Equations

$$Z = \frac{\bar{X}-\mu_0}{S/\sqrt{n}}$$
Use T instead of Z for n < 30, data sampled independently, identically distributed, normally distributed pop

Testing if two samples have the same proportion:
$$p = \frac{n_1p_1 + n_2p_2}{n_1+n_2}$$
$$Z = \frac{p_1-p_2}{\sqrt{pq/n_1 + pq/n_2}}$$

Testing if two dependent samples have the same mean:
$$Z = \frac{\bar{X}_D-\mu_0}{S_D/\sqrt{n_D}}$$

Testing if two independent samples have the same mean:
$$Z = \frac{\bar{X}_A-\bar{X}_B-\mu_0}{\sqrt{S^2_A/n_A+S^2_B/n_B}}$$

Testing if two independent small pooled samples have the same mean:
$$Z = \frac{\bar{X}_A-\bar{X}_B-\mu_0}{\sqrt{S^2_P/n_A+S^2_P/n_B}}$$
$$S^2_p = \frac{(n_A-1)s^2_A+(n_B-1)s^2_B}{n_A+n_B-2}$$

Testing about the real proportion of one sample:
$$Z = \frac{p-p_0}{\sqrt{p_0q_0/n}}$$

Two samples with small n but assume same pop. variance:
$$s^2_p=\frac{(n_A-1)s^2_A+(n_B-1)s^2_B}{n_A+n_B-2}$$
$$df = n_A+n_B-2$$

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

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

Assume random/independent sample, normally distributed pop

1One-Way Anova¶

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

$$TSS = \Sigma_{k=1}^K\Sigma_{i=1}^{n_k}(Y_{ij} - Y)^2 = (n-1)s^2$$

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
$$\alpha = \frac{\alpha_F}{K(K-1)/2}$$

2Two-Way Anova with block design¶

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

3Two-Way Anova with Factors¶

Overall test
$$SSE = \sum_{i=1}^{K}\sum_{j=1}^{J}\sum_{r=1}^{R}(Y_{ijr} - Y_{ij-})^2$$
$$MSE = \frac{SSE}{n-KJ}$$
$$SST = R\sum_{i=1}^{K}\sum_{j=1}^{J}(Y_{ij-}-Y)^2$$
$$MST = \frac{SST}{JK-1}$$

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

4Linear 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:
$$Z = \ln{\frac{1+r}{1-r}}$$
$$Z \pm z_{\alpha/2}/\sqrt{(n-3)} = (c_L, c_U)$$
$$[\frac{exp(2*c_L - 1)}{exp(2*c_L + 1)},\frac{exp(2*c_U - 1)}{exp(2*c_U + 1)}]$$
$$E(\hat{y}(x_0)) = \beta_0 + \beta_1x_0$$
$$s_{\hat y(x_0)} = s\sqrt{\frac{1}{n} + \frac{(x_0 - \bar x)^2}{SS_{XX}}}$$
$$S_{\tilde{y}(x_0)} = s\sqrt{1 + \frac{1}{n} + \frac{(x_0 - \bar{x})^2}{SS_{XX}}}$$

5Multiple 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 $\hat{\beta}_j$.

5.1Measuring the fit of the model¶

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

5.2Comparing Nested Models¶

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

6Categorical 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
$$E(n_{jk}) = n\hat{p}_{j-}\hat{p}_{-k}$$
$$df = rc - 1 - (r+c-2) = (r-1)(c-1)$$

7Non-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 $D_i$

7.1Wilcoxon 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

7.2Wilcoxon Independent Rank Sum¶

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