Exam 1 Study Guide: ANOVA, Non-Parametric Tests, and Experimental Design

Vocabulary flashcards covering one-way ANOVA, two-way ANOVA, non-parametric procedures, confidence intervals, degrees of freedom, and relevant R functions from the Exam 1 Study Guide.

qt()

An R function that finds the t-value associated with a given probability.

Non-parametric test uses

Situations appropriate for non-parametric tests: when data is skewed, there are outliers, normality assumptions are not met, or the data is ordered or ordinal.

Non-parametric assumptions

The core requirements for non-parametric tests: independent observations, independent groups, and random sampling.

kruskal.test()

An R function (kruskal.test(Y ~ X, data = DataSet)) used to test for differences across $3+$ medians nonparametrically.

wilcox.test()

An R function (wilcox.test(Y ~ X, data = DataSet)) used to compare $2$ independent groups nonparametrically.

dunn_test()

An R function (dunn_test(DataSet, Y ~ X, p.adjust.method = "bonferroni")) used to identify differing medians after a Kruskal-Wallis test.

Randomization test

A statistical procedure that calculates an observed test statistic, shuffles the data under the assumption that $H_0$ is true, repeats this process many times, and determines the proportion of simulated statistics as extreme as the observed one.

Proportion (Randomization test)

The proportion of simulated statistics as extreme as the observed test statistic, which serves as the approximate p-value.

P-value evidence guidelines

Standards for interpreting p-values: $p > .10$ indicates not strong evidence; $.05 - .10$ indicates some evidence; $.01 - .05$ indicates strong evidence; and $p < .01$ indicates very strong evidence.

Nearly normal condition

A condition satisfied when the data follows a normal distribution or when there are more than $30$ observations.

Confidence interval null hypothesis rule

If zero is not included in the interval, reject $H_0$; if zero is included in the interval, fail to reject $H_0$.

Confidence interval width factors

A large sample size produces a narrower CI, whereas a higher confidence level and greater variability produce a wider CI.

ANOVA assumptions

The requirements that data must be normally distributed, have constant variability, feature random and independent observations, and consist of independent groups.

Individual conclusion

A statistical conclusion that concerns one comparison.

Familywise conclusion

A statistical conclusion that concerns the entire set of comparisons.

Boxplots (in ANOVA)

Visual plots used to compare variability across groups.

Larger F-statistic

An indicator of more evidence against the null hypothesis.

aov()

An R function used to create an ANOVA model.

anova()

An R function used to display the ANOVA table.

pf()

An R function that finds the probability from an F-statistic.

Fisher comparisons in R

Pairwise comparisons conducted using pairwise.t.test(..., p.adjust.method = "none").

Bonferroni comparisons in R

Pairwise comparisons conducted using pairwise.t.test(..., p.adjust.method = "bonferroni").

TukeyHSD()

An R function used to perform Tukey multiple comparisons and find specific mean differences.

Interaction

A difference of differences, occurring when the effect of one of the variables is different across all levels of the other variable.

No interaction

A similarity of differences, occurring when the effect of one variable is the same across all levels of the other variable.

2-way ANOVA model

A model where $y$ is quantitative and $x_1, x_2$ are categorical, defined by the formula $y = \mu + \alpha_i + \beta_j + \gamma_{ij} + \epsilon$.

2-way ANOVA hypotheses

The hypotheses tested in two-way ANOVA: Factor A ($H_0\text{: all } \alpha_i = 0$ vs. $H_a\text{: at least one } \alpha_i \neq 0$), Factor B ($H_0\text{: all } \beta_j = 0$ vs. $H_a\text{: at least one } \beta_j \neq 0$), and Interaction ($H_0\text{: all } \gamma_{ij} = 0$ vs. $H_a\text{: at least one } \gamma_{ij} \neq 0$).

Crossing lines on an interaction plot

A visual pattern indicating there may be statistically significant interaction, representing differences of differences where the effect of one variable differs across levels of the other.

Parallel lines on an interaction plot

A visual pattern indicating there is likely not significant interaction, representing similarity of differences where the effect of one variable is the same across levels of the other.

ANOVA table results analysis strategy

If the interaction is significant, analyze the data by conditioning on the levels of one other variable; if the interaction is not significant, analyze the main effects of the data.

var.equal argument

An argument within t.test() where var.equal = TRUE assumes equal variances and var.equal = FALSE indicates that equal variances cannot be assumed.

interaction.plot()

An R function used to examine interaction visually.

Degrees of freedom: Factor A and Factor B

Calculated as $i - 1$ for Factor A and $j - 1$ for Factor B, where $i$ and $j$ represent the number of levels for each respective factor.

Degrees of freedom: Interaction

Calculated as $(i - 1)(j - 1)$, where $i$ and $j$ represent the number of levels for each factor.

Degrees of freedom: Error (equal sample sizes)

Calculated as $ij(c - 1)$, where $i$ and $j$ are factor levels and $c$ is the number of observations per combination.

Tukey CI interpretation

If the CI includes $0$, there is no statistically significant difference; if the CI excludes $0$, there is a statistically significant difference (a positive interval means the first group has a higher mean; a negative interval means the first group has a lower mean).

Blocking variable

A variable that groups observations according to a characteristic that may influence the response, helping account for that variation while studying the factor of interest.

2-way ANOVA R code

The standard R syntax for running a two-way ANOVA: model <- aov(Y ~ Factor A * Factor B, data = DataFrame).

Wilcoxon Rank Sum Test

A non-parametric test used for two independent samples when the normality assumption is not met to estimate the difference in population medians.

Cohen's d

A statistic evaluated to determine whether there is a meaningful difference between groups.