) exciton-consuming processes in PS II. On the other hand, 1-q_N equals the (normalized) ratio of the rate constant of photochemistry (k_2b) to the combined rate constant (k_N) of all the nonphotochemical deactivation processes excluding the rate constant k₂₂ of energy transfer between PS II units. It is demonstrated that additional (qualitative) information on the individual rate constants, k_N-k₂₂ and k_2b, is provided by the fluorescence ratios 1/F_M and (1/F₀)–(1/F_M), respectively. Although, in theory, 41-5 is determined by the value of both k_2b and k_N-k₂₂, experimental results presented in this paper show that, under various environmental conditions, 41-6 is modulated largely through changes in k _N, confirming the idea that PS II quantum efficiency is dynamically regulated in vivo by nonphotochemical energy dissipation.Abbreviations Chl chlorophyll - F₀, F_M and F_S initial, maximal and steady-state levels of modulated Chl fluorescence emitted by light-adapted leaves - PS I and II photosystem I and II - q_P and q_N (previously defined) photochemical and nonphotochemical components of Chl fluorescence quenching     

Further Study on the Statistical Test to Detect Spatial Pattern

In the study of spatial patterns, the statistic I' = (n — 1)s²/x was commonly used. In this paper, we found that x — s² has an approximated normal distribution with zero mean if the x_i's (i = 1 to n) are independent identically distributed Poisson variables. Based on this conclusion, the hypothesis that a point pattern is completely random can be tested directly. And a method for the test of spatial patterns was proposed which can be sued as an alternative to the Chi-square based dispersion index test.     

Characterization of the WEIBULL Distribution by Properties of Order Statistics

Let X_1:n, X_2:n, X_3:n…, X_n:n be the order statistics of n independent random variables with the common (absolutely continuous strictly increasing) distribution function F. The main results given in this article are:

• 1 For any fixed r and two distinct numbers s₁ and s₂ (1<r<s₁<s₂≦n) the distributions of V_i and W_i (defined in (1.11) and (1.12) are identical for i = 1,2 iff F(x) is WEIBULL (1.2).
• 2 The statistics D₁ and D₂ (as in (1.8) and (1.9)) are independent iff F(x) is WEIBULL (1.2).
• 3 The statistics U_i (1≦j≦n?1) and X_i:n (i ≦ j) are independent iff F(x) is WEIBULL (1.2).
• 4 Let X, X₁, X₂, …, X_k be random variables such that

These conditions are necessary and sufficient for F(x) to be WEIBULL .     

Deviations from Hardy-Weinberg Proportions: Sampling Variances and Use in Estimation of Inbreeding Coefficients

An analysis is made of the distribution of deviations from Hardy-Weinberg proportions with k alleles and of estimates of inbreeding coefficients (f) obtained from these deviations.—If f is small, the best estimate of f in large samples is shown to be 2Σ _i(T_ii/N_i)/(k - 1), where T_ii is an unbiased measure of the excess of the ith homozygote and N_i the number of the ith allele in the sample [frequency = N_i/(2N)]. No extra information is obtained from the T_ij, where these are departures of numbers of heterozygotes from expectation. Alternatively, the best estimator can be computed from the T_ij, ignoring the T_ii. Also (1) the variance of the estimate of f equals 1/(N(k - 1)) when all individuals in the sample are unrelated, and the test for f = 0 with 1 d.f. is given by the ratio of the estimate to its standard error; (2) the variance is reduced if some alleles are rare; and (3) if the sample consists of full-sib families of size n, the variance is increased by a proportion (n - 1)/4 but is not increased by a half-sib relationship.—If f is not small, the structure of the population is of critical importance. (1) If the inbreeding is due to a proportion of inbred matings in an otherwise random-breeding population, f as determined from homozygote excess is the same for all genes and expressions are given for its sampling variance. (2) If the homozygote excess is due to population admixture, f is not the same for all genes. The above estimator is probably close to the best for all f values.     

Determination of the optimum operating time for batch isothermal performance of enzyme-catalyzed multisubstrate reactions

This communication consists of a mathematical analysis encompassing the maximization of the average rate of monomer production in a batch reactor performing an enzymatic reaction in a system consisting of a multiplicity of polymeric substrates which compete with one another for the active site of a soluble enzyme, under the assumption that the form of the rate expression is consistent with the Michaelis-Menten mechanism. The general form for the functional dependence of the various substrate concentrations on time is obtained in dimensionless form using matrix terminology; the optimum batch time is found for a simpler situation and the effect of various process and system variables thereon is discussed. The reasoning developed here emphasizes, in a quantitative fashion, the fact that the commonly used lumped substrate approaches lead to nonconservative decisions in industrial practice, and hence should be avoided when searching for trustworthy estimates of optimum operation.List of Symbols O 1/s row vector of zeros - a 1/s row vector of rate constants k _i(i = 2,...,N) - A 1/s matrix of rate constants k _i and k_–i (i=2,...,N) - b 1/s row vector of rate constant k ₂ and zeros - C mol/m³ molar concentration of S - C mol/m³ vector of molar concentrations of C _i (i=0, 1, 2, ..., N) - C ₀ mol/m³ column vector of initial molar concentrations of C _i(i=0, 1, 2,.., N) - C _–01 mol/m³ column vector of initial molar concentrations of C _i(i=2,..., N) - C _{E, tot} mol/m³ total molar concentration of enzyme molecules - C _i mol/m³ molar concentration of S _i (i=0,1,2,...,N) - C _{i, o} mol/m³ initial molar concentration of S _i(i=0, 1, 2, ..., N) - E enzyme molecule - I identity matrix - K 1/s matrix of lumped rate constants - k _i 1/s pseudo-first order lumped rate constant associated with the formation of S _i -1 (i=1, 2, ...,N) - k _{cat, i} 1/s first order rate constant associated with the formation of S _i-1 (i=1, 2, ..., N) - K _m mol/m³ Michaelis-Menten constant - L number of distinct eigenvalues - M _i multiplicity of the i-th eigenvalue - N maximum number of monomer residues in a single polymeric molecule - r ₁ mol/m³ s rate of formation of S ₀ - r _i mol/m³ s rate of release of S _i -1 - r _opt maximum average dimensionless rate of production of monomer S₀ - S lumped, pseudo substrate - S₁ inert moiety - S _i substrate containing i monomer residues, each labile to detachment as - S₀ by enzymatic action (i=1,2,...,N) - t s time elapsed since startup of batch reaction - t _lag s time interval required for cleaning, loading, and unloading the batch reactor - t _opt s time interval leading to the maximum average rate of monomer production - v _ij s^1-j eigenvectors associated with eigenvalue imi (i=1, 2, ..., L; j =1, 2, ..., M_i) Greek Symbols _ij mol/m³ arbitrary constant associated with eigenvalue _i (i=1, 2, ..., L; j=1, 2, ..., M _i ) - 1/s generic eigenvalue - _i 1/s i-th eigenvalue     

Multivariate Probability in Terms of Marginal Probability and Correlation Coefficient

This paper is motivated by a practical problem relating to student performance in a number of subjects of equal standing. Its mathematical formulation is to find an approximation to a multivariate probability of the form Pr {X₁⩾a, X₂⩾a, …, X_N⩾a} for arbitrary a and N, in terms of p = Pr {X₁⩾a} and q = Corr (X_i, X_j), i≠j, where X_i, i = 1, …, N are exchangeable random variables with mean 0 and variance unity.     

Phase transition and lateral phase separation in planar lipid membranes as sensed by the lipophilic ion dipicrylamine

The permeation of the lipophilic ion dipicrylamine through planar lipid membranes formed from dipalmitoylphosphatidylcholine in n-decane shows an anomaly near the main phase transition of this system. Both the rate constant, k_i, of ion translocation across the membrane interior and the interfacial concentration, N, of this ion have a maximum at about 36°C. Analogous experiments were performed with tetraphenylborate. A considerably lesser effect of the phase transition was found. The addition of cholesterol leads to a broadening of the maxima for k_i and N. The time course of the current following a voltage jump shows a characteristic change below a temperature of about 45°C, if the molar ratio cholesterol/ phosphatidylcholine in the membrane forming solution exceeds 1. While the current transient decays exponentially above 45°C, a sum of two exponential terms yields an adequate fit below that temperature. This is regarded as evidence for a lateral phase separation below 45°C into structurally different domains, which provide two different pathways for dipicrylamine.     

Purification and Subsite Affinities of exo-lnulinase from Penicillium trzebinskii

An exo-inulinase was highly purified from the culture broth of Penicillium trzebinskii by anion exchange, hydrophobic, and gel filtration chromatographies. The enzyme was homogeneous by disc electrophoresis. The molecular weight was 8.7 × 10⁴, and the isoelectric point was pH 4.3. The enzyme hydrolyzed not only inulin and sucrose but also inulooligosaccharides [1^F(1-β-D-fructofuranosyl)_n-1fructose, F_n (n= 2—5)] and fructooligosaccharides [1^F(1-β-D-fructofuranosyl)_n-1 sucrose, GF_n, (n = 2—8)] liberating the nonreducing terminal fructose of the substrates. The substrate specificity was investigated. The K_m (mM) and k_o (sec^?1were: inulin, 0.042 and 159; sucrose, 6.5 and 169; F₂, 2.1 and 62.8; F₃, 0.40 and 126; F₄, 0.47 and 171; and F₅, 0.47 and 131, respectively. Dependence of K_m and k_o values on the degree of polymerization of substrates was observed. The subsite affinities in the active site were 1.05, 4.57, 1.45, 0.09, and — 0.16kcal/mol for subsite 1, 2, 3, 4, and 5, respectively.     

Kinetic study of the quenching reaction of singlet oxygen by α-, β-, γ-, δ-tocotrienols,and palm oil and soybean extracts in solution

Measurements of the singlet oxygen (¹O₂) quenching rates (k_Q (S)) and the relative singlet oxygen absorption capacity (SOAC) values were performed for 11 antioxidants (AOs) (eight vitamin E homologues (α-, β-, γ-, and δ-tocopherols and -tocotrienols (-Tocs and -Toc-3s)), two vitamin E metabolites (α- and γ-carboxyethyl-6-hydroxychroman), and trolox) in ethanol/chloroform/D₂O (50:50:1, v/v/v) and ethanol solutions at 35?°C. Similar measurements were performed for five palm oil extracts 1–5 and one soybean extract 6, which included different concentrations of Tocs, Toc-3s, and carotenoids. Furthermore, the concentrations (wt%) of Tocs, Toc-3s, and carotenoids included in extracts 1–6 were determined. From the results, it has been clarified that the ¹O₂-quenching rates (k_Q (S)) (that is, the relative SOAC value) obtained for extracts 1–6 may be explained as the sum of the product {Σ k_Q^AO-i (S) [AO-i]/100} of the rate constant (k_Q^AO-i (S)) and the concentration ([AO-i]/100) of AO-i (Tocs, Toc-3s, and carotenoid) included.     

Some Results on Error Rates for Quadratic Discrimination with Known Population Parameters

Let population Π_i, i = 1,2, be characterized by a multivariate normal density function, N(μ_i,∑_i), i = 1, 2, respectively. This paper provides conditions under which simple conditional error rates may be computed for the quadratic discriminant function with known population parameters. Also, a simple bound on the overall error rate is derived. Examples are given which demonstrate the proposed methods.     

Some Simple Statistics to Test Against Ordered Alternatives in Complete Block Designs

Consider testing the hypothesis of no treatment effects against a postulated ranking of the m treatments, given data from n Complete Blocks. A suitable test statistic is the weighted average rank correlation w = σb_QiC_i where C_i is the correlation between the postulated ranking and the ranking observed within the ith block, Q_i is the rank of the ith block with respect to credibility, and the b_i's are weights such that 0 ≦ b¹ ≦ … ≦ bn. In this paper we introduce some simple statistics: the first extends the signed-rank statistic to m ≦ 3, the second uses a simple measure of correlation based on the antirank, and the third a statistic based on Spearman's footrule. Tables for critical values are provided and the normal approximation is investigated.     

Note on psychophysical discrimination with more than two stimuli

H. D. Landahl's well-known theory of psychophysical discrimination between two stimuli (Psychometrica,3, 107–125, 1938) is generalized to the case ofn mutually inhibiting stimuli, such that all the corresponding reactions are mutually incompatible so that only one response at most can occur at a time. It is shown that while in the two-stimulus case a “no-response” situation does not necessarily need to occur, in the case ofn stimuli andn responses a “no-response” situation always occurs with finite probability. Therefore, there is a probabilityP _i of the occurrence of each responsei as well as a probabilityP _e of no response, with . The probabilitiesP _i andP _e are expressed in terms of the intensities of then stimuliS _i and in terms of then distribution functions of the fluctuations at then corresponding connections. The expressions are in the form of sums ofn-tuple integrals of the products of the distribution functions, the limits of integration being determined by the intensities of then stimuli.     

Factors affecting estimates of size at age and growth in grey triggerfish Balistes capriscus from the northern Gulf of Mexico

Growth zones in dorsal spines of grey triggerfish Balistes capriscus from the northern Gulf of Mexico were utilized to estimate growth and examine factors that may affect estimates of size at age. Age was estimated from dorsal‐spine sections of 4687 individuals sampled from U.S. waters during 2003–2013, including both fishery‐independent (n = 1312) and fishery‐dependent (n = 3375) samples. Ninety‐six per cent (n = 4498) of these sections were deemed suitable for ageing; average per cent error between two independent readers was 10·8%. Fork length (L_F) ranged from 65 to 697 mm and age estimates from 0 to 14 years. Both sex and sample source (fishery‐independent v. recreational) significantly affected estimated size at age for 2–6 year‐old fish. Data were pooled between sources to fit sex‐specific von Bertalanffy growth functions. Results for the female model were L_∞ = 387 mm L_F, k = 0·52 year^?1 and t₀ = 0·01 year, while for males L_∞ = 405 mm L_F, k = 0·55 year^?1 and t₀ = 0·02 year. These results were significantly different between sexes and indicate clear sexual dimorphism. Thus, growth should be modelled separately by sex when examining population parameters or conducting stock assessment modelling. The positive bias in estimates of size at age computed for recreational v. fishery‐independent samples also has clear implications for stock assessment as growth functions computed with fishery‐dependent samples would tend to overestimate stock productivity.     

On the Analysis of Variance for Time Sequenced Samples

The 3 way nested ANOVA model y_ijk = μ + a_i + B_ij + γ_k + (αγ)_ik + epsilon_ijk With α (treatment or group effects) and γ (time) both being fixed effects and B (the individual effects) random and nested within α, is introduced and explored. The problems associated with the usual approach are explained. The alternative model Is developed and a method of evaluation via the method of linear contrasts is recomended. The test statistic has the distribution of a convolution of F-distributions. Further, a method of investigating the assumptions of the model is offered and a further generalization using path spaces (of dim. K) is developed. Here again the appropriate test statistic has the distribution of a convolution of F-distributions. This combined with the method of linear contrasts offers an elegant solution to the BEHRENS-FISHER problem. y_ijk = f_ik + B_ij + epsilon_ijk     

The Point Evaluation Method for Approximating Function Means,Variances and Covariances

A perennial problem in statistics is the determination of biases, variances and covariances for functions of random variables X₁, X₂, …, X_n which themselves have a known distribution. A common approach is through equations based upon Taylor series approximations but a “point evaluation” method may sometimes be a useful alternative. This involves approximating the multivariate distribution of the X variables by the 2ⁿ points given by X₁=μ₁±₁, X₂ = μ₂ ±₂, …, X_n = = μ_n μ_n, where μ_i is the mean and σ_i the standard deviation of Xi, with appropriate point weights. An advantage over the Taylor series approach is that function derivatives do not have to be explicitely calculated. The point evaluation method is particularly useful in cases where the X variables are uncorrelated. Then the evaluation of the 2ⁿ points can be replaced by the evaluation of 2n points. The point evaluation method is illustrated with powers of a normally distributed variable, and with estimation of gene frequencies from ABO blood group frequencies.     

Photoinhibition and xanthophyll cycle activity in bayberry (Myrica rubra) leaves induced by high irradiance

The effect of high irradiance (HI, photosynthetically active photon flux density of 1 300 μmol m⁻² s⁻¹) on net photosynthetic rate (P _N), chlorophyll fluorescence parameters, and xanthophyll cycle components were studied in fruit tree bayberry leaves. HI induced the photoinhibition and inactivation of photosystem 2 (PS2) reaction centres (RCs), which was characterized by decreased P _N, maximum yield of fluorescence after dark adaptation (F_m), photochemical efficiency of PS2 (F_v/F_m) and quantum yield of PS2 (Φ_PS2), and increased reduction state of Q_A (1-q_P) and non-photochemical quenching (NPQ). Initial fluorescence (F₀) showed a decrease after the first 2 h, and subsequently increased from the third hour exposure to HI. Furthermore, a greater increase in the ratio (F_i-F₀)/(F_p-F₀) which is an expression of the proportion of the Q_B non-reducing PS2 centres, whereas a remarked decrease in the slope of F_i to F_p which represents the rate of Q_A reduction was observed in leaves after HI exposure. Additionally, HI caused an increase in the pool size of the xanthophyll cycle pigments and sustained elevated contents of zeaxanthin (Z), antheraxanthin (A), and de-epoxidation state (DES) at the end of the irradiation period. During HI, decreased F_m, F_v/F_m, Φ_PS2, NPQ, slope of F_i to F_p, V+A+Z, and DES, and increased F₀, 1-q_P, ratio (F_i-F₀)/(F_p-F₀), and V were observed in dithiothreitol (DTT)-fed leaves compared to control ones under the same conditions. Hence photoinhibition caused by HI in bayberry was probably attributed to inactivation of PS2 RCs, and photoprotection from photodamage were mainly related to the xanthophyll cycle-dependent heat dissipation in excess photons.     

The number of species occurring in a sample of a biotic community and its connections with species-abundance relationship and spatial distribution

Connections among species-abundance (i-m _i ), species-frequency (i-F _i ), and species-sample size (S _n -n) relationships were examined on the basis of the mapping data of a natural forest in Thailand. The spatial distribution of individual trees without any discrimination of species was nearly random. Provided that the spatial distribution of each species was random, thei-m _i and thei-F _i relationship was reconstructed from each other in terms of the total number of species (S) and the total number of individuals (N) in the data. The number of species (S _n ) in a subsample consisting ofn individuals was then obtained from thei-F _i relationship. Logarithm ofS _n increased with logn and showed a convex curve through the origin. The values of diversity indices based onN andS(orn andS _n ) were affected by sample size. These trends were further examined on the basis of 944 data sets of biotic communities and three mathematical models of anS-N relationship. The properties of species-area relation were discussed in the light of these results.     

F-tests for Hypotheses with Block Matrices and Under Conditions of Orthogonality in the General Multivariate Gauss-Markoff Model

The multivariate general Gauss-Markoff (MGM) model (U, XB, ∑?σ²V) when the matrices V ≥ 0 and ∑ > 0 are known and the scalar σ² > 0 is unknown, is considered. The present paper is a continuation of two earlier works (Oktaba, 1988a, b). If XB = X₁Σ + X₂Δ, then the F-test for verification the hypothesis WΣA = 0 is presented. Moreover, under conditions of orthogonality the decomposition of the matrix S_A (?BCA)′L^?(?BCA) into the sum of s = r(L) matrices is given, where ?BCA is the estimator of the parametric estimable functions ?BCA, Cov (?BCA) = A′ ∑?σ²L = ?C₄?′, B? = (X′T^?X)^?X′T^?U, C₄ = (X′T^?X)^?M, where M = M′ is any arbitrary matrix such that R(X) ? R(T), T=V+XMX′; T^? is any c-inverse. R(A) is the linear space generated by the colums of A. Then under additional assumption on normality of U the statistics F for testing ?BA = 0 is deduced. Under conditions of normality of U and decomposition of S_A, the statistics F₁, …, F_s for the hypotheses jⁱ BA = 0 (i = 1,…, s) are established.     

Kinetic study on unzippering of polynucleotides and order-order transition in polypeptides

Solutions are presented for N + 1 sequential and reversible first-order reactions for which the magnitude of the reverse rate constant, k_b, for all steps except the last is identical. Also the magnitude of the forward rate constant, k_f, for all steps except the first and last is identical. The initial and final steps are nucleation reactions; therefore, the initial and final k_f are modified by the factors σ′ and γ respectively. The final k_b is modified by the factor γ σ. The ratio k_b/k_f is defined as s, which has the same meaning as s in the Zimm-Bragg theory. The mathematical model is intended to apply to polymeric molecules of N segments and allows the calculation of the mole fraction of molecules in state i at any time t, C_i(t). A molecule in state i has i unreacted segments and N – i reacted ones. Because the reactions are sequential, all reacted segments are contiguous. Our numerical results show that when σ′ is much less than unity and the forward reaction is favored, the relaxation curve is sigmoidal. If, however, the forward and reverse reactions are equally favored (i.e., s ? 1) the relaxation curve is a straight line. When s and σ′ are near unity, the curve is exponential for a considerably large fraction of the reaction. Further, in the exponential for a considerably large fraction of the reaction. Further, in the exponential phase of the reaction, the relaxation time is proportional to N² for highly cooperative systems (i.e., Nσ ? 1). As found by Pipkin and Gibbs, if N is sufficiently large and s is less than unity (e.g., N ? 50 and s ?0.9) the relaxation curve is largely linear with a slope inversely proportional to N. Applications are given for the unwinding of double-helical poly(A·U) and the order–order transition in poly-L -proline.