查看“︁一致性相關係數”︁的源代码

{{Expert needed|subject=統計學}}

'''一致性相關係數'''是在統計學中評估兩個變數之間的一致性（例如再現性或[[評分者間信度]]）的一種方法。

== 定義 ==
一致性相關係數（<math>\rho_c</math>）定義如下：<ref name="LinL1989Concordance">{{Cite journal |last=Lawrence I-Kuei Lin |author-link=Lawrence I-Kuei Lin |title=A concordance correlation coefficient to evaluate reproducibility |url=https://archive.org/details/sim_biometrics_1989-03_45_1/page/n263 |journal=[[Biometrics (journal)|Biometrics]] |date=March 1989 |volume=45 |issue=1 |page=255&ndash;268 |doi=10.2307/2532051 |jstor=2532051 |pmid=2720055}}</ref>

: <math>\rho_c = \frac{2\rho\sigma_x\sigma_y}{\sigma_x^2 + \sigma_y^2 + (\mu_x - \mu_y)^2},</math>

其中，<math>\mu_x</math>和<math>\mu_y</math>分別是各組變量的平[[平均数|均值]]， <math>\sigma^2_x</math>和<math>\sigma^2_y</math>是相對應的[[方差|變異數]]。 <math>\rho</math>是两个变量之间的[[皮尔逊积矩相关系数|相關係數]]。

這是因為，根據定義：<ref name="LinL1989Concordance">{{Cite journal |last=Lawrence I-Kuei Lin |author-link=Lawrence I-Kuei Lin |title=A concordance correlation coefficient to evaluate reproducibility |journal=[[Biometrics (journal)|Biometrics]] |date=March 1989 |volume=45 |issue=1 |page=255&ndash;268 |doi=10.2307/2532051 |jstor=2532051 |pmid=2720055}}<cite class="citation journal cs1" data-ve-ignore="true" id="CITEREFLawrence_I-Kuei_Lin1989">[[Lawrence I-Kuei Lin]] (March 1989). "A concordance correlation coefficient to evaluate reproducibility". ''[[Biometrics (journal)|Biometrics]]''. '''45''' (1): 255–268. [[DOI|doi]]:[[doi:10.2307/2532051|10.2307/2532051]]. [[JSTOR]]&nbsp;[https://www.jstor.org/stable/2532051 2532051]. [[PubMed|PMID]]&nbsp;[https://pubmed.ncbi.nlm.nih.gov/2720055 2720055].</cite></ref>

: <math>\rho_c = 1 - \frac{{\rm Expected\ orthogonal\ squared\ distance\ from\ the\ diagonal\ }x=y}
{{\rm Expected\ orthogonal\ squared\ distance\ from\ the\ diagonal\ }x=y{\rm \ assuming\ independence}}.</math>

如有<math>N</math>對數據<math>(x_n, y_n)</math> ，<math>n=1,...,N</math> ，則一致性相關係數為：

: <math>\hat{\rho}_c = \frac{2 s_{xy}}{s_x^2 + s_y^2 + (\bar{x} - \bar{y})^2},</math>

其中平均值计算方式如下

: <math>\bar{x} = \frac{1}{N} \sum_{n=1}^N x_n</math>

變異數計算方式如下：

: <math>s_x^2 = \frac{1}{N} \sum_{n=1}^N (x_n - \bar{x})^2</math>

共變異數計算方式如下：

: <math>s_{xy} = \frac{1}{N} \sum_{n=1}^N (x_n - \bar{x})(y_n - \bar{y}) .</math>

普通[[皮尔逊积矩相关系数|相关系数]]（Pearson 相關係數）並不受使用有偏或無偏方法估計變異數所影響，但一致性相關係數並非如此。在Lin所發表的原始論文中建議以1/N 進行標準化，<ref name="LinL1989Concordance">{{Cite journal |last=Lawrence I-Kuei Lin |author-link=Lawrence I-Kuei Lin |title=A concordance correlation coefficient to evaluate reproducibility |journal=[[Biometrics (journal)|Biometrics]] |date=March 1989 |volume=45 |issue=1 |page=255&ndash;268 |doi=10.2307/2532051 |jstor=2532051 |pmid=2720055}}<cite class="citation journal cs1" data-ve-ignore="true" id="CITEREFLawrence_I-Kuei_Lin1989">[[Lawrence I-Kuei Lin]] (March 1989). "A concordance correlation coefficient to evaluate reproducibility". ''[[Biometrics (journal)|Biometrics]]''. '''45''' (1): 255–268. [[DOI|doi]]:[[doi:10.2307/2532051|10.2307/2532051]]. [[JSTOR]]&nbsp;[https://www.jstor.org/stable/2532051 2532051]. [[PubMed|PMID]]&nbsp;[https://pubmed.ncbi.nlm.nih.gov/2720055 2720055].</cite></ref>但在另一篇論文，尼克森似乎以 1/(N-1)進行標準化。<ref name="NickersonC1997Note">{{Cite journal |last=Carol A. E. Nickerson |author-link=Carol A. E. Nickerson |title=A Note on "A Concordance Correlation Coefficient to Evaluate Reproducibility |url=https://archive.org/details/sim_biometrics_1997-12_53_4/page/n323 |journal=[[Biometrics (journal)|Biometrics]] |date=December 1997 |volume=53 |issue=4 |page=1503&ndash;1507 |doi=10.2307/2533516 |jstor=2533516}}</ref>因此以不同方式計算一致性相關係數，結果可能略有不同。

== 與其他相關性指標的關係 ==
一致性相關係數與某些稱為[[組內相關|組内相關]]的相關性指標几乎相同。針對不同資料集，比較其一致性相關係數與「普通」的組內相關，發現兩者之間差異甚微。<ref name="NickersonC1997Note">{{Cite journal |last=Carol A. E. Nickerson |author-link=Carol A. E. Nickerson |title=A Note on "A Concordance Correlation Coefficient to Evaluate Reproducibility |journal=[[Biometrics (journal)|Biometrics]] |date=December 1997 |volume=53 |issue=4 |page=1503&ndash;1507 |doi=10.2307/2533516 |jstor=2533516}}<cite class="citation journal cs1" data-ve-ignore="true" id="CITEREFCarol_A._E._Nickerson1997">[[Carol A. E. Nickerson?action=edit&redlink=1|Carol A. E. Nickerson]] (December 1997). "A Note on "A Concordance Correlation Coefficient to Evaluate Reproducibility". ''[[Biometrics (journal)|Biometrics]]''. '''53''' (4): 1503–1507. [[DOI|doi]]:[[doi:10.2307/2533516|10.2307/2533516]]. [[JSTOR]]&nbsp;[https://www.jstor.org/stable/2533516 2533516].</cite></ref> 也有人指出一致性相關係數的概念與 Krippendorff在1970年發表的結果很類似。<ref>{{Cite journal |last=Reinhold Müller |last2=Petra Büttner |title=A critical discussion of intraclass correlation coefficients |journal=[[Statistics in Medicine (journal)|Statistics in Medicine]] |date=December 1994 |volume=13 |issue=23–24 |page=2465&ndash;2476 |doi=10.1002/sim.4780132310 |pmid=7701147}}</ref><ref>{{Cite journal |last=Klaus Krippendorff |author-link=Klaus Krippendorff |editor=E. F. Borgatta |title=Bivariate Agreement Coefficients for Reliability of Data |journal=Sociological Methodology |location=[[San Francisco]] |publisher=[[Jossey-Bass]] |year=1970 |volume=2 |page=139&ndash;150 |doi=10.2307/270787 |jstor=270787}}</ref>

在Lin的原始論文，曾建議一種適用於多個（不只兩個）類別的型式。<ref name="LinL1989Concordance">{{Cite journal |last=Lawrence I-Kuei Lin |author-link=Lawrence I-Kuei Lin |title=A concordance correlation coefficient to evaluate reproducibility |journal=[[Biometrics (journal)|Biometrics]] |date=March 1989 |volume=45 |issue=1 |page=255&ndash;268 |doi=10.2307/2532051 |jstor=2532051 |pmid=2720055}}<cite class="citation journal cs1" data-ve-ignore="true" id="CITEREFLawrence_I-Kuei_Lin1989">[[Lawrence I-Kuei Lin]] (March 1989). "A concordance correlation coefficient to evaluate reproducibility". ''[[Biometrics (journal)|Biometrics]]''. '''45''' (1): 255–268. [[DOI|doi]]:[[doi:10.2307/2532051|10.2307/2532051]]. [[JSTOR]]&nbsp;[https://www.jstor.org/stable/2532051 2532051]. [[PubMed|PMID]]&nbsp;[https://pubmed.ncbi.nlm.nih.gov/2720055 2720055].</cite></ref>十多年以後該型式被提出修正。<ref>{{Cite journal |last=Lawrence I-Kuei Lin |title=A Note on the Concordance Correlation Coefficient |journal=[[Biometrics (journal)|Biometrics]] |date=March 2000 |volume=56 |page=324&ndash;325 |doi=10.1111/j.0006-341X.2000.00324.x |doi-access=free}}</ref>

使用一致性相關係數的例子之一，是[[功能性磁共振成像]]腦部掃描的分析方法之比較。 <ref>{{Cite Q|Q21012624}}</ref>

== 参考資料 ==
{{Reflist}}
[[Category:统计学比率]]
[[Category:协方差与相关性]]