Statistics Cheat Sheet

Simple Linear Regression

\(\hat{y}=\)	\(b_0+b_1 x\)
Coefficients:             Estimate Std. Error t value Pr(>|t|)   (Intercept) (1)       *** x (2) (3) (4)   *** --- signif codes Residual Standard error: (5) Multiple R-squared: (6), Adjusted R-squared: F-statistic: (7), p-value:	(1) \(b_0\) (2) \(b_1\) (3) \(\hat{\sigma}_{b_1}\) (4) \(t=b_1/\hat{\sigma}_{b_1}\) (5) \(\hat{\sigma}\) (6) \(r^2\) (7) \(\text{F}=\frac{\text{MSR}}{\text{MSE}}\)
SSE, SSR, SST Relation Picture	\(\text{SSE}=\sum(y_i-\hat{y}_i)^2\) \(\text{SSR}=\sum(\hat{y} _i-\overline{y})^2\) \(\text{SST}=\sum(y_i-\overline{y})^2\) \(\text{SSE}+\text{SSR}=\text{SST}\)
\(b_0\) \(b_1\) \(\hat{\sigma}\) \(\hat{\sigma}_{b_1}\) \(r^2\) \(t\) \(F\)	\(b_0=\overline{y}-b_1 \overline{x}\) \(b_1=\frac{\sum(x_i-\overline{x})(y_i-\overline{y})}{\sum(x_i-\overline{x})^2}\) \(\hat{\sigma}=\sqrt{\frac{\text{SSE}}{n-2}}\) \(\hat{\sigma}_{b_1}=\frac{\hat{\sigma}}{\sqrt{\sum(x_i-\overline{x})^2}}\) \(r^2=\frac{\text{SSR}}{\text{SST}}\) \(t=\frac{b_1}{\hat{\sigma}_{b_1}}\) \(F=\frac{\text{MSR}}{\text{MSE}}\)
ANOVA Table \(\text{E}(y)=\beta_0+\beta_1 x\)	source of variation sum of squares degrees of freedom mean square \(F\) Regression \(\text{SSR}\) \(1\) \(\text{MSR}=\frac{\text{SSR}}{1}\) \(F=\frac{\text{MSR}}{\text{MSE}}\) Error \(\text{SSE}\) \(n-2\) \(\text{MSE}=\frac{\text{SSE}}{n-2}\)   Total \(\text{SST}\) \(n-1\)
ANOVA Table \(\text{E}(y)=\beta_0+\beta_1 x_1+\beta_2 x_2+\cdots+\beta_p x_p\)	source of variation sum of squares degrees of freedom mean square \(F\) Regression \(\text{SSR}\) \(p\) \(\text{MSR}=\frac{\text{SSR}}{p}\) \(F=\frac{\text{MSR}}{\text{MSE}}\) Error \(\text{SSE}\) \(n-p-1\) \(\text{MSE}=\frac{\text{SSE}}{n-p-1}\)   Total \(\text{SST}\) \(n-1\)
Confidence Interval of \(\hat{y}\) at \(x=x_0\)	\(b_0+b_1 x_0+t_{\alpha/2, n-2}\hat{\sigma}\sqrt{1+\frac{1}{n}+\frac{(x_0-\overline{x})^2}{\sum(x_i-\overline{x})^2}}\)
Big Picture for \(\text{SSE}, \text{SSR}, \text{SST}, r^2\) and \(s_x, s_y, s_{xy}, r_{xy}\)
SSE, SSTR, SST	\(\text{SSE}=\sum_{j=1}^{k}\sum_{i=1}^{n_j}(x_{ij}-\overline{x}_j)^2=\sum_{j=1}^{k}(n_j-1)s_j^2\) \(\text{SSTR}=\sum_{j=1}^{k}n_j(\overline{x}_j-\bar{\bar{x}})^2\) \(\text{SST}=\sum_{j=1}^{k}\sum_{i=1}^{n_j}(x_{ij}-\bar{\bar{x}})^2\)\[ \begin{array}{cccccc} \text{treatment 1} & \text{treatment 2} & \cdots & \text{treatment k} \\ x_{11} & x_{12} & \cdots & x_{1k} \\ x_{21} & x_{22} & \cdots & x_{2k} \\ \vdots & \vdots & \ddots & \vdots \\ x_{n_1 1} & x_{n_2 2} & \cdots & x_{n_k k} \\ \downarrow & \downarrow & \cdots & \downarrow & \searrow \\ \overline{x}_1 & \overline{x}_2 & \cdots & \overline{x}_k & & \bar{\bar{x}} \end{array} \]
Picture	\[ \begin{array}{cccc} & & & \text{SST}\\ \text{SSE} & \downarrow & \searrow & \\ & & \longrightarrow \\ & & \text{SSTR} \end{array} \]
ANOVA Table	source of variation sum of squares degrees of freedom mean square \(F\) Treatments \(\text{SSTR}\) \(k-1\) \(\text{MSTR}=\frac{\text{SSTR}}{k-1}\) \(F=\frac{\text{MSTR}}{\text{MSE}}\) Error \(\text{SSE}\) \(n-k\) \(\text{MSE}=\frac{\text{SSE}}{n-k}\)   Total \(\text{SST}\) \(n-1\)
Fisher's LSD Test Statistic Compare with	\(\frac{\overline{x}_i-\overline{x}_j}{\sqrt{\text{MSE}\left(\frac{1}{n_i}+\frac{1}{n_j}\right)}}\) \(t_{\alpha/2, n-k}\)