STAT 503 Lesson 2: Simple Comparative Experiments

A comprehensive study set for STAT 503 Lesson 2 covering simple comparative experiments, t-test formulas, power calculations, and paired design concepts.

Simple comparative experiment

An experiment comparing two conditions or groups through hypothesis testing, estimation, sample-size determination, and power.

Dot plot

A graphical display useful for seeing the distribution of observations, especially for very small data sets.

Boxplot

A graphical tool useful for both small and larger data sets that provides a quick snapshot of the distribution, including shape, spread, and outliers.

Box in a boxplot

The region spanning the middle $50\%$ of the data, from the $25^{th}$ percentile ($Q1$) to the $75^{th}$ percentile ($Q3$).

Median in a boxplot

The horizontal line located inside the box representing the center of the data.

Interquartile range (IQR)

$IQR = Q3 - Q1$; the width or range covered by the middle $50\%$ of observations.

Boxplot whiskers

Lines that extend toward the minimum and maximum values, limited by the $1.5 \times \text{IQR}$ rule.

Potential boxplot outliers

Individual observations plotted as dots that fall beyond the whiskers.

Equal-variance assumption

The assumption for the pooled two-sample t procedure that both population variances are equal.

Sample mean formula ($\bar{y}$)

$\bar{y} = \frac{\sum_{i=1}^{n} y_i}{n}$.

Sample variance formula ($s^2$)

$s^2 = \frac{\sum_{i=1}^{n} (y_i - \bar{y})^2}{n - 1}$, which serves as an unbiased estimator of $\sigma^2$.

Signal-to-noise interpretation

An engineering description of the t statistic where the 'signal' is the difference between group means and the 'noise' is the within-group variation.

Student t distribution

The distribution used for the t-test, published by William Sealy Gosset under the pseudonym 'Student' because population variances are usually unknown.

Pooled variance formula ($s_p^2$)

$s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}$.

Standard error of two independent means ($SE$)

$SE(\bar{y}_1 - \bar{y}_2) = s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}$ under the equal-variance pooled model.

Two-sample null hypothesis ($H_0$)

$H_0: \mu_1 = \mu_2$ or $H_0: \mu_1 - \mu_2 = 0$.

Two-sample t statistic

$t = \frac{\bar{y}_1 - \bar{y}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$.

Degrees of freedom ($df$) for pooled t-test

$df = n_1 + n_2 - 2$.

p-value

Under $H_0$, the probability of observing a test statistic at least as extreme as the one obtained during the experiment.

Two-sample pooled confidence interval formula

$(\bar{y}_1 - \bar{y}_2) \pm t_{(1-\alpha/2, df)} \times s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}$.

Margin of error bound ($B$)

The desired bound on the margin of error for estimating the difference in population means used in sample-size determination.

Sample-size equation (equal-n)

$n = \left[ \frac{t_{(1-\alpha/2, df)} \times s \times \sqrt{2}}{B} \right]^2$.

Paired samples

A design where two measurements are taken on the same subject or matched unit, making the observations correlated rather than independent.

Paired t-test analysis

An analysis of within-subject differences (e.g., after - before) to remove or reduce subject-specific variation.

Nuisance factor in paired design

A factor, such as a specific patient or subject, that causes substantial variability but is not the primary interest of the study.

Blocking

A design technique, exemplified by the paired t-test, where subjects are treated as 'blocks' to control for variability.

Type I error ($\alpha$)

The probability of rejecting the null hypothesis ($H_0$) when it is actually true.

Type II error ($\beta$)

The probability of failing to reject or accepting the null hypothesis ($H_0$) when the alternative hypothesis ($H_A$) is true.

Power

The probability of rejecting the null hypothesis when the alternative is true, calculated as $1 - \beta$.

Operating characteristic (OC) curves

Graphical curves used to determine $\beta$ and power for the t-test based on sample size and standardized difference.

Standardized mean difference

The true mean difference measured in standard-deviation units: $\frac{|\mu_1 - \mu_2|}{\sigma}$.