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:
  • $d<em>i = x</em>i - y_i$ 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: $\bar{d}$
  • Standard deviation of differences: $s_d$
  • Unknown standard deviation ext{} for normal distribution of differences.
  • Null Hypothesis (H0): No mean difference between populations.
  • Form: $H<em>0: \bar{d} = d</em>0$, where typically $d_0 = 0$ for equal means.
  • Test Statistics:
  • $TS = \frac{(\bar{d} - d<em>0)}{(s</em>d/\sqrt{n})}$
  • Compute p-Value:
  • Left-sided test: $H<em>1: H</em>0$; p-value = P(T < TS) = tCDF(-E99, TS, df) .
  • Right-sided test: $H<em>1: > d</em>0$; p-value = P(T > TS) = tCDF(TS, E99, df) .
  • Two-sided test: $H<em>1: H</em>0$; $p-value$ is twice the probability of the appropriate one-sided hypothesis.
  • Decision Making:
  • If p-value < \alpha , then reject $H<em>0$; if $p-value > \alpha$, fail to reject $H</em>0$.
  • Important to note that one cannot "accept" $H_0$; a hypothesis test does not prove conventional wisdom.
  • Interpretation:
  • If $H<em>0$ is rejected, evidence supports the research hypothesis $H</em>1$.
  • If $H<em>0$ is not rejected, insufficient evidence to support $H</em>1$.

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:
  • $P.E. = \frac{ ext{Sum of Differences}}{n} = \frac{Σd_i}{n}$.
  • Margin of Error (MOE):
  • $MOE = t<em>{α/2} \frac{s</em>d}{\sqrt{n}}$.
  • 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 $\bar{d}$ lies in the interval $(\bar{d}- MOE, \bar{d}+ MOE)$.

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: $\bar{d} = 4.5$.
  • 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:
  • $H<em>0: M</em>d = 0$ 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: $df<em>1 = n</em>1 - 1$
  • Denominator df: $df<em>2 = n</em>2 - 1$
• Theorem 8.3: Hypothesis Test of Equality of Variances - Bartlett's Test
  • General Form: $H<em>0: σ</em>1^2 = σ<em>2^2$ vs. $H</em>1: σ<em>1^2 ≠ σ</em>2^2$.
  • Test statistic configuration:
  • $TS = \frac{S<em>1^2}{S</em>2^2}$.
  • 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.