Statistical Methods and Analysis

T-Test Assumptions

• Levene’s Test: A formal test for checking equal variance T-test assumptions.

  • Key Assumptions:

  • 1. Each group represents a random sample from its population.

  • 2. Data within and between groups are fully independent.

  • 3. The variable $y$ is normally distributed within the population.

    • Check normality using a QQ norm plot; data should fall along the line of normality.

  • 4. The variance of $y$ in the two populations is equal.

    • Use a formal statistical test to check if the distribution of your data is significantly different from a normal distribution, although it isn't strictly necessary.

    • If conducting this test, the biological question is how well the sample data fit a normal distribution.

    • Null Hypothesis ($H_0$): The data is from a normally distributed population.

    • You want to fail to reject the null, so the p-value should be greater than 0.05 (𝛼).

• Robustness of T-Tests:

  • T-tests are very robust to their assumptions; therefore, one can proceed without stringent adherence to assumptions.

• Data Transformations:

  • Data transformations can shift data from being left or right skewed to normal.

  • Use transformed data for statistical analyses (can back-transform later, particularly in regression).

  • Include the transformation used in the summary statement.

• Nonparametric Tests:

  • Do not make claims about population parameters; they have fewer assumptions.

  • Less formal statements of statistical hypotheses, typically rely on ranks or signs of data values.

  • Pros:

  • More widely applicable.

  • Not sensitive to extreme data points; many sample distributions are acceptable.

  • Cons:

  • Less statistical power compared to their parametric counterparts (when the parametric test is appropriate).

  • Greater probability of a Type II error (false negative).

• Mann-Whitney U Test:

  • Nonparametric test version of a two-sample t-test.

  • 6-Step Process:

  1. Null and alternative hypotheses are less specific; they don’t use population parameters and are more vague.

  2. If two populations have similarly shaped distributions, the test can inform about differences between the medians of populations.

  3. Combine data from the two groups and sort, assigning ranks to each value from smallest to largest.

  4. Sum ranks for each group, denoted as $R<em>1$ and $R</em>2$; test statistics are $U<em>1$ and $U</em>2$. Pick the larger of $U<em>1$ and $U</em>2$.

  5. Paired and one-sample t-tests can both transition to either the Wilcoxon signed-rank test or the sign test.

ANOVA (Analysis of Variance)

• Purpose: Used to compare more than two sample means (not variances).

• Signal/Noise Ratio:

  • Calculated as $MS<em>{group} / MS</em>{error} = F$ statistic.

• Degrees of Freedom (df): A property of a calculation.

• Definitions:

  • Factor: An explanatory or independent/experimental variable; can be categorical or numerical but can still be categorized.

  • Response: The dependent variable, the observed or measured outcome.

• ANOVA Assumptions:

  • Analogous to a t-test but with $k$ samples.

  • Null Hypothesis ($H<em>0$): $𝝁</em>1 = 𝝁<em>2 = … = 𝝁</em>k$.

  • Alternative Hypothesis ($H_A$): At least one pair of means is different, which cannot be simply expressed in symbols.

  • The alternative hypothesis is always nondirectional; rejecting $H_0$ does not specify which means differ from others.

• General Test Statistic:

  • $( ext{Observed data} - ext{Expected if null is true}) / ext{Measure of variation}$.

• Variance Partitioning:

  • Variance between groups / variance within groups.

  • Variation within groups is termed as Sum of Squares Error (SS_{error}).

  • Variation between groups is termed as Sum of Squares Between Groups (SS_{group}), attributable to the effect of group.

  • The total variance $SS<em>{total} = SS</em>{error} + SS_{group}$.

• Computing Mean Squares (MS):

  • $MS<em>{group} = SS</em>{group} / (k-1)$

  • $MS<em>{error} = SS</em>{error} / (N - K)$

  • If $F = 0$, the group means are identical; as $F$ approaches infinity, the larger the difference in group means.

• Decision to Reject $H_0$:

  • Compare $F$ to the $F$-distribution, a family of continuous probability distributions used for variance ratios.

  • Usually, perform another test post-ANOVA to determine the direction of differences.

• Coefficient of Determination ($R^2$):

  • $R^2 = SS<em>{group} / SS</em>{total}$. This represents the proportion of variation attributable to $x$.

• Degrees of Freedom:

  • For variation in groups: $df = k - 1$.

  • For error: $df = N - k$.

  • For total: $df = N - 1$.

• General Conditions:

  • ANOVA is a parametric test with assumptions about the underlying population distribution.

  • Requires random samples and independent observations drawn from $k$ normally distributed populations.

  • Verify conditions graphically for skew, kurtosis, outliers, etc.

  • Easier to use residuals for histograms and QQ plots as they allow combining data from multiple groups.

Normality and Variance Tests

• Shapiro-Wilk Test: Checks for normality utilizing the residuals.

• Levene’s Test: Must be executed on raw data to check if variances are equal.

• Residuals:

  • Deviations of data points from their own group means.

  • Kruskal-Wallis Test: Nonparametric form of ANOVA.

• Post-Hoc Tests:

  • Global F-Test: Determines if there is a “main effect” of the factor when $p < 𝛼$.

  • Planned Comparisons vs. Unplanned Comparisons:

  • Planned Comparisons: Defined before data is collected.

  • Unplanned Comparisons: Compare everything to everything else; must adjust for increased probability of a false positive.

  • Tukey-Kramer Test (Tukey HSD): Tests all possible pairwise comparisons of group means.

  • Bonferroni Test: Manages any subset of all possible comparisons.

• Equal Variance: Check by examining boxplots and residual plots (residuals on y-axis and average of group on the x-axis).

• Assumptions:

  • Drawing from normally distributed populations is akin to assuming residuals are also normally distributed.

  • Assuming equal variance in data mirrors assuming equal variance in residuals.

• Post-Hoc Evaluation:

  • ANOVA does not reveal which group differs; conduct further tests if main effects are found.

  • $H<em>0: 𝝁</em>i - 𝝁<em>j = 0$; $H</em>A: 𝝁<em>i - 𝝁</em>j ≠ 0$ for all pairs of means.

  • Conducted under significant findings in ANOVA.

• Statistical Power:

  • Some tests are less powerful than ANOVA despite finding a significant difference; dependability is lower in post-hoc tests.

  • Lower power tests can lead to higher probabilities of Type I errors when the chances of detecting true differences are low.

• Bonferroni Correction: Utilized to adjust 𝛼 when multiple comparisons are made.

  • Implementation:

  • Determine the number of meaningful correlations, denoting as $x$.

  • Calculate adjusted 𝛼: $𝛼^* = 𝛼 / x$.

  • Use $𝛼^*$ as the cutoff for significance.

Correlation and Regression

• Correlation Coefficient ($r$):

  • Used to associate two variables.

• Correlation Tests:

  • Estimating the population correlation coefficient 𝜌 from sample correlation coefficient $r$.

• Assumptions of Correlation Tests:

  • More powerful than other multiple comparison tests.

  • Number of comparisons is $k - 1$.

  • Test statistics ($Q$) based on average rank differences between two groups.

• Scatterplots: Graphical summaries plotted for $n$ values to check for possible correlations, nonlinear trends, and extreme values.

• Calculation:

  • Sum of Products / Sum of Squares: Calculating a correlation does not test a hypothesis.

  • Coefficient $r$: Ranges from -1 to 1 (unitless).

  • Strength and Direction:

  • If $r > 0$, association is positive; if $r < 0$, association is negative.

  • Points clustering tightly around a line indicate stronger relations between $X$ and $Y$.

• Extreme Points: Can significantly affect $r$; correlation coefficient is not a statistical test (𝜎 is an unknown population parameter).

• Hypotheses for Correlation:

  • $H<em>0: 𝜌 = 0$; $H</em>A: 𝜌 ≠ 0$.

  • Test statistic $t = r / SE_r$.

• Distribution under Null Hypothesis: If null is true, $t$ follows a t-distribution with degrees of freedom $df = n - 2$.

• Assumption Check: Use scatterplots; a roughly elliptical distribution is appreciated, while nonlinear or extreme outliers are discouraged.

Spearman’s Rank Correlation and Regression

• Spearman’s Rank Correlation: Measures strength and direction of association between ranks of two variables.

  • Monotonic Relationships: Suggests consistency in responses of variables.

• Regression: Emphasizes prediction of $x$ based on $y$ and vice versa; finds the line best describing relationships.

  • Ordinary Least Squares (OLS) Line: Minimizes sum of squared residuals.

  • Regression line representation: $Y = a + bX$.

• Regression Assumptions:

  • The relationship is monotonic between $X$ and $Y$.

  • Hypotheses: $H<em>0: x$ and $y$ have no correlation; $H</em>A: x$ and $y$ are correlated.

• Test Statistic: Spearman’s rho.

• Coefficient of Determination ($R^2$): Proportion of variance in one variable attributed to its relationship with another.

  • Example: $R^2 = 0.5$ indicates that 50% of variance in $X$ is explained by its linear relationship with $Y$.

• Adjusting 𝛼 for Correlations: Similar to the Bonferroni adjustment process previously mentioned.

• Regression for Prediction:

  • Useful for fitting predictive models.

  • Correspondingly, used to describe functional relationships between variables.

• Interpretation in Multiple Linear Regression:

  • Relates to dependencies of predictors while holding others constant.

  • $\beta<em>1$, $\beta</em>2$ depict changes in $Y$ for unit increases in $X<em>1$, $X</em>2$ respectively.

• Model Complexity and Interpretability:

  • As models grow more complex, power of prediction increases, but interpretation becomes challenging.

Categorical Tests and Goodness-of-Fit Tests

• Goodness-of-Fit Test: Evaluates if observed counts correspond to expected proportions.

  • Compares expected and observed frequencies.

• Test Statistic:

  • $X^2 = rac{ ext{sum} ( ext{observed} - ext{expected})^2}{ ext{expected}}$; accumulates for $k$ categories.

• Hypotheses:

  • $H<em>0$: Frequencies match expected ratios; $H</em>A$: Frequencies deviate from expected ratio.

• Distribution of Test Statistic:

  • Follows a Chi-square ($X^2$) distribution.

  • Assumptions for Goodness-of-Fit Test include random sampling, with all expected frequencies > 1 and 80% > 5.

• Contingency Test: Assesses independence of two categorical variables through counts.

  • Plotted as a mosaic chart; examines table totals of rows and columns.

  • Hypothesized independence expressed as $H<em>0$ and association as $H</em>A$.

• Expected Frequencies: Must meet minimum criteria for valid test execution.

Important Topics for Exam

• Role of distributions in hypothesis testing; understanding conceptual mechanisms behind tests.

• Statistical power, variation sources, and effect size significance; without requiring calculations.

• Differences between parametric and nonparametric tests; knowledge of specific statistical tests.

• Review of statistical test assumptions and their validations, alongside responding to violations.

• Multiple predictors and interactions: conceptual examination with straightforward application queries.

Sampling and Experimental Design

• Random Sampling: Every population unit holds an equal probability of being included.

• Sampling Error: Variability between different samples.

• Haphazard Sampling: Units collected with an intent of randomness.

• Blind Design: Either the subjects, investigators, or both lack knowledge regarding treatment assignments.

• Replication: Independent application of treatments to units.

• Confounding Variables: Outside influences on the response variable.

• Blocking: Grouping units into homogeneous subsets to control variations affecting response.

• Common Statistics:

  • Central tendency (mean, median, mode).

  • Variability measures (range, IQR, standard deviation).

Probability Theory and Distributions

• Mutually Exclusive: Definition ensuring $P(A ∩ B) = 0$ and $P(A ∪ B) = 1$.

• Independent Events: Probability independence across trials – one event’s outcome doesn’t impact the other.

• Sampling Distribution: Characterization of distributions of sample statistics derived from random samples.

• Confidence Intervals: Limits $L<em>1$ and $L</em>2$ for determining confidence intervals at various confidence levels (with emphasis placed on known criteria and estimating sample sizes for precision).