Introduction to Useful Parametric Tests

Vocabulary flashcards covering parametric tests including the Student's t-test, ANOVA, simple linear regression, and Pearson correlation coefficient.

Parametric tests

Statistical tests that make specific assumptions about the data, requiring variables to be continuous or ordinal and normally distributed (bell curve).

Normal distribution (Bell curve)

A symmetric, bell-shaped probability distribution where data points cluster around the mean ($\mu$), with approximately $34.1\%$ of values within one standard deviation ($\sigma$) on either side, $13.6\%$ between $1\sigma$ and $2\sigma$, $2.1\%$ between $2\sigma$ and $3\sigma$, and $0.1\%$ beyond $3\sigma$.

Student's t-test

A parametric statistical test used when comparing two variables (paired or unpaired) to determine whether their means are significantly different; it is most effective when equal sample sizes are used.

Two-sample t-statistic

The test statistic calculated as $t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$, where $\bar{x}_1$ and $\bar{x}_2$ represent sample means, $s_1$ and $s_2$ represent standard deviations, and $n_1$ and $n_2$ represent sample sizes.

One-sided t-test

A directional hypothesis test utilized when a priori (beforehand) knowledge or assumptions lead the researcher to expect one group's mean to be specifically greater or lesser than the other.

Two-sided t-test

A hypothesis test utilized when the direction of difference between two groups ($A$ and $B$) is unknown beforehand, making the hypothesis $A > B$ equally as reasonable as $B > A$.

Degrees of freedom ($df$) for a two-sample t-test

A parameter calculated in a two-sided test as $k = n_1 + n_2 - 2$ (simplifying to $2n - 2$ when sample sizes are identical), or conservatively determined as the smaller of $n_1 - 1$ and $n_2 - 1$ in certain one-sided procedures.

Analysis of Variance (ANOVA)

A parametric statistical test used when evaluating more than two variables ($2+$) to determine whether population means differ significantly under fixed, random, or mixed effects.

One-way ANOVA

A standard ANOVA design sufficient to determine whether statistically significant differences exist among the means of two or more independent samples.

F-statistic

The test statistic calculated in ANOVA, defined as $F = \frac{\text{variance between treatments}}{\text{variance within treatments}} = \frac{MS_{\text{Treatments}}}{MS_{\text{Error}}} = \frac{SS_{\text{Treatments}} / (I - 1)}{SS_{\text{Error}} / (n_T - I)}$.

Simple linear regression

A statistical method where a dependent response variable ($y$) is regressed on an independent explanatory variable ($x$) to construct a best-fit line reported in the form $y = a + bx$.

Ordinary least squares (OLS)

The simplest form of regression that determines the best-fit line by minimizing the vertical distances (deviations in $y$) between the observed data points and the regression line.

Slope ($b$)

In linear regression, the value representing the rate of change or the estimated increase in the dependent variable ($y$) for every one-unit increase in the independent variable ($x$).

Intercept ($a$)

In linear regression, the baseline value of the dependent variable ($y$) when the independent variable ($x$) equals $0$.

Correlation coefficient ($r$)

A standardized statistical measure of the strength and direction of a linear relationship between two continuous quantitative variables, always falling within the range of $-1$ to $1$.

Correlation does not equal causation

A statistical rule stating that an observed association between two variables does not prove that one variable causes the other, as both may be influenced by a third, unmeasured variable.

Categorical variable

A qualitative variable (such as sex or educational level) representing qualitative categories rather than continuous quantities, making the calculation of Pearson correlation ($r$) invalid and meaningless.