A Bayesian Approach to Surrogacy Assessment Using Principal Stratification in Clinical Trials

Summary A surrogate marker (S) is a variable that can be measured earlier and often more easily than the true endpoint (T) in a clinical trial. Most previous research has been devoted to developing surrogacy measures to quantify how well S can replace T or examining the use of S in predicting the effect of a treatment (Z). However, the research often requires one to fit models for the distribution of T given S and Z. It is well known that such models do not have causal interpretations because the models condition on a postrandomization variable S. In this article, we directly model the relationship among T, S, and Z using a potential outcomes framework introduced by Frangakis and Rubin (2002, Biometrics 58 , 21–29). We propose a Bayesian estimation method to evaluate the causal probabilities associated with the cross‐classification of the potential outcomes of S and T when S and T are both binary. We use a log‐linear model to directly model the association between the potential outcomes of S and T through the odds ratios. The quantities derived from this approach always have causal interpretations. However, this causal model is not identifiable from the data without additional assumptions. To reduce the nonidentifiability problem and increase the precision of statistical inferences, we assume monotonicity and incorporate prior belief that is plausible in the surrogate context by using prior distributions. We also explore the relationship among the surrogacy measures based on traditional models and this counterfactual model. The method is applied to the data from a glaucoma treatment study.     

Commentary on "Principal stratification - a goal or a tool?" by Judea Pearl

This commentary takes up Pearl's welcome challenge to clearly articulate the scientific value of principal stratification estimands that we and colleagues have investigated, in the area of randomized placebo-controlled preventive vaccine efficacy trials, especially trials of HIV vaccines. After briefly arguing that certain principal stratification estimands for studying vaccine effects on post-infection outcomes are of genuine scientific interest, the bulk of our commentary argues that the "causal effect predictiveness" (CEP) principal stratification estimand for evaluating immune biomarkers as surrogate endpoints is not of ultimate scientific interest, because it evaluates surrogacy restricted to the setting of a particular vaccine efficacy trial, but is nevertheless useful for guiding the selection of primary immune biomarker endpoints in Phase I/II vaccine trials and for facilitating assessment of transportability/bridging surrogacy.     

Principal stratification in causal inference

Many scientific problems require that treatment comparisons be adjusted for posttreatment variables, but the estimands underlying standard methods are not causal effects. To address this deficiency, we propose a general framework for comparing treatments adjusting for posttreatment variables that yields principal effects based on principal stratification. Principal stratification with respect to a posttreatment variable is a cross-classification of subjects defined by the joint potential values of that posttreatment variable tinder each of the treatments being compared. Principal effects are causal effects within a principal stratum. The key property of principal strata is that they are not affected by treatment assignment and therefore can be used just as any pretreatment covariate. such as age category. As a result, the central property of our principal effects is that they are always causal effects and do not suffer from the complications of standard posttreatment-adjusted estimands. We discuss briefly that such principal causal effects are the link between three recent applications with adjustment for posttreatment variables: (i) treatment noncompliance, (ii) missing outcomes (dropout) following treatment noncompliance. and (iii) censoring by death. We then attack the problem of surrogate or biomarker endpoints, where we show, using principal causal effects, that all current definitions of surrogacy, even when perfectly true, do not generally have the desired interpretation as causal effects of treatment on outcome. We go on to forrmulate estimands based on principal stratification and principal causal effects and show their superiority.     

Application of the Marcus theory for analysis of the temperature dependence of the reactions leading to photosynthetic water oxidation: results and implications

The temperature dependence of donor side reactions was analysed within the framework of the Marcus theory of nonadiabatic electron transfer. The following results were obtained for PS II membrane fragments from spinach: (1) the reorganisation energy of P680^+? reduction by Y_Z is of the order of 0.5?eV in samples with a functionally fully competent water oxidising complex (WOC); (2) destruction of the WOC by Tris-washing gives rise to a drastic increase of λ to values of the order of 1.6?eV; (3) the reorganisation energies of the oxidation steps in the WOC are dependent, on the redox states S _i with values of about 0.6?eV for the reactions Y_Z ^OX S ₀→Y_Z S ₁ and Y_Z ^OX S ₁→Y_Z S ₂, 1.6?eV for the reaction Y_Z ^OX S ₂→Y_Z S ₃ and 1.1?eV (above a characteristic temperature u_c of about 6??°C) for the reaction Y_Z ^OX S ₃→→Y_Z S ₀+O₂. Using an empirical rate constant-distance relationship, the van der Waals distance between Y_Z and P680 was found to be about 10?Å, independent of the presence or absence of the WOC, whereas the distance between Y_Z and the manganese cluster in the WOC was ≥15?Å. Based on the calculated activation energies the environment of Y_Z is inferred to be almost "dry" and hydrophobic when the WOC is intact but becomes enriched with water molecules after WOC destruction. Furthermore, it is concluded that the transition S ₂→S ₃ is an electron transfer reaction gated by a conformational change, i.e. it comprises significant structural changes of functional relevance. Measurements of kinetic H/D isotope exchange effects support the idea that none of these reactions is gated by the break of a covalent O-H bond. The implications of these findings for the mechanism of water oxidation are discussed.     

Evaluating candidate principal surrogate endpoints

SUMMARY: Frangakis and Rubin (2002, Biometrics 58, 21-29) proposed a new definition of a surrogate endpoint (a "principal" surrogate) based on causal effects. We introduce an estimand for evaluating a principal surrogate, the causal effect predictiveness (CEP) surface, which quantifies how well causal treatment effects on the biomarker predict causal treatment effects on the clinical endpoint. Although the CEP surface is not identifiable due to missing potential outcomes, it can be identified by incorporating a baseline covariate(s) that predicts the biomarker. Given case-cohort sampling of such a baseline predictor and the biomarker in a large blinded randomized clinical trial, we develop an estimated likelihood method for estimating the CEP surface. This estimation assesses the "surrogate value" of the biomarker for reliably predicting clinical treatment effects for the same or similar setting as the trial. A CEP surface plot provides a way to compare the surrogate value of multiple biomarkers. The approach is illustrated by the problem of assessing an immune response to a vaccine as a surrogate endpoint for infection.     

Solutions for surrogacy validation with longitudinal outcomes for a gene therapy

Valid surrogate endpoints S can be used as a substitute for a true outcome of interest T to measure treatment efficacy in a clinical trial. We propose a causal inference approach to validate a surrogate by incorporating longitudinal measurements of the true outcomes using a mixed modeling approach, and we define models and quantities for validation that may vary across the study period using principal surrogacy criteria. We consider a surrogate-dependent treatment efficacy curve that allows us to validate the surrogate at different time points. We extend these methods to accommodate a delayed-start treatment design where all patients eventually receive the treatment. Not all parameters are identified in the general setting. We apply a Bayesian approach for estimation and inference, utilizing more informative prior distributions for selected parameters. We consider the sensitivity of these prior assumptions as well as assumptions of independence among certain counterfactual quantities conditional on pretreatment covariates to improve identifiability. We examine the frequentist properties (bias of point and variance estimates, credible interval coverage) of a Bayesian imputation method. Our work is motivated by a clinical trial of a gene therapy where the functional outcomes are measured repeatedly throughout the trial.     

The origin of split EPR signals in the Ca2+-depleted Photosystem II

A light-driven reaction model for the Ca²⁺-depleted Photosystem (PS) II is proposed to explain the split signal observed in electron paramagnetic resonance (EPR) spectra based on a comparison of EPR assignments with recent x-ray structural data. The split signal has a splitting linewidth of 160 G at around g = 2 and is seen upon illumination of the Ca²⁺-depleted PS II in the S₂ state associated with complete or partial disappearance of the S₂ state multiline signal. Another g=2 broad ESR signal with a 110 G linewidth was produced by 245 K illumination for a short period in the Ca²⁺-depleted PS II in S₁ state. At the same time a normal Y_Z^· radical signal was also efficiently trapped. The g=2 broad signal is attributed to an intermediate S₁X^· state in equilibrium with the trapped Y_Z^· radical. Comparison with x-ray structural data suggests that one of the split signals (doublet signal) is attributable to interaction between His 190 and the Y_Z^· radical, and other signals is attributable to interaction between His 337 and the manganese cluster, providing further clues as to the mechanism of water oxidation in photosynthetic oxygen evolution.     

Search for intermediates of photosynthetic water oxidation

Photosystem II of cyanobacteria and plants incorporates the catalytic centre of water oxidation. Powered and clocked by quanta of light the centre accumulates four oxidising equivalents before oxygen is released. The first three oxidising equivalents are stored on the Mn₄Ca-cluster, raising its formal oxidation state from S₀ to S₃ and the third on Y_Z, producing S₃ Y_Z^ox. From there on water oxidation proceeds in what appears as a single reaction step (S₃ Y_Z^ox(H₂O)₂O₂ + 4H⁺ + S₀. Intermediate oxidation products of bound water had not been detected, until our recent report on the stabilisation of such an intermediate by high oxygen pressure (NATURE 430, 2004, 480–483). Based on the oxygen titration (half-point 2.3 bar) the standard free-energy profile of a reaction sequence with a single intermediate was calculated. It revealed a rather small difference (−3 kJ mol⁻¹) between the starting state [S₃Y_Z^OX and the product state S₀Y_Z + O₂ + 4H⁺ . Here we describe the tests for side effects of exposing core particles to high oxygen pressure. We found the reduction of P₆₈₀^{+ ·} in ns and the reduction/dismutation of quinones at the acceptor side of PSII both unaffected, and the inhibition of the oxygen evolving reaction by exposure to high O₂-pressure was fully reversible by decompression to atmospheric conditions.     

The filamentous ascomycete Sordaria macrospora can survive in ambient air without carbonic anhydrases

The rapid interconversion of carbon dioxide and bicarbonate (hydrogen carbonate) is catalysed by metalloenzymes termed carbonic anhydrases (CAs). CAs have been identified in all three domains of life and can be divided into five evolutionarily unrelated classes (α, β, γ, δ and ζ) that do not share significant sequence similarities. The function of the mammalian, prokaryotic and plant α‐CAs has been intensively studied but the function of CAs in filamentous ascomycetes is mostly unknown. The filamentous ascomycete Sordaria macrospora codes for four CAs, three of the β‐class and one of the α‐class. Here, we present a functional analysis of CAS4, the S. macrospora α‐class CA. The CAS4 protein was post‐translationally glycosylated and secreted. The knockout strain Δcas4 had a significantly reduced rate of ascospore germination. To determine the cas genes required for S. macrospora growth under ambient air conditions, we constructed double and triple mutations of the four cas genes in all possible combinations and a quadruple mutant. Vegetative growth rate of the quadruple mutant lacking all cas genes was drastically reduced compared to the wild type and invaded the agar under normal air conditions. Likewise the fruiting bodies were embedded in the agar and completely devoid of mature ascospores.     

On Mohan's Property of the Poisson Distribution

Two interesting results encountered in the literature concerning the Poisson and the negative binomial distributions are due to Moran (1952) and Patil & Seshadri (1964), respectively. Morans result provided a fundamental property of the Poisson distribution. Roughly speaking, he has shown that if Y, Z are independent, non-negative, integer-valued random variables with X = Y | Z then, under some mild restrictions, the conditional distribution of Y | X is binomial if and only if Y, Z are Poisson random variables. Motivated by Morans result Patil & Seshadri obtained a general characterization. A special case of this characterization suggests that, with conditions similar to those imposed by Moran, Y | X is negative hypergeometric if and only if Y, Z are negative binomials. In this paper we examine the results of Moran and Patil & Seshadri in the case where the conditional distribution of Y | X is truncated at an arbitrary point k – 1 (k = 1, 2, …). In fact we attempt to answer the question as to whether Morans property of the Poisson distribution, and subsequently Patil & Seshadris property of the negative binomial distribution, can be extended, in one form or another, to the case where Y | X is binomial truncated at k – 1 and negative hypergeometric truncated at k – 1 respectively.     

The sufficient-component discrete-cause model and its extension to several risk factors

This paper is concerned with Koopman's (1981) discussion of using the ICDR as a measure of deviation from an additive model of no interaction. This measure has been shown to be zero or slightly negative when the assumptions of no interaction hold in the sufficient-component discrete-cause model. The discussion centers around a theoretical structure of sufficient causes for juvenile-onset diabetes where there is a component cause common to all sufficient causes and two binary factors or risks X and Y are measured (X: Coxsackie B₄ infection; Y: HLA4). In this paper, we have extended Koopman's discussion to the case where three or more binary factors X, Y, Z,… are measured. We have considered interaction between factors in every subset and have proved that the ICDR is again either zero or negative when the assumptions of no interaction hold in the sufficient-component cause model. It is supposed that these factors in conjunction will inevitably result in beta-cell destruction together with a set of immunologic conditions that permit dissemination of the enterovirus infection and permit an autoimmune response.     

Oxidative photosynthetic water splitting: energetics,kinetics and mechanism

This minireview is an attempt to summarize our current knowledge on oxidative water splitting in photosynthesis. Based on the extended Kok model (Kok, Forbush, McGloin (1970) Photochem Photobiol 11:457–476) as a framework, the energetics and kinetics of two different types of reactions comprising the overall process are discussed: (i) P680^+• reduction by the redox active tyrosine Y_Z of polypeptide D1 and (ii) Y_z^ox induced oxidation of the four step sequence in the water oxidizing complex (WOC) leading to the formation of molecular oxygen. The mode of coupling between electron transport (ET) and proton transfer (PT) is of key mechanistic relevance for the redox turnover of Y_Z and the reactions within the WOC. The peculiar energetics of the oxidation steps in the WOC assure that redox state S₁ is thermodynamically most stable. This is a general feature in all oxygen evolving photosynthetic organisms and assumed to be of physiological relevance. The reaction coordinate of oxidative water splitting is discussed on the basis of the available information about the Gibbs energy differences between the individual redox states S _i+1 and S _i and the data reported for the activation energies of the individual oxidation steps in the WOC. Finally, an attempt is made to cast our current state of knowledge into a mechanism of oxidative water splitting with special emphasis on the formation of the essential O–O bond and on the active role of the protein in tuning the local proton activity that depends on time and redox state S _i . The O–O linkage is assumed to take place at the level of a complexed peroxide.     

EPR studies of the water oxidizing complex in the S1 and the higher S states: the manganese cluster and YZ radical

The parallel polarization electron paramagnetic resonance (EPR) method has been applied to investigate manganese EPR signals of native S₁ and S₃ states of the water oxidizing complex (WOC) in photosystem (PS) II. The EPR signals in both states were assigned to thermally excited states with S=1, from which zero-field interaction parameters D and E were derived. Three kinds of signals, the doublet signal, the singlet-like signal and g=11–15 signal, were detected in Ca²⁺-depleted PS II. The g=11–15 signal was observed by parallel and perpendicular modes and assigned to a higher oxidation state beyond S₂ in Ca²⁺-depleted PS II. The singlet-like signal was associated with the g=11–15 signal but not with the Y_Z (the tyrosine residue 161 of the D1 polypeptide in PS II) radical. The doublet signal was associated with the Y_Z radical as proved by pulsed electron nuclear double resonance (ENDOR) and ENDOR-induced EPR. The electron transfer mechanism relevant to the role of Y_Z radical was discussed.     

Lactose continuous fermentation with cells recycled by ultrafiltration and lactate separation by electrodialysis: model identification

Summary An off-line parameter estimation method has been developed to predict the dynamic behaviour of a continuous lactose fermentation system. The model used is an unstructured model taking into account cell growth, substrate consumption, and metabolite production (lactic acid). This method, based on the Hooke-Jeeves non-linear-programming technique, results in a good estimation of the biological parameters of the model, and so gives a better understanding of the different phenomena involved in lactose fermentation.Nomenclature Cp, Cs, Cz, Dp, Ds, Dz coefficients in system (A) - Fe bioreactor influent flow rate (1/h) - I current in the ED unit (A) - J lactate flux in the ED unit (g/h) - Kd mortality constant (h^-1) - Kp product inhibition constant (g/l) - Ks strbstrate saturation constant (g/l) - P ₀ product concentration in the bioreactor (g/l) - P ₁ product concentration in the D tank (g/l) - P _0r estimation of P ₀ (g/l) - Q ₀ retentate flow rate (UF influent) (1/h) - Q ₁ permeate flow rate (1/h) - Q ₂₂ cell bleed flow rate (1/h) - Q ₃ recycling flow rate in the ED (influent) (1/h) - Se substrate concentration in the influent (g/l) - S ₀ supstrate concentration in the bioreactor (g/l) - S ₁ substrate concentration in tank D (g/l) - S _0r estimation of S ₀ (g/l) - t time (h) - V ₀ fermentation broth volume (1) - V ₁ tank D volume (1) - X ₀ biomass concentration in the bioreactor (g/l) - Y _P/S (=1/Y _S/P) lactic acid yield coefficient (g lactic acid/g lactose consumed) - Y _X/S (=1/Y _S/X) cell yield coefficient (g cells produced/g lactose consumed) - Y _X/Z (=1/Y _Z/X) second cell yield coefficient (g cells produced/g nitrogen consumed) - Y _x, Y _m input mathematical parameters of the linear system (M ₂) - Ze nitrogen concentration in the influent (g/l) - Z ₀ nitrogen concentration in the bioreactor (g/l) - Z ₁ nitrogen concentration in tank D (g/l) - Z _0r estimation of Z ₀ (g/l) - , constants of the Luedeking and Piret's model - specific growth rate (h^-1) - _max maximum specific growth rate (h^-1)     

A pseudo-bayesian shrinkage approach to regression with missing covariates

Summary We consider the linear regression of outcome Y on regressors W and Z with some values of W missing, when our main interest is the effect of Z on Y, controlling for W. Three common approaches to regression with missing covariates are (i) complete‐case analysis (CC), which discards the incomplete cases, and (ii) ignorable likelihood methods, which base inference on the likelihood based on the observed data, assuming the missing data are missing at random ( Rubin, 1976b ), and (iii) nonignorable modeling, which posits a joint distribution of the variables and missing data indicators. Another simple practical approach that has not received much theoretical attention is to drop the regressor variables containing missing values from the regression modeling (DV, for drop variables). DV does not lead to bias when either (i) the regression coefficient of W is zero or (ii) W and Z are uncorrelated. We propose a pseudo‐Bayesian approach for regression with missing covariates that compromises between the CC and DV estimates, exploiting information in the incomplete cases when the data support DV assumptions. We illustrate favorable properties of the method by simulation, and apply the proposed method to a liver cancer study. Extension of the method to more than one missing covariate is also discussed.     

S-state dependence of the miss probability in Photosystem II

The oxygen production of dark-adapted Photosystem II upon illumination by a series of single-turnover flashes shows a damped period four oscillation with flash number. The damping is attributed to `misses' resulting from a statistical probability that a reaction center fails to produce a stable charge separation after a saturating flash. The origin of misses is of interest because its probable dependence on flash number, in principle, affects the quantitative interpretation of all measurements on phenomena associated with the period four oscillation. We show that the kinetics of chlorophyll fluorescence yield transients induced by a flash series can be used to estimate the relative amplitudes of the miss probability on each flash. It is concluded that a major part of the misses must be caused by failure of the reduction of the oxidized primary electron donor chlorophyll P680⁺ by the secondary donor tyrosine Y_Z before the charge separation is lost by recombination. The probability of this failure is found to increase with the oxidation state of the oxygen-evolving complex: more than half of it occurs upon charge separation in the S₃ state, which is attributed to the presence of Y_Z ^ox S₂ in Boltzmann equilibrium with Y_ZS₃. This revised version was published online in June 2006 with corrections to the Cover Date.     

Heterochromatin of the Drosophila melanogaster Y chromosome as modifier of position effect variegation: the time of its action.

Summary Addition of heterochromatin suppresses while subtraction enhances position effect variegation. The heterochromatin-sensitive period has been determined in white/white-apricot variegated eyes of Y ^S w ^a /w ^a ; Dp (1;3) w ^265-58 flies. When such larvae, carrying a Y-short (Y ^S ) arm at the distal end of one X chromosome, are X-rayed, mitotic recombination leads to one daughter cell with two Y ^S arms and an adjacent daughter cell with no Y ^S arm. When induced after clonal initiation, the frequency of dark clones developing from daughter cells with two Y ^S arms is significantly higher than the frequency of dark clones in the rest of the eye; and this frequency is. even higher when induced before clonal initiation. The modifying action of the Y-heterochromatin is exerted, therefore, during and after clonal initiation. Surprisingly, the frequency of dark clones developing from cells with no Y ^S arm is not lower than the frequency of dark clones in the rest of the eye.     

Volume changes and electrostriction in the primary photoreactions of various photosynthetic systems: estimation of dielectric coefficient in bacterial reaction centers and of the observed volume changes with the Drude–Nernst equation

Photoacoustics (PA) allows the determination of enthalpy and volume changes of photoreactions in photosynthetic reaction centers on the 0.1–10 μs time scale. These include the bacterial centers from Rb. sphaeroides, PS I and PS II centers from Synechocystis and in whole cells. In vitro and in vivo PA data on PS I and PS II revealed that both the volume change (–26 A³) and reaction enthalpy (–0.4 eV) in PS I are the same as those in the bacterial centers. However the volume change in PS II is small and the enthalpy far larger, –1 eV. Assigning the volume changes to electrostriction allows a coherent explanation of these observations. One can explain the large volume decrease in the bacterial centers with an effective dielectric coefficient of ∼4. This is a unique approach to this parameter so important in estimation of protein energetics. The value of the volume contraction for PS I can only be explained if the acceptor is the super- cluster (Fe₄S₄)(Cys₄) with charge change from –1 to –2. The small volume change in PS II is explained by sub-μs electron transfer from Y_Z anion to P₆₈₀ cation, in which charge is only moved from the Y_Z anion to the Q_A with no charge separation or with rapid proton transfer from oxidized Y_Z to a polar region and thus very little change in electrostriction. At more acid pH equally rapid proton transfer from a neighboring histidine to a polar region may be caused by the electric field of the P₆₈₀ cation. This revised version was published online in June 2006 with corrections to the Cover Date.     

EPR kinetic studies of the S−1 state in spinach thylakoids

The Y_Z decay kinetics in a formal S₋₁ state, regarded as a reduced state of the oxygen evolving complex, was determined using time-resolved EPR spectroscopy. This S₋₁ state was generated by biochemical treatment of thylakoid membranes with hydrazine. The steady-state oxygen evolution of the sample was used to optimize the biochemical procedure for performing EPR experiments. A high yield of the S₋₁ state was generated as judged by the two-flash delay in the first maximum of oxygen evolution in Joliot flash-type experiments. We have shown that the Y_Z re-reduction rate by the S₋₁ state is much slower than that of any other S-state transition in hydrazine-treated samples. This slow reduction rate in the S₋₁ to S₀ transition, which is in the order of the S₃ to S₀ transition rate, suggests that this transition is accompanied by some structural rearrangements. Possible explanations of this unique, slow reduction rate in the S₋₁ to S₀ transition are considered, in light of earlier observations by others on hydrazine/hydroxylamine reduced PS II samples.     

A transmissible dicentric chromosome in Drosophila melanogaster

A transmissible dicentric chromosome was recovered in Drosophila melanogaster. The radiation-induced secondary chromosome rearrangement consists essentially of the entire Y and fourth chromosomes joined by 2R heterochromatin. The Y ^S · Y ^L 2Rh4 · chromosome pairs with the X and the free fourth chromosome to form a trivalent in meiosis that is unusual because it forms few chromosome bridges in primary spermatocytes and is transmitted at high frequency. We suggest that the orientation of the weaker fourth chromosome kinetochore eventually fails when opposing the stronger Y kinetochore so that the Y ^S · Y ^L 2Rh4 · moves to the pole to which the Y kinetochore is oriented. There is however an increased frequency of sex chromosome nondisjunction (14%) and of chromosome laggards (6%) in primary spermatocytes; the frequency of exceptional progeny of males containing the Y ^S · Y ^L 2Rh4 · was 7.44% compared with 0.25% in the controls. Disruption of normal sex chromosome disjunction also occurs in females containing the Y ^S · Y ^L 2Rh4 · and a compound X chromosome; the frequency of exceptional progeny was 2.55% versus 0.91% in the controls. Chromosome nondisjunction appears to occur when orientation of the X and Y kinetochores to the same pole is stabilized through tension by the orientation of one or both fourth chromosome kinetochores to the opposite pole. During anaphase, the orientation of the fourth chromosome kinetochore of the Y ^S · Y ^L 2Rh4 · appears to fail and the X and Y ^S · Y ^L 2Rh4 · chromosomes move to the same pole. Y ^S · Y ^L 2Rh4 · chromosome laggards occur with both the Y and fourth chromosome kinetochores amphitelically oriented. This orientation appears to be stable as a result of equal opposing forces toward opposite poles.