STAT 503 - Lesson 3: One-Way ANOVA in the Completely Randomized Design

Exhaustive vocabulary-style flashcards covering Penn State STAT 503 Lesson 3 on One-Way ANOVA, including the Cotton and Loom examples, assumption checks, multiple-comparison procedures, and power planning.

One-way analysis of variance (ANOVA)

A statistical method used for comparing the mean responses across $k$ or $a$ treatment groups in a Completely Randomized Design.

Completely Randomized Design (CRD)

An experimental design with no additional structure among experimental units; factor levels or treatments are randomly assigned to experimental units.

One-way fixed-effects model

The model Y_{ij} = \text{\mu} + \text{\tau}_i + \text{\epsilon}_{ij}, where \text{\mu} is the overall mean, \text{\tau}_i is the effect of treatment $i$, and \text{\epsilon}_{ij} is the random error.

Fixed-effects error assumptions

Random errors (\text{\epsilon}_{ij}) are independent with a mean of $0$ and a common variance \text{\sigma}^2, with Normality assumed for exact F inference.

Treatment-mean form

The representation of the model where \text{\mu}_i = \text{\mu} + \text{\tau}_i.

Typical fixed-effects constraint

The identification constraint \text{\sum}_i \text{\tau}_i = 0, which defines \text{\mu} as the overall mean.

Cotton Example Response Variable

Tensile strength of the synthetic fiber.

Cotton Treatment Levels

Five levels of cotton weight percentage: $15\%$, $20\%$, $25\%$, $30\%$, and $35\%$, evenly spaced.

Cotton F statistic and p-value

$F = 14.76$ with a p-value reported as $p < 0.0005$ (Minitab reported $0.000$).

Cotton ANOVA Sums of Squares

$\text{SS}_\text{Treatment} = 475.8$ and $\text{SSE} = 161.2$.

Cotton model S

$S = 2.83901$, representing the estimated standard deviation of the error.

Cotton R-squared (R^2)

$74.69\%$, indicating the proportion of variability explained by the cotton weight percentage.

Residual (e_ij)

The difference between the observed and fitted value, calculated as $e_{ij} = y_{ij} - \bar{y}_{i.}$ in a one-way ANOVA.

Total Sum of Squares (SST)

$\text{SST} = \sum_i \sum_j (y_{ij} - \bar{y}_{..})^2$.

Treatment Sum of Squares (SS_Tr)

$\text{SS}_\text{Tr} = \sum_i n_i (\bar{y}_{i.} - \bar{y}_{..})^2$.

Error Sum of Squares (SSE)

$\text{SSE} = \sum_i \sum_j (y_{ij} - \bar{y}_{i.})^2$.

One-way ANOVA F statistic formula

$F = \frac{\text{MS}_\text{Tr}}{\text{MSE}}$, where $\text{MS}_\text{Tr} = \frac{\text{SS}_\text{Tr}}{a-1}$ and $\text{MSE} = \frac{\text{SSE}}{N-a}$.

Bartlett's test

A formal test of the equal variance assumption that is sensitive to non-Normality because it is based on sample variances.

Levene's test

A robust test for equal variances based on an ANOVA of absolute deviations from group medians ($z_{ij} = |y_{ij} - \text{median}_i|$).

Hierarchical polynomial principle

A modeling guideline stating that if a higher-order polynomial term is retained, lower-order terms should also be retained regardless of their individual significance.

Sensitivity (Sample Size)

The size of a difference in means that the experiment is designed to detect; increased replication increases sensitivity.

Delta (\Delta)

\Delta = \max(\text{\mu}_i) - \min(\text{\mu}_i), representing the maximum true difference among treatment means for power planning.

Least Favorable Configuration (LFC)

The arrangement of means that produces the least statistical power for a specified maximum difference ($\Delta$).

LFC identity for sum of squared effects

Under the Least Favorable Configuration (LFC), \sum \text{\tau}_i^2 = \frac{\text{\Delta}^2}{2}.

Fixed-effects noncentrality parameter (\lambda)

\text{\lambda} = \frac{n \times \sum \text{\tau}_i^2}{\text{\sigma}^2} (for equal sample sizes $n$).

Fisher's Least Significant Difference (LSD)

A follow-up procedure that performs ordinary pairwise t-tests after the overall F-test rejects the null hypothesis.

Bonferroni method

A multiple-comparison adjustment that tests each of $g$ planned comparisons at an alpha level of \frac{\text{\alpha}}{g} to control familywise Type I error.

Scheffe's method

A conservative procedure providing familywise error protection for all possible contrasts, including post-hoc comparisons.

Tukey Studentized Range Procedure

A procedure designed specifically for comparing all possible pairs of means using a critical value from the studentized range distribution.

Contrast (L)

A linear combination of means L = \sum c_i \text{\mu}_i where the coefficients sum to zero ($\sum c_i = 0$).

Orthogonal contrasts

Two contrasts whose coefficients $c_i$ and $d_i$ satisfy $\sum c_i d_i = 0$ for equal sample sizes.

Dunnett's procedure

A multiple-comparison method specifically tailored for comparing each of $a-1$ treatments against a single control group.

Optimal Dunnett allocation rule

To maximize efficiency, the control group sample size should be $n_0 = \sqrt{t} \times n$, where $t = a-1$ (the number of treatments).

False Discovery Rate (FDR)

An approach that controls the expected percentage of false significant results among all tests declared significant, often used when testing hundreds of hypotheses.

One-way random-effects model

The model Y_{ij} = \text{\mu} + \text{\tau}_i + \text{\epsilon}_{ij} where treatment effects \text{\tau}_i are random variables with mean $0$ and variance \text{\sigma}_\text{\tau}^2.

Random-effect null hypothesis

H_0: \text{\sigma}_\text{\tau}^2 = 0, testing if there is any variability between the population of treatment levels.

Variance components

\text{\sigma}_\text{\tau}^2 (between-treatment variance) and \text{\sigma}^2 (within-treatment or error variance) in a random-effects model.

Expected Mean Squares for random-effects (E[MSE])

E(\text{MSE}) = \text{\sigma}^2.

Expected Mean Squares for random-effects (E[MS_Tr])

E(\text{MS}_\text{Tr}) = \text{\sigma}^2 + n \text{\sigma}_\text{\tau}^2 (for equal sample sizes $n$).

General Linear Test (GLT)

An F-test comparing a reduced model (R) with a full model (F) to measure the reduction in SSE relative to full-model error variance.

General Linear Test F-formula

$F = \frac{\frac{\text{SSE}_R - \text{SSE}_F}{\text{df}_R - \text{df}_F}}{\text{MSE}_F}$.

Loom Example (Random Effects)

A study of 4 looms randomly selected from a population which concluded that the between-loom variance component (\hat{\text{\sigma}}_\text{\tau}^2 = 6.958) is significantly greater than zero.