The Discrete Distribution Used for the Log-rank Test Can Be Inaccurate

When the number of tumors is small, a significance level for the Cox-Mantel (log-rank) test Z is often computed using a discrete approximation to the permutation distribution. For j = 0,…, J let N_j(t) be the number of animals in group j alive and tumor-free at the start of time t. Make a 2 × (1+J) table for each time t of the number of animals R_j(t) with newly palpated tumor out of the total N_j(t) at risk. There are a total of say K tables, one for each distinct time t with observed death or newly palpated tumor. The usual discrete approximation to the permutation distribution of Z is defined by taking tables to be independent with fixed margins N_j(t) and ΣR_j(t) for all t. However, the N_j(t) are random variables for the actual permutation distribution of Z, resulting in dependence among the tables. Calculations for the exact permutation distribution are explained, and examples are given where the exact significance level differs substantially from the usual discrete approximation. The discrepancy arisis primarily because permutations with different Z-scores under the exact distribution can be equal for the discrete approximation, inflating the approximate P-value.     

ALGORITHM FOR APPLICATION OF FOURIER ANALYSIS FOR BIORHYTHMIC BASELINES OF PHARMACODYNAMIC INDIRECT RESPONSE MODELS

The change of an indirect pharmacological response R(t) can be described by a periodic time-dependent production rate k_in (t) and a first-order loss constant k_out. If k_in(t) follows some biological rhythm (e.g., circadian), then the response R(t) also displays a periodic behavior. A new approach for describing the input function in indirect response models with biorhythmic baselines of physiologic substances is introduced. The present approach uses the baseline (placebo) response R_b(t) to recover the equation for k_in(t). Fourier analysis provides an approximate equation for R_b(t) that consists of terms (usually two or three) of the Fourier series (harmonics) that contribute most to the overall sum. The model differential equation is solved backward for k_in(t), yielding the equation involving R_b(t). A computer program was developed to perform the square L²-norm approximation technique. Fourier analysis was also performed based on nonlinear regression. Cortisol suppression after inhalation of fluticasone propionate (FP) was modeled based on the inhibition of the secretion rate k_in(t) using ADAPT II. The pharmacodynamic parameters k_out and IC₅₀ were estimated from the model equation with k_in(t) derived by the new approach. The proposed method of describing the input function needs no assumption about the behavior of k_in(t), is as efficient as methods used previously, and is more flexible in describing the baseline data than the nonlinear regression method. (Chronobiology International, 17(1), 77–93, 2000)     

Theory of transport in linear biological systems: I. Fundamental integral equation

The conditions under which the output,γ _b (t), of a biological system is related to the input,γ _a (t), by an integral equation of the typeγ _b (t) = ∫ ₀ ^t γ _a (ω)w(t−ω)dω, where ω(t) is a transport functioncharacteristic of the system, are analyzed in detail. Methods of solving this type of integral equation are briefly discussed. The theory is then applied to problems in tracer kinetics in which input and output are sums of exponentials, and explicit formulae, which are applicable whether or not the pool is uniformly mixed, are derived for “turnover time” and “pool” size.     

Cell cycle control at the first restriction point and its effect on tissue growth

Cell cycle is controlled at two restriction points, R ₁ and R ₂. At both points the cell will commit apoptosis if it detects irreparable damage. But at R ₁ an undamaged cell also decides whether to proceed to the S phase or go into a quiescent mode, depending on the environmental conditions (e.g., overpopulation, hypoxia). We consider the effect of this decision at the population level in a spherical tissue {r < R(t)}. We prove that if the cells have full control at R ₁, they can manipulate the size of R(t) to ensure that 0 < c ≤ R(t) ≤ C < ∞; simulations further show that R(t) can be made nearly stationary. In the absence of such control, R(t) will either increase to ∞ or decrease to 0. The mathematical model and analysis involve a system of PDEs in {r < R(t)}.     

The convergence in distribution of some simple epidemics

A group of n susceptible individuals exposed to a contagious disease isconsidered. It is assumed that at each point in time one or more susceptible individuals can contract the disease. The progress of this simple batch epidemic is modeled by a stochastic process X_n(t), t[0, ∞), representing the number of infectiveindividuals at time t. In this paper our analysis is restricted to simple batch epidemics with transition rates given by [α²X_n(t){n −X_n(t) +X_n(0)}]^1/2, t[0, ∞), α(0, ∞). This class of simple batch epidemics generalizes a model used and motivated by McNeil (1972) to describe simple epidemic situations. It is shown for this class of simple batch epidemics, that X_n(t), with suitable standardization, converges in distribution as n→∞ to a normal random variable for all t(0, t₀), and t₀ is evaluated.     

Generalized stable population theory

In generalizing stable population theory we give sufficient, then necessary conditions under which a population subject to time dependent vital rates reaches an asymptotic stable exponential equilibrium (as if mortality and fertility were constant). If x ₀(t) is the positive solution of the characteristic equation associated with the linear birth process at time t, then rapid convergence of x ₀(t) to x ₀ and convergence of mortality rates produce a stable exponential equilibrium with asymptotic growth rate x ₀–1. Convergence of x ₀(t) to x ₀ and convergence of mortality rates are necessary. Therefore the two sets of conditions are very close. Various implications of these results are discussed and a conjecture is made in the continuous case.     

A model of photosystem II for the analysis of fast fluorescence rise in plant leaves

The polyphasic patterns of fluorescence induction rise in pea leaves in vivo and after the treatment with ionophores have been studied using a Plant Efficiency Analyzer. To analyze in detail photosystem II (PS II) electron transfer processes, an extended PS II model was applied, which included the sums of exponential functions to specify explicitly the light-driven formation of the transmembrane electric potential (ΔΨ(t)) as well as pH in the lumen (pH_L(t)) and stroma (pH_S(t)). PS II model parameters and numerical coefficients in ΔΨ(t), pH_L(t), and pH_S(t) were evaluated to fit fluorescence induction data for different experimental conditions: leaf in vivo or after ionophore treatment at low or high light intensity. The model imitated changes in the pattern of fluorescence induction rise due to the elimination of transmembrane potential in the presence of ionophores, when ΔΨ = 0 and pH_L(t), pH_S(t) changed to small extent relative to control values in vivo, with maximum ΔΨ(t) ∼ 90 mV and ΔΨ(t) ∼ 40 mV for the stationary state at ΔpH ≅ 1.8. As the light intensity was increased from 300 to 1200 μmol m⁻² s⁻¹, the heat dissipation rate constants increased threefold for nonradiative recombination of P680⁺Phe⁻ and by ∼30% for P680⁺Q_A⁻. The parameters ΔΨ, pH_S and pH_L were analyzed as factors of PS II redox state populations and fluorescence yield. The kinetic mechanism of fluorescence quenching is discussed, which is related with light-induced lumen acidification, when ⁺Q_A⁻ and P680⁺ recombination probability increases to regulate the Q_A reduction.     

Transition metal complexes with sulfur ligands. XXVIII. Ruthenium complexes of the pentadentate thioether thiolate ligands dpttd (2,3,11,12-dibenzo-1,4,7,10,13-pentathiatridecane(−2)) and the sterically demanding derivative bu4-dpttd (14,16,18,20-tetra (t-butyl)-2,3,11,12-dibenzo-1,4,7,10,13-pentathiatridecane(−2))

The template alkylation of Li₂[Ru(CO)₂(S₂C₆H₄)₂] (S₂C₆H₄²⁻ = 1,2-benzenedithiolate(−2)) by S(C₂H₄Br)₂ yields [Ru(CO)₂(dpttd)] (dpttd²⁻ = 3,11,12-dibenzo-1,4,7,10,13-pentathiatridecane(−2)) which is thermally converted into the monocarbonyl complex [Ru(CO)(dpttd)]. The reactions of dpttd-H₂ or dpttd²⁻ with [RuCl₂(PPh₃)₃], [RuCl₂(DMSO)₄], [RuCl₃(PhSCH₃)₃] and RuCl₃(NO)·xH₂O lead to [Ru(L)(dpttd)] and [Ru(L)(dpttd)]Cl (L = PPh₃, DMSO, PhSCH₃, NO), respectively, which are practically insoluble in all common solvents. Better soluble complexes are obtained with the new sterically demanding ligand ^tbu₄-dpttd²⁻ = 14,16,18,20-tetra(t-butyl)-2,3,11,12-dibenzo-1,4,7, 10,13-pentathiatridecane(−2); it is obtained in isomerically pure form by the reaction of tetrabuthylammonium-3,5-di (t-butyl)-1,2-benzenethiolthiolate, NBu₄[^tbu₂-C₆H₂S(SH), with S(C₂H₄Br)₂ and yields on reaction with [RuCl₂(PPh₃)₃] the very soluble [Ru(PPh₃)₂(^tbu₄-dpttd)] as well as [Ru(PPh₃(^tbu₄-dpttd)]. The ¹H NMR and ³¹P NMR spectra indicate that in solution [Ru(PPh₃)₂(^tbu₄-dpttd)] exists as a mixture of diastereomers, whereas [Ru(PPh₃)(^tbu₄-dpttd)] forms one pair of enantiomers only. This was confirmed by an X-ray structure determination of a single crystal. [Ru(PPh₃)(^tbu₄-dpttd)] crystallizes in space group P2₁/n with a = 10.496(4), b = 14.888(6), c = 32.382(12) Å, β = 98.04(3)°, Z = 4 and D_calc. = 1.27 g/cm³, R = 4.84; R_W = 5.06%; the ruthenium center is coordinated pseudooctahedrally by one phosphorus, two thiolate and three thiother S atoms.     

Consistency of the Asymptotic Quasi-Likelihood Estimate on Linear Models

Consider a model y_t = f_t(θ) + M_t, 0 ⩽ t ⩽ T where θ∈ Θ in an unknown parameter, f_t(θ) is a linear predictable process, M_t is a martingale difference, and the nature of E(M²_t/ℱ_t—1) is unknown. This paper presents an estimating procedure for θ based on the asymptotic quasi-likelihood methodology. Conditions under which the asymptotic quasi-likelihood estimate converges to the true parameter θ₀ are discussed. This method is applied to several simulated examples, and estimates of the unknown parameter are obtained by means of a two-stage technique. Comparison is made between the estimates obtained via this method and those obtained via the ordinary least squares method. Discussion is provided on the application of the model.     

On-line parameter and state estimation of continuous cultivation by extended Kalman filter

Summary The on-line estimation of biomass concentration and of three variable parameters of the non-linear model of continuous cultivation by an extended Kalman filter is demonstrated. Yeast growth in aerobic conditions on an ethanol substrate is represented by an unstructured non-linear stochastic t-variant dynamic model. The filter algorithm uses easily accessible data concerning the input substrate concentration, its concentration in the fermentor and dilution rate, and estimates the biomass concentration, maximum specific growth rate, saturation constant and substrate yield coefficient. The microorganismCandida utilis, strain Vratimov, was cultivated on the ethanol substrate. The filter results obtained with the real data from one cultivation experiment are presented. The practical possibility of using this method for on-line estimation of biomass concentration, which is difficult to measure, is discussed.Nomenclature D dilution rate (h^-1) - DO₂ dissolved oxygen concentration (%) - E identity matrix - F Jacobi matrix of the deterministic part of the system equations g - g continuousn-vector non-linear real function - h m-vector non-linear real function - K Kalman filter gain matrix - K _S saturation constant (kgm^-3) - K_S expectation of the saturation constant estimate - M Jacobi matrix of the deterministic part of the measurement equations h - P(t₀) co-variance matrix of the initial values of the state - P(t_k/t_k) c-variance matrix of the error in (t _k|t _k) - P(t_k+1/t_k) co-variance matrix of the error in (t _k+1|t _k - Q co-variance matrix of the state noise - R co-variance matrix of the output noise - S substrate concentration (kgm^-3) - S _i input substrate concentration - t time - t _k discrete time instant with indexk=0, 1, 2,... - u(t) input vector - v(t_k) measurement (output) noise sequence - w(t) n-vector white Gaussian random process - x(t₀) initial state of the system - (t₀) expectation of the initial state values - x(t) n-dimensional state vector - x(t_k) state vector at the time instantt _k - (t_k|t_k) expectation of the state estimate at timet _k when measurements are known to the timet _k - (t_k+1|t_k) expectation of the state prediction - X biomass concentration (kgm^-3) - expectation of the biomass concentration estimate - y(t_k) m-dimensional output vector at the time instantt _k - Y _XIS substrate yield coefficient - _X|S expectation of the substrate yield coefficient estimate - specific growth rate (h^-1) - _M maximum specific growth rate (h^-1) - expectation of the maximum specific growth rate estimate - state transition matrix     

Dynamic light-scattering studies of internal motions in DNA. I. Applicability of the rouse-zimm model

The intermediate scattering function G(K,t) for any polymer model obeying a linear separable Langevin equation can be expressed in terms of the eigenvalues and eigenvectors of its normal coordinate transformation. An algorithm for the extract numerical evaluation of G(K,t) for linear Rouse-Zimm chains in the presence of hydrodynamic interaction has been developed. The computed G(K,t)² were fit to C(t) = A exp(?t/τA) + B, and apparent diffusion coefficients calculated according to D_app ≡ 1/(2τ_AK²). G(K,t)² was surprisingly well-fit by single-exponential decays, especially at both small and large values of Kb, where K is the scattering vector and b the root-mean-squared subunit extension. Plots of D_app vs K² in-variably showed a sigmoidal rise from D₀ at K² = O up to a constant plateau value at large K²b². Analytical expression for G(K,t), exact in the limit of short times, were obtained for circular Rouse-Zimm chains with and without hydrodynamic interaction, and also for free-draining linear chains, and in addition for the independent-segment-mean-force (ISMF) model. The predicted behaviors for G(K,t) at large Kb (or KR_G) was found in all cases to be single-exponential with 1/τ ∝ K² at large Kb, in agreement with the computational results. A simple procedure for estamating all parameter of the Rouse-Zimm model from a plot of D_app vs K² is proposed. Experimental data for both native and pH-denatured calf-thymus DNA in 1.0M Nacl with and without EDTA clearly plateau behavior of D_app at large values of K, in harmony with the present Rouse-Zimm and ISMF theories, and in sharp contrast to previous predictions based on the Rouse-Zimm model.     

Identification of two major resistance genes against race IE-1k of Magnaporthe oryzae in the indica rice cultivar Zhe733

The race IE-1k of Magnaporthe oryzae recovered from the Southern US overcomes the resistance (R) gene Pita. The objectives of the present study were to identify and tag R genes to IE-1k for rice breeding. TM2, S1, 94071, and B isolates of the race IE-1k were used to identify and map R genes from a resistant indica rice cultivar Zhe733 using a recombinant inbred line population from a cross of the genetic stock KBNTlpa1-1 and Zhe733. The ratio of 3 resistant:1 susceptible in 162 RIL of an F_10-11 KBNTlpa1-1/Zhe733 (K/Z) population indicated that two major R genes in Zhe733 confer resistance to IE-1k. A total of 118 polymorphic simple sequence repeat markers were analyzed in 162 F_10-11 individuals of the K/Z population to determine chromosomal locations of the loci conferring resistance to race IE-1k using composite interval mapping. Two major R genes temporarily designated as Pi42(t) and Pi43(t) each providing complete resistance to IE-1k were identified on chromosomes 8 and 11, respectively. RILs containing Pi42(t) and Pi43(t) were also resistant to other US races IB-1, IB-45, IB-49, IB-54, IC-17, IE-1, IG-1, and IH-1. The Pi42(t) gene was mapped between RM310 and RM72, and the location of Pi43(t) was closely associated with two flanking SSR markers RM1233 and RM224 on chromosome 11 in a chromosomal region carrying the resistance gene Pi1. Two molecular markers RM72 and RM1233 identified in this study should be useful for fine mapping and for facilitating incorporation of Pi42(t) and Pi43(t) into advanced breeding lines by marker-assisted selection. The authors S. Lee and Y. Wamishe contribute equally to this work.     

Malthusian parameters,reproductive values and change under selection in self fertilizing age-structured populations

A population, reproducing wholly by selfing, is assumed to be observed at times . Individuals between x–1 and x units of age at time t are said to be in age class x at that time. The rate of increase in the long run of individuals of type A_iAj is denoted by m_ij+1=m_ji+1. For each genotype there is also a set of reproductive values, corresponding to all age classes and genotypes of individuals having descendants of that genotype. Then, if the number of individuals of each sort of ancestor is multiplied by its reproductive value and the products are summed, the result is the total value, which is V_ij(t) for genotype A_iAj. Then V_ij(t+1)–V_ij(t) is equal to m_ijV_ij(t), where m_ij is the Malthusian parameter for A_iAj. Furthermore, if the mean and variance at time t of the m_ijs, weighted by their corresponding reproductive values, are respectively (t) and _m²(t), then m¯(t+1)–m¯(t)=_m²(t)/(1+m¯(t)).     

Almost-one Parameter Hypothesis of Individuality in Height Growth

Thus far an individual height growth curve h_ij(t) of the i-th person in the j-th period, t being his (or her) age, has been studied as a function of t associated with its velocity curve. In this note we introduce a natural scale X(t) in place of t, which linearizes this personal curve and facilitates its analysis, in the sense that this equation of growth contains apparently two personal parameters for one period but one of them plays an essential role. The effectiveness of this approach will be seen in four figures.     

Identification and fine mapping of a thermo-sensitive chlorophyll deficient mutant in rice (<Emphasis Type="Italic">Oryza sativa </Emphasis>L.)

A thermo-sensitive chlorophyll deficient mutant was isolated from more than 15,000 transgenic rice lines. The mutant displayed normal phenotype at 23°C or lower temperature (permissive temperature). However, when grown at 26°C or higher (nonpermissive temperature) the plant exhibited an abnormal phenotype characterized by yellow green leaves. Genetic analysis revealed that a single nuclear-encoded recessive gene is responsible for the mutation, which is tentatively designed as cde1(t) (chlorophyll deficient 1, temporally). PCR analysis and hygromycin resistance assay indicated the mutation was not caused by T-DNA insertion. To isolate the cde1(t) gene, a map-based cloning strategy was employed and 15 new markers (five SSR and ten InDels markers) were developed. A high-resolution physical map of the chromosomal region around the cde1(t) gene was made using F₂ and F₃ population consisting of 1,858 mutant individuals. Finally, the cde1(t) gene was mapped in 7.5 kb region between marker ID10 and marker ID11 on chromosome 2. Sequence analysis revealed only one candidate gene, OsGluRS, in the 7.5 kb region. Cloning and sequencing of the target region from the cde1(t) mutant showed that a missense mutation occurred in the mutant. So the OsGluRS gene (TIGR locus Os02 g02860) which encode glutamyl-tRNA synthetase was identified as the Cde1(t) gene.     

Analysis of a mathematical model for the growth of tumors

 In this paper we study a recently proposed model for the growth of a nonnecrotic, vascularized tumor. The model is in the form of a free-boundary problem whereby the tumor grows (or shrinks) due to cell proliferation or death according to the level of a diffusing nutrient concentration. The tumor is assumed to be spherically symmetric, and its boundary is an unknown function r=s(t). We concentrate on the case where at the boundary of the tumor the birth rate of cells exceeds their death rate, a necessary condition for the existence of a unique stationary solution with radius r=R ₀ (which depends on the various parameters of the problem). Denoting by c the quotient of the diffusion time scale to the tumor doubling time scale, so that c is small, we rigorously prove that (i) lim inf _t→∞ s(t)>0, i.e. once engendered, tumors persist in time. Indeed, we further show that (ii) If c is sufficiently small then s(t)→R ₀ exponentially fast as t→∞, i.e. the steady state solution is globally asymptotically stable. Further, (iii) If c is not “sufficiently small” but is smaller than some constant γ determined explicitly by the parameters of the problem, then lim sup _t→∞ s(t)<∞; if however c is “somewhat” larger than γ then generally s(t) does not remain bounded and, in fact, s(t)→∞ exponentially fast as t→∞. Received: 25 February 1998 / Revised version: 30 April 1998     

Additive genetic variance within populations derived by single-seed descent and pod-bulk descent

Summary Breeders of self-pollinated legumes commonly use single-seed descent (SSD) or pod-bulk descent (PBD) to produce segregating populations of highly inbred individuals. We presented equations for the expected value of the additive genetic variance within populations derived by SSD (E(V _A)_SSD) and PBD (E(V _A)_PBD) in terms of the initial population size (N ₀), the number of seed harvested per pod (M), the probability of survival of an individual (), and the generation at which the population is evaluated (S _t). Differences between (E(V _A)_SSD) and (E(V _A)_PBD) are due to differences in the expected amount of random drift which occurs with the two methods after the S ₀ generation. With both methods, random drift occurs when progeny are sampled from heterozygous parents. An additional component of random drift occurs when sampled progeny fail to survive during SSD, or when sampling occurs amoung families during PBD. For values of N ₀, M, , and S _t that are typical of soybean (Glycine max (L.) Merr.) breeding programs, (E(V _A)_SSD) will be greater than (E(V _A)_PBD). The ratio of (E(V _A)_SSD) to (E(V _A)_PBD) will: (1) increase as M and increase; (2) approach a value of 1.00 as N ₀ increases; and (3) be a curvilinear function of S _t. Plant breeders should compare SSD and PBD based upon values of (E(V _A)_SSD) and (E(V _A)_PBD) and the expected cost of carrying out the two methods.Contribution No. 2910 of the South Carolina Agricultural Experiment Station, Clemson University     

Interaction of maturation delay and nonlinear birth in population and epidemic models

 A population with birth rate function B(N) N and linear death rate for the adult stage is assumed to have a maturation delay T>0. Thus the growth equation N′(t)=B(N(t−T)) N(t−T) e⁻ d ₁ T−dN(t) governs the adult population, with the death rate in previous life stages d ₁≧0. Standard assumptions are made on B(N) so that a unique equilibrium N _e exists. When B(N) N is not monotone, the delay T can qualitatively change the dynamics. For some fixed values of the parameters with d ₁>0, as T increases the equilibrium N _e can switch from being stable to unstable (with numerically observed periodic solutions) and then back to stable. When disease that does not cause death is introduced into the population, a threshold parameter R ₀ is identified. When R ₀<1, the disease dies out; when R ₀>1, the disease remains endemic, either tending to an equilibrium value or oscillating about this value. Numerical simulations indicate that oscillations can also be induced by disease related death in a model with maturation delay. Received: 2 November 1998 / Revised version: 26 February 1999     

Stereoelectronic Properties of N-acetyl-α,β-dehydroamino acid N′-methylamides

Summary α,β-Dehydroamino acids are useful peptide modifiers. However, their stereoelectronic properties still remain insufficiently recognized. Based on FTIR experiments in the range ofv _s(N-H), AI, AII andv _s(C^α=C^β) and ab initio calculations with B3LYP/6–31G*, we studied the solution conformational preferences and the amide electron density perturbation of Ac-ΔXaa-NHMe, where ΔXaa=ΔAla, (E)-ΔAbu, (Z)-ΔAbu, (Z)-ΔLeu, (Z)-ΔPhe and ΔVal. Each of these dehydroamides adopts a C₅ structure, which in Ac-ΔAla-NHMe is fully extended and accompanied by the strong C₅ hydrogen bond. Interaction with bond C^α=C^β lessens the amidic resonance within the flanking amide groups. TheN-terminal C=O bond is noticeably shorter, both amide bonds are longer than the corresponding bonds in the saturated entities and the N-terminal amide system is distorted. Ac-ΔAla-NHMe constitutes an exception. ItsC-terminal amide bond is shorter than the standard one and both amide systems are ideally planar. Ac-(E)-ΔAbu-NHMe shares stereoelectronic features with both Ac-ΔAla-NHMe and (Z)-dehydroamides.     

Abiotic,bottom-up,and top-down influences on recruitment of Rocky Mountain elk in Oregon: A retrospective analysis

Understanding the relative effects of the many factors that may influence recruitment of ungulates is fundamental to managing their populations. Over the last 4 decades, average recruitment in some populations of elk (Cervus elaphus) in Oregon, USA declined from >50 to <20 juveniles per 100 females, and several competing hypotheses address these declines. We developed a priori models and constructed covariates spanning 1977–2005 from hunter-killed elk, elk population estimates, cougar harvest, and weather statistics to evaluate abiotic, bottom-up, and top-down factors that may explain annual variation and long-term trends of pregnancy, juveniles-at-heel in late autumn, and recruitment of juvenile elk in spring. In models of pregnancy status, August precipitation, age, and cougar index had positive effects, whereas previous year (t − 1) winter severity or winter precipitation_(t−1) and elk density had negative effects. In models of juvenile-at-heel in late autumn, August precipitation, August precipitation_(t−1), cougar index × elk density_(t−1), and age had positive effects, whereas cougar index, elk density_(t−1), and winter precipitation_(t−1) had negative effects. Juvenile recruitment was best explained by positive effects of August precipitation_(t−1), lactation rate, and cougar index × elk density_(t−1) and negative effects of cougar index and elk density_(t−1). Winter severity, precipitation, and temperature were not significant in explaining variation in elk recruitment. Annual variation in pregnancy, juvenile-at-heel, and recruitment was most influenced by August precipitation, whereas long-term trends in recruitment were most influenced by cougar densities with relatively weak effects of elk density. These results provide insight into causes of year-to-year and long-term trends of elk recruitment and provide a basis for more rigorous evaluation of factors affecting recruitment of elk. © 2012 The Wildlife Society.