Biostats Notes on Two Sample Tests of Hypotheses

8 Inferences Based on Two Sample Tests of Hypotheses

8.1 Paired Samples versus Independent Samples

  • Definition 8.1: Two samples are classified based on the relationship of their respective data points:
    • Independent Samples:
    • Definition: One sample is not related to the other.
    • Characteristics: Summary measures from each group are calculated and then compared. Sample sizes do not need to be equal.
    • Example: Men vs. women if they are not related.
    • Dependent Samples (Paired Samples):
    • Definition: Each member of one sample corresponds to a member of the other sample.
    • Characteristics: Each data point in one sample is matched with a unique data point in the other sample.
    • Characteristics: The two samples must have the same number of observations (n₁ = n₂).
    • Example Scenarios:
      • Follow-up measurements on the same person (before/after).
      • Measurements on two populations that are very similar (e.g., husband and wife).

Matched Pairs Analysis

  • Notation: If sample one data points are labeled as x1, x2, …, xn and sample two as y1, y2, …, yn, the analysis considers the differences:
    • di = xi - y_i for i = 1, 2, …, n.

Example 8.1: Identifying Dependent or Independent Samples

  • A: 300 registered voters responded to a questionnaire before and after watching a video.
    • Answer: Dependent Samples
  • B: 30 dogs trained with different methods (reward vs. reward-punishment).
    • Answer: Independent Samples
  • C: Systolic blood pressures of 30 adult females vs. 30 adult males.
    • Answer: Independent Samples
  • D: Weights of 65 college students before and after their freshman year.
    • Answer: Dependent Samples

8.2 Analysis of Paired Samples

  • Theorem 8.1: Hypothesis Test of Paired Samples
    • Assumptions:
    • Data comes from an approximately normal distribution.
    • Random sample of differences from the population of all possible differences.
    • Notation:
    • Mean of differences: ar{d}
    • Standard deviation of differences: s_d
    • Unknown standard deviation ext{} for normal distribution of differences.
    • Null Hypothesis (H0): No mean difference between populations.
    • Form: H0: ar{d} = d0 , where typically d_0 = 0 for equal means.
    • Test Statistics:
    • TS = rac{(ar{d} - d0)}{(sd/\sqrt{n})}
    • Compute p-Value:
    • Left-sided test: H1: H0 ; p-value = P(T < TS) = tCDF(-E99, TS, df) .
    • Right-sided test: H1: > d0 ; p-value = P(T > TS) = tCDF(TS, E99, df) .
    • Two-sided test: H1: H0 ; p-value is twice the probability of the appropriate one-sided hypothesis.
    • Decision Making:
    • If p-value < \alpha , then reject H0 ; if p-value > \alpha , fail to reject H0 .
    • Important to note that one cannot "accept" H_0 ; a hypothesis test does not prove conventional wisdom.
    • Interpretation:
    • If H0 is rejected, evidence supports the research hypothesis H1 .
    • If H0 is not rejected, insufficient evidence to support H1 .

Theorem 8.2: Confidence Interval for Paired Samples

  • Assumptions/Considerations:
    • Same as Hypothesis testing:
    • Data is from approximately normal distribution.
    • Normality is less critical if the sample size n 30 .
    • Point Estimate:
    • P.E. = rac{ ext{Sum of Differences}}{n} = rac{Σd_i}{n} .
    • Margin of Error (MOE):
    • MOE = t{α/2} rac{sd}{\sqrt{n}} .
    • Confidence Interval:
    • CI: ar{d} MOE ; thus, the interval is: ar{d} 0 MOE .
    • Interpretation:
    • We are 100*(1-α)% confident that the mean difference ar{d} lies in the interval (ar{d}- MOE, ar{d}+ MOE) .

Example 8.2: Testing Effectiveness of Educational Video

  • Research on children diagnosed with asthma using a test before and after watching a video:
    • Test Scores Before: 67, 62, 54, 93, 60, 89, 41, 67, 62, 57
    • After viewing the video, differences tracked.
    • Point Estimate: ar{d} = 4.5 .
    • Margin of Error Computation:
    • Letting n = 10 , compute using sample values.
    • Construct the confidence interval: e.g., for 95%, yields results for pre-and post-assessment scores between intervals (0.833, 8.167).

Hypothesis Testing After Viewing Video

  • Restating Hypotheses:
    • H0: Md = 0 vs. H1: Md > 0 (mean score after video exceeds prior):
  • Follow another step-wise testing method yielding its accepted conclusions.

8.3 Testing the Variances of Two Populations - Bartlett F-Test

  • Example 8.6: Variances across two groups assessed with Bartlett's test.
  • Definition of F-distribution:
    • A family of distributions that is right skewed. Total area under the curve equals 1. Random variable values are non-negative.
    • Degrees of Freedom (df):
    • Numerator df: df1 = n1 - 1
    • Denominator df: df2 = n2 - 1
  • Theorem 8.3: Hypothesis Test of Equality of Variances - Bartlett's Test
    • General Form: H0: σ1^2 = σ2^2 vs. H1: σ1^2 ≠ σ2^2 .
    • Test statistic configuration:
    • TS = rac{S1^2}{S2^2} .
    • Determine p-value based on the directed test:
    • e.g., for a right-sided test - based on larger sample variance observed.
    • Decision framework for results yields conclusions that affect supplies for variances across experimental groups.

Interpretation of Results:

  • If rejected, strong evidence that variances differ. If not rejected, proceed assuming equal variances until further information indicates otherwise.