Movatterモバイル変換

[0]ホーム

Jump to content

Spearman's rank correlation coefficient

Edit links

From Wikipedia, the free encyclopedia

Nonparametric measure of rank correlation

A Spearman correlation of ${\textstyle 1}$ results when the two variables being compared are monotonically related, even if their relationship is not linear. This means that all data points with greater ${\textstyle x}$ values than that of a given data point will have greater ${\textstyle y}$ values as well. In contrast, this does not give a perfect Pearson correlation.

A Spearman correlation of ${\textstyle 1}$ results when the two variables being compared are monotonically related, even if their relationship is not linear. This means that all data points with greater ${\textstyle x}$ values than that of a given data point will have greater ${\textstyle y}$ values as well. In contrast, this does not give a perfect Pearson correlation.

When the data are roughly elliptically distributed and there are no prominent outliers, the Spearman correlation and Pearson correlation give similar values.

The Spearman correlation is less sensitive than the Pearson correlation to strong outliers that are in the tails of both samples. That is because Spearman'sρ limits the outlier to the value of its rank.

Instatistics,Spearman's rank correlation coefficient orSpearman'sρ is a number ranging from -1 to 1 that indicates how strongly two sets of ranks are correlated. It could be used in a situation where one only has ranked data, such as a tally of gold, silver, and bronze medals. If a statistician wanted to know whether people who are high ranking in sprinting are also high ranking in long-distance running, they would use a Spearman rank correlation coefficient.

The coefficient is named afterCharles Spearman^[1] and often denoted by the Greek letter $\rho$ (rho) or as $r_{s}$ . It is anonparametric measure ofrank correlation (statistical dependence between therankings of twovariables). It assesses how well the relationship between two variables can be described using amonotonic function.

The Spearman correlation between two variables is equal to thePearson correlation between the rank values of those two variables; while Pearson's correlation assesses linear relationships, Spearman's correlation assesses monotonic relationships (whether linear or not). If there are no repeated data values, a perfect Spearman correlation of +1 or −1 occurs when each of the variables is a perfect monotone function of the other.

Intuitively, the Spearman correlation between two variables will be high when observations have a similar (or identical for a correlation of 1)rank (i.e. relative position label of the observations within the variable: 1st, 2nd, 3rd, etc.) between the two variables, and low when observations have a dissimilar (or fully opposed for a correlation of −1) rank between the two variables.

Spearman's coefficient is appropriate for bothcontinuous and discreteordinal variables.^[2]^[3] Both Spearman's $\rho$ andKendall's $\tau$ can be formulated as special cases of a moregeneral correlation coefficient.

Applications

[edit]

The coefficient can be used to determine how well data fits a model,^[4] like when determining the similarity of text documents.^[5]

Definition and calculation

[edit]

The Spearman correlation coefficient is defined as thePearson correlation coefficient between therank variables.^[6]

For a sample of size $\ n\ ,$ the $\ n\$ pairs ofraw scores $\ \left(X_{i},Y_{i}\right)\$ are converted to ranks $\ \operatorname {R} [{X_{i}}],\operatorname {R} [{Y_{i}}]\ ,$ and $\ r_{s}\$ is computed as

r_{s}=\operatorname {\rho } {\bigl [}\ \operatorname {R} [X],\operatorname {R} [Y]\ {\bigr ]}={\frac {\ \operatorname {\mathsf {cov}} {\bigl [}\ \operatorname {R} [X],\operatorname {R} [Y]\ {\bigr ]}\ }{\ \sigma _{\operatorname {R} [X]}\ \sigma _{\operatorname {R} [Y]}\ }},

where

\operatorname {\rho } \

denotes the conventionalPearson correlation coefficient operator, but applied to the rank variables,

\operatorname {\mathsf {cov}} {\bigl [}\ \operatorname {R} [X],\operatorname {R} [Y]\ {\bigr ]}\

is thecovariance of the rank variables,

\sigma _{\operatorname {R} [X]}\

and

\ \sigma _{\operatorname {R} [Y]}\

are thestandard deviations of the rank variables.

Only when all $\ n\$ ranks aredistinct integers (no ties), it can be computed using the popular formula

r_{s}=1-{\frac {6\sum d_{i}^{2}}{\ n\left(n^{2}-1\right)\ }}\ ,

where

d_{i}\equiv \operatorname {R} [X_{i}]-\operatorname {R} [Y_{i}]\

is the difference between the two ranks of each observation,

\ n\

is the number of observations.

[Proof]

Consider a bivariate sample $\ (X_{i},Y_{i})\ ,\ i=1,\ldots \ n\$ with corresponding rank pairs $\ \left(\operatorname {R} [X_{i}],\operatorname {R} [Y_{i}]\right)=(R_{i},S_{i})~.$ Then the Spearman correlation coefficient of $\ (X,Y)\$ is

r_{s}={\frac {{\frac {\ 1\ }{n}}\ \sum _{i=1}^{n}R_{i}\ S_{i}-{\overline {R}}\ {\overline {S}}}{\sigma _{R}\sigma _{S}}}\ ,

where, as usual,

{\begin{aligned}{\overline {R}}&={\frac {\ 1\ }{n}}\sum _{i=1}^{n}R_{i},\\[6pt]{\overline {S}}&={\frac {\ 1\ }{n}}\sum _{i=1}^{n}S_{i},\\[6pt]\sigma _{R}^{2}&={\frac {\ 1\ }{n}}\sum _{i=1}^{n}\left(R_{i}-{\overline {R}}\right)^{2},\end{aligned}}

and

\sigma _{S}^{2}={\frac {\ 1\ }{n}}\sum _{i=1}^{n}\left(S_{i}-{\overline {S}}\right)^{2}~.

We shall show that $\ r_{s}\$ can be expressed purely in terms of $\ d_{i}\equiv R_{i}-S_{i}\ ,$ provided we assume that there be no ties within each sample.

Under this assumption, we have that $\ R,S\$ can be viewed as random variables distributed like a uniformly distributed discrete random variable $U {\displaystyle U}$ on $\ \{\ 1,2,\ \ldots ,\ n\ \}.$ Hence $\ {\overline {R}}={\overline {S}}=\operatorname {\mathbb {E} } \left[\ U\ \right]\$ and $\ \sigma _{R}^{2}=\sigma _{S}^{2}=\operatorname {\mathsf {Var}} \left[\ U\ \right]=\operatorname {\mathbb {E} } [U^{2}]-\operatorname {\mathbb {E} } \left[\ U\ \right]^{2}\ ,$ where

{\begin{aligned}\operatorname {\mathbb {E} } [U]&={\frac {\ 1\ }{n}}\sum _{i=1}^{n}i={\frac {\ n+1\ }{2}},\\[6pt]\operatorname {\mathbb {E} } [U^{2}]&={\frac {\ 1\ }{n}}\sum _{i=1}^{n}i^{2}={\frac {\ (n+1)(2n+1)\ }{6}},\end{aligned}}

and thus

\operatorname {\mathsf {var}} \left[\ U\right]={\frac {\ (n+1)\ (2n+1)\ }{6}}-\left({\frac {\ n+1\ }{2}}\right)^{2}={\frac {\ n^{2}-1\ }{12}}~.

(These sums can be computed using the formulas for thetriangular numbers andsquare pyramidal numbers, or basicsummation results fromumbral calculus.)

Observe now that

{\begin{aligned}{\frac {\ 1\ }{n}}\ &\sum _{i=1}^{n}R_{i}S_{i}-{\overline {R}}{\overline {S}}\\[6pt]&={\frac {\ 1\ }{n}}\ \sum _{i=1}^{n}{\frac {\ 1\ }{2}}(R_{i}^{2}+S_{i}^{2}-d_{i}^{2})-{\overline {R}}^{2}\\[6pt]&={\frac {\ 1\ }{2}}{\frac {\ 1\ }{n}}\ \sum _{i=1}^{n}R_{i}^{2}+{\frac {\ 1\ }{2}}{\frac {\ 1\ }{n}}\ \sum _{i=1}^{n}S_{i}^{2}-{\frac {\ 1\ }{2n}}\ \sum _{i=1}^{n}d_{i}^{2}-{\overline {R}}^{2}\\[6pt]&=\left({\frac {\ 1\ }{n}}\ \sum _{i=1}^{n}R_{i}^{2}-{\overline {R}}^{2}\right)-{\frac {\ 1\ }{2n}}\ \sum _{i=1}^{n}d_{i}^{2}\\[6pt]&=\sigma _{R}^{2}-{\frac {\ 1\ }{2n}}\ \sum _{i=1}^{n}d_{i}^{2}\\[6pt]&=\sigma _{R}\ \sigma _{S}-{\frac {\ 1\ }{2n}}\ \sum _{i=1}^{n}d_{i}^{2}\end{aligned}}

Putting this all together thus yields

{\begin{aligned}r_{s}&={\frac {\ \sigma _{R}\ \sigma _{S}-{\frac {\ 1\ }{2n}}\ \sum _{i=1}^{n}d_{i}^{2}\ }{\sigma _{R}\ \sigma _{S}}}\\[6pt]&=1-{\frac {\ \sum _{i=1}^{n}d_{i}^{2}\ }{2n\cdot {\frac {\ n^{2}-1\ }{12}}}}\\[6pt]&=1-{\frac {\ 6\ \sum _{i=1}^{n}d_{i}^{2}}{\ n(n^{2}-1)\ }}~.\end{aligned}}

Identical values are usually^[7] each assignedfractional ranks equal to the average of their positions in the ascending order of the values, which is equivalent to averaging over all possible permutations.

If ties are present in the data set, the simplified formula above yields incorrect results: Only if in both variables all ranks are distinct, then $\ \sigma _{\operatorname {R} [X]}\ \sigma _{\operatorname {R} [Y]}=$ $\ \operatorname {{\mathsf {v}}ar} {\bigl [}\ \operatorname {R} [X]\ {\bigr ]}=$ $\ \operatorname {{\mathsf {v}}ar} {\bigl [}\ \operatorname {R} [Y]\ {\bigr ]}=$ $\ {\tfrac {\ 1\ }{12}}\left(n^{2}-1\right)\$ (calculated according to biased variance).The first equation — normalizing by the standard deviation — may be used even when ranks are normalized to [0, 1] ("relative ranks") because it is insensitive both to translation and linear scaling.

The simplified method should also not be used in cases where the data set is truncated; that is, when the Spearman's correlation coefficient is desired for the topX records (whether by pre-change rank or post-change rank, or both), the user should use the Pearson correlation coefficient formula given above.^[8]

Related quantities

[edit]

Main article:Correlation and dependence

There are several other numerical measures that quantify the extent ofstatistical dependence between pairs of observations. The most common of these is thePearson product-moment correlation coefficient, which is a similar correlation method to Spearman's rank, that measures the "linear" relationships between the raw numbers rather than between their ranks.

An alternative name for the Spearmanrank correlation is the "grade correlation";^[9] in this, the "rank" of an observation is replaced by the "grade". In continuous distributions, the grade of an observation is, by convention, always one half less than the rank, and hence the grade and rank correlations are the same in this case. More generally, the "grade" of an observation is proportional to an estimate of the fraction of a population less than a given value, with the half-observation adjustment at observed values. Thus this corresponds to one possible treatment of tied ranks. While unusual, the term "grade correlation" is still in use.^[10]

Interpretation

[edit]

Positive and negative Spearman rank correlations

A positive Spearman correlation coefficient corresponds to an increasing monotonic trend betweenX andY.

A negative Spearman correlation coefficient corresponds to a decreasing monotonic trend betweenX andY.

The sign of the Spearman correlation indicates the direction of association betweenX (the independent variable) andY (the dependent variable). IfY tends to increase whenX increases, the Spearman correlation coefficient is positive. IfY tends to decrease whenX increases, the Spearman correlation coefficient is negative. A Spearman correlation of zero indicates that there is no tendency forY to either increase or decrease whenX increases. The Spearman correlation increases in magnitude asX andY become closer to being perfectly monotonic functions of each other. WhenX andY are perfectly monotonically related, the Spearman correlation coefficient becomes 1. A perfectly monotonic increasing relationship implies that for any two pairs of data valuesX_i,Y_i andX_j,Y_j, thatX_i −X_j andY_i −Y_j always have the same sign. A perfectly monotonic decreasing relationship implies that these differences always have opposite signs.

The Spearman correlation coefficient is often described as being "nonparametric". This can have two meanings. First, a perfect Spearman correlation results whenX andY are related by anymonotonic function. Contrast this with the Pearson correlation, which only gives a perfect value whenX andY are related by alinear function. The other sense in which the Spearman correlation is nonparametric is that its exact sampling distribution can be obtained without requiring knowledge (i.e., knowing the parameters) of thejoint probability distribution ofX andY.

The strength of correlations also are an important consideration for interpretation of correlation analyses but descriptions of the strength of correlation are not universally accepted. A small correlation coefficient calculated for a very large sample may be statistically significant without providing a quantitative relation between variables. Similarly, a large correlation coefficient calculated with a small sample size may indicate that a quantitative equation between variables may be possible even though the correlation coefficient is not statistically significant. Akoglu (2018)^[11] noted the need for consistent descriptions of strength and provides different correlation-strength descriptors for psychology, politics, and medicine that have different descriptors and thresholds. Because there is a general lack of consensus on correlation strength, Granato (2014)^[12] defined operational definitions for the absolute values of correlation coefficients as weak (less than 0.5), moderate (greater than or equal to 0.5 and less than 0.75), semi-strong (greater than or equal to 0.75 and less than 0.85), and strong (greater than or equal to 0.85) for use in hydrologic analyses of stormwater treatment statistics. Similarly, Schober and others (2018)^[13] note that "...cutoff points are arbitrary and inconsistent and should be used judiciously..." Despite their warning Schober and others (2018) provide a table indicating that correlations less than or equal to 0.1 are negligible; correlations greater than 0.1 and less than or equal to 0.39 are weak; correlations greater than 0.39 and less than or equal to 0.69 are moderate; correlations greater than 0.69 and less than or equal to 0.89 are strong; and correlations greater than 0.89 are very strong. Descriptions of strength indicate the potential to develop quantitative relations among variables but these descriptors, as with the correlation coefficients themselves, do not indicate causation.

Example

[edit]

In this example, the arbitrary raw data in the table below is used to calculate the correlation between theIQ of a person with the number of hours spent in front ofTV per week [fictitious values used].

IQ, $X_{i}$	Hours ofTV per week, $Y_{i}$
106	7
100	27
86	2
101	50
99	28
103	29
97	20
113	12
112	6
110	17

Firstly, evaluate $d_{i}^{2}$ . To do so use the following steps, reflected in the table below.

Sort the data by the first column ( $X_{i}$ ). Create a new column $x_{i}$ and assign it the ranked values 1, 2, 3, ...,n.
Next, sort the augmented (with $x_{i}$ ) data by the second column ( $Y_{i}$ ). Create a fourth column $y_{i}$ and similarly assign it the ranked values 1, 2, 3, ...,n.
Create a fifth column $d_{i}$ to hold the differences between the two rank columns ( $x_{i}$ and $y_{i}$ ).
Create one final column $d_{i}^{2}$ to hold the value of column $d_{i}$ squared.

IQ, $X_{i}$	Hours ofTV per week, $Y_{i}$	rank $x_{i}$	rank $y_{i}$	$d_{i}$	$d_{i}^{2}$
86	2	1	1	0	0
97	20	2	6	−4	16
99	28	3	8	−5	25
100	27	4	7	−3	9
101	50	5	10	−5	25
103	29	6	9	−3	9
106	7	7	3	4	16
110	17	8	5	3	9
112	6	9	2	7	49
113	12	10	4	6	36

With $d_{i}^{2}$ found, add them to find $\sum d_{i}^{2}=194$ . The value ofn is 10. These values can now be substituted back into the equation

\rho =1-{\frac {6\sum d_{i}^{2}}{n(n^{2}-1)}}

to give

\rho =1-{\frac {6\times 194}{10(10^{2}-1)}},

which evaluates toρ = −29/165 = −0.175757575... with ap-value = 0.627188 (using thet-distribution).

Chart of the data presented. It can be seen that there might be a negative correlation, but that the relationship does not appear definitive.

That the value is close to zero shows that the correlation between IQ and hours spent watching TV is very low, although the negative value suggests that the longer the time spent watching television the lower the IQ. In the case of ties in the original values, this formula should not be used; instead, the Pearson correlation coefficient should be calculated on the ranks (where ties are given ranks, as described above).

Confidence intervals

[edit]

Confidence intervals for Spearman'sρ can be easily obtained using the Jackknife Euclidean likelihood approach in de Carvalho and Marques (2012).^[14] The confidence interval with level $\alpha$ is based on a Wilks' theorem given in the latter paper, and is given by

\left\{\theta :{\frac {\{\sum _{i=1}^{n}(Z_{i}-\theta )\}^{2}}{\sum _{i=1}^{n}(Z_{i}-\theta )^{2}}}\leq \chi _{1,\alpha }^{2}\right\},

where $\chi _{1,\alpha }^{2}$ is the $\alpha$ quantile of a chi-square distribution with one degree of freedom, and the $Z_{i}$ are jackknife pseudo-values. This approach is implemented in the R packagespearmanCI.

Determining significance

[edit]

One approach to test whether an observed value ofρ is significantly different from zero (r will always maintain−1 ≤r ≤ 1) is to calculate the probability that it would be greater than or equal to the observedr, given thenull hypothesis, by using apermutation test. An advantage of this approach is that it automatically takes into account the number of tied data values in the sample and the way they are treated in computing the rank correlation.

Another approach parallels the use of theFisher transformation in the case of the Pearson product-moment correlation coefficient. That is,confidence intervals andhypothesis tests relating to the population valueρ can be carried out using the Fisher transformation:

F(r)={\frac {1}{2}}\ln {\frac {1+r}{1-r}}=\operatorname {arctanh} r.

IfF(r) is the Fisher transformation ofr, the sample Spearman rank correlation coefficient, andn is the sample size, then

z={\sqrt {\frac {n-3}{1.06}}}F(r)

is az-score forr, which approximately follows a standardnormal distribution under thenull hypothesis ofstatistical independence (ρ = 0).^[15]^[16]

One can also test for significance using

t=r{\sqrt {\frac {n-2}{1-r^{2}}}},

which is distributed approximately asStudent'st-distribution withn − 2 degrees of freedom under thenull hypothesis.^[17] A justification for this result relies on a permutation argument.^[18]

A generalization of the Spearman coefficient is useful in the situation where there are three or more conditions, a number of subjects are all observed in each of them, and it is predicted that the observations will have a particular order. For example, a number of subjects might each be given three trials at the same task, and it is predicted that performance will improve from trial to trial. A test of the significance of the trend between conditions in this situation was developed by E. B. Page^[19] and is usually referred to asPage's trend test for ordered alternatives.

Correspondence analysis based on Spearman'sρ

[edit]

Classiccorrespondence analysis is a statistical method that gives a score to every value of two nominal variables. In this way the Pearsoncorrelation coefficient between them is maximized.

There exists an equivalent of this method, calledgrade correspondence analysis, which maximizes Spearman'sρ orKendall's τ.^[20]

Approximating Spearman'sρ from a stream

[edit]

There are two existing approaches to approximating the Spearman's rank correlation coefficient from streaming data.^[21]^[22] The first approach^[21]involves coarsening the joint distribution of $(X,Y)$ . For continuous $X, Y {\displaystyle X,Y}$ values: $m_{1},m_{2}$ cutpoints are selected for $X {\displaystyle X}$ and $Y {\displaystyle Y}$ respectively, discretizingthese random variables. Default cutpoints are added at $-\infty$ and $\infty$ . A count matrix of size $(m_{1}+1)\times (m_{2}+1)$ , denoted $M {\displaystyle M}$ , is then constructed where $M[i,j]$ stores the number of observations thatfall into the two-dimensional cell indexed by $(i,j)$ . For streaming data, when a new observation arrives, the appropriate $M[i,j]$ element is incremented. The Spearman's rankcorrelation can then be computed, based on the count matrix $M {\displaystyle M}$ , using linear algebra operations (Algorithm 2^[21]). Note that for discrete randomvariables, no discretization procedure is necessary. This method is applicable to stationary streaming data as well as large data sets. For non-stationary streaming data, where the Spearman's rank correlation coefficient may change over time, the same procedure can be applied, but to a moving window of observations. When using a moving window, memory requirements grow linearly with chosen window size.

The second approach to approximating the Spearman's rank correlation coefficient from streaming data involves the use of Hermite series based estimators.^[22] These estimators, based onHermite polynomials,allow sequential estimation of the probability density function and cumulative distribution function in univariate and bivariate cases. Bivariate Hermite series densityestimators and univariate Hermite series based cumulative distribution function estimators are plugged into a large sample version of theSpearman's rank correlation coefficient estimator, to give a sequential Spearman's correlation estimator. This estimator is phrased interms of linear algebra operations for computational efficiency (equation (8) and algorithm 1 and 2^[22]). These algorithms are only applicable to continuous random variable data, but havecertain advantages over the count matrix approach in this setting. The first advantage is improved accuracy when applied to large numbers of observations. The second advantage is that the Spearman's rank correlation coefficient can becomputed on non-stationary streams without relying on a moving window. Instead, the Hermite series based estimator uses an exponential weighting scheme to track time-varying Spearman's rank correlation from streaming data,which has constant memory requirements with respect to "effective" moving window size. A software implementation of these Hermite series based algorithms exists^[23] and is discussed in Software implementations.

Software implementations

[edit]

R's statistics base-package implements the testcor.test(x, y, method = "spearman") in its "stats" package (alsocor(x, y, method = "spearman") will work). The packagespearmanCI computes confidence intervals. The packagehermiter^[23] computes fast batch estimates of the Spearman correlation along with sequential estimates (i.e. estimates that are updated in an online/incremental manner as new observations are incorporated).
Stata implementation: spearmanvarlist calculates all pairwise correlation coefficients for all variables invarlist.
MATLAB implementation:[r,p] = corr(x,y,'Type','Spearman') wherer is the Spearman's rank correlation coefficient,p is the p-value, andx andy are vectors.^[24]
Python has many different implementations of the spearman correlation statistic: it can be computed with thespearmanr function of thescipy.stats module, as well as with theDataFrame.corr(method='spearman') method from thepandas library, and thecorr(x, y, method='spearman') function from the statistical packagepingouin.

References

[edit]

^Spearman, C. (January 1904)."The Proof and Measurement of Association between Two Things"(PDF).The American Journal of Psychology.15 (1):72–101.doi:10.2307/1412159.JSTOR 1412159.
^Scale types.
^Lehman, Ann (2005).Jmp For Basic Univariate And Multivariate Statistics: A Step-by-step Guide. Cary, NC: SAS Press. p. 123.ISBN 978-1-59047-576-8.
^Royal Geographic Society."A Guide to Spearman's Rank"(PDF).
^Nino Arsov; Milan Dukovski; Milan Dukovski; Blagoja Evkoski (November 2019)."A Measure of Similarity in Textual Data Using Spearman's Rank Correlation Coefficient".
^Myers, Jerome L.; Well, Arnold D. (2003).Research Design and Statistical Analysis (2nd ed.). Lawrence Erlbaum. pp. 508.ISBN 978-0-8058-4037-7.
^Dodge, Yadolah, ed. (2010).The Concise Encyclopedia of Statistics. New York, NY: Springer-Verlag. p. 502.ISBN 978-0-387-31742-7.
^al Jaber, Ahmed Odeh; Elayyan, Haifaa Omar (2018).Toward Quality Assurance and Excellence in Higher Education. River Publishers. p. 284.ISBN 978-87-93609-54-9.
^Yule, G. U.; Kendall, M. G. (1968) [1950].An Introduction to the Theory of Statistics (14th ed.). Charles Griffin & Co. p. 268.
^Piantadosi, J.; Howlett, P.; Boland, J. (2007)."Matching the grade correlation coefficient using a copula with maximum disorder".Journal of Industrial and Management Optimization.3 (2):305–312.doi:10.3934/jimo.2007.3.305.
^Akoglu, Haldun, 2018, Review Article--User's guide to correlation coefficients: Turkish Journal of Emergency Medicine, Volume 18, Issue 3, Pages 91-93,https://doi.org/10.1016/j.tjem.2018.08.001
^Granato, G.E., 2014, Statistics for stochastic modeling of volume reduction, hydrograph extension, and water-quality treatment by structural stormwater runoff best management practices (BMPs): U.S. Geological Survey Scientific Investigations Report 2014–5037, 37 p.,https://dx.doi.org/10.3133/sir20145037.
^Schober, Patrick, Boer, Christa, Schwarte, Lothar, 2018, Correlation coefficients: Appropriate use and interpretation: Anesthesia & Analgesia 126(5):p 1763-1768, 10.1213/ANE.0000000000002864
^de Carvalho, M.; Marques, F. (2012)."Jackknife Euclidean likelihood-based inference for Spearman's rho"(PDF).North American Actuarial Journal.16 (4): 487‒492.doi:10.1080/10920277.2012.10597644.S2CID 55046385.
^Choi, S. C. (1977). "Tests of Equality of Dependent Correlation Coefficients".Biometrika.64 (3):645–647.doi:10.1093/biomet/64.3.645.
^Fieller, E. C.; Hartley, H. O.; Pearson, E. S. (1957). "Tests for rank correlation coefficients. I".Biometrika.44 (3–4):470–481.CiteSeerX 10.1.1.474.9634.doi:10.1093/biomet/44.3-4.470.
^Press; Vettering; Teukolsky; Flannery (1992).Numerical Recipes in C: The Art of Scientific Computing (2nd ed.). Cambridge University Press. p. 640.ISBN 9780521437202.
^Kendall, M. G.; Stuart, A. (1973). "Sections 31.19, 31.21".The Advanced Theory of Statistics, Volume 2: Inference and Relationship. Griffin.ISBN 978-0-85264-215-3.
^Page, E. B. (1963). "Ordered hypotheses for multiple treatments: A significance test for linear ranks".Journal of the American Statistical Association.58 (301):216–230.doi:10.2307/2282965.JSTOR 2282965.
^Kowalczyk, T.; Pleszczyńska, E.; Ruland, F., eds. (2004).Grade Models and Methods for Data Analysis with Applications for the Analysis of Data Populations. Studies in Fuzziness and Soft Computing. Vol. 151. Berlin Heidelberg New York: Springer Verlag.ISBN 978-3-540-21120-4.
^^a ^b ^cXiao, W. (2019). "Novel Online Algorithms for Nonparametric Correlations with Application to Analyze Sensor Data".2019 IEEE International Conference on Big Data (Big Data). pp. 404–412.doi:10.1109/BigData47090.2019.9006483.ISBN 978-1-7281-0858-2.S2CID 211298570.
^^a ^b ^cStephanou, Michael; Varughese, Melvin (July 2021). "Sequential estimation of Spearman rank correlation using Hermite series estimators".Journal of Multivariate Analysis.186 104783.arXiv:2012.06287.doi:10.1016/j.jmva.2021.104783.S2CID 235742634.
^^a ^bStephanou, Michaeal; Varughese, Melvin (2023). "Hermiter: R package for sequential nonparametric estimation".Computational Statistics.39 (3):1127–1163.arXiv:2111.14091.doi:10.1007/s00180-023-01382-0.S2CID 244715035.
^"Linear or rank correlation - MATLAB corr".www.mathworks.com.

External links

[edit]

Wikiversity has learning resources aboutSpearman's rank correlation coefficient

Table of critical values ofρ for significance with small samples
Spearman's Rank Correlation Coefficient – Excel Guide: sample data and formulae for Excel, developed by theRoyal Geographical Society.

Statistics

Descriptive statistics

Continuous data

Center	Mean Arithmetic Arithmetic-Geometric Contraharmonic Cubic Generalized/power Geometric Harmonic Heronian Heinz Lehmer Median Mode
Dispersion	Average absolute deviation Coefficient of variation Interquartile range Percentile Range Standard deviation Variance
Shape	Central limit theorem Moments Kurtosis L-moments Skewness

Count data

Index of dispersion

Summary tables

Dependence

Graphics

Data collection

Study design	Effect size Missing data Optimal design Population Replication Sample size determination Statistic Statistical power
Survey methodology	Sampling Cluster Stratified Opinion poll Questionnaire Standard error
Controlled experiments	Blocking Factorial experiment Interaction Random assignment Randomized controlled trial Randomized experiment Scientific control
Adaptive designs	Adaptive clinical trial Stochastic approximation Up-and-down designs
Observational studies	Cohort study Cross-sectional study Natural experiment Quasi-experiment

Statistical inference

Statistical theory

Frequentist inference

Point estimation	Estimating equations Maximum likelihood Method of moments M-estimator Minimum distance Unbiased estimators Mean-unbiased minimum-variance Rao–Blackwellization Lehmann–Scheffé theorem Median unbiased Plug-in
Interval estimation	Confidence interval Pivot Likelihood interval Prediction interval Tolerance interval Resampling Bootstrap Jackknife
Testing hypotheses	1- & 2-tails Power Uniformly most powerful test Permutation test Randomization test Multiple comparisons
Parametric tests	Likelihood-ratio Score/Lagrange multiplier Wald

Specific tests

Z-test(normal) Student'st-test F-test
Goodness of fit	Chi-squared G-test Kolmogorov–Smirnov Anderson–Darling Lilliefors Jarque–Bera Normality(Shapiro–Wilk) Likelihood-ratio test Model selection Cross validation AIC BIC
Rank statistics	Sign Sample median Signed rank(Wilcoxon) Hodges–Lehmann estimator Rank sum(Mann–Whitney) Nonparametric anova 1-way(Kruskal–Wallis) 2-way(Friedman) Ordered alternative(Jonckheere–Terpstra) Van der Waerden test

Bayesian inference

Correlation	Pearson product-moment Partial correlation Confounding variable Coefficient of determination
Regression analysis	Errors and residuals Regression validation Mixed effects models Simultaneous equations models Multivariate adaptive regression splines (MARS) Template:Least squares and regression analysis
Linear regression	Simple linear regression Ordinary least squares General linear model Bayesian regression
Non-standard predictors	Nonlinear regression Nonparametric Semiparametric Isotonic Robust Homoscedasticity and Heteroscedasticity
Generalized linear model	Exponential families Logistic(Bernoulli) / Binomial / Poisson regressions
Partition of variance	Analysis of variance (ANOVA, anova) Analysis of covariance Multivariate ANOVA Degrees of freedom

Categorical / multivariate / time-series / survival analysis

Categorical

Multivariate

Time-series

General	Decomposition Trend Stationarity Seasonal adjustment Exponential smoothing Cointegration Structural break Granger causality
Specific tests	Dickey–Fuller Johansen Q-statistic(Ljung–Box) Durbin–Watson Breusch–Godfrey
Time domain	Autocorrelation (ACF) partial (PACF) Cross-correlation (XCF) ARMA model ARIMA model(Box–Jenkins) Autoregressive conditional heteroskedasticity (ARCH) Vector autoregression (VAR) (Autoregressive model (AR))
Frequency domain	Spectral density estimation Fourier analysis Least-squares spectral analysis Wavelet Whittle likelihood

Survival

Survival function	Kaplan–Meier estimator (product limit) Proportional hazards models Accelerated failure time (AFT) model First hitting time
Hazard function	Nelson–Aalen estimator
Test	Log-rank test

Applications

Biostatistics	Bioinformatics Clinical trials / studies Epidemiology Medical statistics
Engineering statistics	Chemometrics Methods engineering Probabilistic design Process / quality control Reliability System identification
Social statistics	Actuarial science Census Crime statistics Demography Econometrics Jurimetrics National accounts Official statistics Population statistics Psychometrics
Spatial statistics	Cartography Environmental statistics Geographic information system Geostatistics Kriging