Experimental Biology: Differences Between Two Groups

Classifying Groups of Data

• Groups of data can be categorized into three primary types:
    - Unpaired (Independent): Data sets involving separate populations or entities measured in parallel.
    - Paired (Dependent): Data where the same populations or items are measured repeatedly (e.g., at time point 1 and later at time point 2).
    - Repeated: A specific type of paired data.

• The Sample/Measurement Unit: It is vital to think carefully about the unit being measured (as introduced in statistics session 1).

Parametric vs. Non-Parametric Distributions

• Parametric Data ("Normally" distributed):
    - Has a symmetrical distribution around the mean value.
    - Requires quantitative data that follows a specific bell-curve shape.

• Non-Parametric Data ("Non-normally" distributed):
    - Is not symmetrically distributed around the mean (skewed distribution).
    - Often occurs when there is a lack of sufficient data to establish a normal distribution.

Establishing Paired Data Structures

• Definition: Paired data occurs when an individual, a sampling station, or a specific test tube is tested twice or across two different conditions.

• Common Scenario: "Before chemical exposure" versus "After chemical exposure" within the same subject.

The T-Test (Parametric Analysis)

• Core Definition: A parametric test used to compare the means of two groups.

• Mandatory Assumptions for Dependent Variables:
    1. The data must be continuous.
    2. The data must be approximately normally distributed.
    3. The variances of the data must be homogeneously distributed (homoscedasticity); one group should not be significantly more variable than the other.

• Mandatory Assumptions for Independent Variables:
    - The "cause" variable must consist of two categorical, related groups or matched pairs.
    - Note: If there are more than 2 groups, ANOVAs are utilized instead.

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

• Logic of the Test: A t-test evaluates whether the observed difference between two groups is larger than what sampling variation would account for, assuming no true population difference exists.

• Internal Mechanics:
    - The test compares the "Signal" (the size of the difference between the means) against the "Noise" (the variation of values within the groups).
    - If the "Signal" is large relative to the "Noise," the groups likely differ in reality.

Variations of the T-Test

• Independent Two-Sample T-test (Unpaired): Compares means of two entirely separate groups (e.g., Treatment Group vs. Control Group).

• Paired T-test (Paired): Compares means from the same group at two different times (e.g., Before intervention vs. After intervention).

• One-Sample T-test: Compares a single group's mean against a known standard or constant value (less common).

Calculation and Interpretation of T-Values

• The T-value: This is derived as the ratio between the difference of means and the variation in the groups.

• Standard Error ($SE$): Calculated differently depending on whether the test is paired or unpaired. Formula: $SE = \frac{SD}{\sqrt{n}}$.

• Relationship of Difference to T-value:
    - As the difference between the two means increases, the T-value increases.
    - A high T-value suggests a substantial difference compared to variation (noise).
    - A high level of variation in the data decreases the T-value, making it more likely the difference is just random fluctuation.

• Degrees of Freedom and P-value: After the T-value is obtained, a $p$-value is derived using degrees of freedom.
    - If p < 0.05, the Null hypothesis is rejected, indicating a likely real difference.
    - If boxes on box plots overlap significantly, a low T-value and high $p$-value are expected (p > 0.05).
    - Small variance/non-overlapping boxes indicate a statistically significant difference.

Mann-Whitney U Test (Non-Parametric Unpaired)

• Definition: The non-parametric equivalent to the unpaired t-test, used only for two test groups.

• Assumptions (or lack thereof):
    - Does NOT assume normal distribution.
    - Does NOT require homogeneity of variance.

• Data Processing: Data is converted into ranks prior to analysis. This makes the test ideal for datasets containing very extreme values or outliers.

• Comparison Metric: It does not compare means; it compares "rank sums."

• Requirement for Variables:
    - Two independent variables with at least ordinal scaled characteristics.
    - Dependent variables must be ordinal, metric, or continuous (anything that can be ranked).

• Hypotheses:
    - $H_0$: There is no difference in terms of central tendency (medians) between the two groups.
    - $H_1$: There is a difference in terms of central tendency.

• Calculations:
    - Ranks are summed for each group and average rank sums are calculated.
    - $U$-values are derived from rank sums ($T_1$ and $T_2$).
    - A $Z$-value is calculated using Standard Deviation and number of samples.
    - $Z$ close to zero = consistent with $H_0$. $Z$ far from zero = significant difference.
    - If p < 0.05, rejection of $H_0$ occurs.

Wilcoxon Signed Ranks Test (Non-Parametric Paired)

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

• Comparison Metric: Based on the ranks of the paired differences rather than means.

• Requirements:
    - Dependent variable must be ordinal or continuous (must be a number to calculate differences).
    - Independent variable must consist of matched pairs.
    - Minimum of six pairs of data required.

• Step-by-Step Analysis:
    1. Calculate the difference for each pair (e.g., Before - After). This includes negative values.
    2. Rank these absolute differences from smallest (Rank 1) to largest.
    3. Re-assign the original positive or negative signs to the ranks.
    4. Sum the Positive Ranks and sum the Negative Ranks.
    5. Obtain the Wilcoxon Rank Statistic ($W$), which is the minimum value of the Positive or Negative rank sums.
    6. Calculate the Expected $W$ value (what is expected if there were no difference).
    7. Obtain a $Z$-value using $W$, expected $W$, and $SD$.
    8. Derive a $p$-value. If p < 0.05, reject $H_0$.

• Case Example (River Flow): If river flow is measured at specific stations and reported as $Z = -2.197$, $p = 0.028$, since p < 0.05, $H_0$ is rejected.

Key Rule for Test Selection

• Strength of Tests: Parametric tests (like t-tests) are "stronger" than non-parametric tests.

• Priority: Always use a parametric test if the data fits the requirements (normal distribution, etc.). Do not default to Mann-Whitney or Wilcoxon if the data is parametric.

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

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

• Function: It checks if frequencies occurring in a sample differ significantly from expected frequencies (what would happen if there was no association).

• Hypotheses:
    - $H_0$: Observed and expected frequencies (from a known random distribution) are NOT different.
    - $H_1$: Observed and expected frequencies are different.

• Common Applications:
    - Relationship between age group and social media preference.
    - Medical treatment success rates relative to patient gender.
    - Association between political party and occupation.
    - Student performance influenced by teaching methods.

• Calculation of Degrees of Freedom ($Df$):
    - $Df = (\text{no. of rows} - 1) \times (\text{no. of columns} - 1)$

• Critical Chi Value Logic:
    - Use the calculated $Df$ and desired $p$-value (typically $0.05$) with a Chi lookup table.
    - If the Critical Chi Value is larger than the Calculated Chi Value, the Null hypothesis is retained.
    - If the Critical Chi Value is smaller than the Calculated Chi Value, the Null hypothesis is rejected.

• Case Study (Fish Lice): Testing if fish lice attach randomly ($H_0$) or have a preference. Reported as $\chi^2 (4, N = 98) = 3.002$, $p = 0.557$. Since p > 0.05, $H_0$ is accepted; the lice attach randomly.

Statistical Measures of Spread

• Standard Deviation ($SD$):
    - The square root of the variance.
    - A measure of dispersion/variability.
    - Shows how far apart values in a dataset are from the mean.
    - Used to identify outliers and summarize descriptive data.

• Standard Error ($SE$):
    - Specifically the Standard Error of the Mean ($SEM$).
    - Measures sampling error; how accurately the sample mean represents the true population mean.
    - Shows the variation likely between different samples of a population and the population itself.
    - It is an estimation of the random sampling process rather than a definite descriptive value.

Statistical Test Selection Matrix (Reference Table)

• Tests of Differences (Two Groups):
    - Paired Continuous Data: Paired t-test.
    - Unpaired Continuous Data: T-test or One-way ANOVA.
    - Paired Discrete/Ordinal Data: Wilcoxon's signed ranks test.
    - Unpaired Discrete/Ordinal Data: Mann-Whitney U test.
    - Paired Categorical Data: Chi-square test.
    - Unpaired Categorical Data: Chi-square test.

• Tests of Relationships:
    - Correlation (Continuous): Pearson product-moment correlation.
    - Correlation (Discrete/Ordinal): Kendall's rank correlation or Spearman's rank correlation.
    - Regression (Continuous): Linear regression, quadratic, or polynomial regression.
    - Regression (Discrete/Ordinal): Logistic regression, Model II regression, or Kendall's robust line fit.

• Complex Analysis:
    - 1+ factor with Covariates: ANCOVA.
    - Many variables/groups to discriminate: CVA, Discriminant function analysis, MANOVA, DCA.
    - Exploring Variables: PCA (Principal Component Analysis).