Spearman's rank correlation coefficient - Wikipedia

Spearman's rank correlation coefficient - Wikipedia
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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.
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's
limits the outlier to the value of its rank.
In
statistics
Spearman's rank correlation coefficient
or
Spearman's
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 after
Charles Spearman
and often denoted by the Greek letter
{\displaystyle \rho }
(rho) or as
{\displaystyle r_{s}}
. It is a
nonparametric
measure of
rank correlation
statistical dependence
between the
rankings
of two
variables
). It assesses how well the relationship between two variables can be described using a
monotonic function
The Spearman correlation between two variables is equal to the
Pearson 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 both
continuous
and discrete
ordinal variables
Both Spearman's
{\displaystyle \rho }
and
Kendall's
{\displaystyle \tau }
can be formulated as special cases of a more
general correlation coefficient
Applications
edit
The coefficient can be used to determine how well data fits a model,
like when determining the similarity of text documents.
Definition and calculation
edit
The Spearman correlation coefficient is defined as the
Pearson correlation coefficient
between the
rank variables
For a sample of size
{\displaystyle \ n\ ,}
the
{\displaystyle \ n\ }
pairs of
raw scores
{\displaystyle \ \left(X_{i},Y_{i}\right)\ }
are converted to ranks
{\displaystyle \ \operatorname {R} [{X_{i}}],\operatorname {R} [{Y_{i}}]\ ,}
and
{\displaystyle \ r_{s}\ }
is computed as
{\displaystyle 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
{\displaystyle \operatorname {\rho } \ }
denotes the conventional
Pearson correlation coefficient operator
, but applied to the rank variables,
{\displaystyle \operatorname {\mathsf {cov}} {\bigl [}\ \operatorname {R} [X],\operatorname {R} [Y]\ {\bigr ]}\ }
is the
covariance
of the rank variables,
{\displaystyle \sigma _{\operatorname {R} [X]}\ }
and
{\displaystyle \ \sigma _{\operatorname {R} [Y]}\ }
are the
standard deviations
of the rank variables.
Only when all
{\displaystyle \ n\ }
ranks are
distinct integers
(no ties), it can be computed using the popular formula
{\displaystyle r_{s}=1-{\frac {6\sum d_{i}^{2}}{\ n\left(n^{2}-1\right)\ }}\ ,}
where
{\displaystyle d_{i}\equiv \operatorname {R} [X_{i}]-\operatorname {R} [Y_{i}]\ }
is the difference between the two ranks of each observation,
{\displaystyle \ n\ }
is the number of observations.
[Proof]
Consider a bivariate sample
{\displaystyle \ (X_{i},Y_{i})\ ,\ i=1,\ldots \ n\ }
with corresponding rank pairs
{\displaystyle \ \left(\operatorname {R} [X_{i}],\operatorname {R} [Y_{i}]\right)=(R_{i},S_{i})~.}
Then the Spearman correlation coefficient of
{\displaystyle \ (X,Y)\ }
is
{\displaystyle 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,
{\displaystyle {\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
{\displaystyle \sigma _{S}^{2}={\frac {\ 1\ }{n}}\sum _{i=1}^{n}\left(S_{i}-{\overline {S}}\right)^{2}~.}
We shall show that
{\displaystyle \ r_{s}\ }
can be expressed purely in terms of
{\displaystyle \ d_{i}\equiv R_{i}-S_{i}\ ,}
provided we assume that there be no ties within each sample.
Under this assumption, we have that
{\displaystyle \ R,S\ }
can be viewed as random variables distributed like a uniformly distributed discrete random variable
{\displaystyle U}
on
{\displaystyle \ \{\ 1,2,\ \ldots ,\ n\ \}.}
Hence
{\displaystyle \ {\overline {R}}={\overline {S}}=\operatorname {\mathbb {E} } \left[\ U\ \right]\ }
and
{\displaystyle \ \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
{\displaystyle {\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
12
{\displaystyle \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 the
triangular numbers
and
square pyramidal numbers
, or basic
summation results
from
umbral calculus
.)
Observe now that
{\displaystyle {\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
12
{\displaystyle {\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
each assigned
fractional 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
{\displaystyle \ \sigma _{\operatorname {R} [X]}\ \sigma _{\operatorname {R} [Y]}=}
{\displaystyle \ \operatorname {{\mathsf {v}}ar} {\bigl [}\ \operatorname {R} [X]\ {\bigr ]}=}
{\displaystyle \ \operatorname {{\mathsf {v}}ar} {\bigl [}\ \operatorname {R} [Y]\ {\bigr ]}=}
12
{\displaystyle \ {\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 top
records (whether by pre-change rank or post-change rank, or both), the user should use the Pearson correlation coefficient formula given above.
Related quantities
edit
Main article:
Correlation and dependence
There are several other numerical measures that quantify the extent of
statistical dependence
between pairs of observations. The most common of these is the
Pearson 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 Spearman
rank correlation
is the "grade correlation";
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 between
and
A negative Spearman correlation coefficient corresponds to a decreasing monotonic trend between
and
The sign of the Spearman correlation indicates the direction of association between
(the independent variable) and
(the dependent variable). If
tends to increase when
increases, the Spearman correlation coefficient is positive. If
tends to decrease when
increases, the Spearman correlation coefficient is negative. A Spearman correlation of zero indicates that there is no tendency for
to either increase or decrease when
increases. The Spearman correlation increases in magnitude as
and
become closer to being perfectly monotonic functions of each other. When
and
are perfectly monotonically related, the Spearman correlation coefficient becomes 1. A perfectly monotonic increasing relationship implies that for any two pairs of data values
and
, that
and
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 when
and
are related by any
monotonic function
. Contrast this with the Pearson correlation, which only gives a perfect value when
and
are related by a
linear
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 the
joint probability distribution
of
and
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 the
IQ
of a person with the number of hours spent in front of
TV
per week [fictitious values used].
IQ
{\displaystyle X_{i}}
Hours of
TV
per week,
{\displaystyle Y_{i}}
106
100
27
86
101
50
99
28
103
29
97
20
113
12
112
110
17
Firstly, evaluate
{\displaystyle d_{i}^{2}}
. To do so use the following steps, reflected in the table below.
Sort the data by the first column (
{\displaystyle X_{i}}
). Create a new column
{\displaystyle x_{i}}
and assign it the ranked values 1, 2, 3, ...,
Next, sort the augmented (with
{\displaystyle x_{i}}
) data by the second column (
{\displaystyle Y_{i}}
). Create a fourth column
{\displaystyle y_{i}}
and similarly assign it the ranked values 1, 2, 3, ...,
Create a fifth column
{\displaystyle d_{i}}
to hold the differences between the two rank columns (
{\displaystyle x_{i}}
and
{\displaystyle y_{i}}
).
Create one final column
{\displaystyle d_{i}^{2}}
to hold the value of column
{\displaystyle d_{i}}
squared.
IQ
{\displaystyle X_{i}}
Hours of
TV
per week,
{\displaystyle Y_{i}}
rank
{\displaystyle x_{i}}
rank
{\displaystyle y_{i}}
{\displaystyle d_{i}}
{\displaystyle d_{i}^{2}}
86
97
20
−4
16
99
28
−5
25
100
27
−3
101
50
10
−5
25
103
29
−3
106
16
110
17
112
49
113
12
10
36
With
{\displaystyle d_{i}^{2}}
found, add them to find
194
{\displaystyle \sum d_{i}^{2}=194}
. The value of
is 10. These values can now be substituted back into the equation
{\displaystyle \rho =1-{\frac {6\sum d_{i}^{2}}{n(n^{2}-1)}}}
to give
194
10
10
{\displaystyle \rho =1-{\frac {6\times 194}{10(10^{2}-1)}},}
which evaluates to
= −29/165 = −0.175757575...
with a
-value
= 0.627188 (using the
-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's
can be easily obtained using the Jackknife Euclidean likelihood approach in de Carvalho and Marques (2012).
14
The confidence interval with level
{\displaystyle \alpha }
is based on a Wilks' theorem given in the latter paper, and is given by
{\displaystyle \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
{\displaystyle \chi _{1,\alpha }^{2}}
is the
{\displaystyle \alpha }
quantile of a chi-square distribution with one degree of freedom, and the
{\displaystyle Z_{i}}
are jackknife pseudo-values. This approach is implemented in the R package
spearmanCI
Determining significance
edit
One approach to test whether an observed value of
is significantly different from zero (
will always maintain
−1 ≤
≤ 1
) is to calculate the probability that it would be greater than or equal to the observed
, given the
null hypothesis
, by using a
permutation 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 the
Fisher transformation
in the case of the Pearson product-moment correlation coefficient. That is,
confidence intervals
and
hypothesis tests
relating to the population value
can be carried out using the Fisher transformation:
ln
arctanh
{\displaystyle F(r)={\frac {1}{2}}\ln {\frac {1+r}{1-r}}=\operatorname {arctanh} r.}
If
) is the Fisher transformation of
, the sample Spearman rank correlation coefficient, and
is the sample size, then
1.06
{\displaystyle z={\sqrt {\frac {n-3}{1.06}}}F(r)}
is a
-score
for
, which approximately follows a standard
normal distribution
under the
null hypothesis
of
statistical independence
= 0
).
15
16
One can also test for significance using
{\displaystyle t=r{\sqrt {\frac {n-2}{1-r^{2}}}},}
which is distributed approximately as
Student's
-distribution
with
− 2
degrees of freedom under the
null 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 as
Page's trend test
for ordered alternatives.
Correspondence analysis based on Spearman's
edit
Classic
correspondence analysis
is a statistical method that gives a score to every value of two nominal variables. In this way the Pearson
correlation coefficient
between them is maximized.
There exists an equivalent of this method, called
grade correspondence analysis
, which maximizes Spearman's
or
Kendall's τ
20
Approximating Spearman's
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
{\displaystyle (X,Y)}
. For continuous
{\displaystyle X,Y}
values:
{\displaystyle m_{1},m_{2}}
cutpoints are selected for
{\displaystyle X}
and
{\displaystyle Y}
respectively, discretizing
these random variables. Default cutpoints are added at
{\displaystyle -\infty }
and
{\displaystyle \infty }
. A count matrix of size
{\displaystyle (m_{1}+1)\times (m_{2}+1)}
, denoted
{\displaystyle M}
, is then constructed where
{\displaystyle M[i,j]}
stores the number of observations that
fall into the two-dimensional cell indexed by
{\displaystyle (i,j)}
. For streaming data, when a new observation arrives, the appropriate
{\displaystyle M[i,j]}
element is incremented. The Spearman's rank
correlation can then be computed, based on the count matrix
{\displaystyle M}
, using linear algebra operations (Algorithm 2
21
). Note that for discrete random
variables, 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 on
Hermite polynomials
allow sequential estimation of the probability density function and cumulative distribution function in univariate and bivariate cases. Bivariate Hermite series density
estimators and univariate Hermite series based cumulative distribution function estimators are plugged into a large sample version of the
Spearman's rank correlation coefficient estimator, to give a sequential Spearman's correlation estimator. This estimator is phrased in
terms 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 have
certain 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 be
computed 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
's statistics base-package implements the test
cor.test(x, y, method = "spearman")
in its "stats" package (also
cor(x, y, method = "spearman")
will work). The package
spearmanCI
computes confidence intervals. The package
hermiter
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:
spearman
varlist
calculates all pairwise correlation coefficients for all variables in
varlist
MATLAB
implementation:
[r,p] = corr(x,y,'Type','Spearman')
where
is the Spearman's rank correlation coefficient,
is the p-value, and
and
are vectors.
24
Python
has many different implementations of the spearman correlation statistic: it can be computed with the
spearmanr
function of the
scipy.stats
module, as well as with the
DataFrame.corr(method='spearman')
method from the
pandas
library, and the
corr(x, y, method='spearman')
function from the statistical package
pingouin
See also
edit
Mathematics portal
Kendall tau rank correlation coefficient
Chebyshev's sum inequality
rearrangement inequality
(These two articles may shed light on the mathematical properties of Spearman's
.)
Distance correlation
Polychoric correlation
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
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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
(2):
305–
312.
doi
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Akoglu, Haldun, 2018, Review Article--User's guide to correlation coefficients: Turkish Journal of Emergency Medicine, Volume 18, Issue 3, Pages 91-93,
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.,
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
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Biometrika
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4):
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481.
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Press; Vettering; Teukolsky; Flannery (1992).
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(2nd ed.). Cambridge University Press. p. 640.
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Kendall, M. G.; Stuart, A. (1973). "Sections 31.19, 31.21".
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Page, E. B. (1963). "Ordered hypotheses for multiple treatments: A significance test for linear ranks".
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58
(301):
216–
230.
doi
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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
Xiao, 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
Stephanou, 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
Stephanou, Michaeal; Varughese, Melvin (2023). "Hermiter: R package for sequential nonparametric estimation".
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doi
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www.mathworks.com
Further reading
edit
Corder, G. W. & Foreman, D. I. (2014). Nonparametric Statistics: A Step-by-Step Approach, Wiley.
ISBN
978-1118840313
Daniel, Wayne W. (1990).
"Spearman rank correlation coefficient"
Applied Nonparametric Statistics
(2nd ed.). Boston: PWS-Kent. pp.
358–
365.
ISBN
978-0-534-91976-4
Spearman C. (1904).
"The proof and measurement of association between two things"
American Journal of Psychology
15
(1):
72–
101.
doi
10.2307/1412159
JSTOR
1412159
Bonett, D. G.; Wright, T. A. (2000). "Sample size requirements for Pearson, Kendall, and Spearman correlations".
Psychometrika
65
23–
28.
doi
10.1007/bf02294183
S2CID
120558581
Kendall M. G. (1970).
Rank correlation methods
(4th ed.). London: Griffin.
ISBN
978-0-852-6419-96
OCLC
136868
Hollander M., Wolfe D. A. (1973).
Nonparametric statistical methods
. New York: Wiley.
ISBN
978-0-471-40635-8
OCLC
520735
Caruso J. C., Cliff N. (1997). "Empirical size, coverage, and power of confidence intervals for Spearman's Rho".
Educational and Psychological Measurement
57
(4):
637–
654.
doi
10.1177/0013164497057004009
S2CID
120481551
External links
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Table of critical values of
for significance with small samples
Spearman's Rank Correlation Coefficient – Excel Guide
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