STA305 Notation Index

Inference and Regression

  • $F_{m,n}$: F distribution with numerator degrees of freedom $m$ and denominator degrees of freedom $n$
  • $x_{(k)}$: $k$-th order statistic
  • $F_0$: reference distribution used in a QQ plot
  • $t_k$: plotting position used in a QQ plot
  • $y_i$: response for unit $i$
  • $x_i$: predictor for unit $i$
  • $e_i$: regression error for unit $i$

Randomization and Testing

  • $T$: generic test statistic
  • $T_i$: treatment assignment for unit $i$
  • $H_0, H_1$: null and alternative hypotheses
  • $p$-value: tail probability of a statistic at least as extreme as the observed one under $H_0$
  • $\mu_A, \mu_B$: group means in two-group comparisons
  • $\bar{d}$: mean of paired differences

Power and Sample Size

  • $\alpha$: Type I error rate
  • $\beta$: Type II error rate
  • $1-\beta$: power
  • $\delta$: effect size to be detected
  • $n, n_1, n_2$: total sample size and group-specific sample sizes
  • $r$: allocation ratio when $n_1 = r n_2$

ANOVA and Multiple Comparisons

  • $y_{ij}$: $j$-th observation from treatment $i$
  • $n_i$: sample size in treatment $i$
  • $k$: number of treatments
  • $N = \sum_{i=1}^{k} n_i$: total sample size
  • $\bar{y}_{i\cdot}$: mean of treatment $i$
  • $\bar{y}_{\cdot\cdot}$: grand mean
  • $\mu_i$: mean of treatment $i$
  • $\mu$: overall level in the treatment-effects model
  • $\tau_i$: effect of treatment $i$
  • $SST$: total sum of squares
  • $SSTreat$: sum of squares due to treatments
  • $SSE$: sum of squares due to error
  • $MSTreat$: treatment mean square
  • $MSE$: error mean square
  • $c = \binom{k}{2}$: number of pairwise comparisons
  • $q_{k, N-k, \alpha}$: Studentized range critical value
  • $f$: ANOVA effect size

Factorial Designs

  • $2^k$: two-level factorial design with $k$ factors
  • $A, B, C$: main-effect columns in a factorial design
  • $AB, AC, BC$: two-factor interaction columns
  • $ABC$: three-factor interaction column
  • $m$: number of replications per run
  • $s^2$: pooled error variance from replicated runs

Causal Inference

  • $Y_i(1)$: potential outcome for unit $i$ under treatment
  • $Y_i(0)$: potential outcome for unit $i$ under control
  • $Y_i^{\text{obs}}$: observed outcome
  • $T_i$: treatment indicator for unit $i$
  • $P(T \mid Y(0), Y(1))$: assignment mechanism
  • $P(T_i = 1 \mid X_i)$: propensity score