Non-normal Measurement Research Articles

A general novel process capability index for normal and non-normal measurements

In this study, a new process capability index is proposed. The new index is quantile-based, can be used in normal and non-normal measurements, and is a general form of the conventional Cpm. The new index is investigated by relying on the Weibull distribution, which holds significant importance in quality control and reliability studies. Point estimation for the index is discussed with parametric and non-parametric approaches. Moreover, several estimators based on methods such as Bayesian and bootstrap are suggested for interval estimation. A Monte Carlo simulation study is conducted to evaluate the performance of methodologies in both on-target and off-target processes. In addition, numerical examples are provided to evaluate two real-world problems where the measurements follow both the Weibull and normal distributions. The usability of the index is particularly evident when evaluating the ball size data in the first real data analysis, where the classical Cpm index is not used.

Shell extremal eigenvalues of tridiagonal Toeplitz Matrices

The shell of a complex tridiagonal Toeplitz matrix is studied. Closed formulas for all quantities involved in its equation are presented. Necessary and sufficient conditions for a Toeplitz tridiagonal matrix to have shell extremal eigenvalues are given. Several, recently introduced, geometric quantities related to the shell are studied as measures of non-normality of these extremal eigenvalues of such matrices. These quantities are also proposed as measures of non-normality for the matrix itself.

On the shell and the shell-extremal eigenvalues of a square matrix

The shell of a matrix is a cubic curve that provides interesting spectral localization results. We study its geometry in the case where the shell has a closed branch that surrounds a simple extremal eigenvalue of the matrix. Several quantities which are related to the closed branch are introduced and studied as measures of non-normality of that particular eigenvalue.

Assessing and compensating for the confounding factors in Scheimpflug-based corneal densitometry.

Scheimpflug-based corneal densitometry is a clinically verified method for assessing corneal transparency. Nevertheless, the estimates of corneal densitometry appear to be correlated with age and eye biometry parameters, such as the anterior chamber depth or the pupil size, and that ensues a convoluted conditional estimation problem, where it is difficult to interpret the results. This study aims at devising a methodology for compensating for such confounding factors by using, as a research platform, a commercially available Scheimpflug camera that allows exporting images in a dynamic fashion, allowing averaging the results from multiple acquisitions. Two approaches are considered, one based on appropriately normalizing the line densitometry signal and one based on image histogram equalization. Then, three parameters for describing corneal densitometry are derived including the mean value of backscatter and the scale and shape parameters of the Weibull distribution estimated in regions of interest encompassing parts of corneal stroma. The results show that, unlike the non-normalized measures, the proposed approaches lead to parameters that are not correlated with age nor the eye biometry.

Non-Fragile H ∞ Order Reduction of LTI Controllers

The eigenvalue perturbation theorem is used to propose a convex fragility criterion with application to control system design. The criterion can be considered as a non-normality measure of the controller state-space matrix. Non-normality of a matrix is defined as its distance to the nearest real normal matrix within a convex normal subspace. Based on the criterion, an $\mathcal {H}_{\infty }$ method for the order reduction of linear time-invariant (LTI) controllers is developed which leads to non-fragile reduced order controllers.

Nearness results for real tridiagonal 2‐Toeplitz matrices

SummaryIn this article, we extend the results for Toeplitz matrices obtained by Noschese, Pasquini, and Reichel. We study the distance d, measured in the Frobenius norm, of a real tridiagonal 2‐Toeplitz matrix T to the closure of the set formed by the real irreducible tridiagonal normal matrices. The matrices in , whose distance to T is d, are characterized, and the location of their eigenvalues is shown to be in a region determined by the field of values of the operator associated with T, when T is in a certain class of matrices that contains the Toeplitz matrices. When T has an odd dimension, the eigenvalues of the closest matrices to T in are explicitly described. Finally, a measure of nonnormality of T is studied for a certain class of matrices T. The theoretical results are illustrated with examples. In addition, known results in the literature for the case in which T is a Toeplitz matrix are recovered.

Comparison the efficacy of minimally invasive surgery and traditional surgery in the treatment of multiple benign mammary lumps

Objective To compare the clinical efficacy of minimally invasive surgery and traditional surgery in the treatment of multiple benign mammary lumps. Methods A retrospective study was conducted to select 158 patients with multiple benign breast masses admitted to Yijishan Hospital of Wannan Medical College from July 2016 to February 2018, all of them were female, average age was (28.3±6.6) years old, range from 19 to 51 years old. The patients were divided into minimally invasive group (n=122) and traditional group (n=36) according to different surgical methods. The minimally invasive group was treated by minimally invasive rotary excision, while the traditional group was treated by traditional surgery. The operation time, number of incisions, average length of incisions, cumulative length of incisions and healing time of incisions were compared between the two groups. The incidence of local hematoma, local infection, breast deformity and local residual were compared after operation, and the satisfaction of the two groups was aslo compared. Normal distribution measurements were expressed by mean±standard deviation (Mean±SD), independent sample t test was used for inter-group comparison; non-normal distribution measurements were expressed by median (quartile spacing) [M(P25, P75)], Mann-Whitney U test was used for inter-group comparison. Chi-square test or Fisher exact probability test was used to compare the count data between the two groups. Results As compared to the traditional group, the minimally invasive group had shorter operation time (20.0 min vs 40.0 min, Z=-8.590, P<0.001), less number of incisions (1.0 vs 2.0, Z=-4.423, P<0.001), smaller average surgical incision length (3.8 mm vs 35.5 mm, Z=-9.211, P<0.001), smaller cumulative surgical incision length (4.0 mm vs 67.2 mm, Z=-9.130, P<0.001), quicker postoperative recovery (4.0 d vs 7.0 d, Z=-9.334, P<0.001). There were no significant differences between the two groups in incidence of postoperative hematoma (4.1% vs 2.8%, χ2=0. 000, P=1.000), incidence of infection (0 vs 2.8%, P=0.228), incidence of postoperative breast shape change (1.6% vs 2.8%, χ2=0.000, P=1.000), and incidence of residual (0.8% vs 0, P=1.000). Psychological satisfaction of patients in minimally invasive group (95.1%) was significantly higher than that in traditional group (58.3%), the difference was statistically significant (P<0.001). Conclusions Compared with traditional surgery, the application of minimally invasive surgery in the treatment of multiple benign mammary lumps has many advantages, such as shorter operation time, less number of incisions, smaller surgical incision length, quicker postoperative recovery and higher satisfaction of patients after operation. It has not significantly increased postoperative complications. It is worthy of clinical application and promotion. Key words: Breast neoplasms; Vacuum; Ultrasonography; Surgical procedures, minimally invasive

Clinical analysis on outcomes in patients with chronic myeloid leukemia in chronic phase after tyrosine kinase inhibitors discontinuation and their influencing factors

Objective To explore the clinical outcomes and their influencing factors of tyrosine kinase inhibitors (TKI) discontinuation in patients with chronic myeloid leukemia in chronic phase (CML-CP). Methods From January 1999 to April 2018, a total of 16 patients with CML-CP who underwent routine outpatient follow-up in Department of Hematology, West China Hospital of Sichuan University, and discontinued TKI for more than 6 months were selected as subjects. Among them, there were 6 male patients and 10 females; the median age at diagnosis was 39.5 years (24.8-53.8 years). The median age at time of TKI discontinuation was 45.5 years old (30.3-69.0 years old). According to whether molecular recurrence occurred during the follow-up period after TKI discontinuation, the patients were divided into recurrent group (n=5) and non-recurrent group (n=11). The clinical data of patients with TKI before and after discontinuation were collected by retrospective method. The causes of TKI discontinuation, treatment-free remission (TFR) status after TKI discontinuation and its possible influencing factors were analyzed. In this study, composition ratio of gender, stage of CML, and TKI discontinuation causes were compared between the two groups using Fisher′s exact test. The Mann-Whitney U test was used to compare the non-normal distribution measurement data, such as duration of TKI treatment, time from TKI treatment to major molecular response (MMR)/deep molecular response (DMR), time of MMR/DMR maintenance. The TFR rate at 6 and 12 months after TKI discontinuation in patients was calculated by Kaplan-Meier method. The procedure followed in this study was in line with the revised Helsinki Declaration of the World Medical Association in 2013. Results ① Among all the 16 patients with CML-CP, causes of TKI discontinuation included TKI-related adverse reactions (n=5), patients′ expectancy (n=5), pregnancy/planned pregnancy (n=4), financial burden (n=1), and combined solid tumor (n=1). The median time of TKI treatment in 16 patients with CML-CP was 53.0 months (34.0-156.0 months). Among them, 15 patients achieved DMR before TKI discontinuation and 1 patient only obtained MMR. The median time to achieve DMR before TKI discontinuation was 39.0 months (10.0-144.0 months). ② The median follow-up time of 16 patients with CML-CP after TKI discontinuation was 17.0 months (7.0-75.0 months). At the end of follow-up, 11 patients had no molecular recurrence, and the median time of TFR was 12.0 months (2.0-75.0 months). TFR rates at 6 and 12 months after TKI discontinuation were 68.8% and 61.9%, respectively. Molecular recurrence occurred in 5 patients. The median recurrence time was 4.0 months (2.0-5.0 months) after TKI discontinuation, and the BCR-ABLIS level was 0.14%-0.88% at the time of recurrence. Among the 5 patients with molecular recurrence, 4 patients restarted TKI at 3, 4, 6 and 8 months after relapse, obtained MMR after 2, 3, 4, and 5 months of restarting TKI, obtained molecular response (MR)4.5 after 2, 3, 7 and 8 months of restarting TKI, respectively. ③ In this study, 5 patients with molecular recurrence occurred within 6 months after TKI discontinuation. The composition ratio of gender, stage of CML, causes of TKI discontinuation, previous treatment of interferon-α, TKI type, as well as duration of TKI treatment, time of initiation of TKI treatment to MMR/DMR, time of MMR/DMR maintenance, age at TKI discontinuation between two groups were compared. And the differences were not statistically significant (P>0.05). ④ Among 16 patients with CML-CP, 2 patients developed TKI withdrawal syndrome, which was characterized by systemic bone pain or myalgia. Conclusions Patients with CML-CP who receive long-term TKI and maintain continuous DMR, long-term TFR can be obtained in about 50% patients, but long-term MR should be monitored after TKI discontinuation. The above-mentioned influencing factors that may affect the clinical outcomes of TKI discontinuation were not statistically different between the recurrent group and the non-recurrent group, which may be related to small sample size of this study. Key words: Leukemia, myeloid; Leukemia, myeloid, chronic-phase; Leukemia, myelogenous, chronic, BCR-ABL positive; Prognosis; Tyrosine kinase inhibitor; Treatment discontinuation; Treatment free remission

Univariate and multivariate skewness and kurtosis for measuring nonnormality: Prevalence, influence and estimation.

Nonnormality of univariate data has been extensively examined previously (Blanca et al., Methodology: European Journal of Research Methods for the Behavioral and Social Sciences, 9(2), 78-84, 2013; Miceeri, Psychological Bulletin, 105(1), 156, 1989). However, less is known of the potential nonnormality of multivariate data although multivariate analysis is commonly used in psychological and educational research. Using univariate and multivariate skewness and kurtosis as measures of nonnormality, this study examined 1,567 univariate distriubtions and 254 multivariate distributions collected from authors of articles published in Psychological Science and the American Education Research Journal. We found that 74 % of univariate distributions and 68 % multivariate distributions deviated from normal distributions. In a simulation study using typical values of skewness and kurtosis that we collected, we found that the resulting type I error rates were 17 % in a t-test and 30 % in a factor analysis under some conditions. Hence, we argue that it is time to routinely report skewness and kurtosis along with other summary statistics such as means and variances. To facilitate future report of skewness and kurtosis, we provide a tutorial on how to compute univariate and multivariate skewness and kurtosis by SAS, SPSS, R and a newly developed Web application.

Constrained fits with non-Gaussian distributions

Non-normally distributed data are ubiquitous in many areas of science, including high-energy physics. We present a general formalism for constrained fits, also called data reconciliation, with data that are not normally distributed. It is based on Bayesian reasoning and implemented via MCMC sampling. We show how systems of both linear and non-linear constraints can be efficiently treated. We also show how the fit can be made robust against outlying observations. The method is demonstrated on a couple of examples ranging from material flow analysis to the combination of non-normal measurements. Finally, we discuss possible applications in the field of event reconstruction, such as vertex fitting and kinematic fitting with non-normal track errors.

Generalized linear models with covariate measurement error and unknown link function

ABSTRACTGeneralized linear models (GLMs) with error-in-covariates are useful in epidemiological research due to the ubiquity of non-normal response variables and inaccurate measurements. The link function in GLMs is chosen by the user depending on the type of response variable, frequently the canonical link function. When covariates are measured with error, incorrect inference can be made, compounded by incorrect choice of link function. In this article we propose three flexible approaches for handling error-in-covariates and estimating an unknown link simultaneously. The first approach uses a fully Bayesian (FB) hierarchical framework, treating the unobserved covariate as a latent variable to be integrated over. The second and third are approximate Bayesian approach which use a Laplace approximation to marginalize the variables measured with error out of the likelihood. Our simulation results show support that the FB approach is often a better choice than the approximate Bayesian approaches for adjusting for measurement error, particularly when the measurement error distribution is misspecified. These approaches are demonstrated on an application with binary response.

Clinical characteristics and prognostic analysis of 16 cases of systemic lupus erythematosus complicated with acute pancreatitis

Objective To investigate the clinical characteristics and prognosis of patients with systemic lupus erythematosus (SLE) complicated with acute pancreatitis (AP). Methods From January 1999 to December 2013, the clinical data of 16 patients with SLE complicated with AP among the total 2 526 cases of SLE was collected. A retrospective analysis was performed and the clinical data of patients was classified and documented, which included general information, past history, clinical symptoms, laboratory findings, imaging findings, treatment and outcome. The rank sum test was performed for analysis of non-normal distributed measurement data, and the Fisher's exact test was used for count data analysis. Results The incidence of SLE complicated with AP was 0.63% (16/2 526). Among them, ten patients were mild acute pancreatitis (MAP) and six patients were severe acute pancreatitis (SAP). All patients were treated with fasting, gastrointestinal decompression, nutritional support, anti-acid, anti-inflammatory, glucocorticoid and somatostatin and so on. Six patients were cured, seven patients improved and three patients died (two SLE complicated with SAP, one SLE complicated with MAP). Compared with the SLE patients complicated with SAP, the SLE patients complicated with MAP were more easily to have lupus nephritis(6/6 versus 5/10, Fisher's exact test), hematological system injuries (6/6 versus 5/10, Fisher's exact test), liver injuries (5/6 versus 0/10, Fisher's exact test), more organs involved (mean 7 versus 3, Z=-3.225) and higher SLE disease active indexes (DAI) score (mean 13.5 versus 6.5, Z=-2.876); the differences were statistically significant (all P<0.05). Compared with the cured and improved SLE patients complicated with AP, lupus encephalopathy (2/3 versus 1/13, Fisher's exact test), more organs involved (mean 7 versus 5, Z=-2.276) and higher SLE DAI score (mean 21 versus 12, Z=-2.195) was more common in dead SLE patients complicated with AP; the differences were statistically significant (all P<0.05). Conclusions SLE patients complicated with SAP are more easily to get lupus nephritis, hematological system injuries, liver injuries, activity of SLE and multiple-organ systems involved. The prognosis of SLE patients complicated with AP was poor in those with activity of SLE, multiple-organ involved and lupus encephalopathy. Key words: Lupus erythematosus, systemic; Acute pancreatitis; Clinical characteristics; Prognosis

Field-measured drag area is a key correlate of level cycling time trial performance.

Drag area (Ad) is a primary factor determining aerodynamic resistance during level cycling and is therefore a key determinant of level time trial performance. However, Ad has traditionally been difficult to measure. Our purpose was to determine the value of adding field-measured Ad as a correlate of level cycling time trial performance. In the field, 19 male cyclists performed a level (22.1 km) time trial. Separately, field-determined Ad and rolling resistance were calculated for subjects along with projected frontal area assessed directly (AP) and indirectly (Est AP). Also, a graded exercise test was performed to determine n}{}dot {V}{O}_{2} peak, lactate threshold (LT), and economy. n}{}dot {V}{O}_{2} peak (n}{}mathrm{l}~min ^{-1}) and power at LT were significantly correlated to power measured during the time trial (r = 0.83 and 0.69, respectively) but were not significantly correlated to performance time (r = − 0.42 and −0.45). The correlation with performance time improved significantly (p < 0.05) when these variables were normalized to Ad. Of note, Ad alone was better correlated to performance time (r = 0.85, p < 0.001) than any combination of non-normalized physiological measure. The best correlate with performance time was field-measured power output during the time trial normalized to Ad (r = − 0.92). AP only accounted for 54% of the variability in Ad. Accordingly, the correlation to performance time was significantly lower using power normalized to AP (r = − 0.75) or Est AP (r = − 0.71). In conclusion, unless normalized to Ad, level time trial performance in the field was not highly correlated to common laboratory measures. Furthermore, our field-measured Ad is easy to determine and was the single best predictor of level time trial performance.

Clinical characteristics analysis of 2625 acute pancreatitis in Jiangxi Province

Objective To analyze the clinical characteristics of patients with acute pancreatitis (AP) in Jiangxi Province.Methods From 2007 to 2012,the data of 2 625 hospitalized patients diagnosed as AP were retrospectively analyzed.The changes of composition in gender,age and etiology during 2007-2009 time period and 2010-2012 time period were compared.Mann Whitney U test was performed for non-normal distribution measurement data analysis and x2 test was for count data analysis.Results Among the 2 625 patients with AP,from 2007 to 2009 there were 1 028 cases and 1 597 cases in the period 2010 to 2012.The ages of the patients in these two period were 50(38,61) and 50 (40,63) years old,respectively.There was statistically significant difference in the age distribution between these two period (U=783 635.5,P＜0.05).Biliary factor was the most common cause of AP in the two time period 2007 to 2009 and 2010 to 2012,which accounted for 55.9％ (575/1 028) and 62.9％ (1 005/1 597),respectively,and the difference was statistically significant (x2=12.778,P＜ 0.01).Alcoholic AP accounted for 1.9％ (19/1 028) and 7.3％ (117/1 597),respectively,hyperlipidemic AP were 14.0％ (144/1 028) and 17.7％ (283/1 597),and idiopathic AP were 22.0％ (226/1 028) and 5.9％ (94/1 597),all the differences were statistically significant (x2 =38.204,6.330 and 151.416,all P＜0.05).In male patients,biliary AP accounted for 52.4％ (732/1 398),which was lower than that of female patients (69.1％,848/1 227),and the difference was statistically significant (x2 ＝76.524,P＜0.01).Alcoholic AP and hyperlipidemic AP accounted for 9.0％ (126/1 398) and 21.4％ (299/1 398) in male patients,which were significantly higher than those of female patients (0.8％,10/1 227; 10.4％,128/1 227),and the differences were statistically significant (x2 89.396 and 57.585,both P＜0.01).Biliary AP accounted for 79.9％ (631/790) in elderly group,which was higher than that of non elderly group (51.7％,949/1 835),and the difference was statistically significant (x2＝182.720,P＜0.01).In elderly group hyperlipidemic AP and alcoholic AP accounted for 2.5％ (20/790) and 1.3％ (10/790),which were significantly lower than those of non-elderly group (22.2％,407/1 835; 6.9％,126/1 835),and the differences were statistically significant (x2 =156.524 and 35.262,both P＜0.01).Conclusions The middle-aged and elderly people are the vulnerable population of AP.Biliary factor is the most common cause of AP,especially in the female and elderly.The proportions of hyperlipidemic AP and alcoholic AP both significantly increase,in the male higher than that of the female and in the non-elderly higher than that of the elderly.The proportion of idiopathic AP significantly decreases. Key words: Acute pancreatitis; Clinical characteristics; Etiology

Clinical study on transjugular intrahepatic portosystemic shunt in the treatment of hepatorenal syndrome

Objective To explore the efficacy and safety of transjugular intrahepatic portosystemic shunt (TIPS) in the treatment of hepatorenal syndrome (HRS).Methods From January 2009 to December 2012,the clinical data of three type 1 HRS cases and five type 2 HRS cases received TIPS operation were collected.The volume of 24 h urine,liver and renal function,and the occurrence of complications of portal hypertension were recorded and followed-up.Wilcoxon rank sum test was performed for non-normal distributed measurement data.Results TIPS of all the patients was successful and combined with devascularization in four patients.Before operation,the serum creatinine and volume of urine was 204.0 (180.8,229.8) μmol/L and 810 (600,925) mL,seven days after operation was 160.7 (120.2,184.0) μmol/L and 1 300 (975,1 500) mL,respectively,and the differences were statistically significant (Z=-2.1 and-2.2,both P＜0.05).One month after operation,renal function of five patients recovered to normal.At the 7th day after operation,total bilirubin level of three patients increased and that of two patients decreased to the preoperative base line value at one month and three month.Mild or moderate hepatic encephalopathy occurred in three patients after operation and relieved after medicine treatment.At 6th month after operation,two of three type 1 HRS patients survived.Of five type 2 HRS patients,one received liver transplantation at 6th week after operation,three of the left four cases survived at the 6th month after operation.Conclusion After safety and efficacy were evaluated,TIPS may be one of the methods used for the treatment of HRS. Key words: Hepatorenal syndrome; Liver cirrhosis; Portosystemic shunt, transjugular intrahepatic; Prognosis

Current optimization of search of objects for the model of distributed Poisson flow of their appearance

The problem of current optimization of object search when their appearance is simulated by random flow regularities is considered. The equation of evolution in time is obtained for the density of non-normalized probabilistic measure that gives probabilistic characteristics of the search. The optimal control law is synthesized for the search, with its properties analyzed. An example is given.

Basic gait and symmetry measures for primary school-aged children and young adults. II: Walking at slow, free and fast speed

This study recorded basic gait data from 656 healthy primary school-aged children (5–13 years) and 81 young adults (18–27 years) whilst walking over-ground across a level walkway at varying speed. It investigated the effect of gait speed and re-examined the issue of gait maturation. Participants completed 6–8 walks at self-selected slow, free and fast speed along a GAITRite walkway whilst wearing athletic shoes. Outcome measures (non-normalized and normalized) were gait speed, cadence, step and stride length, step and stride time, support base, single and double support (%), stance duration (%), foot angle and associated symmetry measures. Compared to free speed, participants walked 24% slower for the slow speed and 30% faster for the fast speed (p<0.0001). Both normalized and non-normalized measures of cadence, step and stride length increased with speed (p<0.001) whereas step and stride time reduced (p<0.001). As a percentage of the gait cycle, single support and stance duration increased with speed (p<0.001) whereas double support reduced (p<0.001). Foot angle was significantly less (less toe-out) for the fast speed than the free and slow speeds (p<0.001) whereas support base was unaffected by speed. Symmetry measures were unaffected by age or speed. Step and stride symmetry differentials (combining conditions) fell around 0.8cm, whereas symmetry differentials for step and stance time, single and double support fell around 0.7%. This information can be used by clinicians and researchers to assess the gait of children.

Restricted estimation in multivariate measurement error regression model

We study a multivariate ultrastructural measurement error (MUME) model with more than one response variable. This model is a synthesis of multivariate functional and structural models. Three consistent estimators of regression coefficients, satisfying the exact linear restrictions have been proposed. Their asymptotic distributions are derived under the assumption of a non-normal measurement error and random error components. A simulation study is carried out to investigate the small sample properties of the estimators. The effect of departure from normality of the measurement errors on the estimators is assessed.

Basic gait and symmetry measures for primary school-aged children and young adults whilst walking barefoot and with shoes

This study investigated the basic spatio-temporal gait measures of 898 primary school-aged children (5–13 years) and 82 young adults (18–27 years). Participants completed 6–8 walks at preferred speed along a GAITRite walkway whilst barefoot and whilst wearing athletic shoes or runners. Outcome measures (non-normalized and normalized) were gait speed, cadence, step and stride length, support base, single and double support, stance duration, foot angle and associated symmetry measures. Non-normalized measures of speed, step and stride length, support base and foot angle increased with age whereas cadence reduced. Normalized measures remained unchanged with age in children whereas the young adults (both conditions) exhibited a 2.3% reduction in single support, a 5.1% increase in double support and a 2.6% increase in stance duration ( p < 0.0001). For the entire sample, shoes increased walking speed by 8 cm s −1, step length by 5.5 cm, stride length by 11.1 cm and base of support by 0.5 cm. In contrast, foot angle and cadence reduced by 0.1° and 3.9 steps min −1 respectively. Shoes increased both double support (1.6%) and stance time (0.8%), whereas single support reduced by 0.8%. Symmetry remained unaffected by age. On average, measures of step and stride symmetry (combining both conditions) fell around 0.7 cm, whereas measures of symmetry for step and stance time, single and double support fell around 0.6%. Footwear significantly affected gait ( p < 0.0001). Gait may not be mature by age 13. Gait is symmetrical in healthy children and young adults but may change with pathology.

The number needed to treat (NNT) can be adjusted for bias when the outcome is measured with error

Background and objective We consider the number needed to treat (NNT) when the event of interest is defined by dichotomizing a continuous response at a threshold level. If the response is measured with error, the resulting NNT is biased. We consider methods to reduce this bias. Methods Bias adjustment was studied using simulations in which we varied the distributions of the underlying response and measurement error, including both normal and nonnormal distributions. We studied a maximum likelihood estimate (MLE) based on normality assumptions, and also considered a simulation–extrapolation estimate (SIMEX) without such assumptions. The treatment effect across all potential thresholds was summarized using an NNT threshold curve. Results Crude NNT estimation was substantially biased due to measurement error. The MLE performed well under normality, and it continued to perform well with nonnormal measurement error, but when the underlying response was nonnormal the MLE was unacceptably biased and was outperformed by the SIMEX estimate. The simulation results were also reflected in empirical data from a randomized study of cholesterol-lowering therapy. Conclusion Ignoring measurement error can lead to substantial bias in NNT, which can have an important practical effect on the interpretation of analyses. Analysis methods that adjust for measurement error bias can be used to assess the sensitivity of NNT estimates to this effect.