| Topic / Model | Formulas and Descriptions | | :------------------------ | :---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | **Basic Metrics** | **Elasticity:** $\text{Slope} \cdot \frac{x}{y}$**Slope:** $\frac{dy}{dx}$**Log-linear slope:** $b_y$**Semi-elasticity:** $\text{Slope} \cdot \frac{100}{y}$ | | **Linear Model** | Function: $y = \beta_1 + \beta_2 x$Slope ($dy/dx$): $\beta_2$Elasticity: $\beta_2 \cdot \frac{x}{y}$ | | **Quadratic Model** | Function: $y = \beta_1 + \beta_2 x^2$Slope ($dy/dx$): $2 \beta_2 x$Elasticity: $(2 \beta_2 x) \cdot \frac{x}{y}$ | | **Cubic Model** | Function: $y = \beta_1 + \beta_2 x^3$Slope ($dy/dx$): $3 \beta_2 x^2$Elasticity: $(3 \beta_2 x^2) \cdot \frac{x}{y}$ | | **Log-Log Model** | Function: $\ln(y) = \beta_1 + \beta_2 \ln(x)$Slope ($dy/dx$): $\beta_2 \frac{y}{x}$Elasticity: $\beta_2$ | | **Log-Linear Model** | Function: $\ln(y) = \beta_1 + \beta_2 x$Slope: $\beta_2 y$Elasticity: $\beta_2 x$*Note: 1 unit change in x $\approx$ 100 $\cdot \beta_2$% change in y* | | **Linear-Log Model** | Function: $y = \beta_1 + \beta_2 \ln(x)$Slope: $\beta_2 \frac{1}{x}$Elasticity: $\frac{\beta_2}{y}$*Note: 1% change in x $\approx \beta_2/100$ unit change in y* | | **Regression Tests** | **F-test:** $F = \frac{(SSE_R - SSE_U)(N - K)}{J \cdot SSE_U}$**Correlation ($r_{xy}$):** $\frac{S_{xy}}{S_x S_y}$<br>**$R^2$:** $1 - \frac{SSE}{SST}$**SST:** $SSE + SSR$ | | **Logist. Reg. Deviance** | **Null deviance:** $-2 \cdot \ln(L_{null})$**Residual deviance:** $-2 \cdot \ln(L_{model})$**Pseudo $R^2$:** $1 - \frac{residual}{null}$ | | **U test (Mann-Whitney)** | $U = n_1 n_2 + \frac{n_1(n_1 + 1)}{2} - W$Centered around $\frac{n_1 n_2}{2}$. Pick smallest stat. | | **Large Sample U test** | $Z = \frac{U - (n_1 n_2 / 2)}{\sqrt{n_1 n_2 (n_1 + n_2 + 1) / 12}}$ | | **Signed-Rank Tests** | **Signed test paired:** Binomial dist $\binom{n}{k} p^k (1 - p)^{n-k}$**Wilcoxon Signed-Rank:** Sum of differences should be 0. | | **Kruskal-Wallis** | $V = \sum_{i=1}^k n_i (\bar{R}_i - \bar{R})^2$$H = \frac{12V}{n(n+1)} = \frac{12}{n(n+1)} \sum_{i=1}^k \frac{R_i^2}{n_i} - 3(n+1)$ | | **Rank Correlation** | **Spearman ($r_s$):** $1 - \frac{6 \sum_{i=1}^n (R(x_i) - R(y_i))^2}{n(n^2 - 1)}$**Kendall ($\tau$):** $\frac{\text{concordant} - \text{discordant}}{\binom{n}{2}}$ | | **P-value Correlation** | $t = \frac{r \cdot \sqrt{n-2}}{\sqrt{1 - r^2}}$ | | **Bayesian Basics** | **Posterior:** $\pi(\theta \mid x) = \frac{f(x \mid \theta)\pi(\theta)}{g(x)}$**Evidence:** $g(x) = \int_{-\infty}^{\infty} f(x \mid \theta)\pi(\theta) d\theta$ | | **Posterior Normal** | $\mu^* = \frac{\sigma_0^2}{\sigma_0^2 + \sigma^2/n} \bar{x} + \frac{\sigma^2/n}{\sigma_0^2 + \sigma^2/n} \mu_0$$\sigma^* = \sqrt{\frac{\sigma_0^2 \sigma^2}{n\sigma_0^2 + \sigma^2}}$ | | **Credible Intervals** | Interval 100(1-a)%: $\int_{-\infty}^{a} \pi(\theta \mid x) d\theta = \int_{b}^{\infty} \pi(\theta \mid x) d\theta = \frac{\alpha}{2}$Mean normal: $\mu^* \pm z_{\alpha/2} \sigma^*$ | | **Bayes Factor** | $BF[H_1 : H_2] = \frac{P(\text{data} \mid H_1)}{P(\text{data} \mid H_2)}$Posterior odds = Bayes factor $\cdot$ Prior odds. | | **Effect Size** | **Standard effect size ($\delta$):** $\frac{\mu - m_0}{\sigma}$**Required sample size ($n_0$):** $\left( \frac{1.96 \cdot sd}{\text{effect size}} \right)^2$ | | **Exponential Dist.** | PDF: $\lambda e^{-\lambda x}$CDF: $1 - e^{-\lambda x}$Mean: $1/\lambda$Median: $\frac{\ln 2}{\lambda}$Var: $1/\lambda^2$ | | **Poisson Dist.** | PMF: $\frac{\lambda^k e^{-\lambda}}{k!}$Mean: $\lambda$Var: $\lambda$Mode: $\lceil \lambda \rceil - 1, \lfloor \lambda \rfloor$ | | **Normal Dist.** | PDF: $\frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$Mean: $\mu$Var: $\sigma^2$Quantile: $\mu + \sigma \sqrt{2} \text{erf}^{-1}(2p - 1)$ | | **Beta Dist.** | PDF: $\frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha, \beta)}$Mean: $\frac{\alpha}{\alpha + \beta}$Var: $\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}$ | | **Gamma Dist.** | PDF: $\frac{1}{\beta^\alpha \Gamma(\alpha)} x^{\alpha-1} e^{-x/\beta}$Mean: $\alpha\beta$Var: $\alpha\beta^2$ |