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Unit A: Biostatistics

5. Test of Significance
The gold standard of biostatistics for competitive exams. Determining if differences are biological phenomena or random sampling errors.
When we see a difference between two groups in an experiment, how do we know it's a real biological phenomenon and not just random sampling error?
A. Logic of Hypothesis Testing

Null Hypothesis (H₀): States there is no real difference or association. (e.g., "The new drug has no effect on parasite load compared to a placebo"). Any difference is due to chance.

Alternative Hypothesis (H₁): States there is a real difference. (e.g., "The drug reduces parasite load").

Decision Errors

  • Type I Error (α): Rejecting H₀ when it is actually true. (False Positive). p < 0.05 is the standard probability threshold.
  • Type II Error (β): Failing to reject H₀ when it is actually false. (False Negative).
  • Power of the Test (1 - β): Probability of correctly detecting an effect that actually exists.
B. Test Categorization

Parametric Tests

Assume data follows a normal distribution (Gaussian), variances are equal (homoscedasticity), and data is continuous.

e.g., t-test, F-test, ANOVA

Non-Parametric Tests

"Distribution-free" tests used when data is skewed, sample sizes are tiny, or data is nominal/ordinal.

e.g., Chi-square, Mann-Whitney U, Kruskal-Wallis

C. The Four Heavyweight Tests

1. Student’s t-Test
Developed by William Gosset for small samples (n < 30).

Used when population standard deviation is unknown.

One-Sample t-test:

Compares a sample mean to a known population mean.

Independent/Two-Sample t-test:

Compares means of two distinct groups (e.g., wild vs captive frogs).

Paired t-test:

Compares two means from the same individuals measured twice (e.g., weight before/after diet).

2. F-Test (Variance Ratio Test)
Comparing variability between populations.

F = s₁² / s₂²

Note: s₁² > s₂², meaning the larger variance is always the numerator (F ≥ 1).

Crucial Application:

Checks the assumption of equal variances before a t-test and forms the backbone of ANOVA.

3. Analysis of Variance (ANOVA)
Comparing 3 or more groups simultaneously.

Why not multiple t-tests?

Running 6 t-tests for 4 groups makes the cumulative Type I error skyrocket to ~26.5%. ANOVA controls this by testing all means at once.

Mechanism: ANOVA splits variation into:

  • Between Groups: Measures the effect of treatments.
  • Within Groups (Error): Measures random individual differences.
F = MS_between / MS_within
4. Chi-Square Test (χ²)
Non-parametric test for categorical/count data.
χ² = Σ [ (O - E)² / E ]

1. Goodness of Fit: Tests if observed ratios match theoretical Mendelian ratios (e.g., 3:1).

2. Test of Independence: Checks if two categorical variables are linked (e.g., disease vs sex).

Degrees of Freedom (df):

• Goodness of Fit: df = k - 1

• Contingency Table: df = (r - 1)(c - 1)

Higher-Level Examination Check
Select the correct analytical tool for the scenario.

Scenario:

A genetics researcher crosses two heterozygous fruit flies (AaBb × AaBb) and counts the phenotypes of 400 offspring to determine whether the two genes are assorting independently according to standard Mendelian laws (9:3:3:1).

Question 1:

Which specific test of significance should the researcher use to analyze this data?

Question 2:

How many degrees of freedom (df) will be used when looking up the critical value in the statistical tables?

Hint: Categorical count data vs. Ratio.