HOME › TEACHING › EXPERIMENTAL DESIGN › LECTURES › BIOL 683 - CONTINUOUS DATA & STATISTICAL COMPARISONS (PRE‑CLASS READING)

Biol 683 - Continuous Data & Statistical Comparisons (Pre‑class Reading).

This handout mirrors our slide deck and adds context, assumptions, and code snippets for your pre‑class reading. We’ll use class time for practice and discussion. 【7†source】

1) Continuous data, sampling, and the CLT

Many biological variables (e.g., size, gene expression) are right‑skewed; size is often approximately exponential. If you repeatedly sample a population and compute the mean from each sample, the distribution of those means is approximately normal by the Central Limit Theorem (CLT). 【7†source】

Key idea. The normality of the sampling distribution of the mean underpins t‑tests and confidence intervals-even when raw data are non‑normal.

2) What is a p‑value?

A p‑value is the probability of obtaining data as extreme as (or more extreme than) what you observed, assuming the null hypothesis is true. It is not the probability the null is true, nor your probability of being wrong.

3) Comparing means (t‑tests)

3.1 One‑sample t‑test

Assesses whether a sample mean differs from a hypothesized value \(\mu\). Assumes the variable is normally distributed in the population. 【7†source】

Test statistic: \[ t \,=\, \frac{\bar Y - \mu}{\mathrm{SEM}} \]

For proportion data, an arcsine square‑root transform is standard; remember to interpret on the original scale. 【7†source】

3.2 Two‑sample t‑test (independent groups)

Compares means of two groups (e.g., control vs treatment). Welch’s version tolerates unequal variances. 【7†source】

Welch test statistic: \[ t \,=\, \frac{\bar Y_a - \bar Y_b}{\sqrt{\tfrac{s_a^2}{n_a} + \tfrac{s_b^2}{n_b}}} \]

3.3 Paired‑sample t‑test

For before–after or matched designs; analyze differences \(d_i = y_{ai} - y_{bi}\). Assumes the differences are normally distributed. 【7†source】

\[ t \,=\, \frac{\bar d}{\mathrm{SE}d}, \qquad \bar d \,=\, \frac{1}{n}\sum{i=1}^{n} d_i \]

3.4 Confidence intervals and overlap

If two 95% CIs do not overlap, the groups differ at \(\alpha=0.05\). If they overlap, the test might still be significant-run the formal test. 【7†source】

4) Comparing variances (Levene’s test)

Variance can be biologically meaningful (e.g., variance in reproductive success). Levene’s test compares absolute deviations from group centers and is robust when normality is doubtful; it assumes roughly symmetric distributions. 【7†source】

5) Assumptions & diagnostics

  • t‑tests: normality (variable or differences), independence; Welch’s handles unequal variances. 【7†source】
  • Levene’s: symmetry (not strict normality). 【7†source】

Visual checks often beat formal tests: - Histograms and QQ‑plots help spot skew, heavy tails, and odd shapes. - Shapiro–Wilk: low power with small \(n\), oversensitive with large \(n\). Context matters. 【7†source】

6) Transformations (use for assumptions, not p‑hacking)

  • Log: right‑skewed positive data
  • Arcsine(sqrt): proportions in \([0,1]\)
  • Square‑root: counts (often add 0.5 first)

If you use a non‑standard transform, justify it and consider confirming results with a rank‑based method (e.g., Mann–Whitney). 【7†source】

7) Non‑parametric & permutation tests

Mann–Whitney / Wilcoxon rank‑sum

Rank‑based alternative to the two‑sample t‑test. Assumes symmetry and same‑shaped distributions. A significant result implies distributional differences (mean, variance, or skew). 【7†source】

Sign test

Minimal‑assumption alternative to one‑sample or paired t‑tests; uses only the sign of differences. Low power-reserve for special cases. 【7†source】

Permutation tests

Shuffle labels to build a null distribution for a statistic (e.g., mean difference). Highly flexible with minimal assumptions. 【7†source】

8) Correlation and causation

Pearson’s r measures linear association: \( r = \tfrac{\mathrm{cov}(X,Y)}{s_X s_Y} \). Assumes approximate bivariate normality and is sensitive to outliers and clustering. Spearman’s rho is rank‑based and robust to non‑normality and monotone nonlinearity. Time trends can induce spurious correlations-interrogate mechanisms and confounders. 【7†source】

9) Quick selection guide

Question Recommended test Core assumptions
Is a mean ≠ constant? One‑sample t Normality
Do two groups differ in mean? Welch two‑sample t Normality; unequal variances allowed
Before–after (same units)? Paired t Normality of differences
Do groups differ in variance (spread)? Levene’s Symmetry
Distributional difference, weak assumptions Mann–Whitney Symmetry; same shape
Minimal paired/one‑sample alternative Sign test None (very low power)
Flexible, custom statistic Permutation Exchangeability / valid randomization
Linear association Pearson’s r Bivariate normality; linearity
Monotone association, robust Spearman’s \(\rho\) Monotonic relationship

10) Minimal R snippets (non‑evaluated by default)

These are short examples for in‑class practice. By default they do not execute when knitting so the document compiles cleanly. Replace placeholders and run interactively.

One‑sample t

y <- c( )          # your sample numeric vector
mu0 <- 0           # hypothesized mean
t.test(y, mu = mu0)

Welch two‑sample t

# df with numeric y and factor group with 2 levels
t.test(y ~ group, data = df)

Paired t

t.test(y_after, y_before, paired = TRUE)

Levene’s test

# install.packages("car")
library(car)
leveneTest(y ~ group, data = df)

Mann–Whitney (Wilcoxon rank‑sum)

wilcox.test(y ~ group, data = df, exact = FALSE)

Sign test (paired)

# install.packages("BSDA")
library(BSDA)
SIGN.test(y_after - y_before, md = 0)

Permutation test (mean difference, two groups)

set.seed(1)
obs <- with(df, mean(y[group=="A"]) - mean(y[group=="B"]))
B <- 5000
null <- replicate(B, {
  g <- sample(df$group)              # permute labels
  with(df, mean(y[g=="A"]) - mean(y[g=="B"]))
})
pval <- mean(abs(null) >= abs(obs))
c(observed_diff = obs, p_value = pval)

11) Demo‑safe examples (self‑contained; these DO run)

Toggle eval = TRUE below if you need a compiled document with outputs.

set.seed(42)
n <- 25
y <- rlnorm(n, meanlog = 0, sdlog = 0.6)   # skewed positive data
y_log <- log(y)
group <- factor(rep(c("A","B"), each = n))
df <- data.frame(
  y = c(rnorm(n, 10, 2), rnorm(n, 11, 2.5)),
  group = group
)

One‑sample t on log‑scale (example)

mu0 <- 0
t.test(y_log, mu = mu0)
## 
##  One Sample t-test
## 
## data:  y_log
## t = 0.71778, df = 24, p-value = 0.4798
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  -0.2110228  0.4360662
## sample estimates:
## mean of x 
## 0.1125217

Welch two‑sample t (example)

t.test(y ~ group, data = df)
## 
##  Welch Two Sample t-test
## 
## data:  y by group
## t = -3.0439, df = 45.313, p-value = 0.00388
## alternative hypothesis: true difference in means between group A and group B is not equal to 0
## 95 percent confidence interval:
##  -3.1147888 -0.6344437
## sample estimates:
## mean in group A mean in group B 
##        9.482241       11.356857

Levene’s test (example)

if (!requireNamespace("car", quietly = TRUE)) {
  install.packages("car", repos = "https://cloud.r-project.org")
}
library(car)
leveneTest(y ~ group, data = df)

Mann–Whitney (example)

wilcox.test(y ~ group, data = df, exact = FALSE)
## 
##  Wilcoxon rank sum test with continuity correction
## 
## data:  y by group
## W = 145, p-value = 0.001194
## alternative hypothesis: true location shift is not equal to 0

12) Source

This handout follows and elaborates the course slide deck (continuous variables, t‑tests, Levene’s, transformations, non‑parametrics, correlation). 【7†source】

Question copied. Paste it into the NotebookLM tab.