Inferential Statistics Intro Continued

Fundamentals of Hypothesis Testing

• Hypotheses as Statements of Truth: Hypotheses are defined as statements believed to be true regarding a specific phenomenon that have not yet been proven. They are structured to be testable.

• The Null Hypothesis ($H_0$): This hypothesis states that no statistical significance or difference exists between two or more populations or groups. It represents the default position that any observed effect is due to chance.     * Specific Example (Elephant Ears): If average temperature does not affect the size of elephant ears, then species $X$ moving to a different climate zone will not evolve different ear sizes.

• The Alternative Hypothesis ($H_1$ or $H_a$): This states that a phenomenon is occurring due to non-random causes and is not just a matter of chance. It is the statement we accept if the Null Hypothesis is rejected.     * Specific Example (Elephant Ears): If average temperature does affect ear size, then populations moving from warmer to cooler climates will evolve smaller ears.

• Statistical Methodology: Scientists use specific statistical tests to determine whether to reject the Null Hypothesis. Rejecting $H_0$ allows for the acceptance of $H_1$.

Standard Deviation and Data Spread

• Definition of Standard Deviation ($SD$): A measurement used to quantify the dispersion or variability of data points relative to their mean. It reflects how far, on average, individual values deviate from the mean.

• Mathematical Representation ($\sigma$): The standard deviation is denoted by the Greek letter sigma ($\sigma$).

• Calculating $SD$: The lecture defines $SD$ as "the square root of the sum of the squared deviations, divided by the number of values."     * Variables involved: $N = \text{number of values}$, $\mu = \text{the mean}$, $x_i = \text{individual values}$.

• Normal Distribution Dispersion (The Empirical Rule): In a normal, symmetrical distribution:     * There is a $68.2\%$ chance a random sample sits within $1\sigma$ of the mean ($34.1\%$ above and $34.1\%$ below).     * There is a $95.4\%$ chance a random sample sits within $2\sigma$ of the mean.     * There is a $99.7\%$ chance a random sample sits within $3\sigma$ of the mean.

• Interpretation of Spreads:     * Normal distributions can have different means ($\mu = 900$, $1100$, or $1300$) while maintaining the same spread.     * Normal distributions can have different standard deviations ($SD = 80$, $120$, or $190$), resulting in thinner or wider curves regardless of the mean.

Confidence Intervals ($CI$)

• Core Concept: The $CI$ is the range in which the true mean of a population lies with a specific probability, derived from a sample set.

• Sampling Variability: Different sample sets from the same population will likely yield different means by chance. The $CI$ accounts for this.

• Common Thresholds: The most used probabilities are $95\%$ and $99\%$. In a $95\%$ $CI$, if a population is sampled $95$ times, the true mean should fall within the limits $95$ times.

• Calculating Limits:     * Lower Limit Calculation: $x_l = \bar{x} - z \cdot \frac{s}{\sqrt{n}}$     * Upper Limit Calculation: $x_u = \bar{x} + z \cdot \frac{s}{\sqrt{n}}$     * Components: $z$-values are standard numbers from look-up tables; other values come from the specific sample set.

• Visualization: $CIs$ are frequently added to correlation or regression plots to show the reliability of the trend (e.g., $90\%$, $95\%$, or $99\%$ bands).

Parametric vs. Non-Parametric Statistics

• Parametric Tests:     * Definition: These tests have complete information about population parameters or make assumptions about the distribution (usually normal).     * Measurement Level: Metric scales, specifically Interval or Ratio scales.     * Central Tendency: Focuses on the Mean.     * Examples: t-test, ANOVA, Pearson correlation.

• Non-Parametric Tests:     * Definition: Applied when the researcher has no knowledge of population parameters or cannot make specific assumptions about the distribution (arbitrary or non-normal).     * Measurement Level: Nominal or Ordinal scales.     * Central Tendency: Focuses on the Median.     * Examples: Mann-Whitney U test, Spearman correlation, Wilcoxon signed-rank test, Kruskal-Wallis test.

• Variable Types in Biology: Most biological and biochemical data involve continuous variables like length, weight, pH, moles, and reaction speed.     * Independent Variable: The cause (e.g., diet).     * Dependent Variable: The effect/measurement (e.g., weight gain) that is "dependent" on the cause.

Determining Data Normality

• Importance: Normality must be determined before selecting a statistical test; it is the "key" prerequisite.

• Analytical/Statistical Tests for Normality: These tests evaluate the hypothesis that the data follows a normal distribution.     * Shapiro-Wilk Test: Most appropriate for small sample sizes ($n < 50$).     * Kolmogorov-Smirnov Test: Used for larger sample sizes ($n \ge 50$).     * Anderson-Darling Test: An alternative normality test.     * P-value Interpretation for Normality:         * If $p < 0.05$: Reject the Null Hypothesis (Data is non-normally distributed).         * If $p > 0.05$: Accept the Null Hypothesis (Data is normally distributed).

• Sample Size Influence: Normality tests are highly sensitive to sample size. A very large sample may return a $p < 0.05$ (non-normal) even if it is representative of a normal population, while a small sample may return a high p-value even if it is skewed.

• Visual Inspection Methods:     * Histograms: Plotting data density.     * Quantile-Quantile (Q-Q) Plots: A stronger method comparing theoretical quantiles of a normal distribution against observed values.         * Procedure: Sort values, divide a theoretical curve into $n+1$ areas, calculate theoretical quantile values, and plot against actual measurements.         * Interpretation: A straight line indicates the data is normal. Deviations at the edges suggest non-normality.

The Meaning and Strength of P-values

• Definition: The lower the P-value, the more confidence one has in rejecting the Null Hypothesis. It quantifies the strength of evidence for an experimental effect.

• Categorization of Significance:     * $p > 0.1$: No compelling evidence to reject the Null Hypothesis.     * $0.05 < p < 0.10$: Mild or weak evidence to reject the Null Hypothesis.     * $p < 0.05$: Moderate evidence to reject the Null Hypothesis (marked by $ * $).     * $p < 0.01$: Strong evidence to reject the Null Hypothesis (marked by $ ** $).     * $p < 0.001$: Very strong evidence to reject the Null Hypothesis (marked by $ *** $).     * *n.s.:* Over $0.05$ (non-significant).

• The 0.05 Threshold: Known as the "holy grail" of cut-offs, established by classic statisticians as a balance between strictness and leniency. Modern scientists debate using binary cut-offs and prefer describing results as statistical "power."

• Extreme Cases: Research like the Higgs-Boson particle requires ultra-low thresholds, such as $0.0000003$ (1 in 3.5 million).

• P-hacking: The unethical practice of altering tests or data to force a result below the significane threshold (e.g., turning $0.051$ into $0.049$). This is considered scientific misconduct.

Box Plots and Data Visualisation

• Interquartile Range (IQR): Also called the midspread or middle $50\%$.     * Data is divided into four rank-ordered parts (quartiles).     * $IQR$ is the difference between the $75^{th}$ and $25^{th}$ percentiles.

• Whiskers: Typically represent $1.5 \times \text{IQR}$.

• Skewness in Box Plots:     * Positively Skewed: Distribution is skewed "up" or to the right; the median is closer to the bottom ($Q1$).     * Negatively Skewed: Distribution is skewed "down" or to the left; the median is closer to the top ($Q3$).

• Comparison of Visuals:     * Bar Plots: Can hide data distributions and skewness.     * Violin Plots: Overlay box plots with data points to reveal the true density and outliers of the population.

Philosophical Implications in Data Analysis

• Hypothesis as a Liability: Excessive focus on a specific hypothesis may prevent researchers from making unexpected discoveries, especially in large datasets.

• The Gorilla Experiment: An exercise involving a dataset of $BMI$ and step counts for $1786$ people.     * Group 1: Asked to test a specific significant difference between men and women.     * Group 2: Asked to be "hypothesis-free" and draw conclusions generally.     * Outcome: If the data is manually plotted, the image of a gorilla appears in the distribution. This serves as a metaphor for "hidden" things in data that automated number-crunching might miss.

• Conclusion: Researchers must explore data manually and visually to capture biological implications that simple stat tests might obscure.