Analysis of Variance (ANOVA) Review

Vocabulary flashcards covering the key terms, hypotheses, assumptions, and formulas for One-Factor Analysis of Variance (ANOVA) derived from the lecture material.

Analysis of Variance (ANOVA)

A comparison of means that allows for the simultaneous comparison of more than two means to identify sources of variation in a numerical dependent variable.

Response Variable (Y)

The numerical dependent variable in ANOVA for which variation is being explained.

Factors

The categorical independent variables used to explain the variation in the response variable.

Treatment

Each possible value of a factor or a combination of factors in an experimental design.

F distribution

The probability distribution used in ANOVA to describe the ratio of two variances.

One-Factor ANOVA

A specific ANOVA model that compares the means of $c$ groups based on a single independent variable.

Null Hypothesis ($H_0$)

H_0: \text{\mu}_1 = \text{\mu}_2 = \text{\dots} = \text{\mu}_c, which asserts that there is no difference between the mean values at various levels of the test factor.

Alternative Hypothesis ($H_1$)

The hypothesis that not all the means are equal.

Total number of observations ($n$)

The sum of sample sizes within each treatment, calculated as $n = n_1 + n_2 + \text{\dots} + n_c$.

ANOVA Linear Model

Expresses that treatment $j$ came from a population with a common mean ($\mu$) plus a treatment effect ($A_j$) plus random error ($e_{ij}$): y_{ij} = \text{\mu} + A_j + e_{ij}.

Random Error ($e_{ij}$)

The unexplained variation assumed to be normally distributed with zero mean and the same variance for all treatments.

ANOVA Assumptions

The requirements that observations on $Y$ are independent, the populations being sampled are normal, and the populations have equal variances.

Grand Mean ($\bar{\bar{y}}$)

The overall sample mean calculated across all observations in all groups.

SST (Total Sum of Squares)

The total variation in the data, partitioned as $SST = SSA + SSE$.

SSA (Sum of Squares for Treatment)

The variation between treatments, representing the deviation of the column mean from the grand mean.

SSE (Sum of Squares for Error)

The variation within treatments, representing the deviation of an observation from its own column mean; also known as unexplained variation.

Mean Squares (MSA and MSE)

Ratios calculated by dividing the Sum of Squares by their respective degrees of freedom to adjust for group size.

F Statistic

The ratio of the variance due to treatments ($MSA$) to the variance due to error ($MSE$): $F = \frac{MSA}{MSE}$.

Numerator Degrees of Freedom

The degrees of freedom associated with treatments (between groups), calculated as $c - 1$.

Denominator Degrees of Freedom

The degrees of freedom associated with error (within groups), calculated as $n - c$.

Decision Rule

Reject $H_0$ if the test statistic $F$ exceeds the critical value $F_{critical}$ or if the p\text{-value} \text{\le} \text{\alpha}. In ANOVA, this is a right-tailed test.

Replicates

The data values obtained from repeated samplings.