Attenuation Statistics Research Articles

Multimodal assessment of brain stiffness variation in healthy subjects using magnetic resonance elastography and ultrasound time-harmonic elastography

Magnetic resonance elastography (MRE) is a noninvasive brain stiffness mapping method. Ultrasound-based transtemporal time-harmonic elastography (THE) is emerging as a cost-effective, fast alternative that has potential applications for bedside monitoring of intracranial pressure. We aim to investigate the accuracy of THE in comparison to MRE performed in the brain. Ten healthy volunteers (25–40 years old) underwent multifrequency MRE (20–35 Hz) and THE (27–56 Hz). Fiducial-marker-based optical tracking of the ultrasound field of view was used to align THE to 3D MRE. THE- and MRE-derived shear wave speed (SWS) was determined as a measure of brain stiffness and averaged within regions of various depths for cross-modality correlation analysis. MRE-measured SWS ranged from 1.0 to 1.3 m/s and was negatively correlated with age (R2 = 0.44, p = 0.035). After registration of both modalities, SWS values were linearly correlated (MRE: 1.14 ± 0.08 m/s, THE: 1.13 ± 0.10 m/s; R2 = 0.62, p = 0.007). Best agreement between modalities was achieved at depths of 40–60 mm, suggesting this range provides a viable trade-off between ultrasound attenuation and near-field bias. Similar brain regions can be consistently measured with both elastography modalities, despite the regional and individual variations of stiffness. Transtemporal THE yields stiffness values in a range similar to those obtained with more expensive MRE.

How Follow-Up Period in Prospective Cohort Studies Affects Relationship Between Baseline Serum 25(OH)D Concentration and Risk of Stroke and Major Cardiovascular Events.

Background/Objectives: Prospective cohort studies are useful for studying how biomolecular status affects risk of adverse health outcomes. Less well known is that the longer the follow-up time, the lower the association (or "apparent effect") due to "regression dilution". Here, we evaluate how follow-up interval from baseline to "event" affects the relationship between baseline serum 25-hydroxyvitamin D [25(OH)D] concentration and the later incidence of stroke and major cardiovascular events (MACEs). Methods: Findings for the relative risk (RR) of stroke and MACEs with respect to serum 25(OH)D concentrations at baseline from prospective cohort studies were plotted against mean follow-up time. Fifteen studies from mainly European countries and the United States were used for stroke and nine studies for MACEs. Linear regression analyses were used to study data for follow-up periods of up to 10 years and for more than 10 years. Results: For stroke, the linear regression fit for 1-10 years is RR = 0.34 + (0.065 × follow-up [years]), r = 0.84, adjusted r2 = 0.67, p < 0.001. No significant variations in association were found for studies with follow-up periods of 10-20 years. For MACEs, the linear fit for 1-8.1 years is RR = 0.61 + (0.055 × follow-up [years]), r = 0.81, adjusted r2 = 0.59, p = 0.03. Discussion: The shorter the follow-up period, the greater the apparent effect of better vitamin D status in reducing risk of stroke and MACEs. In addition, the apparent effect of higher 25(OH)D concentration found for the shortest follow-up time is more than twice as great as the estimate based on average follow-up intervals for all studies. Mechanisms have been found to explain how higher serum 25(OH)D concentrations could reduce risk of stroke and MACEs. Randomized controlled trials have not shown that vitamin D supplementation significantly reduces risk of either stroke or MACEs, probably because risk of both outcomes increases rapidly below 15 ng/mL (38 nmol/L) and it is difficult in Western developed countries to enroll enough participants with concentrations that low. Nonetheless, vitamin D's role in reducing risk of stroke and MACEs could be considered causal on the basis of an evaluation of the evidence using Hill's criteria for causality in a biological system. Conclusions: Serum 25(OH)D concentrations above 20 ng/mL are associated with significantly reduced risk of stroke and MACEs prospectively and in an apparent causal manner. Raising serum 25(OH)D concentrations to >20 ng/mL should, therefore, be recommended for everyone likely to be at risk for stroke or MACEs and indeed in the general population.

Interactions between latent variables in count regression models

In psychology and the social sciences, researchers often model count outcome variables accounting for latent predictors and their interactions. Even though neglecting measurement error in such count regression models (e.g., Poisson or negative binomial regression) can have unfavorable consequences like attenuation bias, such analyses are often carried out in the generalized linear model (GLM) framework using fallible covariates such as sum scores. An alternative is count regression models based on structural equation modeling, which allow to specify latent covariates and thereby account for measurement error. However, the issue of how and when to include interactions between latent covariates or between latent and manifest covariates is rarely discussed for count regression models. In this paper, we present a latent variable count regression model (LV-CRM) allowing for latent covariates as well as interactions among both latent and manifest covariates. We conducted three simulation studies, investigating the estimation accuracy of the LV-CRM and comparing it to GLM-based count regression models. Interestingly, we found that even in scenarios with high reliabilities, the regression coefficients from a GLM-based model can be severely biased. In contrast, even for moderate sample sizes, the LV-CRM provided virtually unbiased regression coefficients. Additionally, statistical inferences yielded mixed results for the GLM-based models (i.e., low coverage rates, but acceptable empirical detection rates), but were generally acceptable using the LV-CRM. We provide an applied example from clinical psychology illustrating how the LV-CRM framework can be used to model count regressions with latent interactions.

A note on trigonometric regression in the presence of Berkson‐type measurement error

In this note, we study how parameter vector estimation for a trigonometric regression model and the expected squared residual error computed from an estimated model are affected by Berkson‐type measurement error. Closed‐form expressions for the parameter vector and the expected squared residual error are obtained by assuming that the observed covariate data are sampled from an equispaced design and that measurement error is generated from a symmetric probability distribution with a mean of zero. Notably, these results indicate that estimates of the amplitude parameters for a trigonometric regression model suffer from attenuation bias when covariate data are mis‐measured, and that estimates of the phase‐shift parameters are unbiased.

Publication and Attenuation Biases in Measuring Skill Substitution

Abstract A key parameter in the analysis of wage inequality is the elasticity of substitution between skilled and unskilled labor. We show that the empirical literature is consistent with both publication and attenuation bias in the estimated inverse elasticities. Publication bias, which exaggerates the mean reported inverse elasticity, dominates and results in corrected inverse elasticities closer to zero than the typically published estimates. The implied mean elasticity is 4, with a lower bound of 2. Elasticities are smaller for developing countries. To derive these results, we use nonlinear tests for publication bias and model averaging techniques that account for model uncertainty.

Name-Based Estimators of Intergenerational Mobility

Abstract Recent studies use names—first and surnames—to estimate intergenerational mobility in sources that lack direct family links. While generating novel evidence on intergenerational transmission processes, it remains unclear how different estimators compare and how reliable they are. This paper evaluates the most popular name-based methods, using newly digitised records from Finland and US census data. We illustrate that their interpretation depends on sampling properties of the data, such as the overlap between the parent and child samples, which differ widely across studies. We correct for the attenuation bias from limited overlap and address other common problems encountered in applications.

Exponential family measurement error models for single-cell CRISPR screens.

CRISPR genome engineering and single-cell RNA sequencing have accelerated biological discovery. Single-cell CRISPR screens unite these two technologies, linking genetic perturbations in individual cells to changes in gene expression and illuminating regulatory networks underlying diseases. Despite their promise, single-cell CRISPR screens present considerable statistical challenges. We demonstrate through theoretical and real data analyses that a standard method for estimation and inference in single-cell CRISPR screens-"thresholded regression"-exhibits attenuation bias and a bias-variance tradeoff as a function of an intrinsic, challenging-to-select tuning parameter. To overcome these difficulties, we introduce GLM-EIV ("GLM-based errors-in-variables"), a new method for single-cell CRISPR screen analysis. GLM-EIV extends the classical errors-in-variables model to responses and noisy predictors that are exponential family-distributed and potentially impacted by the same set of confounding variables. We develop a computational infrastructure to deploy GLM-EIV across hundreds of processors on clouds (e.g. Microsoft Azure) and high-performance clusters. Leveraging this infrastructure, we apply GLM-EIV to analyze two recent, large-scale, single-cell CRISPR screen datasets, yielding several new insights.

An analysis of the accuracy of retrospective birth location recall using sibling data

Many surveys ask participants to retrospectively record their location of birth. This paper examines the accuracy of such data in the UK Biobank using a sample of full siblings. Comparison of reported birth locations for siblings with different age gaps allows us to estimate the probabilities of household moves and of misreported birth locations. Our first contribution is to show that there are inaccuracies in retrospective birth location data, showing a sizeable probability of misreporting, with 28% of birth coordinates, 16% of local districts and 6% of counties of birth being incorrectly reported. Our second contribution is to show that such error can lead to substantial attenuation bias when investigating the impacts of location-based exposures, especially when there is little spatial correlation and limited time variation in the exposure variable. Sibling fixed effect models are shown to be particularly vulnerable to the attenuation bias. Our third contribution is to highlight possible solutions to the attenuation bias and sensitivity analyses to the reporting error.

Abstract P341: Physical Activity and Dual Energy X-ray Absorptiometry Measured Body Composition in Global Multi-Ethnic Cohorts

Background: Obesity is an important risk factor for the development of heart disease, and unfortunately rates of obesity have been growing worldwide. Obesity is often defined by body mass index (BMI) ≥30kg/m 2 , however, BMI cannot distinguish between fat or lean mass nor differences in fat distribution. Dual energy x-ray absorptiometry (DXA) provides more detailed measurements of body composition, but research assessing the association between physical activity and detailed body composition measures in diverse populations is lacking. This study therefore examined associations between physical activity and DXA-derived measures of adiposity in Chinese, Malay, Indian and White European adults from two large-scale prospective cohort studies. Methods: From 2006-2010, adults from the general population were recruited to both The Malaysian Cohort (TMC) study (N=119 555) and the UK Biobank (N=502 487). Physical activity was self-reported by questionnaire at baseline assessment and approximately 5-yearly resurveys in both cohorts. Between 2020-2023, 5489 participants underwent a DXA scan in TMC; while between 2014-2023, 60 280 participants had DXA scans in UK Biobank. Analyses excluded participants with missing data and prior diseases that may impact body composition. Multivariable linear regression (adjusted for demographic and lifestyle confounders) was used to examine associations between physical activity and body composition measures across ethnic groups, and estimates were corrected for regression dilution bias by using estimates of ‘usual physical activity’ derived from repeat measurements. Results: After exclusions, 1723 Chinese, 1245 Malay and 609 Indian participants were included in the TMC analyses and 56 758 White Europeans in the UK Biobank (mean age 57.1 years, 41.2% male in TMC and 54.7 years, 47.7% male in UK). In TMC, physical activity was not associated with BMI, however it was inversely associated with DXA adiposity measures and positively associated with lean mass. Comparing the highest quintile of physical activity to the lowest, total fat mass was 0.7 kg lower (95%CI: -1.1- -0.3) and total lean mass was 0.6 kg higher (0.3-0.9). There was no evidence of effect modification by ethnicity in TMC, but results in White Europeans were stronger: total fat mass was 3.9 kg lower (-4.1- -3.6) and lean mass was 1.2 kg higher (1.1-1.4) comparing the highest to lowest quintiles of physical activity. Interpretation: This is the first large-scale study to examine associations between physical activity and DXA-derived measures of body composition in multi-ethnic cohorts from Malaysia and the UK. The findings support physical activity interventions to reduce obesity-related disease across diverse populations and highlight the value of using detailed body composition measures.

Estimating the sensitivity of CEO compensation to gross versus net accounting performance

AbstractIn empirically estimating the relation between CEO compensation and accounting‐based firm and peer performance, researchers often define the performance variables net of CEO compensation expense. We analytically show that a researcher's use of CEO compensation as a regression's dependent variable and as an expense in defining a regression's independent variables representing accounting‐based firm and peer performance will bias the researcher's pay‐for‐performance and relative performance evaluation (RPE) regression coefficients. In a panel estimation of CEO compensation, we document an attenuation bias in the coefficients on net firm and net peer performance. This evidence may partially explain inferences of weak CEO incentives and limited usage of RPE in prior work. Our results imply that in CEO compensation regressions, a researcher can remove biases in inferring CEO incentives and RPE usage by using gross rather than net accounting performance variables—that is, by adding back CEO compensation expense to net accounting measures.

Using Digitized Newspapers to Address Measurement Error in Historical Data

This paper shows how to remove attenuation bias in regression analyses due to measurement error in historical data for a given variable of interest by using a secondary measure that can be easily generated from digitized newspapers. We provide three methods for using this secondary variable to deal with non-classical measurement error in a binary treatment: set identification, bias reduction via sample restriction, and a parametric bias correction. We demonstrate the usefulness of our methods by replicating four recent economic history papers. Relative to the initial analyses, our results yield markedly larger coefficient estimates.

Baseline and usual triglyceride-glucose index and the risk of chronic kidney disease: a prospective cohort study

Triglyceride-glucose (TyG) index is an emerging marker of adverse cardiometabolic conditions such as cardiovascular disease and type 2 diabetes. The long-term relevance of TyG index to chronic kidney disease (CKD) is uncertain. We aimed to assess the association of TyG index with CKD risk and its utility in risk prediction in a prospective study. The TyG index was calculated using fasting triglycerides and fasting plasma glucose (FPG) levels measured in 2362 men aged 42–61 years with normal kidney function using the formula: Ln (fasting triglycerides [mg/dL] × FPG [mg/dL]/2). Multivariable adjusted hazard ratios (HRs) (95% confidence intervals, CIs) were estimated for CKD. Correction for within-person variability was made using data from repeat measurements of triglycerides and FPG taken 11 years after baseline. Over a median follow-up duration of 17.5 years, 223 CKD cases were recorded. The age-adjusted regression dilution ratio for the TyG index was 0.54 (95% CI, 0.48–0.60). The risk of CKD increased continuously with increasing TyG index across the range 9.3 to 11.6 (p value for nonlinearity<.001). In analysis adjusted for established risk factors, a unit higher TyG index was associated with an increased risk of CKD (HR 1.59, 95% CI 1.24–2.05). Comparing extreme tertiles of the TyG index, the corresponding adjusted HR (95% CI) for CKD was 1.61 (1.15–2.27). Addition of the TyG index to a CKD risk prediction model containing established risk factors improved risk discrimination and reclassification (p value for difference in −2 log likelihood<.001; NRI=47.66%, p=.014; IDI=0.0164, p<.001). Higher TyG index is associated with an increased risk of CKD and improves the prediction and classification of CKD beyond established risk factors. Using single baseline estimations of the TyG index to investigate its association with CKD risk could considerably under-estimate the true association.

A study on the rain attenuation statistics using the different Ku-band satellite signals measured at multiple locations

A study on the rain attenuation statistics using the different Ku-band satellite signals measured at multiple locations

Total least squares bias in climate fingerprinting regressions with heterogeneous noise variances and correlated explanatory variables

AbstractRegression‐based “fingerprinting” methods in climate science employ total least squares (TLS) or orthogonal regression to remedy attenuation bias arising from measurement error due to reliance on climate model‐generated explanatory variables. Proving the consistency of multivariate TLS requires assuming noise variances are equal across all variables in the model. This assumption has been challenged empirically in the climate context but little is known about TLS biases when the assumption is violated. Monte Carlo analysis is used herein to examine coefficient biases when the noise variances are not equal. The analysis allows the explanatory variables to be negatively correlated which is typical in climate applications. Ordinary least squares (OLS) exhibits the expected attenuation bias which vanishes as the noise variances on the explanatory variables disappear. In some cases, TLS corrects attenuation bias but more typically imparts large and generally positive biases. OLS performs well when the true value of whereas TLS performs quite poorly. This implies that TLS is not well suited for tests of the null. When TLS tends to exhibit opposite biases to OLS. Diagnostic information specific to each data sample should be consulted before using TLS to avoid spurious inferences and replacing OLS attenuation bias with other, potentially larger biases.

Dose error impacts on a collection of realistic dose-response curves based on a NASA sonic boom community noise survey

The National Aeronautics and Space Administration (NASA) plans to conduct surveys of community response to quiet supersonic flight to collect dose-response data for international regulators. Previous models of noise dose versus annoyance response ignored uncertainty in the noise dose experienced by survey respondents. This dose error causes attenuation bias and results in inaccurate dose-response relationships. The impacts of dose errors can be explored by introducing error into the dose estimates of a simulated community noise survey. The simulated population annoyance response is determined by specific noise response characteristics, including the onset of annoyance intercept, rate of increasing annoyance, participant intercept variability, and response rates. This paper explores the effects of dose errors on notional populations created by perturbing the noise response characteristics observed in participants in a recent NASA flight test, Quiet Supersonic Flights 2018 (QSF18). The QSF18-based population parameters were shifted up to ±20% to create the study populations. For the anticipated dose range of the X-59, changes to the onset of annoyance intercept have the greatest impact on erroneous model results. Quantifying point estimate errors illustrates the impact of dose error, with errors up to 14 dB observed for a fixed high annoyance percentage.

Twelve years of rain attenuation statistics of Earth–space propagation experiment at Ka band in Toulouse

SummarySince 2009, ONERA has been running Ka band propagation experiments in Toulouse (France, latitude 43.57°N, longitude 1.47°E). A rain gauge was also deployed on site to collect rainfall rate measurements concurrently to beacon data. Since April 2011, the beacon receiver has been recording the 20.2 GHz (vertical polarisation) Astra 3B beacon signal along a slant path of 35.1° of elevation angle. All years of the experiment had excellent data availability, hence giving 12 years of data at Ka band. First, the propagation experiment and the data processing methodology are described. Second, a statistical analysis of rain attenuation and rainfall rate is conducted. Comparisons are then performed with the prediction methods of Recommendations ITU‐R P.837‐7 (rainfall rate), P.618‐13 (rain attenuation) and P.678‐3 (variability of propagation phenomena).

Prediction of Q-Band Rain Attenuation Statistics Using Frequency Scaling Models for Temperate Location, Vigo, Spain

The prediction of yearly rain attenuation statistics at Q-band is made from Ka-band measurement for a temperate location, Vigo, Spain, using different frequency scaling models. The validity of the different frequency scaling models is tested for the present location by graphical comparison between measurement and prediction and also in terms of mean, standard deviation, and root mean square errors (RMSE). The Battesti model performs better compared to other existing frequency scaling models, RMSE is only 4.16% and 5.52% for the periods July 2016 to June 2017 and July 2017 to June 2018 respectively. The proposed new power law model increases the accuracy of the prediction for which the RMSE error is 1.09% and 2.17% less compared to the Battesti model for the periods July 2016 to June 2017 and July 2017 to June 2018, respectively. In predicting monthly statistics, the proposed model performs better compared to the Battesti model for all the months except December, January, and February. While predicting monthly statistics, for the proposed model, the maximum and minimum prediction errors are 18.50% and 0.39%, respectively. For the prediction of diurnal statistics, the proposed model performs better compared to the Battesti model with a very small prediction error, less than 4% for both days. The present study can be useful in the design and planning of satellite communication links in present or similar climatic regions for estimating rain attenuation at Q band or above, for which the experimental setup's cost is very high and requires a large receiver margin.

Genetic causal association of adiposity, body fat distribution with incident heart failure in a European cohort: A linear and non-linear Mendelian randomisation analysis

Abstract Background Excess adiposity has been associated with higher risk of heart failure in observational studies. However, these observational studies are prone to residual confounding and the shape and strength of the independent associations of adiposity measures with heart failure is uncertain. Purpose This study aimed to compare the associations of usual and genetically determined general and central adiposity measures on incident heart failure in a European population cohort. Methods The MR analysis was performed in a cohort of 326,300 adults (mean age 55.9 years, 56% women) of European ancestry with complete genetic data and no prior cardiac and vascular disease recruited from UK general population in 2006-2010. Resurvey on a subset of 20,346 participants was conducted in 2012-13. Regression dilution ratios (RDR) of the different adiposity measures were calculated using Pearson’s correlation method to obtain usual levels of each adiposity measure. Participants were followed up by linkage to electronic health records for incident heart failure events. Externally weighted genetic risk scores (GRS) for body mass index (BMI) and waist circumference (WC) adjusted for BMI (WCadjBMI) were constructed using single nucleotide polymorphisms (SNPs) that were genome-wide significant for BMI and WC as reported by the Genetic Investigation of ANthropometric Traits (GIANT) consortium. Associations of usual measured adiposity measures and genetically predicted usual adiposity (76-SNP BMI-GRS [pleiotropic for other adiposity measures], 36-SNP BMI-GRS [non-pleiotropic for other adiposity measures] and 68-SNP WCadjBMI-GRS) with incident HF subtypes were determined using Cox proportional hazards regression models. The shapes of the associations of genetically predicted adiposity with heart failure at different levels of adiposity measures were examined using localized average causal effects within strata of residual adiposity measures after adjustment for GRS and correction for regression dilution. Hazard ratios (HRs) are presented as per equivalent 5 usual BMI units higher adiposity measure. Results Over a median follow-up of 12.8 years, there were 6,570 first-ever incident HF events. A positive non-linear association (J-shape) was observed between genetically predicted BMI and heart failure with while positive log-linear association was observed for genetically predicted WCadjBMI. Consistent with the observational analyses, the hazard ratio per 5 BMI units equivalent higher genetically predicted WCadjBMI were notably stronger (68-SNP WCadjBMI GRS: HR 2.03, 95% CI1.04-3.97) than for genetically predicted BMI (36-SNP GRS-BMI: HR 1.78, 95% CI 1.12-2.81 and 76-SNP GRS-BMI: HR 1.99, 95% CI 1.62-2.45). Conclusion The findings provide evidence for increased heart failure risk at extremes of BMI and also confirm the stronger causal role of central adiposity over general adiposity in heart failure across the whole range of waist circumference.Figure 1Figure 2

Mendelian randomization study of the relevance of blood pressure to vascular mortality among 150,000 participants in the mexico city prospective study

Abstract Background Mendelian randomization (MR) studies are commonly used to reduce confounding and reverse-causality when estimating the effects of risk factors on disease and provide estimates of effects over the whole life-course. However, most previous MR studies have been conducted in populations of European-ancestry. We used MR to investigate the effect of systolic blood pressure (SBP) on vascular mortality at ages 35-74 in a large prospective study of Mexican adults. Methods Between 1998 and 2004, 150,000 adults aged ≥35 years were recruited into the Mexico City Prospective Study and followed for median 19 years(1,2). Using 886 externally-selected single nucleotide polymorphisms (SNPs)(3) we created a genetic risk score for SBP (GRS-SBP). Those taking an antihypertensive drug at recruitment had their SBP adjusted by +6 mmHg(2) (adjustment by +15 mmHg rather than +6 mmHg was done in a sensitivity analysis). Cox regression and the MR ‘ratio’ method were then used to estimate the causal effect of SBP on vascular mortality at ages 35-74 years. Analyses were adjusted for age at risk, sex, residential district, education, smoking, alcohol consumption, physical activity, adiposity, diabetes mellitus and the first 7 genetic principal components. They excluded those aged ≥75 years, had incomplete data, or had prior vascular disease or any other chronic condition (except diabetes). Alternative MR approaches (inverse-variance-weighted, weighted median and MR Egger) were used for sensitivity analyses. Conventional epidemiological analyses of the association between SBP and mortality after correction for regression dilution bias were also done. Results Among 127,849 included individuals, the GRS-SBP explained 1.9% of baseline SBP variation. Those in the top fifth of the GRS-SBP distribution had 6.6 mmHg higher SBP and twice the prevalence of self-reported hypertension and antihypertensive drug use than those in the bottom fifth, but the GRS-SBP was uncorrelated with other factors. The strength of association between GRS-SBP and measured SBP was broadly similar in men and women, at different ages, and between people with different proportions of Amerindian genetic ancestry. Using the ratio method and a single estimate of the association of GRS-SBP with SBP, each 10 mmHg higher genetically predicted SBP was associated with 58% higher vascular mortality at ages 35-74 years (RR 1.58, 95% CI 1.27-1.96), including 69% higher ischaemic heart disease mortality (RR 1.69, 95% CI 1.31-2.16) and 65% higher stroke mortality (RR 1.65, 1.15-2.33). These genetic estimates were somewhat larger than conventional observational associations of usual SBP to risk. Alternative MR approaches found no evidence of pleiotropy and yielded similar conclusions. Conclusions These MR analyses confirm that blood pressure is a major cause of premature death from vascular diseases in Mexican adults.

Nonparametric Gini-Frisch bounds

The Gini-Frisch bounds partially identify the constant slope coefficient in a linear equation when the explanatory variable suffers from classical measurement error. This paper generalizes these quintessential bounds to accommodate nonparametric heterogeneous effects. It provides suitable conditions under which the main insights that underlie the Gini-Frisch bounds apply to partially identify the average marginal effect of an error-laden variable in a nonparametric nonseparable equation. To this end, the paper puts forward a nonparametric analogue to the standard “forward” and “reverse” linear regression bounds. The nonparametric forward regression bound generalizes the linear regression “attenuation bias” due to classical measurement error.