![\"FAQ__How_are_the_likelihood_ratio__Wald__and_Lagrange_multiplier__score__tests_different_and_or_similar_\"](\"http:\/\/zhanxw.com\/blog\/wp-content\/uploads\/2014\/12\/FAQ__How_are_the_likelihood_ratio__Wald__and_Lagrange_multiplier__score__tests_different_and_or_similar_.png\")
<\/a>
\n\uff08\u6458\u81ea\uff1ahttp:\/\/www.ats.ucla.edu\/stat\/mult_pkg\/faq\/general\/nested_tests.htm\uff09<\/p>\nLikelihood Ratio Test\u8ba1\u7b97\u7684\u662f[latex] \\xi^R = 2 (l(\\hat{a}) – l(0)) [\/latex], \u5c31\u662f\u4e24\u500d\u7ea2\u8272\u7ad6\u7ebf\u7684\u957f\u5ea6\uff0c\u8fd9\u4e2a\u7edf\u8ba1\u91cf\u8fd1\u4f3c\u6709\u81ea\u7531\u5ea6\u4e3a1\u7684\u5361\u65b9\u5206\u5e03\uff08\u5047\u8bbe\u53ea\u6709\u4e00\u4e2a\u81ea\u7531\u53d8\u91cf\uff09
\n\u8fd9\u91cc\u7684[latex]\\hat{a}[\/latex]\u662f\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\uff08Maximum Likelihood Estimator):<\/p>\n
$$
\n\\xi^R = 2 (l(\\hat{a}) – l(0)) = 2 log(L(\\hat{a})\/L(0)) \\sim \\chi^2_1
\n$$<\/p>\n
Score Test\u53ea\u9700\u8981\u8003\u8651[latex] 0 [\/latex] \u8fd9\u4e00\u70b9\u4e0a\u4f3c\u7136\u51fd\u6570\u7684\u6027\u8d28\uff0c\u5728\u56fe\u50cf\u4e0a\u901a\u8fc7\u84dd\u8272\u90e8\u5206\u8868\u793a\uff08\u901a\u8fc7\u8003\u5bdf\u8fd9\u4e00\u70b9\u7684\u659c\u7387\u548c\u66f2\u7387\uff09\u3002
\n\u5199\u6210\u516c\u5f0f\u662f\uff1a
\n$$
\n\\begin{align}
\n\u659c\u7387 &\uff1d U = l'(0) \\\\
\n|\u66f2\u7387| &\uff1d V = |l”(0)| = -l”(0) \\\\
\n\\xi^S &= U’ * V^{-1} * U = \\frac{l'(0)^2}{|l”(0)|} \\sim \\chi^2_1
\n\\end{align}
\n$$<\/p>\n
\u800cWald Test\u53ea\u9700\u8981\u8003\u8651[latex]\\hat{a}[\/latex]\u8fd9\u4e00\u70b9\u4e0a\u4f3c\u7136\u51fd\u6570\u7684\u6027\u8d28\uff0c\u5728\u56fe\u50cf\u4e0a\u7528\u7eff\u8272\u7684\u6a2a\u7ebf\u8868\u793a\u7684\uff08\u901a\u8fc7\u6bd4\u8f83\u7eff\u8272\u6a2a\u7ebf\u7684\u5bbd\u5ea6\u548c\u5728[latex]\\hat{a}[\/latex]\u7684\u66f2\u7387\uff09\u3002
\n$$
\n\\begin{align}
\n\u659c\u7387 &\uff1d l'(a) \\\\
\n\\xi^W &= \\hat{a}^2 * |l”(a)| \\sim \\chi^2_1
\n\\end{align}
\n$$<\/p>\n
Score Test\u7b49\u4ef7\u4e8eLikelihood Ratio Test\u548cWald Test<\/h2>\n
\u4ece\u6e10\u8fdb\u6027\u8d28\uff08Asymptotic property\uff09\u4e0a\u8bb2\uff0cScore Test\u548cLikelihood Ratio Test\uff08LRT\uff09\uff0cWald Test\u662f\u7b49\u4ef7\u7684\u3002
\n\u8be6\u7ec6\u7684\u4e25\u683c\u7684\u8bc1\u660e\u53ef\u4ee5\u53c2\u89c1\uff3b2\uff3d\uff0c\u6587\u732e\u91cc\u6709\u66f4\u4e00\u822c\u7684\u7ed3\u8bba\uff08\u4e0d\u9650\u4e8e\u81ea\u7531\u5ea6\u7b49\u4e8e\u4e00\uff09\u3002
\n\u4f46\u662f\u5728\u4e0d\u4e25\u683c\u7684\u610f\u601d\u4e0b\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u4e00\u4e2a\u7b80\u5316\u7684\u4f8b\u5b50\u6765\u5c55\u793a\u8fd9\u79cd\u7b49\u4ef7\u6027\u8d28\u3002
\n\u4ecd\u7136\u5047\u5b9a[latex] H0: a = 0 [\/latex] \u4ee5\u53ca [latex] H_a: a \\neq 0 [\/latex]\u3002<\/p>\n
\u5bf9\u4e8eLikelihood Ratio Test \uff08LRT\uff09\uff0c\u7edf\u8ba1\u91cf\uff08test statistics\uff09\u662f\u4e24\u500d\u7ea2\u7ebf\u7684\u957f\u5ea6\uff1a
\n$$
\n\\chi^{R} = 2 * ( l(\\hat{a}) – l(0))
\n$$<\/p>\n
\u9996\u5148\u8bc1\u660eWald Test\u548cLRT\u662f\u7b49\u4ef7\u7684\u3002
\n$$
\nl(0) = l(\\hat{a}) – \\hat{a} l'(\\hat{a}) + \\frac{1}{2} \\hat{a}^2 l”(\\hat{a})
\n$$
\n\u6839\u636eMLE\u7684\u6027\u8d28\uff0c[latex]l'(\\hat{a}) = 0 [\/latex]\u3002\u56e0\u6b64\uff1a
\n$$
\nl(0) = l(\\hat{a}) + \\frac{1}{2} \\hat{a}^2 l”(\\hat{a})
\n$$
\n\u6574\u7406\u4e00\u4e0b\uff0c\u5f97\u5230\uff1a
\n$$
\n\\xi^R = 2 (l(\\hat{a}) – l(0)) = \\hat{a}^2 * (-l”(\\hat{a})) = \\hat{a}^2 * |l”(\\hat{a})|= \\xi^W
\n$$<\/p>\n
\u63a5\u4e0b\u6765\u8bc1\u660eScore Test\u548cLRT\u662f\u7b49\u4ef7\u7684\u3002
\n\u6839\u636e\u6cf0\u52d2\u5c55\u5f00\u548cMLE\u7684\u6027\u8d28\uff1a
\n$$
\n\\begin{align}
\nl(\\hat{a}) &= l(0) + \\hat{a} * l'(0) + \\frac{1}{2}\\hat{a}^2 * l”(0) \\\\
\nl'(\\hat{a}) &= l'(0) + \\hat{a} * l”(0) = 0
\n\\end{align}
\n$$
\n\u7b2c\u4e8c\u884c\u7684\u5f0f\u5b50\u53ef\u4ee5\u5199\u6210\uff1a
\n$$
\n\\hat{a} = – \\frac{l'(0)}{l”(0)}
\n$$
\n\u7b2c\u4e00\u884c\u53ef\u4ee5\u5199\u6210\uff1a
\n$$
\n\\begin{align}
\n\\xi^R &= 2* (l(\\hat{a}) – l(0)) = 2 * \\hat{a} * l'(0) + \\hat{a}^2 * l”(0) \\\\
\n&= – 2 * \\frac{l'(0)^2}{l”(0)} + \\frac{l'(0)^2}{l”(0)} \\\\
\n&= \\frac{l'(0)^2}{-l”(0)} \\\\
\n&= \\frac{l'(0)^2}{|l”(0)|} \\\\
\n&= \\xi^S
\n\\end{align}
\n$$<\/p>\n
\u5728\u4e0a\u9762\u7684\u63a8\u5bfc\u4e2d\uff0c\u6211\u4eec\u7528\u5230\u4e86\u591a\u6b21[latex]l”()[\/latex]\uff0c\u8fd9\u4e2a\u51fd\u6570\u4e00\u822c\u662f\u8d1f\u6570\u3002\u5bf9\u8fd9\u4e2a\u51fd\u6570\u53d6\u671f\u671b\u5c31\u662fFisher Information\u3002
\n\u800cFisher Information\u7684\u5012\u6570\u6216\u8005\u662f\u9006\u77e9\u9635\uff0c\u5c31\u662fScore statistics\u7684\u65b9\u5dee\u3002<\/p>\n
Score Test\u7684\u5f62\u5f0f<\/h2>\n
\u4e0a\u9762\u7684\u4f8b\u5b50\u90fd\u662f\u68c0\u9a8c\u4e00\u4e2a\u53c2\u6570\uff0c\u5982\u679c\u53c2\u6570\u4e2a\u6570\u591a\u4e8e\u4e00\uff0c\u90a3\u4e48\u6709\u66f4\u4e00\u822c\u7684\u7ed3\u8bba\u3002
\n\u5047\u8bbe\u53c2\u6570\u662f[latex]\\theta = (\\theta_1, \\theta_2) [\/latex]\uff0c\u8fd9\u91cc[latex]\\theta_1,\\theta_2[\/latex]\u5206\u522b\u662f\u957f\u5ea6\u4e3a[latex]p_1,p_2[\/latex]\u7684\u5411\u91cf\uff0c
\n\u5047\u8bbe\u68c0\u9a8c\u7684\u662f\uff1a [latex]H_0: \\theta_1 = 0, H_a: \\theta_1 \\neq 0[\/latex].<\/p>\n
\u9996\u5148\u8ba1\u7b97[latex]H_0[\/latex]\u4e0b\u7684MLE,\u5047\u8bbe\u662f\uff1a [latex]\\hat{\\theta} = (0, \\hat{\\theta}_2) [\/latex]\u3002\u7136\u540e\u8ba1\u7b97U\u548cV
\n$$
\n\\begin{align}
\nU &= Score = l'(\\hat{\\theta}) = l'( (\\theta_1 = 0, \\hat{\\theta}_2) ) \\\\
\nV &= I^{11}_{\\hat{\\theta}} = (I_{11} – I_{12} (I_{22})^{-1} I_{22} )_{\\hat{\\theta}}
\n\\end{align}
\n$$
\n\u8fd9\u91cc[latex]I[\/latex]\u662fFisher information\uff1a
\n$$
\nI(\\theta)
\n= \\left[ \\begin{array}{cc}
\nI_{11} & I_{12} \\\\
\nI_{21} & I_{22}
\n\\end{array} \\right]_\\theta
\n= \\left[ \\begin{array}{cc}
\nI_{11}(\\theta) & I_{12}(\\theta) \\\\
\nI_{21}(\\theta) & I_{22}(\\theta)
\n\\end{array} \\right]
\n$$
\n\u6700\u540e\uff0c\u7edf\u8ba1\u91cf[latex] U * V^{-1} * U’ \\sim \\chi_{p_1} [\/latex]\u3002<\/p>\n
\u4e3a\u4ec0\u4e48\u4e0a\u9762\u7684[latex]V[\/latex]\u4e0d\u662f[latex]I_{11}(\\theta)[\/latex]\uff0c\u800c\u662f[latex]I_{11}(\\theta)[\/latex]\u51cf\u53bb\u989d\u5916\u7684\u4e00\u9879\u5462\uff1f
\n\u53ef\u4ee5\u8ba4\u4e3a\u68c0\u9a8c[latex]\\theta_1[\/latex] \u5b9e\u9645\u4e0a\u662f\u5728\u63a7\u5236[latex]\\theta_2[\/latex]\u7684\u540c\u65f6\u68c0\u9a8c[latex]\\theta_1[\/latex]\uff0c
\n\u56e0\u6b64[latex]\\theta_1[\/latex]\u662f\u5728[latex]\\theta_2[\/latex]\u7684residual space\u91cc\u9762\uff0c\u56e0\u6b64\u5b83\u672c\u8eab\u7684\u53d8\u5316\uff08Variance\uff09\u5c31\u5c0f\u4e86\u3002<\/p>\n
\u4e3e\u4f8b\u6765\u8bf4\uff08\u4e0b\u4e00\u8282\u7684\u4f8b\u5b50\u4e5f\u4f1a\u63d0\u5230\uff09\uff0c\u5047\u8bbe[latex]H_0: b = 0, H_a: b \\neq 0[\/latex]\u3002
\n\u5982\u679c\u6a21\u578b\u662f[latex] Y = X b + \\epsilon, \\epsilon_{ii} \\sim N(0,1) [\/latex]\u3002
\n\u90a3\u4e48[latex] V = \\sigma^2 (X’X) [\/latex].
\n\u5982\u679c\u6a21\u578b\u662f[latex] Y = X b + Z r + \\epsilon, \\epsilon_{ii} \\sim N(0,1) [\/latex]\uff0c
\n\u90a3\u4e48\u6309\u7167\u516c\u5f0f\uff0c\u7b2c\u4e00\u79cd\u65b9\u6cd5\u7684\u5f97\u5230\u7684[latex] V = X’X – X’Z (Z’Z)^{-1} Z X = V_{XX} [\/latex]<\/p>\n
\u53e6\u4e00\u79cd\u65b9\u6cd5\u662f\u6c42\u51faX\u7684Residual adjusting for Z:
\n$$
\nX_Z = (I-H) X = (I – Z (Z’Z)^{-1} Z’ ) X
\n$$
\n\u53ef\u4ee5\u5f97\u5230\uff1a
\n$$
\nV_{XX} = X_Z’ X_Z = X’ (I-H)’ (I-H) X = X’ (I – H) X = X’X – X’Z(Z’Z)^{-1}ZX
\n$$
\n\u8fd9\u548c\u7b2c\u4e00\u79cd\u65b9\u6cd5\u5f97\u5230\u7684[latex]V[\/latex]\u662f\u7b49\u4ef7\u7684\u3002<\/p>\n
\u4e0a\u9762\u7684\u4f8b\u5b50\u5047\u8bbe\u8bef\u5dee\u662f\u72ec\u7acb\u540c\u5206\u5e03\u7684\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u63a8\u5e7f\u5230generalized linear regression\u3002
\n\u90a3\u4e48\u76f8\u5e94\u7684[latex]V[\/latex]\u9879\u5c31\u662f[latex]X’WX – X’W (X’W^{-1}X)^{-1} W X'[\/latex]\uff0c
\n\u5176\u4e2d[latex]W[\/latex]\u53ef\u4ee5\u662f\u5df2\u77e5\u7684\uff0c\u6216\u8005\u901a\u8fc7mean function\u6c42\u51fa\u6765\u7684\u3002
\n\u6bd4\u5982Logistic regression\u91cc[latex]W = \\mu * (1-\\mu)[\/latex]\u3002<\/p>\n
\u66f4\u4e00\u822c\u7684\u7ed3\u8bba\u53ef\u4ee5\u53c2\u8003\uff3b1\uff3d\u3002<\/p>\n
\u4e00\u4e9b\u4f8b\u5b50<\/h2>\n
\u4e0b\u9762\u7ed9\u51fa\u4e00\u4e9b\u5177\u4f53\u4f8b\u5b50\uff0c\u90fd\u5047\u8bbe [latex]H_0: \\theta_1 = 0, H_a: \\theta_1 \\neq 0[\/latex]\u3002<\/p>\n
1. \u7b80\u5355\u7684\u7ebf\u6027\u56de\u5f52\uff08Simple Linear Regression) \uff1a[latex] Y = X b + \\epsilon, \\epsilon_{ii} \\sim N(0,\\sigma^2) [\/latex]<\/p>\n
$$
\nl(b, \\sigma^2) = – \\frac{n}{2} \\log(\\sigma^2) – \\frac{(Y-Xb)'(Y-Xb)}{2 \\sigma^2}
\n$$
\n\u53ef\u4ee5\u6c42\u51fa[latex]\\hat{\\sigma}^2 = \\frac{1}{n} Y’Y [\/latex]
\n$$
\nU_b = \\frac{\\partial l}{\\partial b} = Y’X \/ \\hat{\\sigma}^2 \\\\
\nV_{bb} = -\\frac{\\partial l^2}{\\partial^2 b} = X’X \/ \\hat{\\sigma}^2\\\\
\n$$
\n\u56e0\u6b64[latex]\\xi^S = \\frac{(Y’X \/ \\hat{\\sigma}^2)^2}{ X’X \/ \\hat{\\sigma}^2 } = \\frac{ (Y’X)^2 }{(X’X)(Y’Y)\/n} \\sim \\chi^2_1[\/latex].
\n\u5f53[latex]X[\/latex]\u548c[latex]Y[\/latex]\u7684\u5747\u503c\u90fd\u662f\u96f6\u7684\u65f6\u5019\uff08centered\uff09\uff0c[latex]\\xi^S = n r^2[\/latex].
\n\u8fd9\u91cc[latex]r[\/latex]\u662f\u76ae\u5c14\u68ee\u76f8\u5173\u7cfb\u6570\uff08Pearson correlation coefficient\uff09\u3002<\/p>\n
2. \u4e00\u822c\u7684\u7ebf\u6027\u56de\u5f52\uff1a[latex] Y = X b + Z r + \\epsilon, \\epsilon_{ii} \\sim N(0,1) [\/latex]
\n$$
\n\\begin{align}
\nU_b &= \\frac{\\partial l}{\\partial b} = (Y – Z \\hat{r})’X \/ \\hat{\\sigma}^2 \\\\
\nV_{bb} &= -\\frac{\\partial l^2}{\\partial^2 b} = (X’X – X’Z(Z’Z)^{-1}Z’X) \/ \\hat{\\sigma}^2\\\\
\n\\hat{\\sigma}^2 &\uff1d \\frac{ (Y-Z \\hat{r})'(Y-Z \\hat{r})}{n} \\\\
\n\\hat{r} &= (Z’Z)^{-1} Z’Y
\n\\end{align}
\n$$<\/p>\n
\u4e0a\u9762\u7684\u5f62\u5f0f\u4e5f\u8bb8\u4e0d\u592a\u597d\u8bb0\uff0c\u4e0d\u8fc7\u5982\u679c\u5b9a\u4e49[latex]H_Z = Z(Z’Z)^{-1}Z'[\/latex]\uff0c\u5219\u6709\uff1a
\n$$
\n\\begin{align}
\nU_b &= Y'(I-H_z)X \\\\
\nV_{bb} &= X'(I-H_z)X \/\\hat{\\sigma}^2 ]\\\\
\n\\hat{\\sigma}^2 &\uff1d \\frac{ Y’ (I-H_z) Y }{n} \\\\
\n\\end{align}
\n$$
\n\u53ef\u89c1[latex] I-H_z [\/latex]\u5728\u8fd9\u91cc\u6709\u5173\u952e\u4f5c\u7528\uff0c\u5982\u679c\u8ba9:
\n$$
\nX_z = (I-H_z)X \\\\
\nY_z = (I-H_z)Y
\n$$
\n\u90a3\u4e48\u4e0a\u9762\u7684Score test\u7b49\u4ef7\u4e8e\u7b2c\u4e00\u79cd\u60c5\u51b5\uff1a [latex] Y_z = X_z b + \\epsilon, \\epsilon_{ii} \\sim N(0, 1)[\/latex].<\/p>\n
\u6b64\u5916\u8fd8\u53ef\u4ee5\u8bc1\u660e[latex] \\xi^S < \\xi^R < \\xi^W [\/latex]\uff0c\u89c1\uff3b2\uff3d\u3002\n\n3. Logistic\u56de\u5f52\uff1a [latex]logit(E(Y)) = X b [\/latex]\n\n$$\n\\begin{align}\nl &= \\sum_i y_i \\log(p_i) + (1-y_i) * \\log(1-p_i) \\\\\n& = \\sum_i y_i \\log(\\frac{p}{1-p}) + log(1-p_i) \\\\ \n& = Y'X b - \\sum_i \\log(1+\\exp(X_i b))\n\\end{align}\n$$\n\n$$\n\\begin{align}\nU &= \\frac{\\partial l}{\\partial b} = Y'X - (\\frac{\\exp(Xb)}{1+\\exp(Xb)})' X = (Y - \\hat{Y})' X \\\\\nV &= - \\frac{\\partial^2 l}{\\partial b^2} = - \\frac{\\partial -X' \\hat{Y}}{\\partial b} = X' \\frac{\\exp(Xb)}{(1+\\exp(Xb))^2} X = X' W X \\\\\n\\end{align}\n$$\n\u4e0a\u5f0f\u4e2d[latex]W = diag(\\hat{y} * (1-\\hat{y})) [\/latex].\n\u5728[latex]H_0[\/latex]\u4e0b\uff0c[latex]b = 0 [\/latex]\uff0c\u56e0\u6b64\uff1a\n$$\n\\begin{align}\nU_b &= (Y - \\frac{1}{2})' X \\\\\nV_{bb} &= \\frac{1}{4} X' X \\\\\n\\end{align}\n$$\n\n\n4. Logistic\u56de\u5f52\uff1a [latex]logit(E(Y)) = X b + Z r[\/latex]\n\n$$\n\\begin{align}\nl &= \\sum_i y_i \\log(p_i) + (1-y_i) * \\log(1-p_i) \\\\\n& = \\sum_i y_i \\log(\\frac{p}{1-p}) + log(1-p_i) \\\\ \n& = Y'(X b + Zr) - \\sum_i \\log(1+\\exp(X_i b + Z_i r))\n\\end{align}\n$$\n\u7ecf\u8fc7\u4e00\u4e9b\u8df3\u6b65\uff1a\n$$\n\\begin{align}\nU_b &= \\frac{\\partial l}{\\partial b} = (Y - \\hat{Y})' X \\\\\nV_{bb} &= - \\frac{\\partial^2 l}{\\partial b^2} = X' W X \uff0d X'WZ (Z'WZ)^{-1} Z'W X \\\\\n\\hat{Y} &= \\frac{1}{1 + \\exp(- Z \\hat{r} )} \n\\end{align}\n$$\n\u7c7b\u4f3c\u7684\uff0c\u4e0a\u5f0f\u4e2d[latex]W = diag(\\hat{y} * (1-\\hat{y})) [\/latex].\n[latex]\\hat{r}[\/latex]\u662f\u5728[latex]b=0[\/latex]\u65f6\u7684MLE\u4f30\u8ba1\u3002\n\u4e0a\u9762\u8fd9\u4e2a\u5f62\u5f0f\u548c\uff3b3\uff3d\u91cc\u7684\u516c\u5f0f\u662f\u7b49\u4ef7\u7684\u3002\n\n\n[1] Chen, C.-F. Score Tests for Regression Models. Journal of the American Statistical Association (1983).doi:10.1080\/01621459.1983.10477945\n[2] Lin, D.-Y. & Tang, Z.-Z. A General Framework for Detecting Disease Associations with Rare Variants in Sequencing Studies. Am. J. Hum. Genet. 89, 354\u201367 (2011).\n[3] Liu, D. et al. Meta-analysis of gene-level tests for rare variant association. Nature Genetics 46, 200204 (2013).\n<\/p>\n","protected":false},"excerpt":{"rendered":"
\u62c9\u683c\u6717\u65e5\u4e58\u6570\u68c0\u9a8c Lagrange Multiplier test (Score test) \u62c9\u683c\u6717\u65e5\u4e58\u6570\u68c0\u9a8c\uff0c\u82f1\u6587\u662fLagrange multiplier test\uff0c\u6216\u8005\u53eb\u505aScore test\u662f\u4e00\u79cd\u5e38\u7528\u7684\u7edf\u8ba1\u68c0\u9a8c\u3002 \u62c9\u683c\u6717\u65e5\u4e58\u6570\u68c0\u9a8c\u7684\u540d\u79f0\u6765\u6e90\u4e8e\u8fd9\u4e2a\u68c0\u9a8c\u7528\u7684\u662f\u62c9\u683c\u6717\u65e5\u4e58\u6570\u7684\u5206\u5e03\uff0c\u89c1\uff3b2\uff3d\u3002 Score test\u7684\u540d\u79f0\u5219\u6765\u81ea\u4e8eScore\u672c\u8eab\u3002 \u4e3a\u4e86\u5199\u8d77\u6765\u65b9\u4fbf\uff0c\u4e0b\u9762\u90fd\u7528Score Test\u6765\u4ee3\u66ff\u62c9\u683c\u6717\u65e5\u4e58\u6570\u68c0\u9a8c\u3002 Score Test\uff0cLikelihood Ratio Test\u548cWald Test\u7684\u56fe\u5f62\u8868\u793a \u5047\u8bbe\u4f3c\u7136\u51fd\u6570[latex] L(.) [\/latex] \u53ea\u6709\u4e00\u4e2a\u53c2\u6570, \u8fd9\u4e09\u79cd\u68c0\u9a8c\u53ef\u4ee5\u5728\u4e00\u5f20\u56fe\u91cc\u8868\u793a\u51fa\u6765\uff1a \uff08\u6458\u81ea\uff1ahttp:\/\/www.ats.ucla.edu\/stat\/mult_pkg\/faq\/general\/nested_tests.htm\uff09 Likelihood Ratio Test\u8ba1\u7b97\u7684\u662f[latex] \\xi^R = 2 (l(\\hat{a}) – l(0)) [\/latex], \u5c31\u662f\u4e24\u500d\u7ea2\u8272\u7ad6\u7ebf\u7684\u957f\u5ea6\uff0c\u8fd9\u4e2a\u7edf\u8ba1\u91cf\u8fd1\u4f3c\u6709\u81ea\u7531\u5ea6\u4e3a1\u7684\u5361\u65b9\u5206\u5e03\uff08\u5047\u8bbe\u53ea\u6709\u4e00\u4e2a\u81ea\u7531\u53d8\u91cf\uff09 \u8fd9\u91cc\u7684[latex]\\hat{a}[\/latex]\u662f\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\uff08Maximum Likelihood Estimator): $$ \\xi^R = 2 (l(\\hat{a}) – l(0)) = 2 log(L(\\hat{a})\/L(0)) \\sim \\chi^2_1 $$ Score Test\u53ea\u9700\u8981\u8003\u8651[latex] […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[15],"tags":[],"class_list":["post-585","post","type-post","status-publish","format-standard","hentry","category-statistics"],"_links":{"self":[{"href":"https:\/\/zhanxw.com\/blog\/wp-json\/wp\/v2\/posts\/585","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/zhanxw.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/zhanxw.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/zhanxw.com\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/zhanxw.com\/blog\/wp-json\/wp\/v2\/comments?post=585"}],"version-history":[{"count":0,"href":"https:\/\/zhanxw.com\/blog\/wp-json\/wp\/v2\/posts\/585\/revisions"}],"wp:attachment":[{"href":"https:\/\/zhanxw.com\/blog\/wp-json\/wp\/v2\/media?parent=585"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/zhanxw.com\/blog\/wp-json\/wp\/v2\/categories?post=585"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/zhanxw.com\/blog\/wp-json\/wp\/v2\/tags?post=585"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}