Correlation between tensile and indentation behavior of particle-reinforced metal matrix composites

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Correlation

CorrelationXu JiajinNational Research Center for Foreign Language Education Beijing Foreign Studies University2Key points•Why correlation?•What is correlation analysis about?•How to make a correlation analysis?–Case studiesWhy Correlation?4Three things that stats can do •1.Summarizing univariate data •2.Testing the significance of differences •3.Exploring relationships b/t variables5Three things that stats can do •1.Summarizing univariate data •2.Testing the significance of differences •3.Exploring relationships b/t variables6探究事物之间的关联•植物的生长是否浇水的多少有关系，有多大关系•足球成绩好坏是否与身体（体质、人种）有关？•兴趣高、成绩好•元认知策略使用越多，学习进步越快•学好统计学有利于身体健康Key ides of correlationanalysis8•Correlation: co ‐relation . The co ‐relation is represented by a ‘correlation coefficient , r .•The range of the coefficient: ‐1to 1.•Three critical values: ‐1, 0and 1.Strength of correlationPositive correlation Strength of correlation Direction of correlationDirection of correlation Positive correlationDirection of correlation•Less Negative correlation12Two main types of correlation•Pearson : standard type, suitable for interval data (e.g. score, freq.)•Pearson r coefficient•Spearman : suitable for ordinal/rank data•Spearman rho coefficient13Significance•Similar to t ‐test and ANOVA statistics, the correlation coefficients need to be statistically significant.< .05Sig./P 值/alpha （α）值Coefficient of Determination r Ær2Æ% of variance explained15Coefficient of Determination •The squared correlation coefficient is called the coefficient of determination .•Multiplied by 100, this proportion of variance indicates the percentage of variance that is accounted for.•Correlation coefficients of .30 account for about 9% of the variance. Correlation of .70 explains about 49% of variance.Effect sizeCase Study 1Is connector use by Chinese EFL learners correlated with theirwriting quality?SPSS ProceduresAnalyze‐Correlate‐Bivariate1921Reporting correlations•In correlation tables/matrices •Embedded in textCorrelation tablesCorrelation tables(Dörnyei2007: 227)2324Embedded in text •As one would expect from the extensive literature documenting the benefits of intrinsic motivation, there was a significant positive correlation between overall GPA and intrinsic motivation (r = .34, p < .oo1).(Dörnyei 2007: 227)Practice: CET4 and CET6 Correlational analysisHomework英语成绩是否与语文成绩有相关性？28Wrap Up & Look Forward •Correlation coefficients provide a way to determine the strength & the direction of the relationship b/t two variables.•This index does not ... demonstrate a causal association b/t two variables.29Wrap Up & Look Forward •The coefficient of determination determines how much variance in one variable is explained by another variable.•Correlation coefficients are the precursors to the more sophisticated statistics involved in multiple regression (Urdan 2005: 87).30Thank you32。

Temporal correlations of orientations

Neurocomputing52–54(2003)117–123/locate/neucom Temporal correlations of orientationsin natural scenesChristoph Kayser∗,Wolfgang Einh a user,Peter K o nigInstitute of Neuroinformatics,ETH/UNI Z u rich,Winterthurerstrasse190,8057Z u rich,SwitzerlandAbstractThe visual system performs complicated operations such as visual grouping e ciently on its natural input.To study this adaptation to natural stimuli we measure spatio-temporal interactions of orientations in scenes with natural temporal structure recorded using a camera mounted to a cat’s head.Weÿnd long range spatial and long lasting temporal correlations of orientations with collinear interactions being most prevalent and preserved over time.The spatial extent of correlations corresponds to the length of horizontal cortical connections and the temporal duration of the interactions allows co-activation of lateral and bottom up input by the same visual event.c 2003Elsevier Science B.V.All rights reserved.Keywords:Natural scenes;Orientations;Gestalt principles;Image statistics;Temporal coherence1.IntroductionIn recent years processing of natural stimuli by the visual system received increased attention(cf.Ref.[12]).Indeed it was found that early stages of visual processing are speciÿcally adapted to the structure of natural scenes[1,4].Furthermore,laws for object perception and visual grouping,the Gestalt rules[8,14],can be linked to the statistics of natural scenes.As an example the law of good continuation,favouring collinear arrangements of orientations over parallel,was shown to have a counterpart in the interaction of orientations in still images[7,9,11].Similar interactions of ori-entations are also found in contextual e ects in psychophysical experiments[6,10], in surround interactions in V1receptiveÿelds[6]and in lateral connections in V1 [2,5].Therefore it is of particular interest to link them to properties of natural scenes.∗Corresponding author.Tel.:+41-1-6353044;fax:+41-1-6353053.E-mail address:kayser@ini.phys.ethz.ch(C.Kayser).0925-2312/03/$-see front matter c 2003Elsevier Science B.V.All rights reserved.doi:10.1016/S0925-2312(02)00789-0118 C.Kayser et al./Neurocomputing52–54(2003)117–123To our knowledge,however,up to now correlations in natural scenes have only been investigated in still images.This neglects the temporal structure and it remains un-clear whether these correlations persist on time scales relevant for lateral interactions in the cortex.Given the possibly long delays for tangential connections,correlations must extend over substantial temporal periods in order to fully cover the spatial extent of long-range connections.Furthermore,some of the previous studies did not report ÿlter or correlation scales in units of degrees of visual angle leaving possible links to anatomical scales uncertain.Finally,some of the previous studies used still images captured by humans,therefore introducing a possible artistic or anthropocentric bias. Here we address these issues and study spatio-temporal interactions of orientations in a large database of natural movies captured by a camera mounted to a cat’s head.2.MethodsWe recorded movie sequences using a removable lightweight CCD-camera(Conrad electronics,Hirschau,Germany)mounted to the head of cats while taken for walks in di erent local environments like grassland,forest and the university campus.These videos contain a large variety of di erent speeds and accelerations as a result of the natural movements of the cat.Fig.1shows four sample images of our database.For this study a total of three animals was used and all procedures are in agreement with national and institutional guidelines for animal care.Videos were recorded via a cable connected to the leash onto a standard VHS-VCR (Pal)carried by the human experimenter and digitised o ine at a temporal resolution of25Hz,320×240pixels(1pixels≈12min of arc)and16bit color depth.For this study videos were converted to8-bit gray scale and12sequences(about40000frames total)were used.Before further processing the images were normalized to zero mean. The image statistics was investigated using oriented wavelets.Single frames were convolved with pairs of circular Gabor wavelets of90◦relative phase shift.Filters had a envelope of20pixel width and a spatial frequency of7(1/pixels).The amplitude of the orientation was computed by summing the squared amplitudes of two phaseFig.1.Four sample frames of our database are shown on the left.The amplitudes of the oriented energy detectors for the same frames is shown on the right.The bards indicate the orientation of the respective ÿlters used.C.Kayser et al./Neurocomputing 52–54(2003)117–123119shifted ÿlters and subjecting the result to a square root,resembling a two subunit en-ergy model.At each point the amplitudes of eight equally spaced orientations from 0◦(horizontal)to 157:5◦were computed.We deÿne the ‘prominent’orientation of each point by averaging the amplitude vectors (length =amplitude of ÿlter response ;orientation =orientation of the ÿlter)of the eight ÿlters.The resulting vector average has an orientation ,deÿning the prominent orientation of the point,and a length A ( ;x;t ),specifying the magnitude of the local orientation strength.For computa-tional convenience these orientations were binned into 16bin between 0◦and 180◦.The second order statistics of these orientations was calculated assuming transla-tion invariance of natural images.Thus correlations of two prominent orientations 1and 2were computed over all pairs of points with the same spatial separation x and temporal separation t (the mean runs over all points (x;t )with prominent orientation 1).C ( 1; 2; x; t )= (A ( 1;x;t )− A ( 1;x;t ) )∗(A ( 2;x + x;t + t )− A ( 2;x + x;t + t ) ) (A ( 1;x;t )− A ( 1;x;t ) )2 ∗ (A ( 2;x + x;t + t )− A ( 2;x + x;t + t ) )2 :Correlations were computed for temporal lags from t =0to 30frames (1:2s)and on a spatial grid of points spaced about 2◦apart.Therefore the kernels overlapped only for the smallest spatial distance used.As a control we also computed correlations using the maximally active orientation at each point instead of the ‘prominent’orientation yielding similar results as reported below.3.ResultsFirst we investigate temporal correlations at the same point in space.Fig.2A demon-strates that if an orientation is present at one point in time then the amplitude of this orientation in the next frames at the same point is also likely to be high.Temporal correlations are strongest for the cardinal orientations,i.e.horizontal and vertical.For the other orientations correlations decay faster but are still signiÿcant over several hun-dreds of milliseconds (decay time constants for 0◦:¿1s,45◦:490ms,90◦:900ms,135◦:360ms).Thus the presence of an oriented segment gives a strong prediction for the orientation at the same point later in time.Next we look at the two dimensional spatial distribution of correlations as well as correlations of di erent orientations.Fig.2B shows the correlations between segments of 4di erent orientations (0◦;45◦;90◦;135◦)situated at di erent relative locations in the same frame.Iso-orientation correlations (panels on the diagonal)are stronger than cross-orientation correlations.Furthermore the contour lines of the iso-orientation correlations are elongated along the direction of the particular orientation.This shows that collinear structures are more prevalent than parallel shifted contours.Also parallel contours occur more likely than T-junctions since the iso-orientation correlations are at all points stronger than the correlations of this orientation with the orthogonal.An example of how the spatial correlations decay independently of the spatial direction is120 C.Kayser et al./Neurocomputing52–54(2003)117–123Fig.2.(A)The correlation of orientation amplitude over time at the same pixel.Squares:0◦,stars:90◦, dashed:all other orientations(spaced22:5◦).(B)Correlations of di erent combinations of orientations and di erent spatial arrangements of the two points in the same frame.The orientations are(from top to bottom and left to right):0◦;135◦;90◦;45◦.(C)Correlations over time of points with prominent horizontal orientation but which are spatially separated by di erent distances independent of the relative orientation. Squares:2.1deg spatial distance,stars:4:2◦,dashed:6:4◦,diamonds:8:4◦.(D)Same as in B but here the two points are also separated by400ms in time.shown in Fig.2C for the horizontal(90◦)orientation.Correlations decay fastest during theÿrst2◦of spatial distance but extend well up to8◦.Our data set allows analyzing how these spatial correlations evolve over time.Fig.2D shows the same data as in Fig.2B but for segments400ms apart in time.The spatial arrangement of correlations is the same as for zero time lag but the amplitudes decayed by a factor higher than2.For the cardinal orientations again collinear interactions are prevalent.This is in agreement with Fig.2A which shows that these orientations are very stable over time.Since the oblique orientations are less well correlated over time we would expect that collinearity will here be less prominent for larger time lags.Indeed the contour lines of the correlations for the oblique orientations are more circular symmetric.To quantify these changes over time we measure the aspect ratio (length/width)of the contour lines for the di erent time lags.Collinearity means a high aspect ratio and a loss of collinearity therefore is accompanied with a decreaseC.Kayser et al./Neurocomputing52–54(2003)117–123121Fig.3.(A)Relative change of the aspect ratio of the correlation contours in Fig.2C as a function of time. Shows is the aspect ratio at each point in time divided by the aspect ratio at t=0.Squares:0◦,stars:90◦, solid:45◦,dashed:135◦.(B)Areas of strong correlations.We deÿned spatio temporal separations with a correlation over0.4as strong.Theÿgure shows these areas for the correlation diagram of Fig.2C.(C) Shows the size of these areas relative to the total patch size over time.Lines are labeled as in A.in aspect ing this measure,Fig.3A shows that collinearity is preserved over long temporal lags and is strongest for the cardinal orientations.To quantify the change in amplitude of the spatial correlations in a di erent way,we deÿne areas of strong interactions by thresholding correlations.We chose a threshold of0.4to ensure that even for zero time-lag only iso-orientation correlations exceed this threshold(Fig.3B).As expected from Fig.2the decay times are slowest for the cardinal orientations but independent of the orientation there exist points with strong correlations for at least280ms(Fig.3C).We performed controls to see how these results depend on the amount of data used. The above data were averaged over our whole database.Since one feature of our video sequences is their variety in terms of landscapes,etc.we look at the di erences between di erent sequences.In Fig.4we show the correlations for one oblique orien-tation(135◦).The mean and standard deviation over12video sequences is shown in Fig.4A.The error is rather small compared to the correlation values.More importantly, the correlation surface plus minus the error(Fig.4B)shows the same spatial structure as the mean.Also,the distinct pattern of correlations is visible in averages over shorter sequences(data not shown).Thus the distinct patterns of spatial correlations are not introduced by averaging over a large data set.As a further control,we useÿlters of a di erent spatial scale and frequency to measure the orientation content.Theÿlters used for Fig.4C are twice as large as the ones used for the other experiments.The results are basically the same as with the lower frequencyÿlters.Again collinearity is most prevalent.Therefore our results generalize over a wide range ofÿlter parameters.122 C.Kayser et al./Neurocomputing52–54(2003)117–123Fig.4.(A)For an example orientation(45◦)we show the mean(left)over12video sequences together with the standard deviation(right).(B)The mean plus=minus the standard deviation.(C)Cross orientation correlations over space forÿlters of a higher spatial frequency.4.DiscussionWe recorded natural image sequences from a camera mounted to a cat’s head closely matching the animal’s visual input.Thereby our database circumvents possible artistic or anthropocentric biases introduced in pictures and movies taken by humans.The database contains a large set of di erent environments,ranging from forest to grasslands and university campus.Furthermore the used sequences were recorded in di erent seasons and times of day providing a huge variety of lighting conditions.In respect to the temporal analysis it is worth noting that our video sequences contain natural movements of an animal,which might di er considerably from mercial movies ÿlmed by humans.In qualitative agreement with previous studies[7,9,11]weÿnd spatial correlations corresponding to the Gestalt laws.For all orientations collinear contours are more prevalent than parallel contours and correlations between orthogonal orientations are weakest.However weÿnd correlations over distances of up to8degrees of visual angle(Fig.2).This is considerably larger than distances reported in previous studies. For example Kaschube et al.[7]ÿnd that already for small distances correlations are relatively weak(¡0:15in a range from1◦to4◦).However,they do not indicate the size of their kernels in the same units.Sigman et al.[11]report similar correlations usingÿlters of size smaller than10min of arc.Our higher correlation could be due to methodological di erences to other studies besides the use of di erent and possibly larger kernels.We computed the‘prominent’orientation of a point by vector averaging the outputs of8oriented energy detectors.But correlations computed on these promi-nent orientations are very similar to correlations computed on the maximally active orientation(data not shown)a method used in Ref.[11].The spatial distances of the correlations reported hereÿt well with anatomical data on long-range horizontal connections in primary visual cortex.In cat V18◦of visualC.Kayser et al./Neurocomputing52–54(2003)117–123123 angle correspond roughly to8mm[13].This is also the extent of long-range connec-tions which preferentially connect iso-orientation domains[5]and in some mammals preferentially mediate collinear interactions[2].In the temporal domain weÿnd long lasting correlations of orientations to extend several hundreds of milliseconds preserving their spatial structure i.e.collinearity.These persist su ciently long to allow bottom up and long range lateral input to be coactive and driven by the same orientated structure even given the slow speeds of lateral con-nections reported in Ref.[3].Therefore the spatio-temporal interactions of orientations seem to fully cover the range of tangential connections and provide a substrate that could also guide the development of orientation maps and long-range connections in primary visual cortex.AcknowledgementsThis work wasÿnancially supported by the Centre of Neuroscience Zurich, (ZNZ),Honda R&D Europe(Germany)and the Swiss national fund(SNF grant No.31-65415.01).References[1]J.J.Atick,A.N.Redlich,What does the retina know about natural scenes,Neural Comput.4(1992)196–210.[2]W.H.Bosking,Y.Zhang,B.Schoÿeld,D.Fitzpatrick,Orientation selectivity and the arrangement ofhorizontal connections in tree shrew striate cortex,J.Neurosci.17(6)(1997)2112–2127.[3]V.Bringuier,F.Chavane,L.Glaeser,Y.Fregnac,Horizontal propagation of visual activity in thesynaptic integrationÿeld of area17neurons,Sci.283(1999)695–699.[4]Y.Dan,J.J.Atick,R.C.Reid,E cient coding of natural scenes in the lateral geniculate nucleus:experimental test of a computational theory,J.Neurosci.16(10)(1996)3351–3362.[5]C.D.Gilbert,T.N.Wiesel,Columnar speciÿcity of intrinsic horizontal and corticocortical connectionsin cat visual cortex,J.Neurosci.9(7)(1989)2432–2442.[6]M.K.Kapadia,M.Ito,C.D.Gilbert,G.Westheimer,Improvement in visual sensitivity by changesin local context:Parallel studies in human observers and in V1of alert monkeys,Neuron15(1995) 843–856.[7]M.Kaschube,F.Wolf,T.Geisel,S.L o wel,The prevalence of collinear contours in the real world,Neurocomputing38–40(2001)1335–1339.[8]K.Kofka,Principles of Gestalt Psychology,Harcourt&Brace,New York,1935.[9]N.Krueger,Collinearity and parallelism are statistically signiÿcant second order relations of complexcell responses,Neural Process.Lett.8(1998)117–129.[10]U.Polat,D.Sagi,Spatial interactions in human vision:from near to far via experience dependentcascades of connections,A91(1994)1206–1209.[11]M.Sigman,G.A.Cecchi,C.D.Gilbert,M.O.Magnasco,On a common circle:natural scenes and Gestaltrules,Proc.Natl.Acad.Sci.98(4)(2001)1935–1940.[12]E.P.Simoncelli,B.A.Olshausen,Natural image statistics and neural representation,Ann.Rev.Neurosci.24(2001)1193–1215.[13]R.J.Tusa,A.C.Rosenquist,L.A.Palmer,Retinotopic organization of areas18and19in the cat,p.Neurol.185(1979)657–678.[14]M.Wertheimer,Laws of Organization in the Perceptual Form,Harcourt&Brace,USA,1938.。

decay of correlation 数学名词

decay of correlation 数学名词Decay of correlation（相关性的衰减）refers to the decrease in correlation between two variables as the distance between them increases. It is a mathematical concept used to quantify the relationship between two variables across different spatial or temporal distances.1. The decay of correlation between rainfall and crop yield was observed as the distance between the two fields increased.雨量与农作物产量之间的相关性随着两个田地之间的距离增加而减弱。

2. The study analyzed the decay of correlation between interest rates and stock market performance over a one-year timespan.该研究分析了利率和股市表现之间的相关性在一年的时间内是如何衰减的。

3. As the distance between two cities increased, thedecay of correlation between their population sizes became more noticeable.随着两个城市之间的距离增加，它们的人口规模之间的相关性衰减变得更加明显。

4. The researchers used statistical methods to determine the decay of correlation between air pollution andrespiratory diseases in different neighborhoods.研究人员使用统计方法来确定不同社区之间空气污染和呼吸道疾病之间的相关性衰减。

correlation and regression

Calculate the correlation coefficient for the above data

The correlation between two random variables X and Y is a measure of the degree of linear association between the two variables.
Two variables are highly correlated if they move well together. Correlation is indicated by the correlation coefficient. The population correlation coefficient is denoted by ρ. The coefficient ρ can take on any value from -1 to +1. The possible values of ρ and their interpretations are given below. 1. When ρ is equal to zero, there is no correlation. That is, there is no linear relationship between the two random variables.
The range of the correlation coefficient is to 1, inclusive. If x and y have a strong positive linear correlation, r is close to 1. If x and y have a strong negative linear correlation, r is close to If x and y have perfect positive linear correlation or perfect negative linear correlation, r is equal to 1 or , respectively. If there is no linear correlation or a weak linear correlation, r is close to 0. It is important to remember that if r is close to 0, it does not mean that there is no relation between x and y, just that there is no linear relation. Several examples are shown.

协方差

1
18.05 class 7, Covariance and Correlation, Spring 2014
2
2.2
Sums and integrals for computing covariance
Since covariance is deﬁned as an expected value we compute it in the usual way as a sum or integral. Discrete case: If X and Y have joint pmf p(xi , yj ) then
n m
n
m
p(xi , yj )xi yj − µX µY .CovFra bibliotekX, Y ) =
i=1 j =1
p(xi , yj )(xi − µX )(yj − µY ) =
i=1 j =1
Continuous case: If X and Y have joint pdf f (x, y ) over range [a, b] × [c, d] then
18.05 class 7, Covariance and Correlation, Spring 2014
3
So Cov(XY ) = E (XY ) − µX µY =
5 1 −1= . 4 4 Next we redo the computation of Cov(X, Y ) using the properties of covariance. As usual, let Xi be the result of the ith ﬂip, so Xi ∼ Bernoulli(0.5). We have X = X1 + X2 and Y = X2 + X3 . We know E (Xi ) = 1/2 and Var(Xi ) = 1/4. Therefore using Property 2 of covariance, we have Cov(X, Y ) = Cov(X1 +X2 , X2 +X3 ) = Cov(X1 , X2 )+Cov(X1 , X3 )+Cov(X2 , X2 )+Cov(X2 , X3 ). Since the diﬀerent tosses are independent we know Cov(X1 , X2 ) = Cov(X1 , X3 ) = Cov(X2 , X3 ) = 0. Looking at the expression for Cov(X, Y ) there is only one non-zero term Cov(X, Y ) = Cov(X2 , X2 ) = Var(X2 ) = 1 . 4

correlation 标准流程

correlation 标准流程英文回答：Correlation.Correlation is a statistical measure that expresses the extent to which two variables are linearly related. It is a value between -1 and 1, where -1 indicates a perfect negative correlation, 0 indicates no correlation, and 1 indicates a perfect positive correlation.The correlation coefficient is calculated by dividing the covariance of the two variables by the product of their standard deviations. The covariance is a measure of how much the two variables vary together, and the standard deviation is a measure of how much each variable varies on its own.Correlation is a useful tool for understanding the relationship between two variables. It can be used toidentify trends, make predictions, and test hypotheses. However, it is important to note that correlation does not imply causation. Just because two variables are correlated does not mean that one causes the other.Types of Correlation.There are three main types of correlation:Positive correlation: This type of correlation occurs when two variables increase or decrease together. For example, the number of hours you study for a test and your score on the test are positively correlated.Negative correlation: This type of correlation occurs when one variable increases and the other variable decreases. For example, the amount of money you spend ongas and your car's gas mileage are negatively correlated.No correlation: This type of correlation occurs when there is no relationship between two variables. For example, the number of times you flip a coin and the number of headsyou get are not correlated.Strength of Correlation.The strength of a correlation is determined by the absolute value of the correlation coefficient. The closer the correlation coefficient is to 1 or -1, the stronger the correlation. A correlation coefficient of 0 indicates that there is no correlation between the two variables.Significance of Correlation.The significance of a correlation is determined by the p-value. The p-value is the probability of obtaining a correlation coefficient as large as or larger than the one that was observed, assuming that there is no correlation between the two variables. A p-value less than 0.05 is considered to be statistically significant.Correlation Analysis.Correlation analysis is a statistical technique that isused to identify and measure the relationship between two or more variables. Correlation analysis can be used to:Identify trends.Make predictions.Test hypotheses.Control for confounding variables.Correlation analysis is a valuable tool for understanding the relationships between variables. However, it is important to note that correlation does not imply causation.中文回答：相关性。

惭愧，今天才注意到统计上的关联（association）与相关（corelation）是不同的

惭愧，今天才注意到统计上的关联（association）与相关（corelation）是不同的虽然教书多载，以前一直以为关联和相关为同一个意思，只不过国人翻译的不同，今日总觉得哪里不对，于是乎一探究竟，发现两者差别还真是挺大的。

英文原文如下，松哥就不翻译了，怕又翻出歧义来，大家看看吧！以前分析一直忽略下图中红框部分，看完今天的推送，你就能明白下图中那么多选项的意义了！如果您还没明白，也别急，松哥正在撰写《统计思维与SPSS24.0实战解析》，里面会有详细的，全新的解读哦！精鼎35-36期SPSS高级研习班开班通知：（详情点击）精鼎35期（合肥）-36期（昆明）全国SPSS研习班报名啦！/Association vs CorrelationAssociation and correlation are two methods of explaining a relationship between two statistical variables. Association refers to a more generalized term and correlation can be considered as a special case of association, where the relationship between the variables is linear in nature.What is Association?The statistical term association is defined as a relationship between two random variables which makes them statistically dependent. It refers to rather a general relationship without specifics of the relationship being mentioned, and it is not necessary to be a causal relationship.Many statistical methods are used to establish the association between two variables. Pearson’s correlation coefficient, odds ratio, distance correlation, Goodman’s and Kruskal’s Lambda and Spearman’s rho (ρ) are a few examples.What is Correlation?Correlation is a measure of the strength of the relationship between two variables. The correlation coefficient quantifies the degree of change of one variable based on the change of theother variable. In statistics, correlation is connected to the concept of dependence, which is the statistical relationship between two variablesThe Pearson’s correlation coefficient or just the correlation coefficient r is a value between -1 and 1 (-1≤r≤+1). It is the most commonly used correlation coefficient and valid only for a linear relationship between the variables. If r=0, no relationship exist, and if r≥0, the relation is directly proportional; the value of one variable increases with the increase in the other. If r≤0, the relationship is inversely proportional; one variable decreases as the other increases.Because of the linearity condition, correlation coefficient r can also be used to establish the presence of a linear relationship between the variables.Spearman’s rank correlation coefficient and Kendrall’s rank correlation coefficient measure the strength of the relationship, excluding the linear factor. They consider the extent one variable increases or decreases with the other. If both variables increase together the coefficient is going to be positive and if one variable increases while the other decreases the coefficient value is going to be negative.The rank correlation coefficients are used just to establish the type of the relationship, but not to investigate in detail like the Pearson’s correlation coefficient. They are also used to reduce the calculations and make the results more independent of the non-normality of the distributions considered.What is the difference between Association and Correlation?· Association refers to the general relationship between two random variables while the correlation refers to a more or less alinear relationship between the random variables.· Association is a concept, but correlation is a measure of association and mathematical tools are provided to measure the magnitude of the correlation.·Pearson’s product moment correlation coefficient establishes the presence of a linear relationship and determines the nature of the relationship (whether they are proportional or inversely proportional).· Rank correlation coefficients are used to determine the nature of the relationship only, excluding the linearity of the relation (it may or may not be linear, but it will tell whether the variables increase together, decrease together or one increases while the other decreases or vice versa).。

Wavelet Correlation Coefficient of 'strongly correlated' financial time series

and a translation parameter b (−∞ < b < ∞) such that
ψ(x − b)
Ψa, b = a
(2)
The wavelet domain of one dimensional function Ψ is rather two dimensional
in nature; one dimension corresponds to scale and other to translation. The
Abstract : In this paper, wavelet concepts are used to study two ’strongly correlated’
ﬁnancial time series. Apart from obtaining wavelet spectra, We also calculate wavelet correlation coeﬃcient and show that strong correlation or strong anticorrelation depends on scale.
in November 1994 and operations in derivatives in June 2000. NSE index also
known as NIFTY is determined from 50 stocks of companies taken from 23
sectors of economy. Figure 1 shows BSE index and NSE index for the year
arXiv:cond-mat/0303099v2 [cond-mat.stat-mech] 24 Oct 2003

correlation

correlationCorrelationIntroductionCorrelation is a statistical measure that determines the degree to which two variables are related to each other. It is an important concept in many fields, including statistics, economics, social sciences, and healthcare. In this document, we will explore the concept of correlation, its types, and its significance in various applications.What is Correlation?Correlation quantifies the statistical relationship between two variables. It measures how changes in one variable correspond to changes in another variable. Correlation is typically represented by the correlation coefficient, which ranges from -1 to +1. A positive correlation indicates a direct relationship, while a negative correlation indicates an inverse relationship. A correlation coefficient close to zero indicates a weak or no relationship between the variables.Types of CorrelationThere are three main types of correlation: positive correlation, negative correlation, and zero correlation.1. Positive Correlation: When two variables increase or decrease together, they are said to have a positive correlation. For example, there is a positive correlation between the amount of study time and test scores. As the study time increases, the test scores also tend to increase. The correlation coefficient for a positive correlation ranges from 0 to +1.2. Negative Correlation: In contrast to a positive correlation, a negative correlation exists when one variable increases while the other decreases. For instance, there is a negative correlation between the number of hours spent watching TV and academic performance. As the hours spent watching TV increase, the academic performance tends to decrease. The correlation coefficient for a negative correlation ranges from 0 to -1.3. Zero Correlation: Zero correlation, as the name suggests, implies no relationship between the variables. The changes in one variable do not correspond to any changes in the othervariable. When the correlation coefficient is close to zero, it indicates a weak or no correlation.Significance of CorrelationCorrelation has several practical applications in different fields.1. Statistics: Correlation analysis is used to determine the strength and direction of the relationship between variables. It helps statisticians to understand the patterns and trends in data. Correlation coefficients are widely used in regression analysis and predictive modeling.2. Economics: In economics, correlation analysis helps to identify relationships between different economic variables such as inflation and unemployment rates, interest rates and investment, or GDP and consumer spending. Understanding these relationships is essential for making informed economic decisions.3. Social Sciences: Correlation is used in social sciences to study various phenomena, such as the relationship between education and income, crime rates and poverty, or healthbehaviors and disease outcomes. Correlation can provide insights into social trends and patterns.4. Healthcare: Correlation plays a crucial role in healthcare research. It helps to identify risk factors, assess treatment effectiveness, and understand the relationship between lifestyle choices and health outcomes. For example, studying the correlation between smoking and lung cancer can help healthcare professionals develop effective prevention strategies.ConclusionCorrelation is a powerful statistical tool that measures the relationship between two variables. It helps us understand how changes in one variable relate to changes in another variable. By analyzing correlation coefficients, we can determine the strength and direction of the relationship. Correlation has wide-ranging applications in statistics, economics, social sciences, healthcare, and other fields. Understanding correlation is essential for making informed decisions and drawing meaningful conclusions from data.。

SAS系统和数据分析典型相关分析

第三十七课典型相关分析典型相关分析（Canonical Correlation Analysis ）是研究两组变量间相关关系的一种多元统计分析方法。

它能够揭示两组变量之间的内在联系，真正反映两组变量间的线性相关情况。

一、典型相关分析我们研究过两个随机变量间的相关，它们可以用相关系数表示。

然而，在实际中常常会遇到要研究两组随机变量间),,,(21p x x x 和),,,(21q y y y 的相关关系问题。

),,,(21p x x x 和),,,(21q y y y 可能是完全不同的，但是它们的线性函数可能存在密切的关系，这种密切的关系能反映),,,(21p x x x 和),,,(21q y y y 之间的相关关系。

因此，就要找出),,,(21p x x x 的一个线性组合u 及),,,(21q y y y 的一个线性组合v ，希望找到的u 和v 之间有最大可能的相关系数，以充分反映两组变量间的关系。

这样就把研究两组随机变量间相关关系的问题转化为研究两个随机变量间的相关关系。

如果一对变量（u ，v ）还不能完全刻画两组变量间的相关关系时，可以继续找第二对变量，希望这对变量在与第一对变量（u ，v ）不相关的情况下也具有尽可能大的相关系数。

直到进行到找不到相关变量对时为止。

这便引导出典型相关变量的概念。

1. 典型相关系数与典型相关变量设有两组随机变量),,,(21p x x x 和),,,(21q y y y ，假定它们都已经标准化了，即p i x D x E i i ,,2,1= ,1=)(,0=)( ，q i y D y E i i ,,2,1= ,1=)(,0=)( ，若记：⎪⎪⎪⎪⎪⎭⎫ ⎝⎛=⎪⎪⎪⎪⎪⎭⎫ ⎝⎛=p p y y y y x x x x 2121, 此时，它们的协方差矩阵（也是相关系数矩阵）为：R R R R R y x D yy xy yx xx =⎪⎪⎭⎫ ⎝⎛=⎪⎪⎭⎫ ⎝⎛ 其中，()()yx xy yy xx R R y x Cov R y D R x D ====),(,,实际上，我们要找：y m v x l u 1111,'='=使1u 和1v 的相关系数),(11v u ρ达到最大。

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New Evidence on Measuring Financial

New Evidence on Measuring FinancialConstraints:Moving Beyond the KZ Index Charles J.HadlockMichigan State UniversityJoshua R.PierceUniversity of South CarolinaWe collect detailed qualitative information from ﬁnancial ﬁlings to categorize ﬁnancial constraints for a random sample of ﬁrms from 1995to ing this categorization,we estimate ordered logit models predicting constraints as a function of different quantita-tive factors.Our ﬁndings cast serious doubt on the validity of the KZ index as a measure of ﬁnancial constraints,while offering mixed evidence on the validity of other common measures of constraints.We ﬁnd that ﬁrm size and age are particularly useful predictors of ﬁnancial constraint levels,and we propose a measure of ﬁnancial constraints that is based solely on these ﬁrm characteristics.(JEL G31,G32,D92)A large literature in corporate ﬁnance examines how various frictions in the process of raising external capital can generate ﬁnancial constraints for ﬁrms.Researchers have hypothesized that these constraints may have a substantial effect on a variety of decisions,including a ﬁrm’s major investment and cap-ital structure choices (e.g.,Hennessy and Whited 2007).Additional research suggests that ﬁnancial constraints may be related to a ﬁrm’s subsequent stock returns (e.g.,Lamont et al.2001).To study the role of ﬁnancial constraints in ﬁrm behavior,researchers are often in need of a measure of the severity of these constraints.The literature has suggested many possibilities,including investment–cash ﬂow sensitivities (Fazzari et al.1988),the Kaplan and Zingales (KZ)index of constraints (Lamont et al.2001),the Whited and Wu (WW)index of constraints (Whited and Wu 2006),and a variety of different sorting criteria based on ﬁrm characteristics.We describe these approaches in more detail below.While there are many possible methods for measuring ﬁnancial constraints,considerable debate exists with respect to the relative merits of each approach.This is not surprising,since each method relies on certain empirical and/or the-Prior versions of this article circulated under alternative titles.We thank Julian Atanassov,Sreedhar Bharath,Murillo Campello,Jonathan Carmel,Jonathan Cohn,Ted Fee,Jun-Koo Kang,Michael Mazzeo,Uday Rajan,David Scharfstein,Michael Weisbach,two anonymous referees,and seminar participants at George Mason,Michigan,North Carolina,Oregon,Pittsburgh,South Carolina,Texas,Texas Tech,and Wayne State for helpful comments.Tehseen Baweja and Randall Yu provided superb data assistance.All errors remain our own.Send correspondence to Charles J.Hadlock,Department of Finance,Michigan State University,315Eppley Center,East Lansing,MI 48824-1121;telephone:(517)353-9330.E-mail:hadlock@.c The Author 2010.Published by Oxford University Press on behalf of The Society for Financial Studies.All rights reserved.For Permissions,please e-mail:journals.permissions@.doi:10.1093/rfs/hhq009RFS Advance Access published March 1, 2010 at Wuhan University Library on March 12, 2010 Downloaded fromThe Review of Financial Studies/v00n002010oretical assumptions that may or may not be valid.In addition,many of these methods rely on endogenousﬁnancial choices that may not have a straightfor-ward relation to constraints.For example,while an exogenous increase in cash on hand may help alleviate the constraints that a givenﬁrm faces,the fact that aﬁrm chooses to hold a high level of cash may be an indication that theﬁrm is constrained and is holding cash for precautionary reasons.In this article,we studyﬁnancial constraints by exploiting an approachﬁrst advocated by Kaplan and Zingales(1997).In particular,we use qualitative in-formation to categorize aﬁrm’sﬁnancial constraint status by carefully reading statements made by managers in SECﬁlings for a sample of randomly selected ﬁrms from1995to2004.1This direct approach to categorizingﬁnancial con-straints is not practical for large samples,since it requires extensive hand data collection.However,by studying the relation between constraint categories and variousﬁrm characteristics,we can make inferences that are useful for thinking about how to measureﬁnancial constraints in larger samples.We exploit our qualitative data onﬁnancial constraints for two purposes. First,we critically evaluate methods commonly used in the literature to mea-sureﬁnancial constraints.We pay particular attention to the KZ index,given its relative prominence in the literature and the fact that our data are particularly useful for evaluating this measure.Second,after examining past approaches, we propose a simple new approach for measuring constraints that has substan-tial support in the data and considerable intuitive appeal.We then subject thisnew measure to a variety of robustness checks.To evaluate the KZ index,we estimate ordered logit models in which aﬁrm’s categorized level of constraints is modeled as a function ofﬁve Compustat-based variables.This modeling approach parallels the analysis of Lamont et al. (2001),who create the original KZ index by estimating similar models using the original Kaplan and Zingales(1997)sample.The KZ index,which is based on the estimated coefﬁcients from one of the Lamont,Polk,and Saa-Requejo models,loads positively on leverage and Q,and negatively on cashﬂow,cash levels,and dividends.In the ordered logit models we estimate,only two of theﬁve components of the KZ index,cashﬂow and leverage,are consistently signiﬁcant with a sign that agrees with the KZ index.For two of the otherﬁve components, Q and dividends,the coefﬁcientsﬂip signs across estimated models and in many cases are insigniﬁcant,particularly for the dividend variable.Finally,in contrast to its negative loading in the KZ index,weﬁnd that cash holdings generally display a positive and signiﬁcant coefﬁcient in models predicting constraints.This positive relation is consistent with constrainedﬁrms holding cash for precautionary reasons.1The information we use includes statements regarding the strength of aﬁrm’s liquidity position and theﬁrm’s ability to raise any needed external funds.Additional details are provided below.2 at Wuhan University Library on March 12, 2010 Downloaded fromNew Evidence on Measuring Financial ConstraintsOur estimates differ substantially from the KZ index coefﬁcients even though we use a parallel modeling approach.Upon further investigation,we ﬁnd that the differences most likely arise from the fact that the dependent variable in the original modeling underlying the KZ index includes quantita-tive information in addition to qualitative information.This treatment adds a hard-wired element to the estimates underlying the KZ index,since the same information is mechanically built into both the dependent and the independent variables.In our treatment,we are careful to avoid this problem.Once this problem is addressed,ourﬁndings indicate that many of the estimated coefﬁcients change substantially.Clearly our evidence raises serious questions about the use of the KZ index. To explore this issue further,we calculate the KZ index for the entire Com-pustat universe and compare this to an index constructed using the coefﬁcient estimates from one of our models.Weﬁnd that the correlation between the tra-ditional index and our alternative version of this index is approximately zero. This provides compelling evidence that the KZ index is unlikely to be a useful measure ofﬁnancial constraints.Thus,it would appear that researchers should apply extreme caution when using the traditional KZ index or interpreting re-sults based on index sorts.An alternative index ofﬁnancial constraints has been proposed by Whited and Wu(2006),who exploit a Euler equation approach from a structural model of investment to create the WW index.This index loads on six different factorscreated from Compustat data.When we use these six factors as explanatory variables in ordered logit models predicting constraints,only three of the six variables have signiﬁcant coefﬁcients that agree in sign with the WW index. Two of these variables,cashﬂow and leverage,are essentially the same vari-ables thatﬁgure prominently in the KZ index.Thus,the only truly new variable from the WW index that offers marginal explanatory power in our models is ﬁrm size.As one would expect,smallerﬁrms are more likely to be constrained.A more traditional approach to identifyingﬁnancially constrainedﬁrms is to sort by aﬁrm characteristic that is believed to be associated with constraints. To evaluate this approach,we study the relation between several common sort-ing variables and ourﬁnancial constraint categories.Weﬁnd that some of these sorting variables are not signiﬁcantly related to constraint categories.Two vari-ables that do appear to be closely related toﬁnancial constraints areﬁrm size and age.An appealing feature of these variables is that they are much less en-dogenous than most other sorting variables.Once we control forﬁrm size and age,some of the variables that are signiﬁcantly related to constraints in a uni-variate sense become insigniﬁcant.Thus,it appears that some common sorting variables are largely proxies forﬁrm size and/or age.The only variables that consistently predict aﬁrm’s constraint status in our sample after controlling for size and age are aﬁrm’s leverage and cash ﬂow.However,given the endogenous nature of these variables,particularly the leverage variable,we are hesitant to recommend any measure of constraints3 at Wuhan University Library on March 12, 2010 Downloaded fromThe Review of Financial Studies/v00n002010that is derived from a model that relies on these factors.In addition,as we explain below,typical disclosure practices may lead us to under-detect the presence of constraints inﬁrms with low leverage,thus possibly leading to a spurious coefﬁcient on leverage.Given these concerns,we recommend that researchers rely solely onﬁrm size and age,two relatively exogenousﬁrm characteristics,to identify constrainedﬁrms.To provide further guidance on the role of size and age inﬁnancial con-straints,we examine the relation between these factors and constraints for sub-samples grouped byﬁrm characteristics and time period.While there is minor variation across groups,the general form of the relation between size,age,and ﬁnancial constraint categories appears to be robust.Weﬁnd that the role of both size and age in predicting constraints is nonlinear.At certain points,roughly the sample ninety-ﬁfth percentiles($4.5billion in assets,thirty-seven years in age),the relation between constraints and theseﬁrm characteristics is essen-tiallyﬂat.Below these cutoffs,we uncover a quadratic relation between size and constraints and a linear relation between age and constraints.We represent this relation in what we call the size–age or SA index.2This index indicates thatﬁnancial constraints fall sharply as young and smallﬁrms start to mature and grow.Eventually,these relations appear to level off.Since all measures ofﬁnancial constraints have potential shortcomings,we attempt to provide corroboratory evidence regarding our proposed index.In particular,we exploit the cashﬂow sensitivity of cash approach advanced byAlmeida et al.(2004).When we sortﬁrms into constrained and unconstrained groups using the SA index,weﬁnd that the constrainedﬁrms display a sig-niﬁcant sensitivity of cash to cashﬂow,whereas the unconstrainedﬁrms do not.This evidence increases our conﬁdence in the SA index as a reasonable measure of constraints.While we cannot prove that our index is the optimal measure of constraints, it has many advantages over other approaches,including its intuitive appeal, its independence from various theoretical assumptions,and the presence of corroborating evidence from an alternative approach.The correlation between the SA index and the KZ index is negligible,casting additional doubt on the usefulness of the KZ index.The correlation between the SA index and the WW index is much higher,but this largely reﬂects the fact that the WW index includesﬁrm size as one of its six components.For completeness,we use our data to revisit the Kaplan and Zingales (1997)assertion that investment–cashﬂow sensitivities are dubious measures2This index is derived from coefﬁcients in one of our ordered logit models presented below.The index is cal-culated as(−0.737*Size)+(0.043*Size2)−(0.040*Age),where Size equals the log of inﬂation-adjustedbook assets,and Age is the number of years theﬁrm is listed with a non-missing stock price on Compustat. In calculating this index,Size is winsorized(i.e.,capped)at(the log of)$4.5billion,and Age is winsorized at thirty-seven years.4 at Wuhan University Library on March 12, 2010 Downloaded fromNew Evidence on Measuring Financial Constraintsofﬁnancial constraints.3Ourﬁndings here are consistent with what Kaplan and Zingales(1997)report.In particular,using both our direct qualitative categorization of constraints and the SA index,weﬁnd that investment–cash ﬂow sensitivities are not monotonically increasing in aﬁrm’s level ofﬁnancial constraints.The rest of the article is organized as follows.In Section1,we detail our sample selection procedure and our assignment ofﬁrms intoﬁnancial constraint groups using qualitative information.In Section2,we use our data to critically evaluate past approaches for measuringﬁnancial constraints.In Section3,we further explore the relation betweenﬁnancial constraints and the size and age of aﬁrm and propose a simple index based on theseﬁrm characteristics.In Section4,we revisit the prior evidence on investment–cash ﬂow sensitivities.Section5concludes.1.Sample Construction and Categorization of Financial Constraints1.1Sample selection and data collectionOur goal is to study a large and representative sample of modern public ﬁrms.We begin with the set of all Compustatﬁrms in existence at some point between1995and2004.From this universe,we eliminate allﬁnancial ﬁrms(SIC Codes6000–6999),regulated utilities(SIC Codes4900–4949), andﬁrms incorporated outside the United States.We then sortﬁrms byCompustat identiﬁer and select every twenty-fourthﬁrm for further analysis. This procedure results in a random sample of407ﬁrms that should be broadly representative of the overall Compustat universe.After selecting the initial sample,we locate eachﬁrm’s annual reports and 10-Kﬁlings from Lexis-Nexis and SEC EDGAR.We restrict the sample to ﬁrm years for which we can locate at least one of these electronicﬁlings.In addition,we impose the requirement that theﬁrm has nonzero sales and assets in the observation year and sufﬁcient accounting data to calculate all of the components of the KZ index.The resulting sample consists of356uniqueﬁrms and1,848ﬁrm years during the1995–2004period.4To collect qualitative information onﬁnancial constraints,we carefully read annual reports and10-Kﬁlings following the general procedure outlined by Kaplan and Zingales(1997).In particular,for eachﬁrm year,we read the annual letter to shareholders and the management discussion and analysis section.In addition,we perform an electronic search of the entire text of the annual report and/or10-K to identify all sections of text that include 3For critiques of the investment–cashﬂow approach,see Cleary(1999),Kaplan and Zingales(1997),Erickson and Whited(2000),Alti(2003),and Moyen(2004).For a defense,see Fazzari et al.(2000).4While we borrow heavily from Kaplan and Zingales(2000),the sample we study is quite different from theirs. They study a small sample(forty-nineﬁrms)from the1970s and1980s that satisﬁes a variety of sampling requirements pertaining to industry,size,growth,dividend policy,and survival.5 at Wuhan University Library on March 12, 2010 Downloaded fromThe Review of Financial Studies/v00n002010the following keywords:ﬁnancing,ﬁnance,investing,invest,capital,liquid, liquidity,note,covenant,amend,waive,violate,and credit.Using these procedures,we extract every statement that pertains to a ﬁrm’s ability to raise funds orﬁnance its current or future operations.5In manyﬁlings,we identify multiple statements.We assign to each individual statement an integer code from1to5,with higher(lower)numbers being more indicative of the presence(lack)of constraints.These codes are based on the description provided by KZ regarding their categorization scheme. Later,we aggregate these codes to derive a single overall categorization of a ﬁrm’sﬁnancial constraint status in any given year.It is important to note that there are literally hundreds of different types of relevant statements made by sampleﬁrms.Grouping such a large number of statements intoﬁve categories necessarily requires some judgment.Speciﬁc examples of how we code different types of statements are reported in the Appendix.Following the spirit of the KZ algorithm,we assign to category1all state-ments that indicate that aﬁrm has excessive or more than sufﬁcient liquidity to fund all of its capital needs.In category2,we place all statements that in-dicate that aﬁrm has adequate or sufﬁcient liquidity to fund its needs.The main difference between category1and category2is the strength of theﬁrm’s language.In category3,we place all statements that provide some qualiﬁca-tion regarding theﬁrm’s ability to fund future needs,but that do not indicate any type of current problem.Most of these statements are soft warnings,oftengeneric or boilerplate in character,indicating that under some possible future scenario theﬁrm could have difﬁculty raising funds orﬁnancing desired in-vestments.Category3also includes all statements that are opaque and thus not easy to classify into the other groups.We place all statements that indicate some current liquidity problem into category4,but with no direct indication that these problems have led to a substantive change in theﬁrm’s investment policy or to overtﬁnancial stress. This would include difﬁculties in obtainingﬁnancing or the postponing of a security issue.Finally,category5includes all cases of clearﬁnancial prob-lems/constraints including a current and substantive covenant violation,a rev-elation that investment has been affected by liquidity problems,going concern statements,or involuntary losses of usual sources of credit.65We were assisted by two trained accountants in our search and categorization efforts.Allﬁlings were searched independently by at least two individuals to minimize the probability of missing any relevant disclosure.6Someﬁrms indicate that a covenant had been waived or amended.Often theseﬁrms indicate that the violation was technical in nature and not of substantive concern.For example,someﬁrms indicate that a covenant was routinely waived,and others indicate that an accounting ratio fell below a threshold because of a one-time event such as an asset sale or special charge.Since these cases are quite different from and less serious than current violations,in our baseline coding,we ignore waived/amended covenants.Alternative treatments of these cases are discussed below.6 at Wuhan University Library on March 12, 2010 Downloaded fromNew Evidence on Measuring Financial Constraints1.2Categorization of aﬁrm’s overallﬁnancial constraint statusWe proceed to assign eachﬁrm year to a singleﬁnancial constraint group. Borrowing from the KZ algorithm and terminology,we createﬁve mutually exclusive groups:notﬁnancially constrained(NFC),likely notﬁnancially con-strained(LNFC),potentiallyﬁnancially constrained(PFC),likelyﬁnancially constrained(LFC),andﬁnancially constrained(FC).We place in the NFC groupﬁrms with at least one statement coded as a1and no statement coded below a2.These areﬁrms that indicate more than sufﬁcient liquidity and re-veal no evidence to the contrary.In the LNFC category,we place allﬁrms with statements solely coded as2s.These areﬁrms that indicate adequate or sufﬁcient liquidity with no statements of excessive liquidity and no statements indicating any weakness.7We place allﬁrms with mixed information on their constraint status into the PFC category.Speciﬁcally,we include all observations in which theﬁrm re-veals a statement coded as2or better(indicatingﬁnancial strength),but also reveals a statement coded as3or worse(indicating possibleﬁnancial weak-ness).We also include in this category cases in which all of theﬁrm’s state-ments are coded as3.The LFC category includesﬁrms with at least one statement coded as4,no statement coded as5,and no statement coded better than3.These areﬁrms that indicate some current liquidity problems,with no offsetting positive statement and no statement that is so severe that they are brought into the lowest(FC)category.Finally,all observations with at least one statement coded as5and no other statement coded better than3are assigned to the FC category.These are ﬁrms that clearly indicate the presence of constraints with no strong offsetting positive revelation.We refer to this initial categorization scheme as qualitative scheme1and report a sample breakdown in Column1of table1.For comparison purposes, we report in Column4the correspondingﬁgures reported by KZ.One peculiar feature of qualitative scheme1is that a large number ofﬁrms are placed in the PFC category(32.36%versus7.30%in the KZ sample).This elevated rate primarily reﬂects the fact that manyﬁrms in our sample provide boilerplate generic warnings about future uncertainties that could affect aﬁrm’s liquidity position.These statements place manyﬁrms that otherwise report strongﬁ-nancial health into the PFC category.In our estimation,many of these generic warning statements are uninformative.In particular,they appear to be included as a blanket protection against future legal liability and often pertain to unfore-seen or unlikely contingencies that could potentially affect almost anyﬁrm.In light of these observations,we prefer an alternative assignment procedure that ignores all generic or soft nonspeciﬁc warnings regarding aﬁrm’s future liquidity position.This procedure,which we refer to as qualitative scheme2,7We also place in this group the few observations with no useful qualitative disclosure that could be used to ascertain aﬁrm’sﬁnancial constraint status.If we exclude these observations,the ordered logit results we report below in tables3,4,and6are substantively unchanged.7 at Wuhan University Library on March 12, 2010 Downloaded fromThe Review of Financial Studies/v00n002010Table1Frequency of Financial Constraint CategoriesConstraint assignment procedureQualitative Qualitative Qual./quant.KZ samplescheme1scheme2schemeNotﬁnancially const:NFC10.28%10.98%55.84%54.50% Likely notﬁnancially const.:LNFC50.49%71.59%31.01%30.90% Potentiallyﬁnancially const.:PFC32.36%10.55% 6.44%7.30% Likelyﬁnancially const.:LFC0.32%0.32% 2.49% 4.80% Financially const.:FC 6.55% 6.55% 4.22% 2.60% Correlation with qual.scheme1 1.00Correlation with qual.scheme20.89 1.00Correlation with qual./quant.scheme0.750.87 1.00This table reports the fraction of allﬁrm-year observations in which an observation is assigned to the indicated ﬁnancial constraint group.Theﬁgures in Columns1–3pertain to our random sample of1,848Compustatﬁrm years representing356ﬁrms operating during the1995–2004period for observations with non-missing data on theﬁve components of the KZ index.Qualitative scheme1uses only qualitative statements made byﬁrms in their ﬁlings subsequent to theﬁscal year-end regarding theﬁrm’s liquidity position and ability to fund investments. The exact algorithm used in coding and categorizing this information is detailed in the text.Qualitative scheme 2is constructed identically to scheme1except that it ignores all soft and generic nonspeciﬁc warnings made byﬁrms regarding possible future scenarios under which theﬁrm could experience a liquidity problem.The qualitative/quantitative scheme augments scheme2by movingﬁrms upward one category if theﬁrm materially increases dividends,repurchases shares,or has a high(top quartile)level of(cash/capital expenditures)on hand. Additional details concerning the assignment procedures are provided in the text and the Appendix.Theﬁgures in Column4are taken from table2of Kaplan and Zingales(1997)and are based on their sample and algorithm for categorizing constraints.The correlationﬁgures represent simple correlations over the sample between the two constraint assignment procedures in the indicated cell.is identical to qualitative scheme1outlined earlier except that it ignores this one class of statements.As we report in Column2of table1,this modiﬁcation moves many(a few)ﬁrms from the PFC grouping up into the LNFC(NFC) grouping.It is important to emphasize that the categorization schemes outlined above deliberately differ from the KZ procedure in one key respect.In particular, in our categorization,we choose to ignore quantitative information on both the size of aﬁrm’s cash position and its recent dividend/repurchase behavior. We do so this because it seems inappropriate to incorporate this information into categories that will eventually be used for coding our dependent variables, given that this same information will later be used to construct some of the in-dependent variables.Such treatment would lead to uninformative coefﬁcients that are hardwired and potentially misleading in terms of their ability to de-scribe the underlying relation between quantitative variables and qualitative disclosures of constraints.For completeness,we experiment with modifying our qualitative scheme2 categorization to more closely match the exact KZ treatment by incorporating quantitative information on dividends,repurchases,and cash balances.In par-ticular,we move aﬁrm’s constraint status up one notch in a given year(e.g., from PFC to LNFC)if any of the following criteria are met:(i)theﬁrm ini-tiates a dividend;(ii)theﬁrm has a material increase in dividends(change in dividends/assets greater than theﬁfth percentile of dividend increasers);(iii) 8 at Wuhan University Library on March 12, 2010 Downloaded fromNew Evidence on Measuring Financial ConstraintsTable2Sample Characteristics(1)(2)(3)(4)(5)(6) Statistic Mean Mean Mean Median Median Median Cashﬂow/K−2.379−0.915−9.3150.2430.327−0.907 Cash/K 3.689 3.579 4.2080.4390.5080.199 Dividends/K0.0770.0640.1390.0000.0000.000 Tobin’s Q 2.672 2.036 5.686 1.535 1.489 1.809 Debt/total capital0.3380.2770.6290.2750.2240.728 Capital exp./K0.4110.4150.3920.2140.2290.133 Prop.,plant,278.370303.457159.48020.66429.594 2.951 equip.(PPE)Book assets782.928872.877356.647124.627167.08914.800 Age13.92314.71610.1659.0009.0007.000 Sales growth0.2720.2470.3940.0570.070−0.049#of qualitative 3.37 3.32 3.62 3.00 3.00 3.00 statementsWhich observations All Less More All Less Moreconstrained constrained constrained constrained Theﬁgures in each column represent the mean or median of the indicated variable over the indicated set of observations.Theﬁgures in Columns1and4refer to our random sample of1,848Compustatﬁrm years repre-senting356ﬁrms operating during the1995–2004period.Theﬁgures in Columns2and5are calculated over the subset of observations in which theﬁrm was classiﬁed as less constrained(NFC/LNFC)using qualitative scheme2to categorize constraints.Theﬁgures in Columns3and6are calculated over the subset of observa-tions in which theﬁrm was classiﬁed as more constrained(PFC/LFC/FC).All variables are constructed from Compustat information.The PPE and book assets statistics are in millions of inﬂation adjusted year2004dol-lars.All variables that are normalized by K are divided by beginning-of-period PPE.Cashﬂow is deﬁned to be operating income plus depreciation(Compustat item18+item14).Cash is deﬁned to be cash plus marketable securities(item1).Dividends are total annual dividend payments(item21+item19).Tobin’s Q is deﬁned as (book assets minus book common equity minus deferred taxes plus market equity)/book assets calculated as [item6−item60−item74+(item25×item24)]/item6.Debt is deﬁned as short-term plus long-term debt(item9+item34).Total capital is deﬁned as debt plus total stockholders’equity(item9+item34+item216). If stockholders’equity is negative,we set debt/total capital equal to1.Capital expenditures are item128.Age is deﬁned to be the number of years preceding the observation year that theﬁrm has a non-missing stock price on the Compustatﬁle.Sales growth is deﬁned as(sales in year t minus sales in year t−1)/sales in year t−1. Sales areﬁrst inﬂation adjusted before making this growth calculation.The number of statements row refers to the number of qualitative statements from disclosureﬁlings that were used in assigning theﬁrm to a constraint grouping using qualitative scheme2,as outlined in the text.theﬁrm repurchases a material number of shares(repurchases/assets greater than theﬁfth percentile of repurchasers);or(iv)theﬁrm’s balance of cash and marketable securities normalized by capital expenditures falls in the top sample quartile.The resulting categorization is referred to in what follows as the qualitative/quantitative categorization scheme.We report in Column3of table1the percentage ofﬁrms in each of the constraint categories using this alternative scheme.As theﬁgures illustrate,the sample frequencies using the qualitative/quantitative scheme more closely resemble theﬁgures reported by KZ,with the modal category beingﬁrms in the most unconstrained(NFC) category.In table2,we present summary statistics for the sample as a whole and for subsamples grouped by the level of constraints using our preferred constraint assignment procedure,qualitative scheme2.Several interesting differences be-tween the more constrained and less constrainedﬁrms emerge.In particular, comparing both the reported means and medians for the subsamples grouped9 at Wuhan University Library on March 12, 2010 Downloaded from。

corr指标 -回复

corr指标-回复[Correlation Index (CORR)]IntroductionThe correlation index, also known as CORR, is a statistical measure used to determine the relationship between two variables. It helps to understand the strength and direction of the relationship between these variables. CORR is widely used in various fields, including finance, economics, social sciences, and healthcare. In this article, we will explore the concept of CORR, its calculation, interpretation, and how it is used in different disciplines.Understanding CorrelationCorrelation refers to the statistical relationship between two or more variables. In simple terms, it answers the question, "Does a change in one variable correspond to a change in another variable?" Correlation can be positive, negative, or zero.Positive correlation indicates that as one variable increases, the other variable also tends to increase. For example, there is apositive correlation between hours spent studying and exam grades. Negative correlation, on the other hand, implies that as one variable increases, the other variable tends to decrease. An example of negative correlation is the relationship between exercise and body weight. Zero correlation means that there is no relationship between the variables.Calculating CorrelationTo calculate the correlation index, we use statistical techniques such as the Pearson correlation coefficient, Spearman's rank correlation coefficient, or Kendall's tau coefficient. The most common method is Pearson's correlation coefficient, which measures the linear relationship between two continuous variables.Pearson's correlation coefficient (r) ranges from -1 to 1. A value of -1 indicates a perfect negative correlation, +1 signifies a perfect positive correlation, and 0 implies no correlation. The formula for calculating Pearson's correlation coefficient is:r = (NΣxy - ΣxΣy) / sqrt((NΣx^2 - (Σx)^2) * (NΣy^2 - (Σy)^2))Here, N represents the sample size, Σrepresents the summation (sum of all the values), x and y denote the variables being analyzed, and xy represents their respective product.Interpretation of CorrelationAfter calculating the correlation coefficient, the next step is interpretation. A positive correlation coefficient implies that the variables move in the same direction. For example, a value close to +1 indicates a strong positive relationship. On the other hand, a negative correlation coefficient indicates that the variables move in opposite directions. A value close to -1 suggests a strong negative relationship. A value close to 0 means there is little to no correlation between the variables.It is essential to note that correlation does not imply causation. Just because two variables are correlated does not mean that one variable causes the change in the other. Correlation can help identify relationships, but further analysis is needed to establish causation.Applications of CorrelationCORR is utilized in various disciplines to gain insights into the relationship between variables. In finance, correlation helps investors diversify their portfolios by identifying assets that do not move in tandem. By investing in assets with low correlation, investors can reduce risk. For example, a portfolio combining stocks and bonds may be less volatile than solely investing in stocks.In economics, correlation helps analyze the impact of different factors on variables like GDP, inflation, and employment rates. It can provide valuable insights into how changes in interest rates, government spending, or consumer behavior affect the economy.In social sciences, correlation is used to study relationships between variables such as crime rates and socioeconomic indicators, educational attainment, and income levels, or happiness levels and life satisfaction factors. Understanding these relationships can aid policymakers in formulating effective strategies.In healthcare, correlation is used to examine the relationshipsbetween risk factors and health outcomes. For example, researchers might study the correlation between smoking habits and rates of lung cancer, or the correlation between exercise frequency and heart disease. Such studies provide a basis for preventive measures and public health interventions.ConclusionIn conclusion, the correlation index (CORR) is a valuable statistical measure used to understand the relationship between variables in different disciplines. It helps identify the strength and direction of the relationship, but caution should be exercised in interpreting the results to avoid assuming causation. CORR is widely used in finance, economics, social sciences, and healthcare to make informed decisions, develop strategies, and improve outcomes. Understanding the correlation between variables provides valuable insights into various phenomena, ultimately leading to better problem-solving and decision-making.。

老年人身体基本活动能力对失能的影响研究

·3502··论著·老年人身体基本活动能力对失能的影响研究文湘田1，钮文异2*【摘要】　背景　随着人口老龄化进程的加快，预计到2030年我国≥65岁老年人口占总人口比重将达20%以上，其中失能老年人规模将超过7 700万，占总失能人口的比例将≥57%。

老年人身体基本活动能力下降可导致其生活质量下降，并使社会及家庭的养老负担加重。

目的　探究身体基本活动能力对老年人失能状况的影响。

方法　于2021年3月，采用2015年中国健康与养老全国追踪调查（CHARLS）数据，选取接受了身体基本活动能力、失能状况评估且关键变量值（性别、年龄、户口所在地、受教育程度）完整的老年人5 276例作为研究对象。

通过握力、站立测试、起坐测试和步速评价老年人的身体基本活动能力，运用工具性日常生活能力（IADL）量表评估老年人失能发生情况。

比较不同失能发生情况老年人的握力水平、站立测试与起坐测试完成情况、步速，应用二元Logistic 回归探讨身体基本活动能力对老年人失能状况的影响。

结果　5 276例老年人握力为27.92（9.02）kg，其中1 713例（32.47%）握力水平为低，1 770例（33.55%）握力水平为中等，1 793例（33.98%）握力水平为高；3 930例（74.49%）能够完成站立测试；5 128例（97.19%）能够完成起坐测试；步速为0.96（3.12）m/s，其中1 750例（33.17%）步速慢，1 759例（33.34%）步速中等，1 767例（33.49%）步速快；1 419例（26.90%）老年人发生失能。

不同失能发生情况老年人握力水平、站立测试与起坐测试完成情况、步速比较，差异有统计学意义（P<0.05）。

二元Logistic 回归结果显示：无论对于60~74岁还是≥75岁老年人，握力水平、站立测试与起坐测试完成情况、步速均是老年人失能发生的影响因素（P<0.05），特别是无法完成起坐测试的60~74、≥75岁老年人失能的发生风险分别是能够完成者的3.045、4.126倍。

高考英语一轮复习精选提分专练第四周五成功与失败高三全册英语试题0

入舵市安恙阳光实验学校成功与失败Ⅰ.语境填词1.I wasn’t (success)，so they looked down on me.2.The child was found (被抛弃的) but unharmed.3.Iran says its nuclear ambitions are for peaceful (目的).4.They said that they (试图) to finish the task before July.5.We are learning how to confront death instead of (逃避) its reality.Ⅱ.单项填空6.How you the challenge in the second half will determine what you become after the game，whether you are a winner or a loser.A.respond toB.submit tomit toD.stick to7.I have always to China，and now my dream has .A.dream coming；come trueB.dreamt of coming；come trueC.dreamed to come；realizedD.dreamed coming；realizing8.The WFP has launched a new app called Share The Meal， the potential，people believe，is enormous as it allows smart phone users to make donatio ns with a simple tap on their phone.(2017·南通如东、徐州联考)A.whoseB.thatC.whichD.of which9.Even with the gift of 10,000，it was difficult for her to a mortgage，but she eventually found a lender and a house.A.payB.obtainC.cancelD.accept10.A notice was in order to remind the students of the changed lecture time.A.sent upB.given upC.set upD.put upⅠ.阅读理解They say a cat has nine lives，and I think that possible since I am now living my third life and I’m not even a cat.My father died when I was 15，and we had a hard struggle to make a living.And my mother，who was seriously ill in her last years，died while still in her 60s.My sister married soon after，and I followed her example within the year.This was when I began to enjoy my first life.I was very happy，in excellent health.I had a good job in San Jose and a beautiful home up the peninsula(半岛) in San Carlos.Life was a pleasant dream.Then the dream ended.I became afflicted(使苦恼) with a slowly progressive disease of the motor nerves，affecting first my right arm and leg，and then my other side.Thus began my second life...In spite of my disease I still drove to and from work each day，with the aid of special equipment installed in my car.And I managed to keep my health and optimism，to a degree，because of 14 steps.Crazy？Not at all.Our home was an affair with 14 steps leading up from the garage to the kitchen door.Those steps were a standard measure of life.They were my yardstick，my challenge to continue living.I felt that if the day arrived when I was unable to lift one foot up one step and then drag the other painfully after it—repeating the process 14 times，I would be through—I could then admit defeat and lie down and die.Then on a dark night in August,1971，I began my third life.It was raining when I started home that night；strong winds and slashing rain beat down on the car as I drove slowly down one of the lesstrav elled roads.Suddenly the steering wheel jerked(猝然一动).In the same instant I heard the bang of a blowout.It was impossible for me to change that tire！Utterly impossible!I started the engine and thumped slowly along until I came to the dirt road，where I turned in and where I found lighted windows welcomed me to a house and pulled into the driveway and honked the horn.The door opened and a little girl stood there.When she knew what happened to me，she went into the house and a moment later came out，followed by a man who called a cheerful greeting.I sat there comfortable and dry，and felt a bit sorry for the man and the little girl working so hard in the storm.About an hour later，the man’s voice was heard，“This is a bad night for car trouble，but you’re all set now.”“Thanks，” I said，“How much do I owe you？” He shook his head，“Nothing.Cynthia told me you were a cripple.Glad to be of help.I know you’d do the same for me.There’s no charge，friend.” I held out a fivedollar bill，“No！I like to pay my way.” He made no ef fort to take it and the little girl stepped closer to the window and said quietly，“Grandpa can’t see it.”1.“A cat has nine lives” here means “”.A.a cat can live nine times longer than any other animalB.a cat can die ninthC.a lucky man can not die easilyD.the author will live nine times2.What do you think of the man who helped change the tire?A.Old，warmhearted but pitiable.B.Blind，but warmhearted and happy.C.A blind old man that has nothing to do every day.D.A poor old man that is always ready to help others.3.How do you understand the underlined sentence “I followed her example within the year”？A.He listened to his sister carefully.B.Mother told him that he must get the agreement from his sister for whatever he would do.C.His sister got married.He，too.D.His sister was a great woman.He must learn from her.Ⅱ.任务型阅读The most common use of intelligence test scores is to predict degree of academic success.Such scores are used in some communities as bases for admitting able children to schools at ages younger than normal，and they are very generally used to determine admissions to schools beyond public secondary school.Another commonuse in elementary schools involves comparing such scores with performances in various subjects to identify children who are working below capacity.The greatest problem in using intelligence tests for the purpose of prediction is that no dependable criterion of their accuracyexists.The ideal criteria would be objective and reliable achievement tests following instructions in each subject，but there are few such tests，especially at the college level.Studies have shown that correlations(相关性) between intelligence tests and achievement tests in various subjects through secondary school range roughly from 0.5 to 0.8.Such correlations are fairly high，but they do not suggest anywhere near complete agreement.At the college level there are two major tests used as criteria of admission.By far the more important is the College Entrance Examination，constructed by the Educational Testing Service authorized by the College Entrance Examination Board.These tests are returned to the Educational Testing Service for scoring，and the results are then made available to the various colleges authorized by the students to receive them.The second test of this type is the American College Test，which operates in essentially the same fashion.Both tests constitute(组成) measures of certain skills，abilities，and knowledge that have been found to be related to success in college.Their correlations with academic success are limited for three outstanding reasons.First，measures of achievement in college are themselves perhaps no more reliable than those in elementary and secondary schools.Second，intellectual factors do not alone determine academic success，especially at the college level.Many students drop out of schools because they are inadequately motivated or because they dislike the instructional programme.Third，correlations are lowered because the use of such tests for denying admission to some students means that the range of scores for those admitted is restricted，and such restrictions tend to reduce correlations.答案精析高考高频单词与短语识记排查跟踪训练Ⅰ.1.successful 2.abandoned 3.purposes 4.attempted 5.avoiding Ⅱ.6.A [句意为：你对下半场如何反应将会决定比赛后你成为成功者还是失败者。

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Infraspinatus and Supraspinatus Tendon Strain ExplainedUsing Multiple Regression ModelsN ELLY A NDARAWIS -P URI ,1A NDREW F.K UNTZ ,1A BBAS F.J AWAD ,2and L OUIS J.S OSLOWSKY 11McKay Orthopaedic Research Laboratory,University of Pennsylvania,424Stemmler Hall,Philadelphia,PA,USA;and2Department of Pediatrics,University of Pennsylvania,Philadelphia,PA,USA(Received 7January 2010;accepted 23April 2010;published online 11May 2010)Associate Editor Eric M.Darling oversaw the review of this article.Abstract —Supraspinatus tendon tears are complex yet com-mon.We have shown that the supraspinatus and infraspina-tus tendons interact,indicated by parallel changes in strain in the supraspinatus and infraspinatus with increasing size of supraspinatus tear,load applied to the supraspinatus,and changes in glenohumeral rotation but not abduction angle,suggesting disruption in the interaction between the two tendons with increase in abduction angle.While considering these factors individually is valuable,the contribution of each factor in the context of all others on strain in the supraspi-natus,or on the interaction between the two tendons is unknown and has important implications in the management of rotator cuff tears.In this study,regression models using least-square estimation with backward and forward elimina-tion were used to predict strains in the infraspinatus and supraspinatus from joint position,supraspinatus load,and supraspinatus tear size or repair.Interestingly,despite previ-ous ﬁndings showing that supraspinatus tear size signiﬁcantly affects infraspinatus strain,tear size was not a signiﬁcant predictor of infraspinatus strain,emphasizing the importance of other factors evaluated such as joint position and shoulder loading in management of cuff tears and postoperative care.A better understanding of the loading environment in rotator cuff tendons necessitates multifactorial complex models.Keywords —Rotator cuff,Glenohumeral abduction,Gleno-humeral rotation,Principal strain.INTRODUCTIONSupraspinatus tendon tears are a common and not well understood soft tissue injury.6–9The high incidence of rotator cuff tears has motivated evaluation of the likelihood of tear propagation,which would inﬂuence surgical intervention.To address this problem,strainwas used to measure the load-bearing capacity in the torn supraspinatus tendon.4,12However,the inhomo-geneity of the supraspinatus tendon and the complex loading environment of the rotator cuff confound the predictability of rotator cuff tear propagation.Because of its high incidence of injury,several studies have investigated the eﬀect of a supraspinatus tendon tear on the remaining intact portion of the tendon to understand changes in its loading environment due to a tear.4,12Strain,indicative of the load-bearing response of the remaining intact portion of the tendon,was cal-culated in order to infer the risk of tear propagation.However,despite the importance of this data,assessing the likelihood of tear propagation is further compli-cated by the complex loading environment and the tis-sue architecture of the rotator cuff.For instance,the interaction between the humeral head and the torn supraspinatus tendon has been shown to decrease the effect of a supraspinatus tendon tear on tendon strain at certain joint positions.4Therefore,other rotator cuff structures may impact the likelihood of supraspinatus tear propagation and reciprocally be impacted by the existence of a supraspinatus tendon tear.Our previous ﬁndings indicate that the supraspina-tus and infraspinatus tendons mechanically interact such that conditions that result in increased strains in the supraspinatus tendon also result in increased strain in the infraspinatus tendon.1–3More speciﬁcally,such interaction existed with increase in supraspinatus ten-don full-thickness 3and partial-thickness tear size,1two different repair techniques,increase in supraspinatus tendon load,1,3and change in joint rotation angle.We have also shown that the mechanical interaction between the two tendons is interrupted with increase in joint abduction angle.While considering these fac-tors individually is valuable,the contribution of each factor in the context of all others on strain in theAddress correspondence to Louis J.Soslowsky,McKay Ortho-paedic Research Laboratory,University of Pennsylvania,424Stemmler Hall,Philadelphia,PA,USA.Electronic mail:soslowsk@Annals of Biomedical Engineering ,Vol.38,No.9,September 2010(Ó2010)pp.2979–2987DOI:10.1007/s10439-010-0056-90090-6964/10/0900-2979/0Ó2010Biomedical Engineering Society2979supraspinatus,or on the interaction between the two tendons is unknown and has important implications in the management of rotator cuff tears.Regression models provide statistical quantiﬁcation of the relationship between a response(dependent) variable and predictor(independent)variable(s).Such model(s)allow simultaneous evaluation of the rela-tionship between a dependent variable and several independent variables.Development of a multiple regression model to predict the eﬀect of supraspinatus tendon load,tear size,and joint position on strain in the supraspinatus and infraspinatus tendons would have signiﬁcant clinical implications,since it would demonstrate the level of contribution of each factor in the context of all other ing the developed model(s),clinical management of supraspinatus ten-don tears can be eﬃciently and optimally focused to target the factors that have the most impact on strain in the supraspinatus and infraspinatus tendon.Addi-tionally,the developed regression model(s)also allows extrapolation to predict the eﬀect of conditions that have not been experimentally evaluated resulting in more guiding data than can be practically collected experimentally.Therefore,the objective of this study was to develop multivariate regression models of infraspinatus and supraspinatus tendon strain as a function of supraspinatus tendon load,tear type or repair technique,and rotator cuﬀjoint position.We hypothesize that(H1)supraspinatus tendon tear type, load,repair technique,and joint position will all be signiﬁcant predictors of strain in the infraspinatus ten-don;and that(H2)supraspinatus tendon tear type,load, repair technique,and joint position will all be signiﬁ-cant predictors of strain in the supraspinatus tendon.MATERIALS AND METHODS All experimental methods were previously described in detail.1–3Brieﬂy,10healthy,fresh-frozen cadaveric shoulders(average age48.7±15.1years)were care-fully dissected free of soft tissue,retaining only the proximal humerus,supraspinatus,and infraspinatus tendons.The proximal end of the infraspinatus tendon was sutured using a Krakow stitch to allow application of a load.Prior to mechanical testing,the specimens were maintained in a physiologic bath of phosphate-buffered saline(PBS).The experimental setup is shown in Fig.1.The bursal side of both the supraspinatus and the infraspinatus tendons was air-brushed with black paint to create aﬁne speckled texture for subsequent texture correlation strain analysis using Vic2D(Version4.4.1, Correlated Solutions Inc.,Columbia,SC)as previously described.1–3The specimen was mounted in poly-methyl meth-acrylate(PMMA)in custom grips that allowed for repeatable and controlled changes in glenohumeral abduction and rotation.1,2The supraspinatus tendon was attached to a testing machine(5543,Instron, Norwood,MA)to allow for controlled loading and the infraspinatus tendon was attached to a pulley system with a nominal,constant load of9.8N.1–3Supraspinatus tendon loading protocol,previously described,consisted of preconditioning followed by a constant ramp to90N at a strain rate of0.1%per second of the length of the tendon.1–3Loading proto-col was applied for the intact,33%,66%,and100% bursal-side partial thickness tears through the anterior 66%of the width of the supraspinatus tendon and for two supraspinatus repairs(modiﬁed Mason-Allen with transosseous boneﬁxation or4-suture-bridge suture-anchor as often used in arthroscopic repair;Fig.2). Each supraspinatus tear and repair wereevaluated FIGURE1.Schematic representation of experimentalsetup.FIGURE2.Schematic representation of(a)4-suture-bridge suture-anchor as often used in arthroscopic repair,and (b)supraspinatus tendon tears.Bursal-side partial thickness tears through the anterior66%of the width of the supraspi-natus tendon were surgically introduced through the0% (intact),33%,66%,or100%of the thickness of the tendon.A NDARAWIS-P URI et al. 2980at each of the following joint positions:(1)neutral rotation with0°,30°,and60°abduction2;and(2)0°abduction with0°,30°internal,and external rotation.1 These joint positions were evaluated because they provided a good sampling of the interaction between the supraspinatus and infraspinatus tendons as their respective relationships to the humeral head was altered.Both tendons were generously moistened with PBS after each completion of the loading protocol.As previously described,digital images were taken of the insertion site of both the supraspinatus and the infraspinatus tendons at1s intervals during the load-ing ramp phase of the mechanical testing protocol.1–3 For each supraspinatus tendon tear level and joint position,images at5,30,60,and90N of supraspi-natus tendon loads were chosen for evaluation for the infraspinatus and supraspinatus tendons.Similarly,for transosseous and arthroscopic supraspinatus tendon repairs,images at5,30,60,and90N of supraspinatus tendon loads were chosen for evaluation at each joint position for the infraspinatus tendon only.Local strains in the repaired supraspinatus tendon could not be evaluated due to tendon buckling at the repair site.A digital grid of nodes was thenﬁtted to the inser-tion site of the infraspinatus and the consistently intact posterior1/3of the supraspinatus tendons as previously described.3Displacements of the nodes between the5N load and each of the30,60,and90N loads were determined from which two-dimensional Lagrangianﬁnite strain tensor(e xx,e yy,and e xy)and principal strain components were calculated.1,2Aver-age maximum and minimum principal strains were determined for the regions of interest in both tendons, thereby completely and simply depicting the loading environment by absorbing the shear strain component into the orthogonal tensile(maximum principal strain) and compressive(minimum principal strain)compo-nents through a coordinate rotation.Using the collected strain data,the following mul-tiple regression equation was used.EPij¼C ijþb1ijðAbd angleÞþb2ijðRot angleÞþb3ijðLoadÞþb4ijðTear sizeÞ;ð1Þwhere i=1,2represents average maximum and min-imum principal strain,respectively;j=1,2represents supraspinatus and infraspinatus tendons,respectively.C ij represents constant,b1ij,b2ij b3ij,and b4ij are the regression coefﬁcients calculated from each of the regression models.Abduction angle and rotation angle are represented by Abd_angle(0°,30°,or60°)and Rot_angle(internal rotation:230°,and external rota-tion:30°),respectively.Load was inputted in Newtons, and tear size was inputted on a scale of1–100. Equation(1)produces four different regression models (four models from two tendons and two mechanical parameters:maximum and minimum principal strain). In addition,the following multiple regression equation was used:EPi2¼C i2þb1i2ðAbd angleÞþb2i2ðRot angleÞþb3i2ðLoadÞþb4i2ðrepairÞð2ÞEquation(2)produces two different regression models(for the infraspinatus tendon and two mechanical parameters:maximum and minimum principal strain).Summary statistics of all variables were examined and described by mean,median,standard deviation, minimum and maximum to ensure that assumptions necessary for linear regression analysis were met.For each experimental condition,average maximum and minimum strain value of all10specimens was calcu-lated each tendon,yielding a total of60y-values inputted into regression model1,and30y-values inputted into regression model2.Standard deviation for each data point was calculated and used for model assessment as will be discussed below.Multiple linear regression models were used to quantify the relation-ship between the dependent variables(principal strain) and the predictor variables(supraspinatus load,tear size or repair technique,joint rotation,and joint abduction).Regression models utilizing least-square estimation with backward and forward elimination were used to predict strain values.10In this method,a simple linear regression model isﬁrstﬁt for each of the potential independent variables.For each simple linear regression model,partial F statistic(F*)is cal-culated to determine whether or not the slope is zero. An independent variable with the largest F*is iden-tiﬁed and compared to a predetermined value of F to enter.If F*exceeds F then the variable is added to the regression model,and the analysis is repeated for the additional independent variables.If F*does not exceed F to enter,stepwise analysis terminates.After each independent variable is added,the analysis is repeated for all the independent variables with revised model.F*is calculated for the previously included variables,and compared to a predetermined value of F to remove.If F*falls below the predetermined value of F,the variable is removed from the model.Toler-ance level was set at0.01,preventing entry of highly correlated independent variables in the model.Based on the degrees of freedom,F to enter and to be removed were,respectively,set to4and3.9,corre-sponding to a signiﬁcance level of0.05for a single test.Signiﬁcance was set at p£0.05and a trend at p£0.1.Additionally,Durbin Watson(DW)statistic was calculated to identify the existence of correlationMultiple Regression Model of Rotator Cuff Tendon Strain2981between independent variables included in the model.For Eq.(1)(n =60),based on established guidelines for interpretation of DW statistic,11a DW value lower than 1.55indicated correlation between the indepen-dent variables.Similarly,for Eq.(2),a DW value lower than 1.35indicated correlation between the indepen-dent variables.11The robustness of the model was evaluated by randomly selecting and removing 3of 60data points (for the supraspinatus tendon tear cases)or 2of 30data points (for the supraspinatus tendon repair cases)at a time and recalculating the regression coeﬃcients for a total number of ﬁve trials.This number of data points was removed at a time because regression coeﬃcients calculated without this number of data points did not diﬀer from those calculated including these data points.The model was then used to predict the removed data points,and the results were com-pared with the actual values as a way to assess model stability.The prediction was considered accurate if the predicted value was within 1standard deviation of the actual value.RESULTSIn general,glenohumeral abduction angle and supraspinatus tendon load consistently correlated with both,supraspinatus and infraspinatus tendon strain.Results shown in Tables 1and 2will be discussed in detail below.Shown R 2(coefﬁcient of determination)values represent the contribution of the set of inde-pendent variables in describing the variability in the dependent variable.The signiﬁcance of the overall model ﬁt is indicated by the p values.Correlations Between the Independent VariablesIncluding Supraspinatus Tendon Tear,and Infraspinatus Tendon Strain Glenohumeral abduction angle,joint rotation and supraspinatus tendon load were all signiﬁcantpredictors of average maximum principal strain in the infraspinatus tendon.DW statistic of 2.15was calcu-lated,indicating no correlation between the indepen-dent variables included in the model.Similarly,glenohumeral abduction angle,and supraspinatus tendon load were predictors of average minimum principal strain in the infraspinatus.A calculated DW statistic of 1.84indicates no correlation between the independent variables.Results are shown in Table 1.Correlations Between the Independent VariablesIncluding Supraspinatus Tendon Tear,and Supraspinatus Tendon Strain Glenohumeral abduction angle and supraspinatus tendon load were signiﬁcant predictors of average maximum principal strain in the supraspinatus tendon.A model R 2of 0.35implies that the dependent variable is not completely modeled by the signiﬁcant indepen-dent variables.A calculated DW statistic of 0.73indicates correlation between the independent vari-ables.Glenohumeral abduction angle,joint rotation,supraspinatus tendon load,and partial-thickness tear size were signiﬁcant predictors of average minimum principal strain in the infraspinatus tendon.R 2of 0.72for this model indicates that the dependent variable is well modeled by the signiﬁcant independent variables.A calculated DW statistic of 1.11indicates some cor-relation between the independent variables.Results are shown in Table 1.Correlations Between the Independent VariablesIncluding Supraspinatus Tendon Repair,and Infraspinatus Tendon Strain Glenohumeral abduction angle,joint rotation,and supraspinatus tendon load and repair were signiﬁcant predictors of average maximum principal strain in the infraspinatus tendon.DW statistic of 1.27was calculated,indicating some correlation between the independent variables in this model.Similarly,TABLE 1.Signiﬁcant coefﬁcient (b )results using multiple regression analysis for strain in the infraspinatus and supraspinatustendon with supraspinatus partial-thickness tear.Dependent variable Model P Model R 2Glenohumeral abduction angle (b )Joint rotation angle (b )SS load (b )SS partial-thicknesstear size (b )Average maximum principal strain in IS 0.0010.6820.050.030.019–Average minimum principal strain in IS 0.0010.660.05–0.02–Average maximum principal strain in SS 0.0010.350.04–0.06–Averageminimum principal strain in SS0.0010.720.0620.0520.0520.01P =model signiﬁcance;R 2=correlation coefﬁcient of model.SS Supraspinatus tendon;IS Infraspinatus tendon.A NDARAWIS -P URI et al.2982glenohumeral abduction angle,and supraspinatus tendon load were predictors of average minimum principal strain in the infraspinatus tendon.A calcu-lated DW statistic of1.2indicates some correlation between the independent variables.Results are shown in Table2.Results shown in Tables1and2can be represented in equation form where the dependent variable values are equal to the signiﬁcant independent predictors multiplied by their individual coefﬁcient.Note that based on model1,four-ﬁtted independent equations were produced.Equations(3)and(4)represent prin-cipal strain in the infraspinatus tendon with supra-spinatus tears,and Eqs.(5)and(6)represent principal strain in the supraspinatus tendon.Based on model2, twoﬁtted independent equations were produced. Equations(7)and(8)represent principal strain in the infraspinatus tendon with supraspinatus tendon repair.Theﬁtted equations developed using models1 and2are:EP1Inf¼1:9À0:05ÂðAbd angleÞ½þ0:03ÂðRot angleÞ½þ0:02ÂðLoadÞ½ ð3ÞEP2Inf¼À1:88þ½0:05ÂðAbd angleÞÀ½0:02ÂðLoadÞ ð4ÞEP1Sup¼2:98þ0:04ðAbd angleÞþ½0:06ÂðLoadÞð5ÞEP2Sup¼À1:9þ½0:06ÂðAbd angleÞÀ½0:05ÂðRot angleÞ À½0:05ÂðLoadÞÀ½0:01ÂðTear sizeÞ ð6ÞEP1Inf¼0:6À½0:04ÂðAbd angleÞþ½0:03ÂðRot angleÞ þ½0:02ÂðLoadÞþ½0:47ÂðrepairÞ ð7ÞEP2Inf¼À1:17þ½0:03ÂðAbd angleÞÀ½0:01ÂðLoadÞ ð8ÞModel AssessmentInfraspinatus Tendon Strain with Supraspinatus Tendon TearRandom removal of three data points at a time from the data set and re-calculating the regression coeﬃ-cients did not alter the regression coeﬃcients calcu-lated from the overall model.Results ensure that removing this number of data points for model assessment was acceptable.Regression coeﬃcients from allﬁve trials are shown in Table3.For both average maximum and minimum princi-pal strain,the developed regression model accurately predicted strains.The actual and predicted average maximum and minimum principal strain values from allﬁve evaluation trials are shown in Table4.Pre-dicted values generally fell within1standard deviation of the actual value,supporting the stability and robustness of the model.However,predicted strain did not accurately mirror actual strain for small strain values near0%.Supraspinatus Tendon Strain with Supraspinatus Tendon TearRandom removal of three data points at a time from the data set and re-calculating the regression coeﬃ-cients did not signiﬁcantly alter the regression coeﬃ-cient for abduction,rotation,and load.The correlation coeﬃcient for tear ranged between0and0.02(for maximum principal strain)and0and20.01(for minimum principal strain)through theﬁve trials.The high(but signiﬁcant)p-values associated with tear (p~0.05)suggests that supraspinatus tendon tear size may be a weak predictor of supraspinatus tendon strain.The regression coefﬁcients from allﬁve trials are shown in Table3.In the supraspinatus tendon,the developed regres-sion model accurately predicted strains within1stan-dard deviation of actual strain values for80%and 93%of all the conditions for maximum and minimum principal strain,respectively.Actual and predicted maximum and minimum principal strain values from allﬁve evaluation trials are shown in Table5.SinceTABLE2.Signiﬁcant coefﬁcient results using multiple regression analysis for strain in the infraspinatus tendon with supra-spinatus repair.Dependent variable Model P Model R2Glenohumeralabduction angle(b)Joint rotationangle(b)SS load(b)SS repairtechnique(b)Average maximum principal strain in IS0.0010.8520.040.030.020.47 Average minimum principal strain in IS0.0010.770.03–20.01–P=model signiﬁcance,R2=correlation coefﬁcient of model.SS Supraspinatus tendon;IS infraspinatus tendon.Multiple Regression Model of Rotator Cuff Tendon Strain2983this ﬁt for maximum principal strain in the supraspi-natus tendon is associated with a low R 2,it is likely that the inclusion of additional factors would improve the ﬁt to the data.T A B L E 3.S i g n i ﬁc a n t c o e f ﬁc i e n t (b -v a l u e s )r e s u l t s u s i n g m u l t i p l e r e g r e s s i o n a n a l y s i s f o r s t r a i n i n t h e i n f r a s p i n a t u s a n d s u p r a s p i n a t u s t e n d o n s w i t h s u p r a s p i n a t u s t e a r f o r t h e o v e r a l l m o d e l a n d ﬁv e v a l i d a t i o n t r i a l s .A s s o c i a t e d b v a l u e s f o r m a x i m u m p r i n c i p a l s t r a i nA s s o c i a t e d b v a l u e s f o r m i n i m u m p r i n c i p a l s t r a i nC o n s t a n tA b dR o tS S L o a dS S T e a rC o n s t a n tA b dR o tS S L o a dS S T e a rI n f r a s p i n a t u s t e n d o n O v e r a l l m o d e l 1.9020.050.030.0221.880.050.02A v g o f ﬁv e t r i a l s 1.90±0.0620.05±0.000.03±0.000.02±0.0021.86+0.070.05±0.0020.02±0.00S u p r a s p i n a t u s t e n d o n O v e r a l l m o d e l 2.980.04–0.06–21.90.0620.0520.0520.01A v g o f ﬁv e t r i a l s 2.85±0.100.04±0.01–0.06±0.000.00±0.0122.21±0.270.06±0.0020.05+0.0020.05+0.0020.00+0.01A b d G l e n o h u m e r a l a b d u c t i o n a n g l e ;R o t J o i n t r o t a t i o n a n g l e ;S S T e a r S S p a r t i a l -t h i c k n e s s t e a r s i z e .TABLE 4.Actual and predicted maximum and minimum principal strain values in the infraspinatus tendon for asupraspinatus tendon tear.Actual maximum principal strainPredicted maximum principal strain Actual minimum principal strain Predicted minimum principal strain Trial 11.43 1.70*21.8222.39*4.26 4.46*23.2023.41*0.1720.76***20.270.61***Trial 23.50 2.43*23.5022.32*4.60 4.41*22.3823.34*0.3220.09***20.360.11***Trial 33.47 3.89*22.9422.88*0.440.95**20.2520.93***0.380.53*20.4420.42*Trial 42.19 2.48*22.6322.34*1.733.02**22.2222.91*0.95 2.15**20.5320.87**Trial 52.293.35*21.8422.42*2.09 2.87*22.3823.44*0.731.52**20.4021.46****The predicted value is within 1standard deviation from the actual value;**the predicted values is within 2standard deviations from the actual values;***the predicted value is not a good prediction of the actual value.TABLE 5.Actual and predicted maximum and minimum principal strain values in the supraspinatus tendon for asupraspinatus tendon tear.Actual maximum principal strainPredicted maximum principal strain Actual minimum principal strain Predicted minimum principal strain Trial 15.58 4.62*23.0822.58*5.306.33*24.2625.61*7.018.04*27.4827.23*Trial 23.584.63*22.4924.06*4.077.51**21.8924.03**3.339.28***24.0725.65*Trial 35.14 4.58*24.2925.15*7.236.22*27.6326.71*8.597.88*210.2728.27*Trial 42.10 6.03***21.8622.51*9.187.41*23.4221.69*13.339.87*23.7124.04*Trial 55.94 4.52*24.1825.40*7.296.17*27.4925.58*9.027.82*25.5225.76**The predicted value is within 1standard deviation from the actualvalue;**the predicted values is within 2standard deviations from the actual values;and ***the predicted value is not a good pre-diction of the actual value.A NDARAWIS -P URI et al.2984Infraspinatus Tendon Strain with Supraspinatus Tendon RepairRandom removal of two data points at a time from the data set and re-calculating the regression coeﬃ-cients did not alter the values of the regression coeﬃ-cients calculated from the overall model.Results support removing this number of data points for model assessment.The regression coeﬃcients from all ﬁve trials are shown in Table6.For both maximum and minimum principal strain, results showed that the developed regression model accurately predicted strains.The actual and predicted maximum and minimum principal strain values from allﬁve evaluation trials are shown in Table7.Gener-ally,predicted values fell within1standard deviation of the actual value,supporting that the model very accurately predicted the removed conditions.As was found with the supraspinatus tendon tear model,for supraspinatus tendon repair,the predicted strain in the infraspinatus tendon did not accurately mirror actual strain for small actual strain values near0%.DISCUSSIONIn this study,multiple regression analysis was used to simultaneously evaluate the eﬀect of glenohumeral joint position,supraspinatus tendon load,tear and repair on both,supraspinatus and infraspinatus ten-don strain.This approach demonstrates the contribu-tion of each independent variable in the context of the remaining variables,more appropriately mimicking the complex in vivo environment where multiple factors simultaneously coexist.All independent variables evaluated were expected to be signiﬁcant predictors of supraspinatus and infraspinatus tendon strain.Interestingly,supraspina-tus tendon load and glenohumeral abduction angle were consistently signiﬁcant predictors of strain.Pre-viously,we have shown that glenohumeral abduction angle signiﬁcantly impacts supraspinatus tendon strain,5where higher strain in the supraspinatus ten-don were associated with abduction angles that mini-mized contact between the supraspinatus tendon and the humeral head.Unexpectedly,infraspinatus tendon strain dropped to negligible amounts with increase in abduction angle indicating a disruption in the inter-action between the two with increase in abduction angle.The signiﬁcance of abduction angle as predictor of both principal strain components in the supraspi-natus tendon even in the context of all other factors evaluated,emphasizes the importance of joint position in management of rotator cuff tears and postoperative care.Thisﬁnding is further supported by the effect of rotation angle since it was a signiﬁcant predictor of average maximum principal strain in the infraspinatus tendon(for both supraspinatus tendon tear and repair) and average minimum principal strain in the supra-spinatus tendon(for supraspinatus tendon tear).Previously,we have shown that introduction of supraspinatus tendon tear resulted in increase in supraspinatus and infraspinatus tendon strain.1,3 However,in the context of all other factors evaluated, supraspinatus tendon tear size was a signiﬁcant pre-dictor of minimum principal strain in the supraspina-tus tendon but of neither strain component in theTABLE6.Signiﬁcant coefﬁcient(b-values)results using multiple regression analysis for strain in the infraspinatus tendon with supraspinatus repair for the overall model and each of theﬁve validation trials.Associated b values for maximum principal strain Associated b values for minimum principalstrainConstant Abd Rot Load Repair Constant Abd Load Infraspinatus tendonOverall model0.6020.040.030.020.4721.170.0320.01 Avg.ofﬁve trials0.59±0.1220.04±0.000.03±0.000.02±0.000.49±0.0521.17±0.050.03±0.0020.01±0.00 Abd Glenohumeral abduction angle;Rot joint rotation angle;SS Tear SS partial-thickness tear size.TABLE7.Actual and predicted maximum and minimumprincipal strain values in the infraspinatus tendon for asupraspinatus tendon repair.Actual maximum principal strain PredictedmaximumprincipalstrainActualminimumprincipalstrainPredictedminimumprincipalstrainTrial1 2.93 2.48*22.6821.96*0.470.98**20.2820.67**Trial2 1.62 1.77*22.3522.46*0.2520.81***20.230.43***Trial3 1.79 2.1*21.7721.57*0.73 1.55**20.4421.09**Trial4 1.44 1.31*21.9221.99*4.52 3.61*21.9622.41*Trial5 2.39 2.92*21.1321.64*0.570.31*20.5220.05***The predicted value is within1standard deviation from the actualvalue;**the predicted values is within2standard deviations fromthe actual values;and***the predicted value is not a good pre-diction of the actual value.Multiple Regression Model of Rotator Cuff Tendon Strain2985。

斯皮尔曼相关系数英文

斯皮尔曼相关系数英文全文共四篇示例，供读者参考第一篇示例：Spearman’s rank correlation coefficient, also known as Spearman’s rho or simply Spearman’s correlation, is a statistical measure used to determine the strength and direction of a monotonic relationship between two variables. Unlike Pearson’s correlatio n coefficient, which measures the strength and direction of a linear relationship between two variables, Spearman’s correlation coefficient measures the strength and direction of a monotonic relationship between two variables.第二篇示例：Spearman's rank correlation coefficient, named after Charles Spearman, is a statistical measure of the degree of association between two variables. It is a non-parametric measure of the strength and direction of the relationship between two ranked variables. The Spearman's rank correlation coefficient is denoted by the symbol ρ.ρ = 1 - [(6 Σd^2) / (n^3 - n)]The Spearman's rank correlation coefficient ranges from -1 to 1, where:第三篇示例：Spearman's rank correlation coefficient, also known as Spearman's rho, is a statistical measure of the strength and direction of association between two ranked variables. It was named after Charles Spearman, who developed the concept in 1904.第四篇示例：Spearman's rank correlation coefficient, also known as Spearman's rho, is a statistical measure of the strength and direction of the relationship between two variables. It assesses how well the relationship between two variables can be described by a monotonic function. Unlike Pearson's correlation coefficient, which measures the linear relationship between variables, Spearman's correlation coefficient is used when the relationship between variables is not necessarily linear.rho = 1 - 6 * Σd^2 / (n^3 - n)。