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	$\frac{\overline{x}_i-\overline{x}_j}{\sqrt{\text{MSE}\left(\frac{1}{n_i}+\frac{1}{n_j}\right)}}$ $t_{\alpha/2, n-k}$