Independent Samples t-Tests, One-Way ANOVA, and Within-Subjects Designs

Vocabulary practice flashcards covering independent-samples t-tests, one-way between-subjects ANOVA, and methodological principles of within-subjects designs.

Independent Samples t-Test

A parametric hypothesis test used to evaluate whether the difference between the sample means of two separate, independent groups is statistically significant, requiring an interval or ratio dependent variable and a between-subjects independent variable with two levels.

Sampling Distribution of the Difference Between Two Means

The theoretical distribution created by repeatedly drawing pairs of random samples of size $n$ from two populations, calculating the difference between their sample means ($\bar{X}_1 - \bar{X}_2$), and plotting all resulting differences; its mean equals $\mu_1 - \mu_2 = 0$ under the null hypothesis.

Estimated Standard Error of the Difference ($s_{\bar{X}_1 - \bar{X}_2}$)

A measure of the dispersion or sampling error expected by chance between two sample means from identical populations, calculated as $s_{\bar{X}_1 - \bar{X}_2} = \sqrt{\frac{s_p^2}{n_1} + \frac{s_p^2}{n_2}}$ and serving as the denominator in the independent samples $t$-test statistic.

Pooled Variance ($s_p^2$)

A weighted average of two sample variances that weights each sample's variance by its degrees of freedom ($s_p^2 = \frac{s_1^2(df_1) + s_2^2(df_2)}{df_1 + df_2}$), providing an unbiased estimate of the shared population variance under the homogeneity of variance assumption.

Homogeneity of Variance Assumption

The assumption in independent groups tests that the populations from which samples are drawn possess equal variances; the independent $t$-test is considered robust to violations as long as the larger sample variance does not exceed two times the smaller sample variance.

Confidence Interval for Independent Samples t-Test

The calculated interval around the obtained mean difference ($(\bar{X}_1 - \bar{X}_2) \pm |t_{crit}|(s_{\bar{X}_1 - \bar{X}_2})$) within which researchers are 95% (or 99%) confident the true population mean difference ($\mu_1 - \mu_2$) resides; if zero is included in the interval, the null hypothesis cannot be rejected.

Cohen's d

A standardized effect size metric for $t$-tests calculated as $d = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{s_p^2}}$, expressing the difference between two sample means in units of pooled standard deviation, categorized as small ($|d| \le .2$), medium ($.2 < |d| < .8$), or large ($|d| \ge .8$).

Eta-Squared ($\eta^2$)

A standardized measure of effect size that quantifies the proportion of total variability in the dependent variable explained by the independent variable ($\eta^2 = \frac{t_{obs}^2}{t_{obs}^2 + df}$ in a $t$-test, or $\eta^2 = \frac{SS_B}{SS_{total}}$ in an ANOVA), interpreted as trivial ($\eta^2 < .01$), small ($.01 \le \eta^2 \le .09$), medium ($.10 \le \eta^2 \le .25$), or large ($\eta^2 > .25$).

Levene's Test for Equality of Variances

An inferential statistical test reported in SPSS that evaluates whether the assumption of homogeneity of variance has been met; if statistically significant ($p < .05$), equal variances cannot be assumed, requiring the use of corrected degrees of freedom and output rows.

One-Way Between-Subjects ANOVA

A parametric hypothesis test used to compare means across three or more levels of a single between-subjects independent variable to examine whether at least one group mean significantly differs from the others on an interval or ratio dependent variable.

Experimentwise Alpha Inflation

The cumulative increase in the probability of committing at least one Type I error across a family of multiple separate hypothesis tests on the same data, avoided in ANOVA by conducting a single omnibus test that maintains the designated alpha level across all group comparisons simultaneously.

Between-Groups Variability ($MS_B$)

A variance estimate in ANOVA reflecting the dispersion among sample group means relative to the grand mean, driven by treatment effects produced by the independent variable in addition to random sampling error.

Within-Groups Variability ($MS_W$)

A variance estimate in ANOVA reflecting the dispersion of individual participant scores within each group around their own group mean, caused purely by random sampling error, including individual differences and experimental measurement error.

F-Ratio

The test statistic for an ANOVA calculated by dividing between-groups variance by within-groups variance ($F = \frac{MS_B}{MS_W}$), which compares the treatment effect plus sampling error to sampling error alone and has an expected value of approximately 1 when the null hypothesis is true.

F Distribution

A theoretical, positively skewed probability distribution of $F$-ratios that is bounded by 0 on the left, extends to positive infinity, features a mode of approximately 1 under the null hypothesis, and always maintains a one-tailed rejection region in its upper tail.

ANOVA Source Table

A standardized format for organizing ANOVA calculations into rows for Between Groups, Within Groups, and Total variation, alongside columns for Sum of Squares ($SS$), degrees of freedom ($df$), Mean Squares ($MS = \frac{SS}{df}$), and the final $F$-ratio.

Sum of Squares Total ($SS_{total}$)

The total variability present in an entire ANOVA dataset, computed definitionally as the sum of squared deviations of each individual score from the grand mean: $SS_{total} = \sum (X - \bar{X}_G)^2$.

Sum of Squares Between ($SS_B$)

The variability attributable to differences among treatment group means, calculated definitionally as $SS_B = n \sum (\bar{X}_j - \bar{X}_G)^2$, where each group mean's squared deviation from the grand mean is weighted by the group sample size.

Sum of Squares Within ($SS_W$)

The variability attributable to individual differences and unsystematic error within groups, calculated definitionally as the sum of squared deviations of each individual score from its respective group mean: $SS_W = \sum (X - \bar{X}_j)^2$.

Post-Hoc Test

Follow-up statistical comparisons conducted after obtaining a statistically significant omnibus $F$-test in an ANOVA to determine exactly which specific pairs of group means differ significantly while controlling the experimentwise error rate.

Tukey's Honestly Significant Difference (HSD) Test

A specific post-hoc test that systematically evaluates all pairwise mean differences and confidence intervals following a significant ANOVA while preventing the inflation of Type I error rate.

Between-Subjects Design

An experimental research design in which each participant is assigned to only one treatment condition, meaning different individuals are tested across the different levels of the independent variable.

Within-Subjects Design

An experimental research design (also known as a repeated-measures design) in which every participant experiences all levels of the independent variable, substantially reducing error variance by controlling for individual differences.

Concurrent Measures Design

A within-subjects experimental design where participants are presented with two or more stimuli simultaneously and a single preference, choice, or behavioral response is recorded.

Sensitization Effects

A threat to internal validity in within-subjects designs where exposure to multiple experimental conditions allows participants to discern demand characteristics or guess the study hypothesis, altering their natural behavior.

Order Effects

Confounds in within-subjects designs that occur when the sequence in which treatments are presented affects participant responses, comprising progressive effects and carryover effects.

Progressive Effects

Order effects in within-subjects designs that accumulate systematically over time as a function of completing repeated tasks, specifically manifesting as practice effects (performance gains) or fatigue effects (performance declines).

Carryover Effects

Order effects that happen when the physiological, psychological, or contextual influence of one condition fails to dissipate before the participant is tested in a subsequent condition.

Counterbalancing

A methodological procedure used in within-subjects designs where participants are assigned to experience treatment conditions in different sequence orders, which does not eliminate order effects but distributes them equally across conditions.