Computation and uses of central trend lines

1984 ◽  
Vol 62 (10) ◽  
pp. 1897-1905 ◽  
Author(s):  
W. E. Ricker

A bivariate array of naturally variable observations can take many different forms, depending on the relative lengths of the measurement units used. Each of these has a different central trend or major axis. In a standard presentation the major axis has a slope of ± 1 obtained when 1 standard deviation (s) of each variate, Y and X, occupies the same distance on its coordinate axis. With any other presentation the position of the standard trend is indicated by a line whose slope is the ratio of the standard deviations; it is called the standard (or reduced) major axis, or geometric mean regression line (GMR). The GMR is symmetrical, invariant with change of scale, and "robust." Besides indicating the central trend, it is a suitable line for estimating Y from X, or X from Y, in two common situations where ordinary regressions fail: (i) when the sampling procedure was not random with respect to the entire population (but was random with respect to its standard trend); (ii) when the population sampled departs seriously from a bivariate normal configuration. In the latter case an alternative "Schnute" line is appropriate if components of the population may have different sY/sX ratios.

Author(s):  
William V. Harper ◽  
David J. Stucki ◽  
Thomas A. Bubenik ◽  
Clifford J. Maier ◽  
David A. R. Shanks ◽  
...  

The importance of comparing in-line inspection (ILI) calls to excavation data should not be underestimated. Neither should it be undertaken without a solid understanding of the methodologies being employed. Such a comparison is not only a key part of assessing how well the tool performed, but also for an API 1163 evaluation and any subsequent use of the ILI data. The development of unity (1-1) plots and the associated regression analysis are commonly used to provide the basis for predicting the likelihood of leaks or failures from unexcavated ILI calls. Combining such analysis with statistically active corrosion methods into perhaps a probability of exceedance (POE) study helps develop an integrity maintenance plan for the years ahead. The theoretical underpinnings of standard regression analysis are based on the assumption that the independent variable (often thought of as x) is measured without error as a design variable. The dependent variable (often labeled y) is modeled as having uncertainty or error. Pipeline companies may run their regressions differently, but ILI to field excavation regressions often use the ILI depth as the x variable and field depth as the y variable. This is especially the case in which a probability of exceedance analysis is desired involving transforming ILI calls to predicted depths for a comparison to a threshold of interest such as 80% wall thickness. However, in ILI to field depth regressions, both the measured depths can have error. Thus, the underlying least squares regression assumptions are violated. Often one common result is a regression line that has a slope much less than the ideal 1-1 relationship. Reduced Major Axis (RMA) Regression is specifically formulated to handle errors in both the x and y variables. It is not commonly found in the standard literature but has a long pedigree including the 1995 text book Biometry by Sokal and Rohlf in which it appears under the title of Model II regression. In this paper we demonstrate the potential improvements brought about by RMA regression. Building on a solid comparison between ILI data and excavations provides the foundation for more accurate predictions and management plans that reliably provide longer range planning. This may also result in cost savings as the time between ILI runs might be lengthened due to a better analysis of such important data.


2017 ◽  
Author(s):  
Erin Dunne ◽  
Ian E. Galbally ◽  
Min Cheng ◽  
Paul Selleck ◽  
Suzie B. Molloy ◽  
...  

Abstract. Understanding uncertainty is essential for utilizing atmospheric VOC measurements in robust ways to develop atmospheric science. This study describes an inter-comparison of the VOC data, and the derived uncertainty estimates, measured with three independent techniques (PTR-MS, AT-GC-FID and DNPH-HPLC) during the Sydney Particle Study campaigns in 2012. The compounds and compound classes compared, based on objective selection criteria from the available data, were: benzene, toluene, C8 aromatics, isoprene, formaldehyde, acetaldehyde and acetone. Bottom-up uncertainty analyses were undertaken for each compound and each measurement system. Top-down uncertainties were quantified via the inter-comparisons. Four metrics were used for the inter-comparisons: the slope and intercept as determined by reduced major axis regression, the correlation, and the root mean standard deviation of the observation from the regression line. In all seven comparisons the correlations between independent measurement techniques were high with R2 values of median 0.93 (range: 0.72–0.98) and small root mean standard deviations of the observations from the regression line with a median of 0.13 (range: 0.04–0.23 ppb). These results give a high degree of confidence that for each comparison the response of the two independent techniques are dominated by the same constituents. The slope and intercept as determined by reduced major axis regression gives a different story. The slopes varied considerably with a median of 1.23 and range 1.08 to 2.03. The intercepts varied with a median of 0.02 and range −0.07 to 0.31 ppb. An ideal comparison would give a slope of 1.00 and an intercept of zero. This analysis identified some poorly understood and poorly quantified sources of uncertainty in the measurement techniques including: the contributions of non-target compounds to the measurement of the target compound for benzene, toluene and isoprene by PTR-MS; and, the under-reporting of formaldehyde, acetaldehyde and acetone by the DNPH technique. As well as these, this study has identified a specific interference of liquid water with acetone measurements by the DNPH technique. These relationships reported for Sydney 2012 were incorporated into a larger analysis with 61 other published inter-comparison studies for the same compounds. Overall for the light aromatics, isoprene and the C1–C3 carbonyls the uncertainty in a set of measurements varies by a factor of between 1.5 and two. These uncertainties (~50 %) are significantly higher than uncertainties estimated using standard propagation of error methods, which in this case were ~22 % or less, and are the result of the presence of poorly understood or neglected processes that affect the measurement and its uncertainty. The uncertainties in VOC measurements identified here should be considered when: assessing the reliability of VOC measurements from individual instruments; when utilising VOC data to constrain and inform air quality and climate models; when using VOC observations for human exposure studies; and, when comparing ambient VOC data with satellite retrievals.


1988 ◽  
Vol 66 (11) ◽  
pp. 2329-2339 ◽  
Author(s):  
B. H. McArdle

Most biologists are now aware that ordinary least square regression is not appropriate when the X and Y variables are both subject to random error. When there is no information about their error variances, there is no correct unbiased solution. Although the major axis and reduced major axis (geometric mean) methods are widely recommended for this situation, they make different, equally restrictive assumptions about the error variances. By using simulated data sets that violate these assumptions, the reduced major axis method is shown to be generally more efficient and less biased than the major axis method. It is concluded that if the error rate of the X variable is thought to be more than a third of that on the Y variable, then the reduced major axis method is preferable; otherwise the least squares technique is acceptable. An analogous technique, the standard minor axis method, is described for use in place of least squares multiple regression when all of the variables are subject to error.


1982 ◽  
Vol 45 (6) ◽  
pp. 561-565 ◽  
Author(s):  
R. T. MARSHALL ◽  
Y. H. LEE ◽  
B. L. O'BRIEN ◽  
W. A. MOATS

Samples of skim milk and nonfat dry milk (NDM) made from it were collected, paired and tested for pyruvate concentration, [P], and Direct Microscopic count (DMC). The skim milk was tested for Standard Plate Count (SPC) and Psychrotrophic Plate Count (PPC). The geometric average DMC of skim milk was more than three times higher than that of the paired NDM samples. However, [P] of NDM was not significantly different from that of the skim milk. Although [P] of skim milk was poorly correlated with SPC and PPC, r = .31 and .26, respectively, it was relatively well correlated with DMC, r = .64. Data were widely dispersed around the regression line when [P] was ≤ 4.0 mg/L. However, [P] increased rapidly when DMCs were > 106/ml. A limit of 10 mg/L of [P] in NDM reconstituted 1:9 was chosen to represent the current U.S. Department of Agriculture Standard for DMC in NDM. This limit failed to classify about 10% of the samples correctly, assuming that each geometric mean DMC was correct. However, the probability that samples meeting the DMC standard would be rejected by the pyruvate test was quite low and the probability was moderate that samples which would be acceptable by the pyruvate test would be rejected by the DMC. For the latter, 28% of the samples having DMCs of ≥ 107/ml contained < 10 mg/L of pyruvate. No sample having ≥ 10 mg/L of pyruvate had a DMC of ≤ 107/ml. Pyruvate concentration in NDM did not change during storage at 5 or 32°C for 90 days.


1984 ◽  
Vol 56 (2) ◽  
pp. 536-539 ◽  
Author(s):  
D. L. Sherrill ◽  
G. D. Swanson

The ventilatory response to changes in alveolar (arterial) CO2 is widely used as an index of respiratory control behavior. Methods for estimating these response slopes should incorporate the possibility that there may be errors in both the independent (partial pressure of CO2) and dependent (ventilation) variables. In a recent paper Daubenspeck and Ogden (J. Appl. Physiol. Respirat. Environ. Exercise Physiol. 45:823–829, 1978) have suggested problems inherent in the traditional technique of reduced major axis and have suggested a more contemporary technique of directional statistics. We have previously analyzed both techniques and developed a method to overcome the problems of reduced major axis and problems inherent in the use of directional statistics. Under the assumption of a bivariate normal distribution, we demonstrate that our slope estimate is similar to the maximum likelihood estimate proposed by Mardia et al. (J. Appl. Physiol.: Respirat. Environ. Exercise Physiol. 54: 309–313, 1983) for this problem. In addition, we demonstrate a bootstrap statistical approach when the distributions are not normally distributed. These concepts are illustrated using O2-CO2 interaction data.


Author(s):  
Brahim Boussidi ◽  
Peter Cornillon ◽  
Gavino Puggioni ◽  
Chelle Gentemann

This study was undertaken to derive and analyze the Advanced Microwave Scanning Radiometer - EOS (AMSR-E) sea surface temperature (SST) footprint associated with the Remote Sensing Systems (RSS) Level-2 (L2) product. The footprint, in this case, is characterized by the weight attributed to each 4 4 km square contributing to the SST value of a given AMSR-E pixel. High-resolution L2 SST fields obtained from the MODerate-resolution Imaging Spectroradiometer (MODIS), carried on the same spacecraft as AMSR-E, are used as the sub-resolution “ground truth“ from which the AMSR-E footprint is determined. Mathematically, the approach is equivalent to a linear inversion problem, and its solution is pursued by means of a constrained least square approximation based on the bootstrap sampling procedure. The method yielded an elliptic-like Gaussian kernel with an aspect ratio 1.58, very close to the AMSR-E 6.93GHz channel aspect ratio, 1.7. (The 6.93GHz channel is the primary spectral frequency used to determine SST.) The semi-major axis of the estimated footprint is found to be alignedwith the instantaneous field-of-view of the sensor as expected fromthe geometric characteristics of AMSR-E. Footprintswere also analyzed year-by-year and as a function of latitude and found to be stable – no dependence on latitude or on time. Precise knowledge of the footprint is central for any satellite-derived product characterization and, in particular, for efforts to deconvolve the heavily oversampled AMSR-E SST fields and for studies devoted to product validation and comparison. A preliminarly analysis suggests that use of the derived footprint will reduce the variance between AMSR-E and MODIS fields compared to the results obtained.


2012 ◽  
Vol 27 (2) ◽  
pp. 13
Author(s):  
L. A. Salcido -Guevara ◽  
F. Arreguín -Sánchez ◽  
L. Palmeri ◽  
A. Barausse

We tested the hypothesis that ecosystem metabolism follows a quarter power scaling relation, analogous to organisms. Logarithm of Biomass/Production (B/P) to Trophic Level (TL) relationship was estimated to 98 trophic models of aquatic ecosystems. A normal distribution of the slopes gives a modal value of 0.64, which was significantly different of the theoretical value of 0.75 (p0.05). We also tested for error in both variables, Log (B/P) and TL, through a Reduced Major Axis regression with similar results, with a modal value of 0.756 (p>0.05). We also explored a geographic distribution showing no significant relation (p>0.05) to latitude and between different regions of the world. We conclude that: a) ecosystem metabolism follows the quarter-power scaling rule; b) transfer efficiency between TL plays a relevant role characterizing local attributes to ecosystem metabolism; and c) there is neither latitudinal nor geographic differences. These findings confirm the existence of a metabolic scaling regularity in aquatic ecosystems. Regularidad del escalamiento metabólico en ecosistemas acuáticos Se contrastó la hipótesis de que el metabolismo de un ecosistema sigue una relación de escalamiento análoga a la existente en los organismos. La relación entre el logaritmo de la razón Producción/Biomasa (B/P) y el nivel trófico (TL) se estimó para 98 modelos tróficos de los ecosistemas acuáticos. Una distribución normal de las pendientes de esta relación produjo un valor modal de 0.64 que es significativamente diferente del valor teórico de 0.75 (p0.05) similar al teórico esperado. También se contrastó la hipótesis de existencia de error en ambas variables, logaritmo (B/P) y TL, a través de la técnica de regresión denominada “Reduced Major Axis”, con resultados similares según el valor modal de 0.756, sin diferencia estadísticamente significativa (p>0.05) del valor teórico. Se exploró la existencia de algún patrón en la distribución geográfica, sin obtenerse relación significativa (p>0.05) con la latitud, o con diferentes regiones del mundo. Las conclusiones son: a) el metabolismo del ecosistema sigue la regla de escalamiento metabólico de 3/4; b) la eficiencia de la transferencia entre TL desempeña un papel relevante, representando los atributos locales del metabolismo del ecosistema; c) no hay una diferencias latitudinal o geográfica. Estos resultados confirman la existencia de una regularidad en el escalamiento metabólico en ecosistemas acuáticos.


The Condor ◽  
2007 ◽  
Vol 109 (3) ◽  
pp. 705-714 ◽  
Author(s):  
Todd W. Arnold ◽  
Andy J. Green

AbstractAbstract. Numerous investigators have used allometric regression to characterize the relationship between proportional egg composition and egg size, which is a potentially important characterization for assessing maternal investment in reproduction. Herein, we document two important shortcomings of this approach. First, regressing log component mass against log egg mass involves regressing Y on itself, since each component (Y) is necessarily a part of the whole egg (X). This creates correlated errors, which leads to biased estimates of the regression slope. To circumvent this problem, we recommend regressing egg component masses on a relatively inert component like total water mass. Secondly, investigators routinely use ordinary least squares regression to estimate the slope of allometric relationships, which assumes that all error resides in Y. We demonstrate that this assumption is false, but so are the underlying error assumptions of commonly used alternatives such as reduced major axis and major axis regression. Because each egg is unique and determining composition involves destructive sampling, there is no obvious way to assess measurement error in Y versus X. As a solution, we recommend that investigators analyze multiple eggs per clutch whenever possible and fit a reduced major axis based on the among-female component of variability.


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