Biostatistics (BIOG 2000): Inference, Hypothesis Testing, Nonparametric Tests, Correlation, and Categorical Data Analysis

Vocabulary practice flashcards covering biostatistical inference, one- and two-sample parametric tests, nonparametric tests, correlation coefficients, single and two-sample proportions, and paired categorical tests from BIOG 2000.

Point Estimation

The inferential process of calculating a single numerical value from sample data to serve as the best guess for an unknown population parameter.

Interval Estimation

The inferential process of determining a range of plausible values (a confidence interval) that is claimed to contain the true population parameter with a specified degree of confidence.

Testing of Hypothesis

A formal statistical procedure used to evaluate a specific claim or hypothesis about a population parameter based on sample data.

Accuracy (in Estimation)

A property of an estimator describing how close sample estimates are to the true population parameter ($\mu$); it requires the sample to be representative of the target population.

Precision (in Estimation)

The degree of consistency or stability of an estimate ($\bar{X}$) across repeated random samples of size $n$, quantified inversely by the magnitude of the error or standard error.

Standard Error ($SE$)

The standard deviation of the sampling distribution of a statistic (such as $\frac{\sigma}{\sqrt{n}}$ or $\frac{s}{\sqrt{n}}$ for the sample mean), representing variability between sample averages rather than variability among individuals.

Central Limit Theorem

The statistical principle stating that for a sufficiently large sample size $n$, the sampling distribution of the sample mean ($\bar{X}$) is approximately normal, regardless of the shape of the original population distribution.

Margin of Error ($MOE$)

The maximum expected difference between the observed sample estimate and the true population parameter, calculated as the critical value times the standard error and inversely proportional to $\sqrt{n}$.

Student's $t$-Distribution

A symmetric, bell-shaped probability distribution centered at zero with heavier tails than the standard normal ($Z$) distribution, belonging to a family indexed by degrees of freedom ($df$) that approaches the $Z$-distribution as $df$ increases.

Null Hypothesis ($H_0$)

The baseline statistical claim assumed to be true during testing, stating no effect, no difference, or the exact opposite of the research hypothesis.

Alternative Hypothesis ($H_1$)

The research hypothesis representing the claim, effect, or difference that the investigator aims to prove or find evidence to support.

Test Statistic

A standardized numerical value calculated from sample data (such as $t = \frac{\bar{X} - \mu_0}{s / \sqrt{n}}$) that measures how far the sample estimate deviates from the null hypothesis value.

$p$-Value

The probability, calculated under the assumption that the null hypothesis ($H_0$) is true, of obtaining a test statistic as extreme as or more extreme than the observed value.

Level of Significance ($\alpha$)

The predetermined threshold probability of committing a Type I error (commonly set at $0.05$ or $0.01$), below which the $p$-value is considered statistically significant enough to reject $H_0$.

Type I Error

The error committed when a true null hypothesis is rejected, resulting in supporting a false research hypothesis (a false positive / false significant result); its probability equals $\alpha$.

Type II Error

The error committed when a false null hypothesis is not rejected, resulting in failing to support a true research hypothesis (a false negative / nonsignificant result); its probability equals $\beta$.

Power of a Test

The probability of correctly rejecting a false null hypothesis to support a true research hypothesis, calculated as $1 - \beta$ and typically targeted at $80\%$ or $90\%$.

One-Sample $t$-Test

A parametric hypothesis test used to evaluate whether the mean ($\mu$) of a single population differs significantly from a hypothesized reference value ($\mu_0$), using $df = n - 1$.

Levene's Test for Homogeneity of Variances

An inferential test evaluating $H_0: \sigma_1^2 = \sigma_2^2$ versus $H_1: \sigma_1^2 \ne \sigma_2^2$ to determine whether equal variances can be assumed prior to running an independent samples $t$-test.

Student's Independent Samples $t$-Test

The version of the two-sample independent $t$-test applied when population variances are assumed equal (Levene's $p > \alpha$), utilizing a pooled variance ($s_p^2$) and degrees of freedom $df = n_1 + n_2 - 2$.

Welch's $t$-Test

The version of the two-sample independent $t$-test applied when population variances cannot be assumed equal (Levene's $p \le \alpha$), utilizing unpooled standard errors and Satterthwaite-approximated degrees of freedom.

Pooled Variance ($s_p^2$)

The weighted average variance across two independent samples, computed as $s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}$, simplifying to $\frac{s_1^2 + s_2^2}{2}$ when sample sizes are equal.

Paired $t$-Test

A parametric hypothesis test that evaluates the mean difference ($\mu_d$) within paired observations (pre/post or matched subjects) by analyzing individual differences $d = \text{before} - \text{after}$ with $df = n - 1$.

Clinical Significance

The practical importance and health impact of a study's findings on patients, judged by the magnitude of the observed effect rather than solely by the statistical $p$-value.

Nonparametric Tests

Statistical distribution-free hypothesis tests that do not rely on the assumption of normality for the underlying random variable, making them ideal for small or highly skewed samples.

Wilcoxon Signed-Rank Test

A nonparametric test for paired continuous data or single-sample medians that ranks absolute non-zero differences ($|Z_i|$) and calculates test statistics from the sum of positive ranks ($T^+$) or negative ranks ($T^-$).

Wilcoxon Rank Sum Test (Mann-Whitney Test)

A nonparametric counterpart to the independent two-sample $t$-test used to test for a location shift ($\delta$) between two independent distributions by ranking all combined observations together.

Pearson's Correlation Coefficient ($r$)

A unitless measure ranging between $-1$ and $+1$ that quantifies the strength and direction of the linear association between two continuous random variables.

Coefficient of Determination ($R^2$)

A metric ranging from $0$ to $1$ (the square of Pearson's $r$) that represents the proportion of total variability in the dependent variable explained by its linear relationship with the independent variable.

Spearman's Rank Correlation Coefficient ($\rho$)

A nonparametric correlation coefficient computed by applying Pearson's formula to the ranked values of two variables, assessing monotonic relationships rather than strictly linear ones.

Monotonic Relationship

A relationship between two variables that consistently moves in one direction—either continuously increasing or continuously decreasing—without needing to follow a straight line.

$Z$-Test for One Population Proportion

A large-sample parametric hypothesis test evaluating a single proportion $H_0: \pi = \pi_0$ using $Z = \frac{p - \pi_0}{\sqrt{\frac{\pi_0(1 - \pi_0)}{n}}}$, valid when $n\pi_0 \ge 5$ and $n(1 - \pi_0) \ge 5$.

Binomial Test

An exact test for a single population proportion based directly on the binomial distribution, serving as the required method when sample size rules for the $Z$-test fail.

Chi-Square Test ($\chi^2$) of Independence

A statistical test for comparing two independent categorical proportions in a $2 \times 2$ table, mathematically equivalent to the two-sample independent $Z$-test ($\chi^2 = Z^2$) with $df = 1$.

Expected Cell Count

The theoretical frequency expected in a contingency table cell under the assumption of independence, computed as $\frac{\text{Row Total} \times \text{Column Total}}{\text{Overall Total}}$; all cells must be $\ge 5$ for a valid Chi-square test.

Fisher's Exact Test

An exact non-asymptotic test used to compare two independent proportions or analyze $2 \times 2$ contingency tables when at least one expected cell count is less than $5$.

Concordant Pairs

Pairs in matched or paired categorical data where both subjects exhibit the exact same outcome (cells $a$ and $d$ in a $2 \times 2$ matched table).

Discordant Pairs

Pairs in matched or paired categorical data where the two subjects display opposite outcomes (cells $b$ and $c$ in a $2 \times 2$ matched table), forming the sole basis for comparing treatments in McNemar's test.

McNemar's Test

A hypothesis test for paired binary proportions that tests $H_0: \pi_1 = \pi_2$ strictly using discordant counts ($b$ and $c$), either via the binomial distribution with $p = 0.50$ or via the large-sample statistic $\chi^2 = \frac{(b - c)^2}{b + c}$.

Reliability

The consistency or stability of measurements or diagnostic classifications between two raters evaluating the same subjects, irrespective of whether the diagnoses are correct.

Validity

The accuracy of a measurement or diagnosis, reflecting how correctly it measures the true state of health or disease compared to an established reference standard.

Cohen's Kappa Measure of Agreement ($\hat{\kappa}$)

An index ranging from $-\frac{p_e}{1 - p_e}$ to $+1$ that quantifies inter-rater agreement on a categorical scale adjusted for chance: $\hat{\kappa} = \frac{p_o - p_e}{1 - p_e}$.

Expected Agreement ($p_e$)

The proportion of total agreement between two raters that would be expected purely by chance, computed from marginal totals as $p_e = \frac{(a + c)(a + b) + (b + d)(c + d)}{N^2}$.