Experimental Biology: Statistical Differences Between Two Groups

Core Educational Resources

• Primary Textbook Recommendation: Choosing and Using Statistics: A Biologist's Guide, 3rd Edition.     * Author: Calvin Dytham.     * ISBN: $978-1-405-19839-4$.     * Publication Date: January $2011$.     * Publisher: Wiley-Blackwell.     * Length: $320\,\text{pages}$.     * Availability: Accessible at the University library in both physical and ebook formats.

Learning Objectives

• Explicate the fundamental differences between paired versus unpaired data structures.

• Determine the appropriate statistical tests for paired versus unpaired data sets.

• Identify the correct tests to apply for parametric versus non-parametric data distributions.

• Interpret the outputs generated by various statistical tests.

• Note on mathematical depth: While the structural logic of calculating tests will be covered (e.g., the meaning of a $Z\text{ value}$ or intermediate figures), the primary focus for students is the interpretation of the $p\text{ value}$.

Classification of Data Groups

• Unpaired Data (Independent): Refers to separate populations or distinct entities measured in parallel.

• Paired Data (Dependent): Refers to the same populations or entities measured repeatedly (e.g., at an initial time point and a later time point).

• Repeated Data: This is categorized as a specific type of paired data.

• Measurement Unit: It is critical to consider the sample or measurement unit carefully, as established in the introductory statistics session.

Distribution and Parametric Status

• Parametric Data: Characterized as "normally" distributed data. This involves a symmetrical distribution centered around the mean.

• Non-parametric Data: Characterized as "non-normally" distributed. This data is not symmetrical and is typically skewed in some manner, or the sample size is insufficient to establish a normal distribution.

Paired Data Characteristics

• Paired data occurs when a single individual, specific sampling station, or individual test tube is tested twice.

• Example Scenarios:     * Measurements taken "Before chemical exposure."     * Measurements taken "After chemical exposure."

Functional Overview of T-Tests

• Primary Purpose: To compare two means.

• Data Requirements: Requires parametric data.

• Application: Can be applied to either paired or unpaired data sets.

• Assumptions and Pre-conditions:     * The dependent variable must be continuous.     * The data must be approximately normally distributed.     * Variances must be homogeneously distributed (homogeneity of variance); for instance, one group cannot exhibit extreme variability while the other is stable.     * The independent variable (the cause) must consist of two categorical related groups or matched pairs.

• Note on Scale: If there are more than $2$ groups, alternative tests like ANOVAs must be used.

• Null Hypothesis ($H_0$): For the paired t-test, the null hypothesis posits that there is NO difference between the two data series (e.g., comparing the heights of two populations).

Sub-Types of T-Tests

• Independent Two-Sample T-test (Unpaired): Compares the means of two entirely separate groups, such as a treatment group versus a control group.

• Paired T-test (Paired): Compares means from the same group at two different time intervals, such as a "Before" versus "After" intervention.

• One-Sample T-test: Compares the mean of a single group against a known standard or specific value (noted as being less common).

T-Test Calculation Logic and Mechanics

• Signal vs. Noise: The test evaluates whether the difference between means (the "signal") is larger than the natural variation within the groups (the "noise").

• Real-World Significance: If the difference of means is large relative to natural variation, it suggests the groups differ in reality, not just within the sample.

• The T-value: This is derived as the ratio between the difference of means and the variation in the groups.     * Formula for Standard Error ($SE$): $SE = \frac{SD}{n}$.     * The $t-value$ is directly linked to the difference between means; as the difference increases, the $t-value$ increases.     * A large $t-value$ suggests a substantial difference compared to noise.     * A smaller $t-value$ occurs when there is high variation. Greater variation makes mean differences more likely to be the result of random fluctuation.

• Degrees of Freedom: Used to derive the $p\text{ value}$ once the $t-value$ is obtained.

• Significance Threshold: If p < 0.05, the Null Hypothesis ($H_0$) is rejected, suggesting a likely real difference between groups.

• Visual Logic:     * If two box plots show large overlapping variation, they are unlikely to be statistically different.     * If the boxes do not overlap, they are likely to be statistically different.     * Low variation in results can reveal significant differences that high variation would otherwise obscure.

Mann-Whitney U Test (Non-parametric Unpaired)

• Purpose: The non-parametric equivalent to the unpaired t-test.

• Scope: Only functions for two test groups.

• Assumptions: Unlike parametric tests, it does not assume normal distribution or homogeneity of variance.

• Mechanism: It compares "rank sums" rather than means.     * Data is first converted into ranks.     * Ideal for datasets with extreme values or outliers because ranking reduces the impact of such values.

• Variable Requirements:     * Two independent variables with at least ordinal scaled characteristics.     * Dependent variables that are ordinal, metric, or continuous (allowing for ranking).

• Hypotheses:     * Null Hypothesis ($H_0$): No difference in terms of central tendency (medians) between the two groups.     * Alternative Hypothesis ($H_1$): There is a difference with respect to the central tendency between the groups.

• Step-by-Step Calculation:     * Ranks are assigned and summed for each group ($T_1$ and $T_2$).     * Average rank sums are calculated.     * $U\text{-values}$ are calculated from the rank sums.     * A $Z\text{ value}$ is derived using Standard Deviation ($SD$) and number of samples ($n$).     * A $Z\text{ value}$ close to zero supports the null hypothesis; a $Z\text{ value}$ far from zero suggests significance.     * $p\text{ value}$ is derived: If p < 0.05, the null hypothesis is rejected.

Wilcoxon Signed Ranks Test (Non-parametric Paired)

• Purpose: The non-parametric version of the paired t-test.

• Mechanism: Based on the ranks of the paired differences rather than comparison of means.

• Requirements:     * Dependent variable (the effect) must be on an ordinal or continuous scale.     * Independent variable (the cause) must consist of matched pairs.     * A minimum of $6$ pairs of data are required for analysis.

• Procedural Steps:     * Calculate the difference in values for each pair (e.g., "Before" minus "After"). This can result in negative or positive values.     * Rank these differences from smallest to largest regardless of the sign (positive/negative).     * Re-assign the positive or negative signs to the rank scale (e.g., $1$ to $7$ if there are $7$ samples).     * Sum the positive ranks and sum the negative ranks.     * Calculate the Wilcoxon Rank Statistic ($W$), which is the minimum value of the positive or negative rank sums.     * Calculate the Expected $W\text{ value}$ (the value expected if no difference exists).     * Derive a $Z\text{ value}$ using $W$, Expected $W$, and $SD$.     * Derive the $p\text{ value}$. Reject $H_0$ if p < 0.05.

• Example Case Study:     * River flow ($m/sec$) measured along several stations in a single river.     * Reported Result: $Z = -2.197$, $p = 0.028$.     * Interpretation: Since p < 0.05, the Null Hypothesis belongs to the unlikely category.

The Hierarchy of Tests

• Fundamental Rule: Always prefer a parametric test (like a t-test) if the data fits the normal distribution requirements.

• Reasoning: Parametric tests are considered "stronger" than non-parametric alternatives like Mann-Whitney or Wilcoxon.

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

• Purpose: A non-parametric test used to determine if there is a relationship between two categorical variables.

• Frequency Comparison: Checks if frequencies occurring in the sample differ significantly from expected frequencies.

• Hypotheses:     * Null Hypothesis ($H_0$): Observed and expected frequencies (from a known random distribution) are NOT different from each other (no association).     * Alternative Hypothesis ($H_1$): Observed and expected frequencies are different (an association exists).

• Applications/Examples:     * Age group versus preferred social media platform.     * Medical treatment success rates versus patient gender.     * Political party affiliation versus occupation.     * Student performance versus teaching method.

• Calculation Components:     * Contingency Table: The raw categorical data.     * Expected Frequency Table: Converts raw data to a distribution showing what numbers would look like if there were no difference between groups.     * Degrees of Freedom ($Df$): Calculated as $Df = (\text{no. of rows} - 1) \times (\text{no. of columns} - 1)$.     * Critical Chi Value: Obtained from a "Chi lookup table" using $Df$ and the desired $p\text{ value}$ (usually $0.05$).     * Decision Rule: If the Critical Chi Value is larger than the Calculated Chi Value, the Null Hypothesis is retained. If smaller, the Null Hypothesis is rejected.

• Case Study: Investigation into whether fish lice attach to fish randomly or have a preference.     * Reported Result: $\chi^2 (4, N = 98) = 3.002$, $p = 0.557$.     * Interpretation: Since p > 0.05, the Null Hypothesis is likely true (lice attach randomly).

Distinguishing Standard Deviation and Standard Error

• Standard Deviation ($SD$):     * Definition: The square root of the variance.     * Function: A measure of dispersion or variability; shows how much values in a dataset vary from each other and how far apart values are from the mean.     * Use Case: Identifying outliers and providing descriptive statistics to summarize data.

• Standard Error ($SE$):     * Definition: A measure of spread, most commonly the "standard error of the mean."     * Function: Measures sampling error; evaluates how accurately the sample mean represents the population mean.     * Use Case: Inferential statistics; estimating the variation likely to exist between different samples of a population and the population itself.

Comprehensive Statistical Test Decision Matrix

• One-Sample Tests (1 group):     * Categorical Data: G-test (fit to uniform); Chi-square test (fit to Poisson).     * Discrete/Ordinal Data: Wilcoxon's one-sample test (e.g., median of $0$?).     * Continuous Data: One-sample t-test (e.g., mean of $0$?); Kolmogorov-Smirnov or Anderson-Darling tests (to test fit to normal).

• Tests of Relationship (2 variables):     * Categorical: Chi-square for association.     * Discrete/Ordinal: Kendall’s rank correlation; Spearman’s rank correlation.     * Continuous: Pearson product-moment correlation.

• Regression (1 cause, 1 effect):     * Discrete/Ordinal: Logistic regression; Model II regression; Kendall’s robust line fit.     * Continuous: Linear regression; Quadratic or Polynomial regression.

• Comparing Two Groups:     * Unpaired + Categorical: Chi-square test.     * Unpaired + Discrete: Mann-Whitney U test.     * Unpaired + Continuous: T-test; One-way ANOVA.     * Paired + Categorical: Chi-square test.     * Paired + Discrete: Wilcoxon's signed ranks test.     * Paired + Continuous: Paired t-test.

• Comparing >2 Groups (1 cause):     * Categorical: Chi-square test.     * Discrete/Ordinal: Kruskal-Wallis test.     * Continuous: One-way ANOVA.

• Multifactorial Tests:     * Discrete + 2 factors (no replication): Friedman test.     * Discrete + 2 factors (weak option): Scheirer-Ray-Hare.     * Continuous + 2 factors: Two-way or multi-way ANOVA.     * Continuous + Repeated Measures: Repeated-measures ANOVA.

• Advanced/Multivariate Tests:     * Many causes/effects: Path analysis.     * Exploring variables: PCA (Principal Component Analysis).     * Discriminating groups with many variables: CVA (Canonical Variate Analysis); Discriminant function analysis; MANOVA; Multiple regression; DCA (Detrended Correspondence Analysis).     * Analysis with Covariates: ANCOVA (for $1$ variable); MANCOVA (for >1 variable).