Student t-Test in Biostatistics

Student t-Test in Biostatistics

1. Definition, history & rationale

• The Student t-test is a parametric significance test for comparing one or two means of continuous (numerical) data, telling us whether to accept or reject a null hypothesis about means (Medhi Ch.11, pp.129–30).
• It is the small-sample modification of the large-sample z-test: for large samples a difference ≥ 1·96 times its standard error (SE) occurs by chance ≤ 1 in 20 (P ≤ 0·05) and ≥ 2·576 SE ≤ 1 in 100 (P ≤ 0·01); with small samples the uncertainty in estimating the SE has been ignored, so a modified procedure — the t-test — is needed (Swinscow & Campbell 10e Ch.7, p.62).
• Foundations laid by W. S. Gosset, who published under the pseudonym "Student" in 1908 while working as a statistician at the Guinness brewery in Dublin — hence "Student's t test" (Swinscow & Campbell 10e Ch.7, p.62; Medhi Ch.11, p.129).
• The t-test is preferable when the number of observations is < 60, and certainly when ≤ 30; above this the t and z (Normal) results converge (Swinscow & Campbell 10e Ch.7, p.62).
  • As sample size → ∞ the t-distribution multiples shrink to the Normal (z) values in Table A; the smaller the sample, the larger the multiple of SE required for a given probability (Swinscow & Campbell 10e Ch.7, pp.63–4).
• In the Indian/PG framing: the t-test is used when sample size is < 30 (n < 30); n > 30 is analysed by the z-test, and the ratio of two group variances by the F-test (variance₁/variance₂). The t-distribution curve resembles the Normal curve (Medhi Ch.11, p.129).
• The t-test (a.k.a. Student's t-test) is the method of choice when making a single comparison between two groups of normally-distributed, parametric data (Shargel 8e Ch.3, p.17 of chapter / "Statistical inference techniques").
• Four classic applications of the t-distribution (Swinscow & Campbell 10e Ch.7, pp.62–3):
  • (1) calculating a confidence interval (CI) for a sample mean;
  • (2) testing how significantly a sample mean differs from a postulated population mean (one-sample t);
  • (3) testing whether two samples could have come from the same population (two-sample / unpaired t);
  • (4) testing the significance of the difference between the means of two sets of paired observations (paired t).