Independent Measurement Errors Research Articles

Joint models for a primary endpoint and multiple longitudinal covariate processes.

Joint models are formulated to investigate the association between a primary endpoint and features of multiple longitudinal processes. In particular, the subject-specific random effects in a multivariate linear random-effects model for multiple longitudinal processes are predictors in a generalized linear model for primary endpoints. Li, Zhang, and Davidian (2004, Biometrics60, 1-7) proposed an estimation procedure that makes no distributional assumption on the random effects but assumes independent within-subject measurement errors in the longitudinal covariate process. Based on an asymptotic bias analysis, we found that their estimators can be biased when random effects do not fully explain the within-subject correlations among longitudinal covariate measurements. Specifically, the existing procedure is fairly sensitive to the independent measurement error assumption. To overcome this limitation, we propose new estimation procedures that require neither a distributional or covariance structural assumption on covariate random effects nor an independence assumption on within-subject measurement errors. These new procedures are more flexible, readily cover scenarios that have multivariate longitudinal covariate processes, and can be implemented using available software. Through simulations and an analysis of data from a hypertension study, we evaluate and illustrate the numerical performances of the new estimators.

Measuring stair use in two office buildings: a comparison between an objective and a self‐reported method

Measuring stair use reliably and objectively is complicated and difficult. In this study, stair use was measured at an individual level by using an innovative registration system and was compared with self-reported data. The purpose of this study was to gain an insight into the comparability of self-reported stair use vs objectively measured stair use. Self-reported and objective stair use was measured in two worksites and was operationalized as how often a subject uses the stairs per week (i.e., stair-use frequency) and the number of floors covered (up or down) in a week with each use. Analyses were performed by means of the intraclass correlation coefficients (ICCs). A number of significant differences in stair use between worksites were found. ICCs of 0.55 and 0.24 for stair-use frequency were found in worksites 1 and 2, respectively. The ICCs for the number of floors covered were lower at 0.39 and 0.19 for worksites 1 and 2, respectively. The comparability of self-reported and objectively measured stair use is moderate to poor, and given the independent measurement errors of both methods, this might have been expected. Comparability seemed to be dependent on worksite characteristics.

Some Variations on Standard Income Measurement Error Models

Measurement error can have a significant impact on measures of inequality. Using a fairly flexible parametric specification of an independent multiplicative measurement error (IMME) model we explore the relationship between changes in the variance of measurement error, for a given mean of measurement error, on the Gini Coefficient. While the measured Gini is greater than the true Gini, the difference decreases as the variance of measurement error decreases. Copulas are used to relax the assumption of independence of measurement error and true income. In this case the measured Gini can be larger or smaller than the true Gini, depending on the correlation between true income and measurement error. Using the same approach with simulations the effect of a different distribution of measurement error is investigated.

Bivariate linear mixed models using SAS proc MIXED

Bivariate linear mixed models are useful when analyzing longitudinal data of two associated markers. In this paper, we present a bivariate linear mixed model including random effects or first-order auto-regressive process and independent measurement error for both markers. Codes and tricks to fit these models using SAS Proc MIXED are provided. Limitations of this program are discussed and an example in the field of HIV infection is shown. Despite some limitations, SAS Proc MIXED is a useful tool that may be easily extendable to multivariate response in longitudinal studies.

Estimating mixing density from a system of observations

Let Y11,…Y1m,Y21…,Yn1,…Ynm be observations such that Yij=Xi+εij, where xi are i.i.d. unobservable random variables with density f and εij are independent measurement errors, with density fε. First, this paper study the estimation of f from the system of observations {Yij}n,m i=1,j=1 when fε is known. Upper bounds on the rate of quadratic mean convergence are obtained. The upper bounds depend on the geometry of the problem and on the rate of growing of m as function of n. Then, we study two special situations inside the system of observations. The first case is when a parameter of the characteristic function of ε is unknown. The second case corresponds to εij non identically distributed. A simulation study has been conducted to show the sampling behavior of the estimators for distinct values of n and m.

Longitudinal models for AIDS marker data.

Over the past decade, researchers have put a great amount of effort into developing suitable models for the analysis of longitudinal CD4 data and other markers of AIDS progression. These models must be general enough to allow for different patterns of change in the marker data. In this paper, we review the existing literature including our preferred models which involve mixed effects, stochastic terms and independent measurement error. Adding stochastic terms to standard mixed effects models gives an interpretable and parsimonious method for generalizing the covariance structure of the measurement error and short-term variability. We focus on univariate and bivariate models with integrated Ornstein-Uhlenbeck (IOU) stochastic terms. The IOU process allows for a range of biologically plausible derivative tracking that encompasses both random trajectory and Brownian motion behaviour. We illustrate these modelling techniques on longitudinal CD4 and viral RNA data.

A multivariate model to relate hydrological, chemical and biological parameters to salmonid biomass in Italian Alpine rivers

A multivariate statistical analysis was performed to assess the relationships between a series of river habitat parameters and the variation in salmonid biomass in 13 stations in Italian Alpine salmonid rivers. Various river data were collected and principal component analysis was used to identify the relevant parameters. The best regression model was calculated using two statistics: Mallows’Cp and R2 adjusted for degrees of freedom. The best model only uses six parameters and explained 89% of variation in fish biomass. Various statistical indices were calculated to evaluate model quality and robustness. A sensitivity analysis was carried out to determine how the model slope would change due to independent measurement errors in the parameters. These effects were mostly negligible. This demonstrated that it is possible to simplify habitat quality evaluation using a subset of environmental parameters which should be useful for river management.

Density Estimation for the Case of Supersmooth Measurement Error

The problem is to estimate the probability density of a random variable contaminated by an independent measurement error. I explore one of the worst-case scenario when the characteristic function of this measurement error decreases exponentially and thus optimal estimators converge only with logarithmic rate. The particular example of such measurement error is any random variable contaminated by normal, Cauchy, or another stable random variable. For this setting and circular data, I suggest an asymptotically efficient data-driven estimator that is adaptive to both smoothness of estimated density and distribution of measurement error. Moreover, this estimator is universal in sense that its derivatives and integral are sharp estimators of the corresponding derivatives and the cumulative distribution function, and these estimators are sharp both globally and pointwise. For the case of small sample sizes, I suggest a modified estimator that mimics an optimal linear pseudoestimator. I explore this estimator theoretically, via intensive Monte Carlo simulations and practical examples.

Markov Chains With Measurement Error: Estimating the `True' Course of a Marker of the Progression of Human Immunodeficiency Virus Disease

SUMMARY A Markov chain is a useful way of describing cohort data. Longitudinal observations of a marker of the progression of the human immunodeficiency virus (HIV), such as CD4 cell count, measured on members of a cohort study, can be analysed as a continuous time Markov chain by categorizing the CD4 cell counts into stages. Unfortunately, CD4 cell counts are subject to substantial measurement error and short timescale variability. Thus, fitting a Markov chain to raw CD4 cell count measurements does not determine the transition probabilities for the true or underlying CD4 cell counts; the measurement error results in a process that is too rough. Assuming independent measurement errors, we propose a likelihood-based method for estimating the 'true'or underlying transition probabilities. The Markov structure allows efficient calculation of the likelihood by using hidden Markov model methodology. As an example, we consider CD4 cell count data from 430 HIV-infected participants in the San Francisco Men's Health Study by categofizing the marker data into seven stages; up to 17 observations are available for each individual.We find that including measurement error both produces a significantly better fit and provides a model for CD4 progression that is more biologically reasonable.

Assessing monoplane camera calibration performance with off-the-shelf hardware

This work discusses how to assess the performance of a monoplane camera calibration procedure with off-the-shelf, low-cost hardware. Four aspects are considered: repeatability, congruence of the estimated parameters with independent mechanical measurement, back-projection and projection errors in realistic conditions, i.e. outside the calibration plane. The relevance, in low-cost equipment, of systematic errors, represented by non-unity gain factors, and of non-radial distortion is pointed out. Knowledge of gain factors may even be brought to advantage, e.g. for improving the estimates of target distance and lens focal length.

Analysis of many short time sequences: Forecast improvements achieved by shrinkage

AbstractCredibility models in actuarial science deal with multiple short time series where each series represents claim amounts of different insurance groups. Commonly used credibility models imply shrinkage of group‐specific estimates towards their average. In this paper we model the claim size yu in group i and at time t as the sum of three independent components: yit = μr + δi + ϵit. The first component, μt = μt−1 + mt, represents time‐varying levels that are common to all groups. The second component, δi, represents random group offsets that are the same in all periods, and the third component represents independent measurement errors. In this paper we show how to obtain forecasts from this model and we discuss the nature of the forecasts, with particular emphasis on shrinkage. We also assess the forecast improvements that can be expected from such a model. Finally, we discuss an extension of the above model which also allows the group offsets to change over time. We assume that the offsets for different groups follow independent random walks.

Uncorrelated measurement error in flood frequency inference

When measurement error (ME) is present, the true value of the annual maximum discharge and of the censoring threshold discharge is unknown. The impact of using estimated rather than true discharges on the inference of flood quantiles is considered. The likelihood function is formulated for data which consist of annual maximum discharges (systematic data) and occurrences of floods above some threshold discharge (binomial‐censored data). The formulation makes explicit allowance for the corruption of discharges by statistically independent ME. This formulation can be used to get maximum likelihood estimates or perform a Bayesian analysis. A limited sampling experiment was conducted to assess the performance of two‐parameter lognormal maximum likelihood quantile estimators in the presence of simple uncorreiated discontinuous ME. It was found that ME reduces the information content of systematic and binomial‐censored data with the reduction being greatest for distributions with low coefficients of variation. When only using systematic data it was found that the performance of estimators making allowance for ME was, at best, only modestly better than that of estimators ignoring ME. The principal finding, however, was that the performance of maximum likelihood estimators using binomial‐censored data but ignoring ME can be substantially worse than that of estimators making explicit allowance for ME. In fact, the loss in performance can be so great that in some cases, it may be preferable to ignore the binomial‐censored data. Finally, it is shown that a Bayesian analysis of flood data affected by ME provides insights not offered by maximum likelihood estimation. When the threshold discharge is in error, the posterior distribution of the lognormal parameters can become bimodal as the length of the binomial‐censored record increases, indicating an inconsistency between the information in the systematic and binomial‐censored data.

Effect of regression to the mean in the presence of within‐subject variability

Regression to the mean arises often in statistical applications where the units chosen for study relate to some observed characteristic in the extreme of its distribution. Gardner and Heady attribute the effect of regression to the mean to measurement errors. They assume the model Yi = U + ei, where U is a fixed within-subject component and ei is the random measurement error. They suggest several replicate measurements to reduce the regression effect under the assumption that the measurement errors ei are independent within subjects. While measurement errors play an important role in regression to the mean, one should not overlook within-subject variation. In this paper, we consider a model to estimate the regression effect in the presence of correlated within-subject effects as well as independent measurement errors.

Rates of convergence of some estimators in a class of deconvolution problems

This paper studies the problem of estimating the density of U when only independent copies of X = U + Z is observable where Z is an independent measurement error. Convergence rates of a family of deconvolved kernel density estimators are obtained under different assumptions on the density of Z.

Score Tests in Generalized Linear Measurement Error Models

SUMMARY Hypothesis tests in generalized linear models are studied under the condition that a surrogate w is observed in place of the true predictor x. The efficient score test for the hypothesis of no association depends on the conditional expectation E(x|w) which is generally unknown. The usual test substitutes w for E(x|w) and is asymptotically valid but not efficient. We investigate two new test statistics appropriate when w = x + z where z is an independent measurement error. The first is a Wald test based on estimators corrected for measurement error. Despite the correction for attenuation in the estimator, this test has the same local power as the usual test. The second test employs an estimator of E(x|w) and is both asymptotically efficient for normal errors and approximately efficient when the measurement error variance is small.

Probability analysis for multistage single-parameter tolerance monitoring subject to independent measurement errors

Probability analysis for multistage single-parameter tolerance monitoring subject to independent measurement errors

On the errors-in-variables problem for time series

The usual assumption in the classical errors-in-variables problem of independent measurement errors cannot necessarily be maintained when the data are time series; errors may be strongly serially correlated, possibly containing seasonal effects and trends. When it is possible to identify frequency bands over which the signal-to-noise ratio is large, an approximate solution to the errors-in-variables problem is to omit the remaining frequencies from a time series regression. We draw attention to the danger of “leakage” from the omitted frequencies, and show that the consequent bias can be reduced by means of tapering.

Structural model estimation with correlated measurement errors

Maximum likelihood estimators for model parameters in linear structural relationships having correlated measurement errors are presented. Comparisons between asymptotic mean squared errors of least squares and maximum likelihood estimators of the slope parameter indicate that the least squares estimator is only preferable when the slope parameter is sufficiently small relative to the ratio of error standard deviations; however, unlike structural relationships having independent measurement errors, the region of preference is not simply given by an upper bound on this scaled slope parameter.

Track fitting with multiple scattering: A new method

An analytical calculation of the variance is performed, in some simple case, for standard least-squares estimators of track parameters (accounting for independent measurement errors only); comparison is made with optimal estimators (accounting also for scattering errors, correlated between one point and the following ones). A new method is proposed for optimal estimation: the points measured on the track are included backwards, one by one, in the fitting algorithm, and the scattering is handled locally at each step. The feasibility of the method is shown on real events, for which the geometrical resolution is improved. The algorithm is very flexible and allows fast programmation; moreover the computation time is merely proportional to the number of measured points, contrary to the other optimal estimators.

The Two-Scale Plot: An Exploratory Display of Data with Heterogeneous Variances

A two-scale plot is a display of measurements Xi, i = 1, 2, . . . , N with heterogeneous variances. Often, as in the example below, a provisional estimate S,2 of the variance of each measurement is available; however, it is prudent to check whether the behavior of the data agrees with the provisional variance estimates. A two-scale plot provides such a check and in addition displays both the marginal distribution of the measurements Xi and the marginal distribution of the measurements Zi = XilSi standardized to have approximately unit variance. The plot uses three interrelated coordinates in two dimensions, so a single point represents both a measurement Xi and its standardized value Zi = XilSi. The coordinates of a measurement are (Xi, Zi, Si-) where the measurement Xi determines the diagonal coordinate, the standardized measurement Zi determines the horizontal coordinate, and the scaling factor Si-' determines the vertical coordinate. The example in Figure 1 uses data from an experiment to calibrate a film device designed to measure radiation. Certain atomic particles which strike the film leave marks; the number Yi of marks counted on a specified area of film is used to estimate the radiation level. Physical arguments, which ignore potential problems due to measurement error, suggest the number Yi of marks should be Poisson distributed with expectation proportional to the total dose. The data used here are readings from 135 devices, divided into groups of 50, 50, 25, and 10 devices, and exposed to a known, constant level of radiation for 3 days, 4 days, 7 days, and 14 days, respectively. The total dose is proportional to the number of days of exposure. For each dose Figure 1 contains a boxplot (Tukey, 1977) representing both the residuals Xi = Yi Yi and the standardized residuals Zi = XilSi = (Yi Yi)l/VYi from the proportional Poisson model described above, fitted by maximum likelihood, where Yi denotes the fitted count. Of course, S,2 is taken equal to Yi because under the Poisson model var (Yi) = E(Yj), where var (Yi) denotes the variance of Yi. At each of the four doses, the median residual is slightly negative because the Poisson distribution is skewed. Since the spread of the standardized residuals Zi does not vary with the scale factors Si-', the standardization with S,2 = 1i has been effective in producing standardized residuals with nearly constant variance, which is consistent with the assumption that the counts Yi are Poisson distributed. Departures from the Poisson model can take several forms in a two-scale plot. For example, if the counts Yi had been distributed as the sum of a Poisson variable and an independent measurement error with zero mean and constant variance, so var (Yi) = E( Yi) + 0.2, say, then S,2 = Y, would underestimate the variance of Yi, particularly for small E( Y); as a result, the interquartile range of the standardized residuals Zi = ( Yi Yi)l/VYi would have been greater than in Figure 1, and the spread of the standardized residuals Zi would have tended to increase with Si-'. Alternatively, if the count Yi had been distributed as the sum of a Poisson variable and an independent counting error with zero mean and variance proportional to the expected count, so var (Yi) = E( Y1) (1 + f3), with f3> 0, then S,2 = Yi would underestimate the variance of Yi by a constant factor; as a result, the interquartile range of the standardized residuals Zi would have been greater than in Figure 1, but the spread of the standardized residuals would not have tended to increase with Si-'. In short, the behavior of the standardized residuals Zi as a function of Si-' pro-