Statnote 14: the correlation of two variables (Pearson's 'r')

Pearson's correlation coefficient (‘r’) is one of the most widely used of all statistics. Nevertheless, care needs to be used in interpreting the results because with large numbers of observations, quite small values of ‘r’ become significant and the X variable may only account for a small proportion of the variance in Y. Hence, ‘r squared’ should always be calculated and included in a discussion of the significance of ‘r’. The use of ‘r’ also assumes that the data follow a bivariate normal distribution (see Statnote 17) and this assumption should be examined prior to the study. If the data do not conform to such a distribution, the use of a non-parametric correlation coefficient should be considered. A significant correlation should not be interpreted as indicating ‘causation’ especially in observational studies, in which the two variables may be correlated because of their mutual correlations with other confounding variables.

The use of data analysis methods: basic concepts for optometrists

This article explains first, the reasons why a knowledge of statistics is necessary and describes the role that statistics plays in an experimental investigation. Second, the normal distribution is introduced which describes the natural variability shown by many measurements in optometry and vision sciences. Third, the application of the normal distribution to some common statistical problems including how to determine whether an individual observation is a typical member of a population and how to determine the confidence interval for a sample mean is described.

Data analysis methods in optometry: is there a difference between two samples? Part 2.

In this second article, statistical ideas are extended to the problem of testing whether there is a true difference between two samples of measurements. First, it will be shown that the difference between the means of two samples comes from a population of such differences which is normally distributed. Second, the 't' distribution, one of the most important in statistics, will be applied to a test of the difference between two means using a simple data set drawn from a clinical experiment in optometry. Third, in making a t-test, a statistical judgement is made as to whether there is a significant difference between the means of two samples. Before the widespread use of statistical software, this judgement was made with reference to a statistical table. Even if such tables are not used, it is useful to understand their logical structure and how to use them. Finally, the analysis of data, which are known to depart significantly from the normal distribution, will be described.

Data analysis methods in optometry. Part 5: correlation

1. Pearson's correlation coefficient only tests whether the data fit a linear model. With large numbers of observations, quite small values of r become significant and the X variable may only account for a minute proportion of the variance in Y. Hence, the value of r squared should always be calculated and included in a discussion of the significance of r. 2. The use of r assumes that a bivariate normal distribution is present and this assumption should be examined prior to the study. If Pearson's r is not appropriate, then a non-parametric correlation coefficient such as Spearman's rs may be used. 3. A significant correlation should not be interpreted as indicating causation especially in observational studies in which there is a high probability that the two variables are correlated because of their mutual correlations with other variables. 4. In studies of measurement error, there are problems in using r as a test of reliability and the ‘intra-class correlation coefficient’ should be used as an alternative. A correlation test provides only limited information as to the relationship between two variables. Fitting a regression line to the data using the method known as ‘least square’ provides much more information and the methods of regression and their application in optometry will be discussed in the next article.

The choice of the model in inventory control and the cost of sophistication

This thesis is concerned with the inventory control of items that can be considered independent of one another. The decisions when to order and in what quantity, are the controllable or independent variables in cost expressions which are minimised. The four systems considered are referred to as (Q, R), (nQ,R,T), (M,T) and (M,R,T). Wiith ((Q,R) a fixed quantity Q is ordered each time the order cover (i.e. stock in hand plus on order ) equals or falls below R, the re-order level. With the other three systems reviews are made only at intervals of T. With (nQ,R,T) an order for nQ is placed if on review the inventory cover is less than or equal to R, where n, which is an integer, is chosen at the time so that the new order cover just exceeds R. In (M, T) each order increases the order cover to M. Fnally in (M, R, T) when on review, order cover does not exceed R, enough is ordered to increase it to M. The (Q, R) system is examined at several levels of complexity, so that the theoretical savings in inventory costs obtained with more exact models could be compared with the increases in computational costs. Since the exact model was preferable for the (Q,R) system only exact models were derived for theoretical systems for the other three. Several methods of optimization were tried, but most were found inappropriate for the exact models because of non-convergence. However one method did work for each of the exact models. Demand is considered continuous, and with one exception, the distribution assumed is the normal distribution truncated so that demand is never less than zero. Shortages are assumed to result in backorders, not lost sales. However, the shortage cost is a function of three items, one of which, the backorder cost, may be either a linear, quadratic or an exponential function of the length of time of a backorder, with or without period of grace. Lead times are assumed constant or gamma distributed. Lastly, the actual supply quantity is allowed to be distributed. All the sets of equations were programmed for a KDF 9 computer and the computed performances of the four inventory control procedures are compared under each assurnption.

Within-site variation in lichen growth rates and its implications for direct lichenometry

Variation in lichen growth rates poses a significant challenge for the application of direct lichenometry, i.e. the construction of lichen dating curves from direct measurement of growth rates. To examine the magnitude and possible causes of within-site growth variation, radial growth rates (RaGRs) of thalli of the fast-growing foliose lichen Melanelia fuliginosa ssp. fuliginosa (Fr. ex Duby) Essl. and the slow-growing crustose lichen Rhizocarpon geographicum (L.) DC. were studied on two S-facing slate rock surfaces in north Wales, UK using digital photography and an image analysis system (Image-J). RaGRs of M. fuliginosa ssp. fuliginosa varied from 0.44 to 2.63 mmyr-1 and R. geographicum from 0.10 to 1.50 mmyr-1.5. Analysis of variance suggested no significant variation in RaGRs with vertical or horizontal location on the rock, thallus diameter, aspect, slope, light intensity, rock porosity, rock surface texture, distance to nearest lichen neighbour or distance to vegetation on the rock surface. The frequency distribution of RaGR did not deviate from a normal distribution. It was concluded that despite considerable growth rate variation in both species studied, growth curves could be constructed with sufficient precision to be useful for direct lichenometry. © 2014 Swedish Society for Anthropology and Geography.

Individual variability analysis of fluorescence parameters measured in skin with different levels of nutritive blood flow

Fluorescence spectroscopy has recently become more common in clinical medicine. However, there are still many unresolved issues related to the methodology and implementation of instruments with this technology. In this study, we aimed to assess individual variability of fluorescence parameters of endogenous markers (NADH, FAD, etc.) measured by fluorescent spectroscopy (FS) in situ and to analyse the factors that lead to a significant scatter of results. Most studied fluorophores have an acceptable scatter of values (mostly up to 30%) for diagnostic purposes. Here we provide evidence that the level of blood volume in tissue impacts FS data with a significant inverse correlation. The distribution function of the fluorescence intensity and the fluorescent contrast coefficient values are a function of the normal distribution for most of the studied fluorophores and the redox ratio. The effects of various physiological (different content of skin melanin) and technical (characteristics of optical filters) factors on the measurement results were additionally studied.The data on the variability of the measurement results in FS should be considered when interpreting the diagnostic parameters, as well as when developing new algorithms for data processing and FS devices.

A laser induced breakdown spectroscopy quantitative analysis method based on the robust least squares support vector machine regression model

Data fluctuation in multiple measurements of Laser Induced Breakdown Spectroscopy (LIBS) greatly affects the accuracy of quantitative analysis. A new LIBS quantitative analysis method based on the Robust Least Squares Support Vector Machine (RLS-SVM) regression model is proposed. The usual way to enhance the analysis accuracy is to improve the quality and consistency of the emission signal, such as by averaging the spectral signals or spectrum standardization over a number of laser shots. The proposed method focuses more on how to enhance the robustness of the quantitative analysis regression model. The proposed RLS-SVM regression model originates from the Weighted Least Squares Support Vector Machine (WLS-SVM) but has an improved segmented weighting function and residual error calculation according to the statistical distribution of measured spectral data. Through the improved segmented weighting function, the information on the spectral data in the normal distribution will be retained in the regression model while the information on the outliers will be restrained or removed. Copper elemental concentration analysis experiments of 16 certified standard brass samples were carried out. The average value of relative standard deviation obtained from the RLS-SVM model was 3.06% and the root mean square error was 1.537%. The experimental results showed that the proposed method achieved better prediction accuracy and better modeling robustness compared with the quantitative analysis methods based on Partial Least Squares (PLS) regression, standard Support Vector Machine (SVM) and WLS-SVM. It was also demonstrated that the improved weighting function had better comprehensive performance in model robustness and convergence speed, compared with the four known weighting functions.

An Estimate of the Probability Pr(X

2000 Mathematics Subject Classification: 33C90, 62E99

Minimum Description Length Principle in Discriminating Marginal Distributions

2010 Mathematics Subject Classification: 94A17, 62B10, 62F03.

Beautiful, but dangerous

The normal distribution is a useful tool for the statistician, but not everyone knows how to wield it. In an extract from his new book, Chancing It, Robert Matthews explains what can happen when things are far from normal.

Corporate cash-pool valuation in a multi-firm context: a closed formula

Following our earlier paper on the subject, we present a general closed formula to value the interest savings due to a multi-firm cash-pool system. Assuming normal distribution of the accounts the total savings can be expressed as the product of three independent factors representing the interest spread, the number and the correlation of the firms, and the time-dependent distribution of the cash accounts. We derive analytic results for two special processes one characterizing the initial build-up period and the other describing the mature period. The value gained in the stationary system can be thought of as the interest, paid at the net interest spread rate on the standard deviation of the account. We show that pooling has substantial value already in the transient period. In order to increase the practical relevance of our analysis we discuss possible extensions of our model and we show how real option pricing technics can be applied here.

A methodology to estimate time varying user responses to travel time and travel time reliability in a road pricing environment

Road pricing has emerged as an effective means of managing road traffic demand while simultaneously raising additional revenues to transportation agencies. Research on the factors that govern travel decisions has shown that user preferences may be a function of the demographic characteristics of the individuals and the perceived trip attributes. However, it is not clear what are the actual trip attributes considered in the travel decision- making process, how these attributes are perceived by travelers, and how the set of trip attributes change as a function of the time of the day or from day to day. In this study, operational Intelligent Transportation Systems (ITS) archives are mined and the aggregated preferences for a priced system are extracted at a fine time aggregation level for an extended number of days. The resulting information is related to corresponding time-varying trip attributes such as travel time, travel time reliability, charged toll, and other parameters. The time-varying user preferences and trip attributes are linked together by means of a binary choice model (Logit) with a linear utility function on trip attributes. The trip attributes weights in the utility function are then dynamically estimated for each time of day by means of an adaptive, limited-memory discrete Kalman filter (ALMF). The relationship between traveler choices and travel time is assessed using different rules to capture the logic that best represents the traveler perception and the effect of the real-time information on the observed preferences. The impact of travel time reliability on traveler choices is investigated considering its multiple definitions. It can be concluded based on the results that using the ALMF algorithm allows a robust estimation of time-varying weights in the utility function at fine time aggregation levels. The high correlations among the trip attributes severely constrain the simultaneous estimation of their weights in the utility function. Despite the data limitations, it is found that, the ALMF algorithm can provide stable estimates of the choice parameters for some periods of the day. Finally, it is found that the daily variation of the user sensitivities for different periods of the day resembles a well-defined normal distribution.

Hazard mitigation element quality in coastal comprehensive plans in a state with strong requirements for hazard mitigation plan elements

This dissertation examines the quality of hazard mitigation elements in a coastal, hazard prone state. I answer two questions. First, in a state with a strong mandate for hazard mitigation elements in comprehensive plans, does plan quality differ among county governments? Second, if such variation exists, what drives this variation? My research focuses primarily on Florida's 35 coastal counties, which are all at risk for hurricane and flood hazards, and all fall under Florida's mandate to have a comprehensive plan that includes a hazard mitigation element. Research methods included document review to rate the hazard mitigation elements of all 35 coastal county plans and subsequent analysis against demographic and hazard history factors. Following this, I conducted an electronic, nationwide survey of planning professionals and academics, informed by interviews of planning leaders in Florida counties. I found that hazard mitigation element quality varied widely among the 35 Florida coastal counties, but were close to a normal distribution. No plans were of exceptionally high quality. Overall, historical hazard effects did not correlate with hazard mitigation element quality, but some demographic variables that are associated with urban populations did. The variance in hazard mitigation element quality indicates that while state law may mandate, and even prescribe, hazard mitigation in local comprehensive plans, not all plans will result in equal, or even adequate, protection for people. Furthermore, the mixed correlations with demographic variables representing social and disaster vulnerability shows that, at least at the county level, vulnerability to hazards does not have a strong effect on hazard mitigation element quality. From a theory perspective, my research is significant because it compares assumptions about vulnerability based on hazard history and demographics to plan quality. The only vulnerability-related variables that appeared to correlate, and at that mildly so, with hazard mitigation element quality, were those typically representing more urban areas. In terms of the theory of Neo-Institutionalism and theories related to learning organizations, my research shows that planning departments appear to have set norms and rules of operating that preclude both significant public involvement and learning from prior hazard events.

Resiliência e apoio social em gestantes tardias

The gestation process, in general, is a very important event on a woman’s life and it brings phisical, phisiological and emotional changes, which by itself is an experience full of intense feelings. By late-aged pregnancy we mean those which occurs at the age of 35 or further. The occurance of this type of pregnancy is rising in Brasil and throughout the world, factors such as, better access to birth control resources and the search for financial stability explains the pregnancy delay. Important processes like resilience and social support can help late-aged pregnant women, in a benefical way, to adapt to the gestation process. Resilience is the capacity that a certain individual or group of individuals have to go through an adverse situation, be able to overcome it and become streghtened, transforming it in motivation for its biopsichosocial development. Social support is a complex and dinamic process that involves transactions between individuals and their social networks, meeting the social needs, promoting and complementing the personal resources that they have to face new demands. This research has the intention of raising information about the issues of late-aged pregnant women in the County of Natal- RN, the main objective was to evaluate the resilience indicators and the social support on late-aged pregnant women in the Natal-RN County. A transversal cut, correlational and descriptive research that was done with 150 lateaged pregnant women. The tools that were used were: A form with sociodemographic and gestation info, the scale of resilience and social support. An eletronic spreadsheet sofware (Excel e SPSS 21.0) was used to analize data which helped on the statistics according to its variables and the objective of this work. For the nominal variables, relative frequencies were used and for continuous the Pearson correlation and determination coefficient were used, regarding that; the sample had a normal distribution. The project fulfilled the ethnic aspects prescribed by Resolution 466/12 of the National Health Council, with a favorable decision (356.436/ 2013) of the UFRN Ethics on Research Committee. Most of the pregnant women had a low money income and education level, born in the state of Rio Grande do Norte they had an average age of 37,49 (±2,577), catholic, married, house wives, they had more than one child and were on their third trimester of pregnancy; they also had a low past abortion rate, not having planned their pregnancy, with an average of 4,22 (±2,506) pre-natal appointments, residing with an average of 3,673 (±1,397) people, having used any sort of birth control device and having high indicators of resilience and social support. The correlations kept between resilience, social support and some of the social demographics and gestation variables were considered low. Such data points out the fact that most of these women were in a stable relationship; they hadn’t had a past of abortion, they were involved with some kind of religion, they were not first pregnancy mothers, had an age on which they are not considered inexperienced mothers and even had scored high on the social support scale, these may all possibly be the most contributing factors on development and resilience building on these 35 years or more mothers. We expect that the data and information from this research may add up knowledge, actions and improvements regarding late-aged pregnant women and the pregnancy phenomena in general.