【统计学笔记】方差分析表和回归分析表的解读

F=MS⊙MSE∼F(df(⊙),df(E))(⊙表示误差来源中因素的简写，MS⊙表示MSA、MSR或MSC等，df(⊙)表示因素⊙的自由度)F = \frac{MS⊙}{MSE} \sim F(\quad df( ⊙),df(E)\quad) \\ \qquad \\

(⊙表示误差来源中因素的简写，MS⊙表示MSA、MSR或MSC等，df(⊙)表示因素⊙的自由度)

F=MSEMS⊙​∼F(df(⊙),df(E))(⊙表示误差来源中因素的简写，MS⊙表示MSA、MSR或MSC等，df(⊙)表示因素⊙的自由度)

MS⊙=SS⊙df(⊙)MS⊙ = \frac{SS⊙}{df(⊙)}MS⊙=df(⊙)SS⊙​

自由度 (dfdfdf)：degree of freedom

平方和 (SSSSSS)：Sum of Square

均方 (MSMSMS)：Mean Square

1. 方差分析表

1.1 单因素方差分析表

k：因素总体的个数n：观测值个数

误差来源平方和SSSSSS自由度dfdfdf均方MSMSMSFFF值PPP值FFF临界值Significance FSignificance \; FSignificanceF组间（因素影响）factor Afactor \; \bold AfactorASSASSASSAk−1k-1k−1MSA=SSAk−1MSA = \frac{SSA}{k-1}MSA=k−1SSA​MSAMSE\frac{MSA}{MSE}MSEMSA​根据显著性水平α\alphaα确定组内（误差）Error\bold{E}rrorErrorSSESSESSEn−kn-kn−kMSE=SSEn−kMSE = \frac{SSE}{n-k}MSE=n−kSSE​总和Total\bold TotalTotalSSTSSTSSTn−1n-1n−11.2 双因素方差分析表

kkk：行因素个数rrr：列因素个数

（为什么不是rrr为行因素个数，ccc是列因素个数呢？哼？）

1.2.1 无交互作用的双因素方差分析表

误差来源平方和SSSSSS自由度dfdfdf均方MSMSMSFFF值P值F临界值行因素Row\bold RowRowSSRSSRSSRk−1k-1k−1MSR=SSRk−1MSR = \frac{SSR}{k-1}MSR=k−1SSR​FR=MSRMSEF_R = \frac{MSR}{MSE}FR​=MSEMSR​根据显著性水平α\alphaα确定列因素Column\bold ColumnColumnSSCSSCSSCr−1r-1r−1MSC=SSCr−1MSC = \frac{SSC}{r-1}MSC=r−1SSC​FC=MSCMSEF_C = \frac{MSC}{MSE}FC​=MSEMSC​误差Error\bold{E}rrorErrorSSESSESSE(k−1)×(r−1)(k-1)\times(r-1)(k−1)×(r−1)MSE=SSE(k−1)×(r−1)MSE = \frac{SSE}{(k-1)\times(r-1)}MSE=(k−1)×(r−1)SSE​总和Total\bold TotalTotalSSTSSTSSTkr−1kr-1kr−11.2.2 有交互作用的双因素方差分析表

2. 回归分析表

2.1 一元回归分析表

回归统计：

统计量公式Multiple RMultiple \; RMultipleR相关系数 r=R2r = \sqrt{R^2}r=R2​R SquareR \; SquareRSquare判定系数 R2=SSRSST=r2R^2 = \frac{SSR}{SST} = r^2R2=SSTSSR​=r2Adjusted R SquareAdjusted \; R \; SquareAdjustedRSquare调整的 R2=1−(1−R2)n−1n−k−1R^2 = 1-(1-R^2)\frac{n-1}{n-k-1}R2=1−(1−R2)n−k−1n−1​标准误差se=MSEs_e = \sqrt{MSE}se​=MSE​方差分析：回归统计需要用到方差分析里的数据

误差来源SSSSSSdfdfdfMSMSMSFFF值Significance FSignificance \; FSignificanceF回归Regression\bold RegressionRegressionSSRSSRSSR111MSR=SSR1MSR = \frac{SSR}{1}MSR=1SSR​F=MSRMSE∼F(1,n−2)F = \frac{MSR}{MSE} \sim F(1, n-2)F=MSEMSR​∼F(1,n−2)根据显著性水平α\alphaα确定残差Error\bold{E}rrorErrorSSESSESSEn−2n-2n−2MSE=SSEn−2MSE = \frac{SSE}{n-2}MSE=n−2SSE​总计Total\bold TotalTotalSSTSSTSSTn−1n-1n−1回归分析估计：

估计量系数CoefficientsCoefficientsCoefficients标准误差ttt 统计量t Statt \; StattStatP值P−valueP-valueP−value置信区间Lower 95%Lower \; 95\%Lower95%置信区间Upper 95%Upper \; 95\%Upper95%截距InterceptInterceptInterceptβ^0\hat \beta_0β^​0​sβ^0s_{\hat \beta_0}sβ^​0​​t=β^0sβ^0t = \frac{\hat \beta_0}{s_{\hat \beta_0}}t=sβ^​0​​β^​0​​斜率X V ⁣ariable 1X \; V\!ariable \;1XVariable1β^1\hat \beta_1β^​1​sβ^1s_{\hat \beta_1}sβ^​1​​t=β^1sβ^1t = \frac{\hat \beta_1}{s_{\hat \beta_1}}t=sβ^​1​​β^​1​​2.2 多元回归分析表（其实只用看这个就好了，当k=1时就是一元回归分析）

k：自变量x的个数

回归统计：

统计量公式Multiple RMultiple \; RMultipleR相关系数 r=R2r = \sqrt{R^2}r=R2​R SquareR \; SquareRSquare判定系数 R2=SSRSST=r2R^2 = \frac{SSR}{SST} = r^2R2=SSTSSR​=r2Adjusted R SquareAdjusted \; R \; SquareAdjustedRSquare调整的 R2=1−(1−R2)n−1n−k−1R^2 = 1-(1-R^2)\frac{n-1}{n-k-1}R2=1−(1−R2)n−k−1n−1​标准误差se=MSEs_e = \sqrt{MSE}se​=MSE​方差分析：回归统计需要用到方差分析里的数据

误差来源SSSSSSdfdfdfMSMSMSFFF值Significance FSignificance \; FSignificanceF回归Regression\bold RegressionRegressionSSRSSRSSRk(自变量x的个数)k(自变量x的个数)k(自变量x的个数)MSR=SSRkMSR = \frac{SSR}{k}MSR=kSSR​F=MSRMSE∼F(k,n−k−1)F = \frac{MSR}{MSE} \sim F(k, n-k-1)F=MSEMSR​∼F(k,n−k−1)根据显著性水平α\alphaα确定残差Error\bold{E}rrorErrorSSESSESSEn−k−1n-k-1n−k−1MSE=SSEn−k−1MSE = \frac{SSE}{n-k-1}MSE=n−k−1SSE​总计Total\bold TotalTotalSSTSSTSSTn−1n-1n−1回归分析估计：

估计量系数Coefficients(β^i)Coefficients(\hat \beta_i)Coefficients(β^​i​)标准误差(sβ^is_{\hat \beta_i}sβ^​i​​)检验统计量( ttt )t Statt \; StattStatP值P−valueP-valueP−value置信区间Lower 95%Lower \; 95\%Lower95%置信区间Upper 95%Upper \; 95\%Upper95%截距InterceptInterceptInterceptβ^0\hat \beta_0β^​0​sβ^0s_{\hat \beta_0}sβ^​0​​t0=β^0sβ^0t_0 = \frac{\hat \beta_0}{s_{\hat \beta_0}}t0​=sβ^​0​​β^​0​​x1x_1x1​X V ⁣ariable 1X \; V\!ariable \;1XVariable1β^1\hat \beta_1β^​1​sβ^1s_{\hat \beta_1}sβ^​1​​t1=β^1sβ^1t_1 = \frac{\hat \beta_1}{s_{\hat \beta_1}}t1​=sβ^​1​​β^​1​​x2x_2x2​X V ⁣ariable 2X \; V\!ariable \;2XVariable2β^2\hat \beta_2β^​2​sβ^2s_{\hat \beta_2}sβ^​2​​t2=β^2sβ^2t_2 = \frac{\hat \beta_2}{s_{\hat \beta_2}}t2​=sβ^​2​​β^​2​​..................xkx_kxk​X V ⁣ariable kX \; V\!ariable \;kXVariablekβ^k\hat \beta_kβ^​k​sβ^ks_{\hat \beta_k}sβ^​k​​tk=β^ksβ^kt_k = \frac{\hat \beta_k}{s_{\hat \beta_k}}tk​=sβ^​k​​β^​k​​