Calculation of the minimum number of replicate spots required for detection of significant gene expression fold change in microarray experiments

MOTIVATION: We present statistical methods for determining the number of per gene replicate spots required in microarray experiments. The purpose of these methods is to obtain an estimate of the sampling variability present in microarray data, and to determine the number of replicate spots required to achieve a high probability of detecting a significant fold change in gene expression, while maintaining a low error rate. Our approach is based on data from control microarrays, and involves the use of standard statistical estimation techniques. RESULTS: After analyzing two experimental data sets containing control array data, we were able to determine the statistical power available for the detection of significant differential expression given differing levels of replication. The inclusion of replicate spots on microarrays not only allows more accurate estimation of the variability present in an experiment, but more importantly increases the probability of detecting genes undergoing significant fold changes in expression, while substantially decreasing the probability of observing fold changes due to chance rather than true differential expression.     

A longitudinal measurement error model with a semicontinuous covariate

Covariate measurement error in regression is typically assumed to act in an additive or multiplicative manner on the true covariate value. However, such an assumption does not hold for the measurement error of sleep-disordered breathing (SDB) in the Wisconsin Sleep Cohort Study (WSCS). The true covariate is the severity of SDB, and the observed surrogate is the number of breathing pauses per unit time of sleep, which has a nonnegative semicontinuous distribution with a point mass at zero. We propose a latent variable measurement error model for the error structure in this situation and implement it in a linear mixed model. The estimation procedure is similar to regression calibration but involves a distributional assumption for the latent variable. Modeling and model-fitting strategies are explored and illustrated through an example from the WSCS.     

A comparative analysis of alternative approaches to fitting species-abundance models

Aims Fits of species-abundance distributions to empirical data are increasingly used to evaluate models of diversity maintenance and community structure and to infer properties of communities, such as species richness. Two distributions predicted by several models are the Poisson lognormal (PLN) and the negative binomial (NB) distribution; however, at least three different ways to parameterize the PLN have been proposed, which differ in whether unobserved species contribute to the likelihood and in whether the likelihood is conditional upon the total number of individuals in the sample. Each of these has an analogue for the NB. Here, we propose a new formulation of the PLN and NB that includes the number of unobserved species as one of the estimated parameters. We investigate the performance of parameter estimates obtained from this reformulation, as well as the existing alternatives, for drawing inferences about the shape of species abundance distributions and estimation of species richness.Methods We simulate the random sampling of a fixed number of individuals from lognormal and gamma community relative abundance distributions, using a previously developed 'individual-based' bootstrap algorithm. We use a range of sample sizes, community species richness levels and shape parameters for the species abundance distributions that span much of the realistic range for empirical data, generating 1?000 simulated data sets for each parameter combination. We then fit each of the alternative likelihoods to each of the simulated data sets, and we assess the bias, sampling variance and estimation error for each method.Important findings Parameter estimates behave reasonably well for most parameter values, exhibiting modest levels of median error. However, for the NB, median error becomes extremely large as the NB approaches either of two limiting cases. For both the NB and PLN,>90% of the variation in the error in model parameters across parameter sets is explained by three quantities that corresponded to the proportion of species not observed in the sample, the expected number of species observed in the sample and the discrepancy between the true NB or PLN distribution and a Poisson distribution with the same mean. There are relatively few systematic differences between the four alternative likelihoods. In particular, failing to condition the likelihood on the total sample sizes does not appear to systematically increase the bias in parameter estimates. Indeed, overall, the classical likelihood performs slightly better than the alternatives. However, our reparameterized likelihood, for which species richness is a fitted parameter, has important advantages over existing approaches for estimating species richness from fitted species-abundance models.     

Planning the suicide experiments.

Suicide experiments involve a great degree of uncertainty in the counting of cell colonies. This work studies from a statistical point of view the precision of the estimation as a function of the number of experimental units which are used. Assuming that the colony numbers in recipients follow a Poisson distribution, we give the necessary number of recipients (a) to determine with a given accuracy the percentage of DNA synthesizing cells (S cells), (b) to test whether or not a cell population is quiescent, and (c) to compare the percentages of S cells in two cell populations.     

Cytometric method of determining the parameters of the interaction of synthetic immunostimulants with the lymphocytic surface

The cytofluorimetric method was suggested for the determination of parameters of polymer adsorption on the cellular surface. Multiple equilibrium in the homogeneous cellular population with N independent binding sites and in the heterogeneous population of cells with the Poisson distribution of adsorption sites was analysed. A possibility was shown of cytofluorimetric drawing of the adsorption isotherm using mean fluorescence intensity of cells in the equilibrium suspension. The equilibrium association constant and the mean number of binding sites were estimated for adsorption of poly-L-lysin on the surface of human myelogenous leukemic cells (cellular line K-252).     

6-(2,4-Dinitrophenyl)-mercaptopurine, a stronger immunosuppressive drug than 6-mercaptopurine to primary humoral T-cell dependent immune response in mice]

The suppressive effect of 6-(2,4-Dinitrophenyl)-mercatopurine (DNP-MP) and 6-mercaptopurine (MP) was investigated on the early primary immune response of mice against the T-cell dependent antigens DNP49-bovine gamma globuline (BGG), sheep red blood cells (SRBC) or FITC8-BCG and the T-cell independent DNP22-Ficoll. The number of IgM antibody forming cells (AbFC) to the hapten determinants and to the SRBCs per 10(6) spleen cells was determined. DNP-MP reduced the number of AbFCs after the immunisation with the T cell dependent antigens always stronger than the MP, independently of the antigen type by which the mice had been immunised. The Anti-DNP22-Ficoll immune response was suppressed equally by both immunosuppressive drugs. DNP-MP is not a specific immunosuppressive drug for the anti-DNP-B-lymphocytes. Helper T-cells and macrophages are discussed as target cells for the stronger unspecific action of DNP-MP.     

Cell Growth and Division: I. A Mathematical Model with Applications to Cell Volume Distributions in Mammalian Suspension Cultures

A mathematical model is formulated for the development of a population of cells in which the individual members may grow and divide or die. A given cell is characterized by its age and volume, and these parameters are assumed to determine the rate of volume growth and the probability per unit time of division or death. The initial value problem is formulated, and it is shown that if cell growth rate is proportional to cell volume, then the volume distribution will not converge to a time-invariant shape without an added dispersive mechanism. Mathematical simplications which are possible for the special case of populations in the exponential phase or in the steady state are considered in some detail. Experimental volume distributions of mammalian cells in exponentially growing suspension cultures are analyzed, and growth rates and division probabilities are deduced. It is concluded that the cell volume growth rate is approximately proportional to cell volume and that the division probability increases with volume above a critical threshold. The effects on volume distribution of division into daughter cells of unequal volumes are examined in computer models.     

Motor unit number estimation--a Bayesian approach

All muscle contractions are dependent on the functioning of motor units. In diseases such as amyotrophic lateral sclerosis (ALS), progressive loss of motor units leads to gradual paralysis. A major difficulty in the search for a treatment for these diseases has been the lack of a reliable measure of disease progression. One possible measure would be an estimate of the number of surviving motor units. Despite over 30 years of motor unit number estimation (MUNE), all proposed methods have been met with practical and theoretical objections. Our aim is to develop a method of MUNE that overcomes these objections. We record the compound muscle action potential (CMAP) from a selected muscle in response to a graded electrical stimulation applied to the nerve. As the stimulus increases, the threshold of each motor unit is exceeded, and the size of the CMAP increases until a maximum response is obtained. However, the threshold potential required to excite an axon is not a precise value but fluctuates over a small range leading to probabilistic activation of motor units in response to a given stimulus. When the threshold ranges of motor units overlap, there may be alternation where the number of motor units that fire in response to the stimulus is variable. This means that increments in the value of the CMAP correspond to the firing of different combinations of motor units. At a fixed stimulus, variability in the CMAP, measured as variance, can be used to conduct MUNE using the "statistical" or the "Poisson" method. However, this method relies on the assumptions that the numbers of motor units that are firing probabilistically have the Poisson distribution and that all single motor unit action potentials (MUAP) have a fixed and identical size. These assumptions are not necessarily correct. We propose to develop a Bayesian statistical methodology to analyze electrophysiological data to provide an estimate of motor unit numbers. Our method of MUNE incorporates the variability of the threshold, the variability between and within single MUAPs, and baseline variability. Our model not only gives the most probable number of motor units but also provides information about both the population of units and individual units. We use Markov chain Monte Carlo to obtain information about the characteristics of individual motor units and about the population of motor units and the Bayesian information criterion for MUNE. We test our method of MUNE on three subjects. Our method provides a reproducible estimate for a patient with stable but severe ALS. In a serial study, we demonstrate a decline in the number of motor unit numbers with a patient with rapidly advancing disease. Finally, with our last patient, we show that our method has the capacity to estimate a larger number of motor units.     

PLANNING THE SUICIDE EXPERIMENTS

Suicide experiments involve a great degree of uncertainty in the counting of cell colonies. This work studies from a statistical point of view the precision of the estimation as a function of the number of experimental units which are used. Assuming that the colony numbers in recipients follow a Poisson distribution, we give the necessary number of recipients (a) to determine with a given accuracy the percentage of DNA synthesizing cells (S cells), (b) to test whether or not a cell population is quiescent, and (c) to compare the percentages of S cells in two cell populations.     

Determination of the content of nonfilterable cells in erythrocyte suspensions as a function of the medium osmolality

The results of most filtration assays for deformability of erythrocytes do not distinguish whether the entire population or only its small fraction exhibits abnormal rheological properties. We developed a simple filtration method for determination of the percentage of nonfilterable cells in erythrocyte suspension using membrane filters with mean pore diameter of 3.1 microns. This method makes it possible to detect even minor abnormal subpopulations in erythrocyte suspensions. The flow rate of buffer depends on the number of free pores of a filter. The plot of the number of pores clogged by nonfilterable cells vs the total number of erythrocytes that were allowed to pass through the filter had a linear portion, with a slope representing the relative content, Z%, of nonfilterable cells in the suspension. We determined Z% for various medium osmolalities u and used the data to derive the distribution of erythrocytes in ucr (ucr is the maximum value of u at which an erythrocyte cannot pass through a pore of a given filter because of geometric limitations). The distribution of ucr in suspension of normal erythrocytes has a maximum of about 200 mOsm/kg and a half-width of about 20 mOsm/kg. The distributions of ucr are altered in normal erythrocyte suspensions at decreased pH values, in cryopreserved and ATP-depleted erythrocyte suspensions and in erythrocytes from a xerocytosis patient.     

Uncertainty in Calculations of Net Primary Production for Grasslands

Net primary production (NPP) is a fundamental characteristic of all ecosystems and foundational to understanding the fluxes of energy and nutrients. Because NPP cannot be measured directly, researchers use field-measured surrogates as input variables in various equations designed to estimate ‘true NPP’. This has led to considerable debate concerning which equations most accurately estimate ‘true NPP’. This debate has influenced efforts to assess NPP in grasslands, with researchers often advocating more complex equations to avoid underestimation. However, this approach ignores the increase in statistical error associated with NPP estimates as a greater number of parameters and more complex mathematical functions are introduced into the equation. Using published grassland data and Monte Carlo simulation techniques, we assessed the relative variability in NPP estimates obtained using six different NPP estimation equations that varied in both the number of parameters and intricacy of mathematical operations. Our results indicated that more complex equations may result in greater uncertainty without reducing the probability of underestimation. The amount of uncertainty associated with estimates of NPP was influenced by the number of parameters as well as the variability in the data and the nature of the mathematical operations. For example, due to greater variability in the field-measured belowground data than aboveground data, estimates of belowground NPP tended to have more uncertainty than estimates of aboveground NPP. An analysis in which the input data were standardized allowed us to isolate the details of the calculations from the variability in the data in assessing the propagation of uncertainty. This analysis made clear that equations with product terms have the potential to magnify the uncertainty of the inputs in the estimates of NPP although this relationship was complicated by interactions with data variability and number of parameters. Our results suggest that more complex NPP estimation equations can increase uncertainty without necessarily reducing risk of underestimation. Because estimates can never be tested by comparison to “true NPP”, we recommend that researchers include an assessment of propagation of statistical error when evaluating the ‘best’ estimation method.     

Curve fitting approach for measurement of cellular osmotic properties by the electrical sensing zone method. II. Membrane water permeability

In a companion paper, we demonstrated that dynamic range limitations can confound measurement of the osmotically inactive volume using electrical sensing zone instruments (e.g., Coulter counters), and presented an improved parameter estimation method in which a lognormal function was fit to the cell volume distribution to allow extrapolation beyond the bounds of the data. Presently, we have investigated the effect of dynamic range limitations on measurement of the cell membrane water permeability (L_p), and adapted the lognormal extrapolation method for estimation of L_p from transient volume data. An alternative strategy (the volume limit adjustment method, in which the measured isotonic volume distribution is used to generate model predictions for curve fitting, and the bounds of the dynamic range are adjusted such that extrapolation is not required) was also developed. The performance of these new algorithms was compared to that of a conventional parameter estimation method. The best-fit L_p values from in vitro experiments with mouse insulinoma (MIN6) cells differed significantly for the different parameter estimation techniques (p < 0.001). Using in silico experiments, the volume limit adjustment method was shown to be the most accurate (relative error 0.4 ± 3.2%), whereas the conventional method underestimated L_p by 19 ± 2% for MIN6 cells. Parametric analysis revealed that the error associated with the conventional method was sensitive to the dynamic range and the width of the volume distribution. Our initial implementation of the lognormal extrapolation method also yielded significant errors, whereas accuracy of this algorithm improved after including a normalization scheme.     

非正态分布预测模型误差的估计

提出了模型预测误差在其分布为非正态分布时的区间估计方法,研究了模型预测误差的估计问题,给出了应用实例.     

A comparison between Poisson and zero-inflated Poisson regression models with an application to number of black spots in Corriedale sheep

Dark spots in the fleece area are often associated with dark fibres in wool, which limits its competitiveness with other textile fibres. Field data from a sheep experiment in Uruguay revealed an excess number of zeros for dark spots. We compared the performance of four Poisson and zero-inflated Poisson (ZIP) models under four simulation scenarios. All models performed reasonably well under the same scenario for which the data were simulated. The deviance information criterion favoured a Poisson model with residual, while the ZIP model with a residual gave estimates closer to their true values under all simulation scenarios. Both Poisson and ZIP models with an error term at the regression level performed better than their counterparts without such an error. Field data from Corriedale sheep were analysed with Poisson and ZIP models with residuals. Parameter estimates were similar for both models. Although the posterior distribution of the sire variance was skewed due to a small number of rams in the dataset, the median of this variance suggested a scope for genetic selection. The main environmental factor was the age of the sheep at shearing. In summary, age related processes seem to drive the number of dark spots in this breed of sheep.     

The Spleen Colony Technique. Iii. Comparison of the Overlap Effect and of Errors In Cfu-S Determination and the [3H]-Thymidine Suicide Data For Several Strains of Mice

A mathematical model of errors of the spleen colony technique is applied to data obtained from four mouse strains and F₁ hybrids. the variance of the colony counts was close to the Poisson distribution in inbred mice and F¹ hybrids. However, it should be checked regularly. the magnitude of the error in CFU-s determination and of the estimations of the S phase fraction was derived, and is presented relative to the mean colony counts for all mouse strains studied. the optimum spleen colony counts are generally higher than those which are commonly used. However, the utilization of the optimum spleen colony counts requires a correction for the effect of colony overlap.     

Autoradiographic studies of the synthesis of RNA and protein as a function of cell volume in Streptococcus faecium

Mid-exponential-phase cultures were either labeled continuously with tritiated leucine and uracil or pulse-labeled with tritiated leucine. The amount of leucine and uracil incorporated into protein or RNA per cell was determined by grain counts of autoradiographs of cells seen in electron micrographs; the volume of each cell was determined by three-dimensional reconstruction. The average number of autoradiographic grains around cells continuously labeled with uracil and leucine increased linearly with cell volume. In contrast, while the average grain count around cells pulse-labeled with leucine increased in a near-linear fashion over most of the volume classes, less than the expected number of grains were seen around cells in large- and small-size classes. The distribution of grains around cells from both the continuously and pulse-labeled populations could be fit at the 5% confidence level with a Poisson distribution modified to take into consideration the volume distribution of each population of cells analyzed. These findings suggested that large changes in the density of RNA and protein do not occur in most cells as they increase in size; however, there may be decreases in the rate of protein synthesis in some large and small cells. The decrease in the rate of protein synthesis appears consistent with the hypothesis that new sites of envelope growth must be introduced into cells that are close to the division event to restore rapid growth.     

Statistical estimation of gene expression using multiple laser scans of microarrays

We propose a statistical model for estimating gene expression using data from multiple laser scans at different settings of hybridized microarrays. A functional regression model is used, based on a non-linear relationship with both additive and multiplicative error terms. The function is derived as the expected value of a pixel, given that values are censored at 65 535, the maximum detectable intensity for double precision scanning software. Maximum likelihood estimation based on a Cauchy distribution is used to fit the model, which is able to estimate gene expressions taking account of outliers and the systematic bias caused by signal censoring of highly expressed genes. We have applied the method to experimental data. Simulation studies suggest that the model can estimate the true gene expression with negligible bias. AVAILABILITY: FORTRAN 90 code for implementing the method can be obtained from the authors.     

A modification of the filtration method for studying the content of nonfilterable cells in erythrocyte suspension

The results of filtration assays provide estimates of the deformability of erythrocytes averaged over the entire suspension. These assays do not distinguish whether the entire population or only its small fraction exhibits abnormal rheological properties. We developed a simple method using a filtrometer to determine the percentage of non-filterable (under given conditions) cells in the erythrocyte suspension. Membrane filters made of a polyethylene terphthalate film had the mean pore diameter of 3.1 microns and the length of cylindrical micropores of 7 microns. The buffer flow rate tb depends on the number of free pores in a filter. The plot of the number of pores clogged by non-filterable cells versus the total number of erythrocytes passed through the filter had a linear portion whose slope represents the relative content Z of non-filterable cells in the suspension. We determined Z for various medium osmolarities u. These data were used to derive the distribution of erythrocytes in ucr, the value of u at which an erythrocyte cannot pass through a pore of a given filter because of geometric limitations. The distribution maximum corresponded to 190-200 mOsm/kg for erythrocytes from the normal blood. This means that normal erythrocytes have the median values of their surface area and area-to-volume ratio of 155-151 microns2 and 1.72-1.68 microns-1, respectively. The half-width of the distribution was approximately 30 mOsm/kg. This finding suggests that the normal blood contains a certain fraction of erythrocytes with a decreased area-to-volume ratio. Our results showed that the distribution is altered in various forms of anemia and in ATP-depleted erythrocyte suspensions.     

Estimation in a Continuous State Space Branching Process

In some biological studies it is impossible to count the total number of individuals, but the size of the population can be measured by it's weight or volume. The probability of extinction of such populations is an interesting parameter, e.g. in modelling of epidemics. In this paper, the estimation of extinction probability and the moments of the offspring distribution is studied and the asymptotic properties of the estimators are discussed.     

Protein fluorescence decay: discrete components or distribution of lifetimes? Really no way out of the dilemma?

A new methodology of fluorescence decay analysis by iterative reconvolution is presented. It is based on the recent finding that the statistics of single-photon time-correlated data are best described by a compound Poisson law and requires the recording of a sample of at least 20 decays. Application of multivariate statistical methods to the analysis of the recovered decay parameters results in improved accuracy and better estimation of the uncertainties of mono- and multiexponential decays. If it is, of course, not possible to distinguish unambiguously between discrete components and a continuous distribution of lifetimes, it is, however, possible to determine a higher limit of the width of such a distribution should it be present. With our methodology, the presence of a distribution of lifetimes with a width of approximately 20% of its center value inevitably leads to a failure in the deconvolution procedure, a fact of crucial importance in protein conformational studies, for example.