Cribsheet

\documentclass[landscape]{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage[T1]{fontenc}
\usepackage{multicol}
\usepackage[margin=1cm]{geometry}
\usepackage[margin=1cm]{geometry}
\setlength{\parindent}{0in}
\begin{document}
\begin{multicols}{3}

Testing the real standard dev of a sample:
$$W=\frac{(n-1)S^2}{\sigma^2_0}, W \sim \chi^2\:df=n-1$$

Assume random/independent sample, normally distributed pop. 

Testing if two population variances are the same:
$$F= \frac{S^2_1}{S^2_2}, \:df=n_1-1,n_2-1$$

Assume two samples are randomly and independently selected from their two populations, which are both normally distributed

One-Way Anova

$$SST = \sum_{k=1}^K n_k(Y_k - \hat Y)^2$$
$$MST = \frac{SST}{K-1}$$
$$SSE = \sum_{k=1}^K\sum_{i=1}^{n_k}(Y_{ij} - Y_k)^2$$
$$s_p^2 = MSE = \frac{SSE}{n-K}$$
$$F=\frac{MST}{MSE}$$
$$TSS = \Sigma_{k=1}^K\Sigma_{i=1}^{n_k}(Y_{ij} - Y)^2 = (n-1)s^2$$

Assume random/independent selection, and each group is normally distributed

Confidence interval for difference between 2 groups if K = 2
$$y_1 - y_2 \pm t_{\alpha/2,n_1+n_2-2}\sqrt{\frac{s^2_p}{n_1}+\frac{s^2_p}{n_2}},\:df=n_1+n_2+2$$
$$y_1-y_2\pm t_{\alpha/2,n-K}\sqrt{\frac{MSE}{n_i}+\frac{MSE}{n_j}}\:df=n-K$$
Bonferroni correction
$$\alpha = \frac{\alpha_F}{K(K-1)/2}$$

Two-Way Anova with block design

$$TSS = \sum_{i=1}^K\sum_{j=1}^B(Y_{ij}-Y)^2$$
$$SST = B\sum_{i=1}^K(Y_{i-} - Y)^2$$
$$SSB = K\sum_{j=1}^B(Y_{-j}-Y)^2$$
$$SSE = TSS - SSB - SST = \sum_{i_1}^{K}\sum_{j=1}^{B}(Y_{ij}-Y_{i-}-Y_{-j}+Y)^2$$

Assume errors are normally distributed, blocks are as homogenous as possible. Treatments are randomly assigned to the units in each block, treatment and block effects are all constants.

Two-Way Anova with Factors

Overall test
$$SSE = \sum_{i=1}^{K}\sum_{j=1}^{J}\sum_{r=1}^{R}(Y_{ijr} - Y_{ij-})^2$$

$$SST = R\sum_{i=1}^{K}\sum_{j=1}^{J}(Y_{ij-}-Y)^2$$
\begin{center}

$MST = \frac{SST}{JK-1}$  $MSE = \frac{SSE}{n-KJ}$
\end{center}
Decomposed tests
$$SS(A) = RJ\sum_{i=1}^{K}(\bar y_{i--} - \bar y_{---})^2$$
$$SS(B) = RK\sum_{j=1}^{J}(\bar y_{-j-} - \bar y_{---})^2$$
\begin{center}
$MS(B) = \frac{SS(B)}{J-1}$ $MS(A) = \frac{SS(A)}{K-1}$
\end{center}
$$SS(AB) = R\sum_{j=1}^J\sum_{i=1}^{K}(\bar y_{ij-} - \bar y_{i--} - \bar y_{-j-} + \bar y_{---})^2$$
$$MS(AB) = \frac{SS(AB)}{KJ-J-K+1}$$
$$F_{AB} = \frac{MS(AB)}{MSE}\:,df=KJ-J-K+1,n-JK$$
if not rejected, then:
$$F_A = \frac{MS(A)}{MSE}\:,df=K-1,n-JK$$

Assume: samples are random/independent, each group normally distributed. 
same \# of experimental units randomly assigned to each $R\times J$ possible 
factor combinations. Errors are normally distributed with the same var as population. 

Linear Regression

$$\sigma_{\hat{\beta}_1} = \frac{\sigma}{\sqrt{S_{XX}}}$$
$$s^2 = \frac{\sigma_{i=1}^{n}(y_i-\hat{y}_i)^2}{n-2} = \frac{SSE}{n-2}$$
$$SSE = SS_{YY} - \hat{\beta}_1SS_{XY}$$
$$T = \frac{\hat{\beta}_1 - 0}{\sigma/\sqrt{SS_{XX}}}$$
$$\hat{\beta}_1 \pm t_{n-2,\alpha/2}\frac{s}{\sqrt{SS_{XX}}}$$
$$r = \frac{SS_{XY}}{\sqrt{SS_{XX}SS_{YY}}}$$
$$SS_{XY} = \sum_{i=1}^n(y_i - \bar{y})(x_i - \bar{x})$$
$$SS_{XX} = \sum_{i=1}^n(x_i - \bar{x})^2$$
$$SS_{YY} = \sum_{i=1}^n(y_i - \bar{y})^2$$
$$r = \hat{\beta}_1\frac{s_x}{s_y}$$
$$SSE = \sum_{i=1}^n(y_i - \hat{y}_i)^2$$
$$r^2 = 1 - \frac{SSE}{SS_{YY}}$$
$$r \pm t_{\alpha/2,n-2}\sqrt{(1-r^2)/(n-2)}$$
In small samples:
$$Z = \ln{\frac{1+r}{1-r}}$$
$$Z \pm z_{\alpha/2}/\sqrt{(n-3)} = (c_L, c_U)$$
$$[\frac{exp(2*c_L - 1)}{exp(2*c_L + 1)},\frac{exp(2*c_U - 1)}{exp(2*c_U + 1)}]$$
$$E(\hat{y}(x_0)) = \beta_0 + \beta_1x_0$$
$$s_{\hat y(x_0)} = s\sqrt{\frac{1}{n} + \frac{(x_0 - \bar x)^2}{SS_{XX}}}$$
$$S_{\tilde{y}(x_0)} = s\sqrt{1 + \frac{1}{n} + \frac{(x_0 - \bar{x})^2}{SS_{XX}}}$$

Assume x as constant(error around the measurement of x is negligible), errors are independent random variables with the same variance as the population(Y), mean of zero. Y is a random variable with the same variance as errors.
Assume that $\hat \beta_1$ and $\hat \beta_0$ are normally distributed.

Multiple Regression
Assume: $E(\epsilon_i) = 0$ for all i, $Var(\epsilon_i) = \sigma^2$, normally and independently distributed errors.
$$s^2 = \frac{\sum_{i=1}^n(y_i-\hat{y}_i)^2}{n-(K+1)}$$
$$t = \frac{\hat{\beta}_j - \beta^*_j}{s\sqrt{c_{jj}}}$$
$c_{jj}$ is the variance of $\hat{\beta}_j$.

Assume x as constant(error around the measurement of x is negligible), errors are independent random variables with the same variance as the population(Y), mean of zero. Y is a random variable with the same variance as errors.

Measuring the fit of the model

$$ SSE = \sum_{i=1}^n(y_i-\hat{y}_i)^2$$
$$ SS_{yy} = \sum_{i=1}^n(y_i - \bar{y})^2$$
$$R^2 = 1 - \frac{SSE}{SS_{yy}}$$
$$R^2_a = 1 - [\frac{n-1}{n-(K+1)}](\frac{SSE}{SS_{yy}})$$
$$F = \frac{(SS_{YY} - SSE)/k}{SSE/[n-(K+1)]} = \frac{R^2/K}{(1-R^2)/(n-(K+1))}$$

Comparing Nested Models

$$F = \frac{(SSE_{M_0} - SSE_{M_1})/(k-g)}{SSE_{M1}/(n-(k+1))},\:df=k-g,n-k-1$$
$$e^{std}_i = \frac{e_i}{s}$$

Categorical Data

$$\chi^2 = \sum_{j=1}^k\frac{(n_j - np_j^{(0)})^2}{np_j^{(0)}}$$
$$\chi^2 = \sum_{j=1}^k\frac{(observed - expected)^2}{expected}$$
The degrees of freedom is the difference between k - 1 and the number of unspecified probabilities in $H_0$.

Test for independence
$E(n_{jk}) = n\hat{p}_{j-}\hat{p}_{-k}$
$df = rc - 1 - (r+c-2) = (r-1)(c-1)$

Non-parametric statistics

$$D_i = X_i - \eta_0$$
use binomial test on $D_i$, $p=0.5$
$$z = \frac{X-np}{\sqrt{npq}}$$

For matched pairs, $D_i = X_i - Y_i$, then do test on $D_i$

Wilcoxon Paired Rank Sum

$T^+$ is rank sum of positive $D_i$
$$E(T^+) = \frac{n(n+1)}{4}$$ 
$$Var(T^+) = \frac{n(n+1)(2n+1)}{24}$$
Use Z statistic

Wilcoxon Independent Rank Sum
$$U = n_1n_2 + \frac{n_1(n_1 + 1)}{2} - W$$
Where W is the rank sum of first sample
$$Z = \frac{U-(n_1n_2/2)}{\sqrt{n_1n_2(n_1+n_2+1)/12}}$$

Assume independent and identically distributed data.

\end{multicols}

\begin{multicols}{2}
\begin{tabular}{c | c| c|c|c|c}

Source & df & SS & MS & F & p-value \\
\hline
Treatments&$K-1$&$SST$&$MST=\frac{SST}{K-1}$&$\frac{MST}{MSE}$&Pr(F\* > F)\\
\hline
Error&$n-K$&$SSE$&$MSE=\frac{SST}{K-1}$&&\\
\hline
Total&$n-1$&$TSS$&&&\\

\end{tabular}                          

\begin{tabular}{c | c| c|c|c}
Source& df & SS & MS & F \\
\hline
Treatments&$K-1$&$SST$&$MST=\frac{SST}{K-1}$&$\frac{MST}{MSE}$\\
\hline
Blocks&$B-1$&$SSB$&$MSB=\frac{SSB}{B-1}$&$\frac{MSB}{MSE}$\\
\hline
Error&$n-K-B+1$&$SSE$&$MSE=\frac{SSE}{n-K-B+1}$&\\
\hline
Total&$n-1$&$TSS$&&\\

\end{tabular}

\end{multicols}

\begin{tabular}{c | c| c|c|c}
Source&df&SS&MS&F\\
\hline
$A$&$K-1$&$SS(A)$&$MS(A)=\frac{SS(A)}{K-1}$&$\frac{MS(A)}{MSE}$\\
\hline
$B$&$J-1$&$SS(B)$&$MS(B)=\frac{SS(B)}{J-1}$&$\frac{MS(B)}{MSE}$\\
\hline
$A \times B$&$KJ - K - J +
1$&$SS(AB)$&$MS(AB)=\frac{SS(AB)}{KJ-K-J+1}$&$\frac{MS(AB)}{MSE}$\\
\hline
Error&$n-KJ$&$SSE$&$MSE = \frac{SSE}{n-KJ}$&\\
\hline
 Total&$n-1$&$TSS$&&\\
\end{tabular}
\end{document}