Contrastes no paramétricos

• Sea \(X\) una variable aleatoria continua.
• Sea \(\mediana\) la mediana de \(X\). \[ H_0\colon\mediana=M_0\iff F_X(M_0)=\frac12 \qquad H_1\colon\mediana\ne M_0 \]
• La distribución empírica \(F_n(\mediana^-)=\frac{\#\{X_i < \mediana\}}n\) verifica \[F_n(\mediana^-)\xrightarrow{\text{cs}}F_X(\mediana^-)=\frac12\]
• Sea \(U=\#\{X_i < M_0\} = nF_n(M_0^-)\); entonces, \(U\hookrightarrow\mathcal B\left(n,\frac12\right)\).

• para variables discretas finitas con valores \(C_1,\dots,C_k\)
• muestras aleatorias simples \(\vec X\) = \((X_1,\dots,X_{n_X})\); \(\vec Y\) = \((Y_1,\dots,Y_{n_Y})\)
• tamaño muestral conjunto \(n\) = \(n_X\) + \(n_Y\)
• sean \(p_{Xj}=\PR{X=C_j}\), \(p_{Yj}=\PR{Y=C_j}\)
• \(H_0\colon \forall\,j,\, p_{Xj}=p_{Yj}=p_{j}\), \(H_1\colon \exists\,j,\,p_{Xj}\ne p_{Yj}\)

• tabla de frecuencias observadas \(O_{ij}=n_{ij}\), \(i\in\{X,Y\}\), \(j\in\{1,\dots,k\}\)

  	\(C_1\)	\(C_2\)	\(\dots\)	\(C_k\)	totales
  \(X\)	\(n_{X1}\)	\(n_{X2}\)	\(\dots\)	\(n_{Xk}\)	\(n_X\)
  \(Y\)	\(n_{Y1}\)	\(n_{Y2}\)	\(\dots\)	\(n_{Yk}\)	\(n_Y\)
  totales	\(n_1\)	\(n_2\)	\(\dots\)	\(n_k\)	\(n\)

• verosimilitud bajo \(H_1\) \[ \mathcal L(\vec n, \vec p_X, \vec p_Y) \propto \prod_{i\in\{X,Y\}}\prod_{j=1}^{k}p_{ij}^{n_{ij}} \] estimaciones máximo-verosímiles \[ \hat p_{ij} = \frac{n_{ij}}{n_i} \]
• verosimilitud bajo \(H_0\) \[ \mathcal L(\vec n, \vec p) \propto \prod_{j=1}^{k}p_{j}^{n_{j}} \] estimaciones máximo-verosímiles \[ \hat p_{j} = \frac{n_j}{n} \]
• frecuencias absolutas esperadas \[ E_{ij} = n_i\hat p_j = \frac{n_in_j}n \]
• razón de verosimilitudes \( \Lambda(\vec n) \) = \( \dfrac {\mathcal L(\vec n, \hat{\vec p})} {\mathcal L(\vec n, \hat{\vec p}_X, \hat{\vec p}_Y)}\) = \( \displaystyle\prod_{i\in\{X,Y\}}\prod_{j=1}^k \left(\dfrac{n_j/n}{n_{ij}/n_i}\right)^{n_{ij}} \) = \( \displaystyle\prod_{i\in\{X,Y\}}\prod_{j=1}^k \left(\dfrac{n_i n_j/n}{n_{ij}}\right)^{n_{ij}} \) = \( \displaystyle\prod_{i\in\{X,Y\}}\prod_{j=1}^k \left(\dfrac{E_{ij}}{O_{ij}}\right)^{O_{ij}} \)
• \(G\) = \(-2\ln\Lambda\) = \(\displaystyle\sum_{i\in\{X,Y\}}\sum_{j=1}^k O_{ij}\ln\frac{O_{ij}}{E_{ij}}\)
• RC = \([G > c]\)
• \(G\) \(\stackrel{H_0}{\hookrightarrow}\) \(\chi^2_{2(k-1)-(k-1)}\) = \(\chi^2_{k-1}\)
• \(D\) = \(\displaystyle\sum_{i\in\{X,Y\}}\sum_{j=1}^k \frac{(O_{ij}-E_{ij})^2}{E_{ij}}\)
• RC = \([D > c]\)
• \(D\) \(\stackrel{H_0}{\hookrightarrow}\) = \(\chi^2_{k-1}\)

• \(H_0\colon F_X=F_Y=F_0\), \(H_1\colon F_X\ne F_Y\)
• muestras aleatorias simples \(\vec X=(X_1,\dots,X_{n_X})\), \(\vec Y=(Y_1,\dots,Y_{n_Y})\)
• \(D_{n_X n_Y}\) = \(\displaystyle \sup_{x\in\mathbb R} |F_{n_X}(x)-F_{n_Y}(x)|\) = \(\displaystyle \sup_{x\in\mathbb R} \left|\frac{\#\{X_i\le x\}}{n_X}-\frac{\#\{Y_i\le x\}}{n_Y}\right|\) = \(\displaystyle \sup_{x\in\mathbb R} \left|\frac{\#\{F_0(X_i)\le F_0(x)\}}{n_X}-\frac{\#\{F_0(Y_i)\le F_0(x)\}}{n_Y}\right|\) = \(\displaystyle \sup_{0 < u < 1} \left|\frac{\#\{U_i\le u\}}{n_X}-\frac{\#\{U_i\le u\}}{n_Y}\right|\)
• de libre distribución bajo \(H_0\) pues \[F_0(X)\;\stackrel{H_0}{\equiv}\;F_0(Y)\;\stackrel{H_0}{\equiv}\;\mathcal U(0;1)\]
• para variables continuas; puede haber problemas si hay empates

• ejemplo

  X <- c (2, 1, 2.5, 4)
  Y <- c (3, 3.1, 2.3, 3.7, 4.5)
  XY <- c (X, Y) # muestra conjunta
  ## XY ordenada: 1 2 2.3 2.5 3 3.1 3.7 4 4.5
  ## FnX = (1, 2, 2, 3, 3, 3, 3, 4, 4) / 4
  ## FnY = (0, 0, 1, 1, 2, 3, 4, 4, 5) / 5
  D <- max (abs (ecdf(X)(XY) - ecdf(Y)(XY))) # 3/4 - 1/5 = 0,55
  1 - psmirnov (D, c(length(X),length(Y)))   # 0.4285714
  ks.test (X, Y)
  ##      Two-sample Kolmogorov-Smirnov test
  ## 
  ## data:  X and Y
  ## D = 0.55, p-value = 0.4286
  ## alternative hypothesis: two-sided
  

• Muy usado.
• \(H_0\colon X\,\stackrel{\mathcal L}{\equiv}\,Y\), \(H_1\colon X\,\stackrel{\mathcal L}{\not\equiv}\,Y\)
• Muestras \(\vec x=(x_1,\dots,x_{n_X})\) y \(\vec y=(y_1,\dots,y_{n_Y})\)
• \(Z_{ij}\) = \([X_i < Y_j]\) \(\hookrightarrow\) \(\mathcal B\left(1,\PR{X < Y}\right)\)
• \(Z_{ij}\) \(\stackrel{H_0}\hookrightarrow\) \(\mathcal B\left(1,\frac12\right)\) pues \(\PR{X < Y\mid H_0}\) = \(\int_{\mathbb R} \int_{x}^{\infty}f(x)f(y)\dif y \dif x\) = \(\int_{\mathbb R} f(x)\int_{x}^{\infty}f(y)\dif y \dif x\) = \(\int_{\mathbb R} f(x)[1-F(x)]\dif x\) = \(\left[-\frac12[1-F(x)]^2\right]_{x=-\infty}^{x=\infty}\) = \(\frac12\)
• \(U\) = \(\sum_{i=1}^{n_X}\sum_{j=1}^{n_Y}Z_{ij}\) \(\in\) \(\{0,\dots,n_Xn_Y\}\)
• \(H_0\) \(\implies\) \(\PR{X < Y}=\frac12\) \(\implies\) \(E(U)=\sum_{i=1}^{n_X}\sum_{j=1}^{n_Y}E(Z_{ij})\) = \(\frac{n_X n_Y}2\) \(\implies\) \(U\approx \frac{n_X n_Y}{2}\) \(\implies\) RC = \([U < c_1] \cup [U > c_2]\)
• ejemplo
  • \(\vec x\) = \((2;1;2.5)\), \(\vec y\) = \((2.4;3.1;2.3;3.7)\)
  • \(n_X = 3\), \(n_Y = 4\)
  • muestra conjunta ordenada: \((1x;2x;2.3y;2.4y;2.5x;3.1y;3.7y)\) \(\implies\) \((x,x,y,y,x,y,y)\)
  • \(u\) = 2 + 2 + 3 + 3 = 4 + 4 + 2 = 10
  • bajo \(H_0\)
    • el número de posibles ordenaciones de tres \(x\) y cuatro \(y\) es \(\binom{7}{3}\) = \(\frac{7!}{3!4!}\) = 35
    • todas las ordenaciones son igualmente probables: \(\PR{xxxyyyy}\) = \(\dots\) = \(\PR{yyyyxxx}\)
    • \(0\le u\le12\)
      • \(u=0\) \(\iff\) \((yyyyxxx)\)
      • \(u=1\) \(\iff\) \((yyyxyxx)\)
      • \(u=2\) \(\iff\) \((yyxyyxx)\) \(\cup\) \((yyyxxyx)\)
      • \(u=3\) \(\iff\) \((yxyyyxx)\) \(\cup\) \((yyxyxyx)\) \(\cup\) \((yyyxxxy)\)
      • …

    • la distribución de \(U\) es simétrica respecto a \(6\)

      • \(U(xyxxyyy)\) = 10 \(\iff\) \(U(yyyxxyx)\) = \(2\) = \(12-10\)

      > dwilcox (0:12, 3, 4)
       [1] 0.02857143 0.02857143 0.05714286 0.08571429 0.11428571 0.11428571
       [7] 0.14285714 0.11428571 0.11428571 0.08571429 0.05714286 0.02857143
      [13] 0.02857143
      > dwilcox (0:12, 3, 4) * 35
       [1] 1 1 2 3 4 4 5 4 4 3 2 1 1
      

• bajo \(H_0\)
  • \(Z_{ij}\) depende de \(Z_{ik}\): \(E(Z_{ij}Z_{ik})\) = \(\PR{Z_{ij}=1\cap Z_{ik}=1}\) = \(\PR{X_i < \min\{Y_j,Y_k\}}\) = \(\displaystyle\int_{-\infty}^{\infty} f(x)\int_x^{\infty}2f(y)[1-F(y)]\dif y\dif x\) = \(\displaystyle\int_{-\infty}^{\infty}f(x) \left[-[1-F(y)]^2\right]_{y=x}^{y=\infty}\dif x\) = \(\displaystyle\int_{-\infty}^{\infty}f(x)[1-F(x)]^2\dif x\) = \(\left[-\frac13[1-F(x)]^3\right]_{x=-\infty}^{x=\infty}\) = \(\frac13\) \(\neq\) \(\frac12\cdot\frac12\) = \(E(Z_{ij})\cdot E(Z_{ik})\)
  • \(Z_{ij}\) depende de \(Z_{lj}\): \(E(Z_{ij}Z_{lj})\) = \(\PR{\max\{X_i,X_l\} < Y_j}\) = \(\frac13\)
  • \(Z_{ij}\) no depende de \(Z_{lk}\): \(E(Z_{ij}Z_{lk})\) = \(\PR{X_i < Y_j\cap X_l < Y_k}\) = \(\PR{X_i < Y_j}\cdot\PR{X_l < Y_k}\) = \(E(Z_{ij})\cdot E(Z_{lk})\) = \(\frac14\)
  • \(\text{Var}(U)\) = \(E(U^2)-E^2(U)\) = \(\displaystyle E\left[\left(\sum_{i=1}^{n_X} \sum_{j=1}^{n_Y}Z_{ij}\right)^2\right]-E^2(U)\) = \(\displaystyle E\left[\left(\sum_{i=1}^{n_X} \sum_{j=1}^{n_Y}Z_{ij}\right) \cdot \left(\sum_{l=1}^{n_X} \sum_{k=1}^{n_Y}Z_{lk}\right)\right]-E^2(U)\) = \(\displaystyle E\left[\left(\sum_{i=1}^{n_X} \sum_{j=1}^{n_Y}Z_{ij}\right) \cdot \left( Z_{ij}+\sum_{k\ne j}Z_{ik}+\sum_{l\ne i}Z_{lj} +\sum_{l\ne i}\sum_{k\ne j}Z_{lk} \right)\right]-E^2(U)\)= \(\displaystyle \sum_{i=1}^{n_X} \sum_{j=1}^{n_Y}E\left[Z_{ij}^2\right]\)+ \(\displaystyle\sum_{i=1}^{n_X} \sum_{j=1}^{n_Y}\sum_{k\ne j}E[Z_{ij}Z_{ik}]\)+ \(\displaystyle\sum_{i=1}^{n_X} \sum_{j=1}^{n_Y}\sum_{l\ne i}E[Z_{ij}Z_{lj}]\)+ \(\displaystyle\sum_{i=1}^{n_X} \sum_{j=1}^{n_Y}\sum_{l\ne i}\sum_{k\ne j} E[Z_{ij}Z_{lk}]\) \(-\) \(E^2(U)\)= \(n_X n_Y\frac12\) + \(n_X n_Y (n_Y-1)\frac13\)+ \(n_X n_Y (n_X-1)\frac13\) + \(n_X n_Y (n_X-1) (n_Y-1)\frac14\) \(-\) \(\left(\frac{n_X n_Y}2\right)^2\)= \(\frac{n_X n_Y (n+1)}{12}\)

Maxima 5.44.0 http://maxima.sourceforge.net
using Lisp GNU Common Lisp (GCL) GCL 2.6.12
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
(%i1) EZij : 1/2 $

(%i2) EZij2 : EZij $

(%i3) EU : nX*nY*EZij $

(%i4) EZijZik : 1/3 $

(%i5) EZijZlj : 1/3 $

(%i6) EZijZlk : 1/4 $

(%i7) VarU : nX*nY*EZij2 + nX*nY*(nY-1)*EZijZik +
  nX*nY*(nX-1)*EZijZlj + nX*nY*(nX-1)*(nY-1)*EZijZlk - EU^2 $

(%i8) factor(VarU) ;
                              nX nY (nY + nX + 1)
(%o8)                         -------------------
                                      12

• \(U\) es asintóticamente gausiana

  x <- rnorm (50)
  y <- rnorm (50, 0.6)
  wilcox.test (x, y)
  wilcox.test (x, y, exact=TRUE)
  

  
  	Wilcoxon rank sum test with continuity correction
  
  data:  x and y
  W = 667, p-value = 5.928e-05
  alternative hypothesis: true location shift is not equal to 0
  
  
  	Wilcoxon rank sum exact test
  
  data:  x and y
  W = 667, p-value = 4.074e-05
  alternative hypothesis: true location shift is not equal to 0
  
  

  (De ?wilcox.test) By default (if ‘exact’ is not specified), an exact p-value is computed if the samples contain less than 50 finite values and there are no ties. Otherwise, a normal approximation is used.

• Asignar rangos a la muestra conjunta y sumarlos: \(R_X\) = \(\displaystyle \sum_{i=1}^{n_X}\text{rango}(X_i)\) ; \(R_Y\) = \(\displaystyle \sum_{i=1}^{n_Y}\text{rango}(Y_i)\) ; \(R_X\) + \(R_Y\) = \(\frac{n (n+1)}2\)
• Si \(\forall\,i,j\), \(x_i < y_j\), entonces \(R_X\) = \(r_{\min}\) = \(1\) + \(\dots\) + \(n_X\) = \(\frac{n_X(n_X+1)}2\)
• Si \(\forall\,i,j\), \(x_i > y_j\), entonces \(R_X\) = \(r_{\max}\) = \((n_Y+1)\) + \(\dots\) + \((n_Y+n_X)\) = \(n_X n_Y+\frac{n_X(n_X+1)}2\) = \(n_X \left(n_Y+\frac{n_X+1}2\right)\)
• Es una trasformación biyectiva de la \(U\) de Mann y Whitney: \(R_Y\) = \(R_{Y_{(1)}}\) + \(R_{Y_{(2)}}\) + \(\dots\) + \(R_{Y_{(n_Y)}}\) = \(U_1+1\) + \(U_2+2\) + \(\dots\) + \(U_{n_Y}+n_Y\) = \(U\) + \(1\) + \(2\) + \(\dots\) + \(n_Y\) = \(U\) + \(\displaystyle\frac{n_Y(n_Y+1)}2\) donde \(R_{Y_{(j)}}\) es el rango de la \(j\)-ésima observación de \(\vec Y\) ordenada y \(U_j\) = \(\#\{X_i < Y_{(j)}\}\)
• \(H_0\colon X\,\stackrel{\mathcal L}{\equiv}\,Y\), \(H_1\colon X\,\stackrel{\mathcal L}{\not\equiv}\,Y\) \(\implies\) RC = \([R_X < c_1]\cup[R_X > c_2]\)

> X <- c(2,1,2.5,4) ; Y <- c(3,3.1,2.3,3.7,4.5)
> nX <- length(X) ; nY <- length(Y)
> names(X) <- rep("X",nX) ; names(Y) <- rep("Y",nY)
> (XY <- names (sort (c (X, Y))))
[1] "X" "X" "Y" "X" "Y" "Y" "Y" "X" "Y"
> (Rx <- sum (which (XY == "X")))
[1] 15
> (Ry <- sum (which (XY == "Y")))
[1] 30
> (U <- sum (sapply (X, function(Xi) sapply (Y, function(Yj) Xi < Yj))))
[1] 15
> ## otra forma de calcular U
> sum(sapply(setdiff(which(XY=="Y"),1),function(j)sum(XY[1:(j-1)]=="X")))
[1] 15
> Ry - nY*(nY+1)/2
[1] 15
> (Umin <- nX*(nX+1)/2)
[1] 10
> (W <- U - Umin)
[1] 5
> wilcox.test (X, Y)

        Wilcoxon rank sum exact test

data:  X and Y
W = 5, p-value = 0.2857
alternative hypothesis: true location shift is not equal to 0

• \(X\) variable discreta finita (o agrupada)
• \(X\) toma valores \(C_1\), …, \(C_r\)
• \(p_{ij}\) = \(\Pr[X_i = C_j]\)
• \(H_0:\forall\,j\in\{1,\dots,r\}\), \(p_{1j}=\dots=p_{kj}\)

• tabla de frecuencias observadas \(O_{ij}\) = \(n_{ij}\)

  	\(C_1\)	\(\dots\)	\(C_r\)	totales
  \(X_1\)	\(n_{11}\)	\(\dots\)	\(n_{1r}\)	\(n_{1\cdot}\)
  \(\vdots\)	\(\vdots\)	\(\vdots\)	\(\vdots\)	\(\vdots\)
  \(X_k\)	\(n_{k1}\)	\(\dots\)	\(n_{kr}\)	\(n_{k\cdot}\)
  totales	\(n_{\cdot1}\)	\(\dots\)	\(n_{\cdot r}\)	\(n\)

• verosimilitud bajo \(H_1\) \[\mathcal L(\vec n,\vec p_1,\dots,\vec p_k) \propto \prod_{i=1}^k\prod_{j=1}^r p_{ij}^{n_{ij}}\] EMV \(\hat p_{ij}\) = \(\frac{n_{ij}}{n_{i\cdot}}\)
• verosimilitud bajo \(H_0\) \[\mathcal L(\vec n,\vec p) \propto \prod_{j=1}^r p_{ij}^{n_{\cdot j}}\] EMV \(\hat p_{j}\) = \(\frac{n_{\cdot j}}{n}\)
• frecuencia esperada bajo \(H_0\): \(E_{ij}\) = \(n_i\hat p_j\) = \(\frac{n_{i\cdot}n_{\cdot j}}n\)
• razón de verosimilitudes \(\Lambda(\vec n)\) = \(\displaystyle \frac{\mathcal L(\vec n,\hat{\vec p})} {\mathcal L(\vec n,\hat{\vec p}_1,\dots,\hat{\vec p}_k)}\) = \(\displaystyle \prod_i\prod_j \left(\frac{n_{\cdot j}/n}{n_{ij}/n_{i\cdot}}\right)^{n_{ij}}\) = \(\displaystyle \prod_i\prod_j \left(\frac{n_{i\cdot}n_{\cdot j}/n}{n_{ij}}\right)^{n_{ij}}\) = \(\displaystyle \prod_i\prod_j \left(\frac{E_{ij}}{O_{ij}}\right)^{O_{ij}}\)
• \(G\) = \(-2\ln\Lambda\) = \(2\sum_i\sum_j O_{ij}\ln{\frac{O_{ij}}{E_{ij}}}\)
  • RC = \([G > c]\)
  • \(G\) \(\xrightarrow[\mathcal L]{H_0}\) \(\chi^2_{k(r-1)-(r-1)}\) = \(\chi^2_{(k-1)(r-1)}\) asintóticamente; la aproximación es buena si \(E_{ij}\ge5\) \(\forall\,i,j\).
• \(D\) = \(\sum_i\sum_j\frac{(O_{ij}-E_{ij})^2}{E_{ij}}\)
  • RC = \([D > c]\)
  • \(D\) \(\xrightarrow[\mathcal L]{H_0}\) \(\chi^2_{(k-1)(r-1)}\) asintóticamente; la aproximación es buena si \(E_{ij}\ge5\) \(\forall\,i,j\).

• \(H_0:\mediana_1=\dots=\mediana_k\), \(H_1:\exists\,i,j,\;\mediana_i\ne\mediana_j\)
• Sea \(M\) la mediana de la muestra conjunta.
• Sea \(U_i\) = \(\#\{X_{ij} < M\}\)
• Sea \(t\) = \(\sum_i U_i\) = \(\lfloor \frac n 2\rfloor\)

• Tabla de frecuencias

  	\(X_1\)	\(\dots\)	\(X_k\)	totales
  \(\#\{x_{ij} < M\}\)	\(u_1\)	\(\dots\)	\(u_k\)	\(t\)
  \(\#\{x_{ij}\ge M\}\)	\(n_1-u_1\)	\(\dots\)	\(n_k-u_k\)	\(n-t\)
  totales	\(n_1\)	\(\dots\)	\(n_k\)	\(n\)

• Se aplica un \(\chi^2\) de Pearson
  • Denótese \(O_{i1}=u_i\) y \(O_{i2}=n_i-u_i\)
  • \(E_{i1}\) = \(n_i\widehat\Pr_0[X_i < M]\) = \(n_i\frac tn\), \(E_{i2}\) = \(n_i\widehat\Pr_0[X_i\ge M]\) = \(n_i\frac {n-t}n\)
  • \(O_{i2}-E_{i2}\) = \(n_i-u_i-n_i\frac{n-t}n\) = \(-u_i+n_i\frac tn\) = \(-(O_{i1}-E_{i1})\)
  • \(D\) = \(\sum_{i=1}^k\sum_{j=1}^2\frac{(O_{ij}-E_{ij})^2}{E_{ij}}\) = \(\sum_i\left(U_i-n_i\frac tn\right)^2 \left(\frac1{E_{i1}}+\frac1{E_{i2}}\right)\) = \(\sum_i\left(U_i-n_i\frac tn\right)^2 \left(\frac1{n_i\frac tn}+\frac1{n_i\frac{n-t}n}\right)\) = \(n^2\sum_i\frac{\left(U_i-\frac{n_it}n\right)}{n_i t(n-t)}\)
    • RC = \([D > c]\)
    • \(D\) \(\xrightarrow[\mathcal L]{H_0}\) \(\chi^2_{k-1}\)

• Muy usado.
• \(H_0:F_1=\dots=F_k\), \(H_1:\exists\,i,j,\,F_i\ne F_j\)
• Sea \(R_{ij}\) el rango de \(x_{ij}\) en la muestra conjunta.
  • \(R_i\) = \(\sum_{j=1}^{n_i}R_{ij}\) suma de rangos para \(i\)
  • \(\bar R_i\) = \(\frac{R_i}{n_i}\) rango medio para \(i\)
  • \(\bar R\) = \(\frac1n\sum_i n_i\bar R_i\) = \(\frac{n+1}2\) rango medio global
• Bajo \(H_0\)
  • Los rangos son invariantes con la trasformación \(U_i\) = \(F_0(X_i)\) \(\hookrightarrow\) \(\mathcal U(0,1)\)
  • Se debe cumplir que \(\bar R_i\) \(\approx\) \(\bar R\) = \(\frac{n+1}2\) \(\forall\,i\)
• Análisis de varianza sobre los rangos
  • SCT = \(\sum_i\sum_j(R_{ij}-\bar R)^2\) = \(\sum_i\sum_j\left((R_{ij}-\bar R_i) + (\bar R_i-\bar R)\right)^2\) = \(\sum_i\sum_j\left(R_{ij}-\bar R_i\right)^2\) + \(\sum_i\sum_j\left(\bar R_i-\bar R\right)^2\) = \(\sum_i\sum_j\left(R_{ij}-\bar R_i\right)^2\) + \(\sum_i n_i\left(\bar R_i-\bar R\right)^2\) = SCE + SCF
  • SCT es constante; depende sólo de \(n\): SCT = \(\sum_{i=1}^n\left(i-\frac{n+1}2\right)^2\) = \(\sum_{i=1}^n i^2 - n\left(\frac{n+1}2\right)^2\) = \(n\frac{n^2-1}{12}\)
  • Estadístico del contraste \(H\) = \(\displaystyle\frac{12}{n(n+1)}\sum_{i=1}^kn_i\left(\bar R_i-\frac{n+1}2\right)^2\) = \(\displaystyle \frac{\text{SCF}}{\frac{\text{SCT}}{n-1}}\)
    • RC = \([H > c]\)
    • \(H\) es de libre distribución bajo \(H_0\)
• Generaliza la prueba de Wilcoxon
  • Considerando dos muestras (la \(i\) frente al resto): \(R_i\) \(\xrightarrow[\mathcal L]{H_0}\) \(\mathcal N\left(\frac{n_i(n+1)}2,\sqrt{\frac{n_i(n-n_i)(n+1)}{12}}\right)\) \(\implies\) \(\frac{\left(\bar R_i-\frac{n+1}2\right)^2}{\frac{(n-n_i)(n+1)}{12n_i}}\) = \(\frac{n_i\left(\bar R_i-\frac{n+1}2\right)^2}{\frac{(n-n_i)(n+1)}{12}}\) \(\xrightarrow[\mathcal L]{H_0}\) \(\chi_1^2\)

  • Las \(\bar R_i\) son dependientes pues \(\sum_i n_i \bar R_i\) = \(\frac{n(n+1)}2\) luego se pierde un grado de libertad: \(H\) = \(\displaystyle \frac{12}{n(n+1)} \sum_{i=1}^k n_i \left(\bar R_i-\frac{n+1}2\right)^2\) \(\xrightarrow[\mathcal L]{H_0}\) \(\chi^2_{k-1}\)

    ## comprobación de la distro asintótica mediante montecarlo
    m <- 100; k <- 4; g <- rep(1:k,each=m)
    n <- m*k; n. <- rep(m,k) # == table(g) 
    H <- function (x, g)
    {
        rangos <- rank(x)
        rango. <- tapply(rangos, g, mean)
        12/n/(n+1)*sum(n.*(rango.-(n+1)/2)^2)
    }
    distri <- replicate (1e5,
    {
        x <- runif(n)
        H(x,g)
    })
    rbind (quantile(distri, pr <- c(1,5,10,50,90,95,99)/100),
           qchisq(pr, k-1))
    

    
                1%        5%       10%      50%      90%      95%      99%
    [1,] 0.1182838 0.3566484 0.5914939 2.369102 6.263054 7.833209 11.21446
    [2,] 0.1148318 0.3518463 0.5843744 2.365974 6.251389 7.814728 11.34487
    

• Ejemplo

  • \(\vec x_1\) = \((2;1;2.5;4)\) \(\implies\) \(n_1=4\)
  • \(\vec x_2\) = \((3;3.1;2.3;3.7;4.5)\) \(\implies\) \(n_2=5\)
  • \(\vec x_3\) = \((2.1;1.3;2.4;4.1)\) \(\implies\) \(n_3=4\)

  x1 = c (2, 1, 2.5, 4)
  x2 = c (3, 3.1, 2.3, 3.7, 4.5)
  x3 = c (2.1, 1.3, 2.4, 4.1)
  conjunta = stack (list (a=x1, b=x2, c=x3))
  n.i = table (conjunta$ind)
  k = length (n.i)
  n = nrow (conjunta)
  r = rank (conjunta$values)
  r.medios = tapply (r, conjunta$ind, mean)
  (H = 12 / (n*(n+1)) * sum (n.i * (r.medios - (n+1)/2) ^ 2))
  1 - pchisq (H, k-1) # p-pvalor
  kruskal.test (values ~ ind, conjunta)
  

  
  [1] 2.175824
  
  [1] 0.3369192
  
  	Kruskal-Wallis rank sum test
  
  data:  values by ind
  Kruskal-Wallis chi-squared = 2.1758, df = 2, p-value = 0.3369
  

• Empates
  • Pueden aparecer por redondeos.
  • A \(t\) observaciones empatadas, con rangos \(i\), \(i+1\), …, \(i+t-1\) se les asigna su rango medio \(\dfrac{i+(i+1)+\dots+(i+t-1)}t\) = \(\dfrac{i\cdot t+(1+2+\dots+t-1)}t\) = \(\dfrac{i\cdot t+\dfrac{(t-1)t}2}t\) = \(\dfrac{2i+t-1}2\)
  • El rango medio global \(\bar R\) no cambia.

  • La corrección por los \(t\) empates en la SCT vale \(\displaystyle\sum_{j=0}^{t-1}(i+j)^2-\sum_{j=0}^{t-1}\left({2i+t-1\over2}\right)^2\) = \(\displaystyle\sum_{j=0}^{t-1}(i+j)^2-t\left({2i+t-1\over2}\right)^2\) = \(\dfrac{t(t^2-1)}{12}\) = \(\dfrac{t^3-t}{12}\)

    (%i7) factor (nusum ((i+j)^2, j, 0, t-1) - t * ((2*i+t-1)/2)^2) ;
                                   (t - 1) t (t + 1)
    (%o7)                          -----------------
                                          12
    

  • Si hay \(g\) grupos de empates, con \(t_j\) observaciones empatadas cada uno, la SCT corregida vale SCT\(^*\) = SCT \(-\) \(\displaystyle\frac1{12}\sum_{j=1}^gt_j^3-t_j\)
  • Los \(t_1,\dots,t_g\) pueden obtenerse en R mediante table
  • Cuando los empates se producen dentro de la misma población \(i\in\{1,\dots,k\}\), el estadístico \(H\) no se modifica.
  • Estadístico corregido \(H^*\) = \((n-1)\dfrac{\text{SCF}}{\text{SCT}^*}\) = \(\dfrac{H}{1-\dfrac{\sum_{j=1}^g(t_j^3-t_j)}{n^3-n}}\)

• Contrastes a posteriori: prueba de Dunn

  • \(H_0:F_1=\dots=F_k\), \(\iff\) \(\displaystyle\bigcap_{i,j}H_0^{ij}:F_i=F_j\)

  \[\bar R_i\xrightarrow[\mathcal L]{H_0} \mathcal N\left(\frac{n+1}2,\sqrt{\frac{(n-n_i)(n+1)}{12n_i}}\right)\]

  • RC para \(H_0^{ij}:F_i=F_j\) \[\displaystyle \left[\frac {\bar R_i-\bar R_j} {\sqrt{\frac{n(n+1)}{12}\left(\frac1{n_i}+\frac1{n_j}\right)}} > z_{1-\alpha^*}\right]\] con \(\alpha^*=\dfrac\alpha{\binom k 2}\)

  dunn.test::dunn.test (list (x1, x2, x3))
  FSA::dunnTest (values ~ ind, conjunta)
  

    Kruskal-Wallis rank sum test
  
  data: x and group
  Kruskal-Wallis chi-squared = 2.1758, df = 2, p-value = 0.34
  
  
  			   Comparison of x by group                            
  				(No adjustment)                                
  Col Mean-|
  Row Mean |          1          2
  ---------+----------------------
         2 |  -1.339728
  	 |     0.0902
  	 |
         3 |  -0.181568   1.148338
  	 |     0.4280     0.1254
  
  alpha = 0.05
  Reject Ho if p <= alpha/2
  
  Dunn (1964) Kruskal-Wallis multiple comparison
    p-values adjusted with the Holm method.
  
    Comparison          Z   P.unadj     P.adj
  1      a - b -1.3397283 0.1803337 0.5410011
  2      a - c -0.1815683 0.8559216 0.8559216
  3      b - c  1.1483385 0.2508288 0.5016577
  

• Contrastes a posteriori: Wilcoxon por parejas

  pairwise.wilcox.test (conjunta$values, conjunta$ind)
  

  
  	Pairwise comparisons using Wilcoxon rank sum exact test 
  
  data:  conjunta$values and conjunta$ind 
  
    a    b   
  b 0.86 -   
  c 0.89 0.86
  
  P value adjustment method: holm
  

• Sea \(\vec X\) = \((X_1,\dots,X_k)\) un vector aleatorio.
• \(H_0:F_1=\dots=F_k\), \(H_1:\exists\,i,j,\,F_i\ne F_j\)

• Muestra \(\vec X_1,\dots,\vec X_n\) aleatoria simple de \(\vec X\) con \(\vec X_i\) = \((X_{i1},\dots,X_{ik})\)

  individuo	\(X_1\)	\(\dots\)	\(X_k\)
  \(1\)	\(x_{11}\)	\(\dots\)	\(x_{1k}\)
  \(\vdots\)	\(\vdots\)	\(\vdots\)	\(\vdots\)
  \(n\)	\(x_{n1}\)	\(\dots\)	\(x_{nk}\)

• Rangos intraindividuo

  individuo	\(X_1\)	\(\dots\)	\(X_k\)	sumas
  \(1\)	\(R_{11}\)	\(\dots\)	\(R_{1k}\)	\(R_{1\cdot}\)
  \(\vdots\)	\(\vdots\)	\(\vdots\)	\(\vdots\)	\(\vdots\)
  \(n\)	\(R_{n1}\)	\(\dots\)	\(R_{nk}\)	\(R_{n\cdot}\)
  sumas	\(R_{\cdot 1}\)	\(\dots\)	\(R_{\cdot k}\)	\(R_{\cdot\cdot}\)

• \(R_{\cdot\cdot}\) = \(\dfrac{nk(k+1)}2\)
• \(\text{SCT}=\sum_{i=1}^n\sum_{j=1}^k\left(R_{ij}-\frac{k+1}2\right)^2 =\frac{n\,(k-1)\,k\,(k+1)}{12}\)
• Estadístico del contraste basado en la dispersión de \(R_{\cdot j}\): \(F\) = \(\displaystyle \frac{12}{nk(k+1)} \sum_{j=1}^k\left(R_{\cdot j}-\frac{n(k+1)}2\right)^2\) = \(\displaystyle \frac {\sum_{j=1}^k\left(R_{\cdot j}-\frac{n(k+1)}2\right)^2} {\frac{\text{SCT}}{k-1}}\)
• RC = \([F > c]\)
• Bajo \(H_0\)
  • \(R_{\cdot j}\) deberían ser similares

  • \(F\) es de libre distribución, \(F\) \(\xrightarrow[\mathcal L]{H_0}\) \(\chi^2_{k-1}\)

    ## comprobación de la distribución asintótica
    k <- 3 # número de variables
    cova <- matrix(c(1.000, 0.872, 0.818,
                     0.872, 1.000, 0.963,
                     0.818, 0.963, 1.000),
                   3)
    mu <- c(1.0, 1.1, 1.1) # medias teóricas
    n  <- 100 # tamaño muestral
    X  <- mvtnorm::rmvnorm(n, mu, cova)
    R0 <- t(apply(X, 1, rank))
    ## estadístico de Friedman
    friedman <- function (R)
    {
        R.j <- apply(R, 2, sum)
        12/(n*k*(k+1)) * sum((R.j - n*(k+1)/2) ^ 2)
    }
    F0 <- friedman(R0)
    ## montecarlo
    distri <- replicate (1e4,
                         friedman(t(replicate(n,sample(k)))))
    c(Friedman=F0, pval=mean(distri>=F0))
    friedman.test(X)
    

    
    Friedman     pval 
      0.5600   0.7578
    
    	Friedman rank sum test
    
    data:  X
    Friedman chi-squared = 0.56, df = 2, p-value = 0.7558
    

    plot(ecdf(distri))
    plot(function(x) pchisq(x,k-1), 0, max(distri), col=2, lwd=2, add=TRUE)
    

    

• Distribución de \(F\)
  • mínimo: \(F\) = \(0\) cuando \(R_{\cdot1}\) = \(\dots\) = \(R_{\cdot k}\) = \(n\dfrac{k+1}2\)

  • máximo: \(F\) = \(n(k-1)\) cuando los rangos son constantes por columnas; por ejemplo,

    individuo	\(X_1\)	\(X_2\)	\(\dots\)	\(X_k\)	sumas
    \(1\)	\(1\)	\(2\)	\(\dots\)	\(k\)	\(R_{1\cdot}\)
    \(\vdots\)	\(\vdots\)	\(\vdots\)	\(\vdots\)	\(\vdots\)	\(\vdots\)
    \(n\)	\(1\)	\(2\)	\(\dots\)	\(k\)	\(R_{n\cdot}\)
    sumas	\(n\)	\(2n\)	\(\dots\)	\(kn\)	\(n\dfrac{k(k+1)}2\)

• Ejemplo

  • datos crudos

    individuo	\(X_1\)	\(X_2\)	\(X_3\)	\(X_4\)
    1	\(5.6\)	\(8.7\)	\(7.2\)	\(6.6\)
    2	\(3.0\)	\(5.6\)	\(5.4\)	\(5.5\)
    3	\(6.6\)	\(3.9\)	\(9.2\)	\(6.7\)
    4	\(3.5\)	\(5.8\)	\(9.6\)	\(7.8\)
    5	\(3.3\)	\(7.8\)	\(4.0\)	\(3.5\)

  • rangos

    individuo	\(X_1\)	\(X_2\)	\(X_3\)	\(X_4\)
    1	1	4	3	2
    2	1	4	2	3
    3	2	1	4	3
    4	1	2	4	3
    5	1	4	3	2
    \(R_{\cdot j}\)	6	15	16	13

  • \(F\) = \(\dfrac {12} {5\cdot4\cdot(4+1)} \sum_{j=1}^4(R_{\cdot j}-12.5)^2\) = \(7.32\)

  X = cbind (X1 = c(5.6,3.0,6.6,3.5,3.3), X2 = c(8.7,5.6,3.9,5.8,7.8),
             X3 = c(7.2,5.4,9.2,9.6,4.0), X4 = c(6.6,5.5,6.7,7.8,3.5))
  (Rij = t (apply (X, 1, rank)))
  (R.j = colSums (Rij))
  n = nrow (X); k = ncol (X)
  (F = 12/n/k/(k+1) * sum ((R.j - n*(k+1)/2) ^ 2))
  1 - pchisq (F, k-1)
  friedman.test (X)
  

       X1 X2 X3 X4
  [1,]  1  4  3  2
  [2,]  1  4  2  3
  [3,]  2  1  4  3
  [4,]  1  2  4  3
  [5,]  1  4  3  2
  X1 X2 X3 X4 
   6 15 16 13 
  [1] 7.32
  [1] 0.06236834
  
  	Friedman rank sum test
  
  data:  X
  Friedman chi-squared = 7.32, df = 3, p-value = 0.06237
  
  

• versión para casos con empates (es la implementada en R):
  • \(F\) = \(\displaystyle \frac{12}{nk(k+1)-\dfrac{\sum_{i=1}^n\sum_{j=1}^{g_i}(t_{ij}^3-t_{ij})}{k-1}} \sum_{j=1}^k\left(R_{\cdot j}-\frac{n(k+1)}2\right)^2\)
• en realidad, la prueba de Friedman tiene como hipótesis nula \(H_0:(X_1,\dots,X_k)\) intercambiable \(\equiv\) \(H_0:F_{X_1,\dots,X_k}=F_{X_{\sigma(1)},\dots,X_{\sigma(k)}}\) \(\forall\,\sigma\in S_k\), donde \(S_k\) es el grupo de permutaciones de orden \(k\)

  • si la distribución conjunta no es intercambiable, no funciona bien:

    ## distro conjunta:
    ##    ##
    ##    ##
    ##  ##
    ##  ##
    ##      ##
    ##      ##
    n <- 30
    dist <- replicate(1e4,
    {
        x <- runif(n,0,3)
        y <- runif(n,0,1) - (x<1) + 2*(x<2)
        friedman.test(cbind(x,y))$p.value
    })
    summary(dist)
    mean(dist<0.05)
    ## > summary(dist)
    ##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max.
    ## 0.0000021 0.0105871 0.0678892 0.1708853 0.2733217 1.0000000
    ## > mean(dist<0.05)
    ## [1] 0.4301
    

    la distribución de los p-valores no es uniforme \(U(0,1)\) bajo \(H_0:F_1=\dots=F_k\)

• Concordancia entre \(n\) jueces al ordenar \(k\) elementos
• \(W\) = \(\dfrac{F}{n(k-1)}\) \(\in\) \([0;1]\)
  • \(W=1\) \(\iff\) concordancia total)
  • \(H_0\): no existe concordancia \(\iff\) \(H_0\): \(W\approx0\)
  • \(H_1\): sí existe concordancia
  • RC = \([F > c]\) = \([W > h]\)

• Equivale a aplicar la \(F\) de Friedman a la tabla binaria (muchos empates): Friedman's (1937) statistic, corrected for ties (…) reduces to Cochran's statistic for only two categories (see Lehmann, p. 267).
• \(\vec X\) = \((X_1,\dots,X_k)\) vector aleatorio con \(X_i\) \(\hookrightarrow\) \(\mathcal B(p_i)\)
• \(H_0\) : \(\forall\,i,j\), \(p_i=p_j\) ; \(H_1\) : \(\exists\,i,j\), \(p_i\ne p_j\)

• \(\hat p_j\) = \(\dfrac{C_j}n\)
• Se mide la dispersión de las \(\hat p_j\) o de las \(C_j\) \[S = \sum_{j=1}^k\left(C_j-\frac Tk\right)^2\] RC = \([S > c]\)
• \(\dfrac{\hat p_j-p}{\sqrt{\dfrac{p(1-p)}n}}\) \(\xrightarrow[\mathcal L]{H_0}\) \(\mathcal N(0;1)\) \(\implies\) \(\dfrac{C_j-\dfrac Tk}{\sqrt{\text{Var}(C_j)}}\) \(\xrightarrow[\mathcal L]{H_0}\) \(\mathcal N(0;1)\)
• \({\text{Var}}(C_j)\) = \(\displaystyle\sum_{i=1}^n{\text{Var}}(X_{ij})\) = \(\displaystyle\sum_{i=1}^n\dfrac{R_i}k \left(1-\dfrac{R_i}k\right)\)
• \(\widehat{\text{Var}}(C_j)\) = \(\displaystyle\sum_{i=1}^n\widehat{\text{Var}}(X_{ij})\) = \(\displaystyle\sum_{i=1}^n\dfrac{R_i}k \left(1-\dfrac{R_i}k\right)\dfrac k{k-1}\) \(\implies\) \(\displaystyle \dfrac{\left(C_j-\dfrac Tk\right)^2} {\sum_i R_i(k-R_i)\dfrac1{k(k-1)}}\) \(\xrightarrow[\mathcal L]{H_0}\) \(\chi^2_1\) \(\implies\) \(Q\) = \(k(k-1)\dfrac S{\sum_i R_i(k-R_i)}\) = \((k-1)\dfrac{\sum_{j=1}^k\left(C_j-\dfrac Tk\right)^2} {T-\sum_{i=1}^n\dfrac{R_i^2}k}\) \(\xrightarrow[\mathcal L]{H_0}\) \(\chi^2_{k-1}\) (las \(C_j\) son dependientes, \(\sum_j C_j\) = \(T\))
• RC = \([Q > c]\)
• Distribución de \(Q\)
  • aproximadamente \(\chi^2_{k-1}\) si \(n > 6\) y \(nk > 24\)
  • Montecarlo
    • mantener \(R_i\) constantes permutando dentro del individuo

• Ejemplo

  individuo	\(X_1\)	\(X_2\)	\(X_3\)	\(X_4\)	\(R_i\)
  1	1	0	0	0	1
  2	0	0	1	1	2
  3	1	0	1	0	2
  4	0	0	0	0	0
  5	1	1	0	1	3
  \(C_{j}\)	3	1	2	2	\(T=8\)

  X <- matrix (c(1,0,1,0,1, 0,0,0,0,1, 0,1,1,0,0, 0,1,0,0,1), 5)
  k <- ncol (X)
  R <- apply (X, 1, sum) # rowSums (X)
  C <- apply (X, 2, sum) # colSums (X)
  T <- sum (X) # == sum (C) == sum (R)
  (Q <-  (k-1) * sum ((C - T/k) ^ 2) / (T - sum (R^2 / k)))
  1 - pchisq (Q, k-1) # p-valor asintótico
  DescTools::CochranQTest (X)
  friedman.test (X) # es lo mismo
  
  Q <- function (X) {
      k <- ncol (X)
      R <- apply (X, 1, sum) # rowSums (X)
      C <- apply (X, 2, sum) # colSums (X)
      T <- sum (X) # == sum (C) == sum (R)
      (k-1) * sum ((C - T/k) ^ 2) / (T - sum (R^2 / k))
  }
  ## P-valor Montecarlo
  mean (replicate (1e4, Q (t (apply (X, 1, sample)))) >= Q(X))
  

  [1] 1.714286
  [1] 0.6337624
  
  	Cochran's Q test
  
  data:  y
  Q = 1.7143, df = 3, p-value = 0.6338
  
  
  	Friedman rank sum test
  
  data:  X
  Friedman chi-squared = 1.7143, df = 3, p-value = 0.6338
  
  [1] 0.9193
  

• Experimento Bernoulli en dos situaciones diferentes para datos pareados.
• Equivale a
  • Q de Cochran con \(k=2\)
  • prueba de los signos con \(2\) muestras

• Ejemplo: proporción de individuos que presenta una característica \(A\), antes y después de cierto tratamiento.

  antes \ después	\(A\)	\(\bar A\)	sumas
  \(A\)	\(a\)	\(b\)	\(n_{A\cdot}\)
  \(\bar A\)	\(c\)	\(d\)	\(n_{\bar A\cdot}\)
  sumas	\(n_{\cdot A}\)	\(n_{\cdot\bar A}\)	\(n\)

• \(H_0\) : \(\Pr[A\mid\text{antes}]\) = \(\Pr[A\mid\text{después}]\)
• \(H_1\) : \(\Pr[A\mid\text{antes}]\) \(\ne\) \(\Pr[A\mid\text{después}]\)
• Estimaciones: \(\widehat\Pr[A\mid\text{antes}]\) = \(\dfrac{a+b}n\), \(\widehat\Pr[A\mid\text{después}]\) = \(\dfrac{a+c}n\)
• \(H_0\) \(\implies\) \(b\approx c\)
  • promedio \(m\) = \(\dfrac{b+c}2\)
  • estadístico \(M\) = \(\dfrac{(b-m)^2+(c-m)^2}m\) = \(\dfrac{2(b-m)^2}m\) = \(\dfrac{\left(b-\dfrac{b+c}2\right)^2}{\dfrac{b+c}4}\) = \(\dfrac{(b-c)^2}{b+c}\)
  • RC = \([M > c]\)
• Si se supone conocido \(b+c\) (el número de individuos que cambian) entonces \(b\) \(\stackrel{H_0}{\hookrightarrow}\) \(\mathcal B(b+c,1/2)\) \(\xrightarrow[b+c\to\infty]{\mathcal L}\) \(\mathcal N\left(\dfrac{b+c}2;\sqrt{\dfrac{b+c}4}\right)\) \(\implies\) \(\dfrac{b-\dfrac{b+c}2}{\sqrt{\dfrac{b+c}4}}\) \(\stackrel{\sim}{\hookrightarrow}\) \(\mathcal N(0;1)\) \(\implies\) \(\dfrac{\left(b-\dfrac{b+c}2\right)^2}{\dfrac{b+c}4}\) = \(M\) \(\stackrel{\sim}{\hookrightarrow}\) \(\chi^2_1\)

• Ejemplo médico

  • respuesta: sufrir o no hipoglucemias
  • cada individuo:
    • una noche con control normal
    • otra noche con control automático

  normal \ autom.	no hipo.	hipoglucemia
  no hipo.	20	7
  hipoglucemia	22	5

  • \(H_0\) : \(\Pr[\text{hipo}\mid\text{normal}]\) = \(\Pr[\text{hipo}\mid\text{auto}]\) ; \(H_1\) : \(\Pr[\text{hipo}\mid\text{normal}]\) \(\ne\) \(\Pr[\text{hipo}\mid\text{auto}]\)
  • \(M\) = \(\dfrac{(22-7)^2}{22+7}\) \(\approx\) \(7.76\)

  1 - pchisq (7.76, 1) # aproximación asintótica
  mcnemar.test (matrix(c(20,22,7,5),2), correct=FALSE) # ídem
  mcnemar.test (matrix(c(20,22,7,5),2)) # corrección por continuidad
  2 * pbinom (7, 22+7, 1/2) # P-valor exacto
  ## lo mismo que correct=FALSE
  friedman.test(do.call(rbind,rep(list(c(0,0),c(0,1),c(1,0),c(1,1)),c(20,22,7,5))))
  

  [1] 0.005341596
  
  	McNemar's Chi-squared test
  
  data:  matrix(c(20, 22, 7, 5), 2)
  McNemar's chi-squared = 7.7586, df = 1, p-value = 0.005346
  
  
  	McNemar's Chi-squared test with continuity correction
  
  data:  matrix(c(20, 22, 7, 5), 2)
  McNemar's chi-squared = 6.7586, df = 1, p-value = 0.00933
  
  [1] 0.008130059
  
  	Friedman rank sum test
  
  data:  do.call(rbind, rep(list(c(0, 0), c(0, 1), c(1, 0), c(1, 1)), c(20, 22, 7, 5)))
  Friedman chi-squared = 7.7586, df = 1, p-value = 0.005346
  
  

• \((X,Y)\) \(\hookrightarrow\) \(\mathcal N_2(\vec\mu;\Sigma)\) \(\implies\) independencia \(\equiv\) correlación lineal nula\
• \(H_0\) : \(X\) y \(Y\) independientes \(\iff\) \(\rho_{XY}=0\)
• EMV de \(\rho\): \(R\) = \(\dfrac{S_{XY}}{S_{X}S_Y}\)
• RC = \([R^2 > c]\) = \(\left[\dfrac{(n-2)R^2}{1-R^2} > h\right]\)
• \((X,Y)\) \(\hookrightarrow\) \(\mathcal N_2(\vec\mu;\Sigma)\) \(\implies\) \(\dfrac{(n-2)R^2}{1-R^2} \) \(\stackrel{H_0}{\hookrightarrow}\) \(F_{1,n-2}\) \(\iff\) \(\sqrt{\dfrac{(n-2)R^2}{1-R^2}}\) \(\stackrel{H_0}{\hookrightarrow}\) \(t_{n-2}\)

plot (mtcars$qsec, mtcars$mpg)

cor.test (mtcars$qsec, mtcars$mpg)

	Pearson's product-moment correlation

data:  mtcars$qsec and mtcars$mpg
t = 2.5252, df = 30, p-value = 0.01708
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 0.08195487 0.66961864
sample estimates:
     cor 
0.418684 

• Para variables
  • cualitativas ordinales
  • rangos de cuantitativas \(\implies\) robustez
• Es el coeficiente de Pearson aplicado a rangos: \(R_s\) = \(\dfrac{\text{Cov}(R_X,R_Y)}{\sqrt{\text{Var}(R_X)\text{Var}(R_Y)}}\) = \(\dfrac{6\sum_{i=1}^n(R_{X,i}-R_{Y,i})^2}{n(n^2-1)}\)
• RC = \([|R_s| > c]\)
• \(\sqrt{n-1}R_s\) \(\xrightarrow[\mathcal L]{H_0}\) \(\mathcal N(0;1)\)

plot (rank(mtcars$qsec), rank(mtcars$mpg))

cor.test (mtcars$qsec, mtcars$mpg, method="spearman")

	Spearman's rank correlation rho

data:  mtcars$qsec and mtcars$mpg
S = 2908.4, p-value = 0.007056
alternative hypothesis: true rho is not equal to 0
sample estimates:
      rho 
0.4669358