Week 6 (biostats) - T-test

Paired t-test: Key ideas

  • Use when you have paired/matched observations (e.g., before-after on the same subjects).
  • Compare mean of differences, not the two raw groups.
  • Let D_i = observation in group 1 − observation in group 2 for each pair.
  • Null hypothesis: H0:\ \mud = 0
  • Alternative: Ha:\ \mud \neq 0 (two-sided) or one-sided as planned.
  • Assumptions: differences are independent and approximately normally distributed.

Key formulas

  • Difference data:
    • Mean difference: \bar{d} = \frac{1}{n}\sum{i=1}^n di
    • Standard deviation of differences: SDd = \sqrt{\frac{1}{n-1}\sum{i=1}^n (d_i - \bar{d})^2}
  • Standard error of the mean difference: SE = \frac{SD_d}{\sqrt{n}}
  • t-statistic: t = \frac{\bar{d}}{SE}
  • Degrees of freedom: df = n - 1
  • 95% CI for the mean difference: \bar{d} \pm t_{df,0.025} \cdot SE
  • Decision rule: if p-value < 0.05 (two-sided), reject H0; otherwise fail to reject.

Example 2: Sleep study (Paired t-test)

  • Design: 10 patients; each measured on drug and on placebo (paired).
  • Differences: di = \text{Drug}i - \text{Placebo}_i
  • Summary results (example):
    • Number of pairs: n = 10
    • Mean difference: \bar{d} = 1.08 hours
    • SD of differences: SD_d = 2.31 hours
    • Standard error: SE = \frac{SD_d}{\sqrt{n}} = \frac{2.31}{\sqrt{10}} \approx 0.73
    • t-statistic: t = \frac{\bar{d}}{SE} = \frac{1.08}{0.73} \approx 1.48
    • Degrees of freedom: df = n - 1 = 9
    • Two-sided p-value: p \in (0.10, 0.20)
    • 95% CI for the mean difference: \bar{d} \pm t_{9,0.025} \cdot SE = 1.08 \pm 2.26 \cdot 0.73 \approx [-0.57, \ 2.73]
  • GraphPad result example: p = 0.1731\,,\; \text{Mean difference} = \bar{d} = 1.08\,,\; \text{95% CI} = [-0.57, 2.73]\,

Interpretation

  • If 95% CI for the mean difference includes 0, the difference is not statistically significant at the 0.05 level.
  • In this example: CI includes 0, and p > 0.05 → do not reject H0; the drug does not have a significant effect on sleep duration.

Steps to report (brief workflow)

  • State objective and study design (paired observations).
  • State hypotheses: H0: \mud = 0\quad vs\quad Ha: \mud \neq 0.
  • Check assumptions (normality of differences, independence).
  • Compute: \bar{d},\ SD_d,\ SE,\ t,\ df,\ p\text{-value},\ 95\%\ CI.
  • Interpret: p-value and CI; conclude whether there is a significant difference.
  • Report results succinctly with means, difference of means, CI, and p-value.

Quick reference: decision when to use which t-test

  • Two groups, continuous data, independent observations: independent-samples t-test.
  • Two measurements on the same subjects (paired): paired t-test.
  • Check equal variances if using two-sample t-test; otherwise use unequal-variances version.