Biostats Midterm II Review

1. Normal Distribution

• a. Empirical Rule

  • X is a normal random variable with mean ($\mu$) and SD ($\sigma$), then:

    • 68% » 1 SD of the mean ($\mu\pm\sigma$ )

    • 95% » 2 SD of the mean ($\mu\pm2\sigma$ )

    • 99.7% » 3 SD of the mean ($\mu\pm3\sigma$)

c. Calculating probabilities and percentiles

• Probabilities: Z-Score to determine the area from a CDF table, then adjust based on left-sided, right-sided (1-CDF value), or between (Area_A - Area_B)

• Percentiles: $Area=\frac{\%}{100}$ , find Z-Score, convert Z-Score to Raw Score: $X=\mu+Z\cdot\sigma$

d. Central Limit Theorem for the sampling distribution of the sample mean

• Concepts:

  • Law of large numbers (sample size n is large enough)

  • Repeated sampling

  • Law of Averages (n increases » closer to truth)

• Summary

  • Universal Rule: $n\ge30$ if population is not normal

    • Under 30 is okay if assuming normality

2. Confidence Intervals for means, proportions, and variances

• Confidence level:$1-\alpha$

  • Lower Critical Value: $\frac{\alpha}{2}$

  • Upper Critical Value:$1-\frac{\alpha}{2}$

• Common Lvls of Confidence

  • 90% » 1.645

  • 95% » 1.960

  • 99% » 2.576

• 2 Means

  • $\mu_1-\mu_2=0$ » $\mu_1=\mu_2$

    • 0 in interval » two means may be the same

  • Pop. Variances: $\sigma_1^2=\sigma_2^2$ or $\sigma_1^2\ne\sigma_2^2$
    
    

  • 1 Variance

    • X² (Chi-Square): $Z^2=\left(\frac{x-\mu}{\sigma}\right)^2$

      • df = $\frac{\left(n-1\right)s^2}{\sigma^2}$ ~ $x^2n-1$

    • 
      

  • 2 Variances (F Test)

    • $\sigma^2x=\sigma^2y$ if CI equals 1
      

3. Hypothesis Testing Concepts

• a. Types I and II error

  • Type I: $\alpha$ » wrongly reject $H_0$

  • Type II: $\beta$ » wrongly accept $H_0$

• b. Null and alternative hypotheses

  • $H_0$ = chance alone, assumed to be true

  • $H_{A}$ = contradiction to $H_0$

    • left-sided: $\mu_1<\mu_2$

    • right-sided: $\mu_1>\mu_2$

    • two-sided: $\mu_1\ne\mu_2$

• d. P-values

  • probability of obtaining a result at least as extreme as the observed result from the data if the $H_0$ was true

• e. When to reject or not reject the null hypothesis

  • reject: when p < $\alpha$

  • fail to reject: when p > $\alpha$
    
    

4. Significance Tests:

• a. T-test: one or two population mean(s), and paired data (pre vs post)

  • One Population:

    • assumptions: normal, continuous, randomly sampled

    • $H_0:\mu=\mu_0$

  • Two Populations

    • assumptions: independent, randomly sampled, normal

    • variances:

      • homogeneity » independent (pooled)

      • heterogeneity » Welch’s

    • $H_0:\mu_{x}=\mu_{y}$

  • Paired Data

    • assumptions: not independent

    • $H_0:\mu_{d}=0$

      • < 0 » increasing effect

      • > 0 » decreasing effect

      • $\ne$ » no effect

  • 
    

• b. Nonparametric tests (Wilcoxon or Mann-Whitney): one or two population median(s), and paired data (pre vs post)

  • Wilcoxon

    • signed-rank

    • one sample and paired test

    • assumptions: random and continuous

    • $H_0:M=M_0$

      
      

  • Mann-Whitney

    • rank-sum

    • two sample

    • d = x1 - x2

    • assumptions: independent, random and continuous, differ by means (if they differ at all)

    • $H_0:M_{x}=M_{y}$

• c. Shapiro Wilk test for normality (useful for small n)

  • $H_0$ : do not deviate from normality

    • p $>\alpha$ » conclude data do not deviate from normality

  • 
    

• d. F Test for comparing two variances to determine if they are significantly different.

  • $H_0$ : $\frac{\sigma_1^2}{\sigma_2^2}=1$

• e. Z-Test for one and two population proportions

  • one prop

    • $H_0:p=p_0$

  • two prop

    • $H_0:p_1=p_2$

• f. Chi-squared test for K proportions

  • $H_0:p_1=p_2\ldots$

    • more likely to accept when O ~ E

    • likely to reject when $X_{obs}^2$ » infinity

  • assumptions: E $\ge5$

    • never want E < 1

    • $1\le E\le5$ is okay
      
      

5. Statistical Power

• a. Definition, interpretation, and purpose of power

  • Definition: detecting something when there really is something ($1-\beta$ )

  • 80% minimum

  • Interpretation: probability that a statistical test will correctly detect an effect if there is one to be found

• b. Power analysis for a t-test

  • power: 80%

  • sample size: n » what we want

  • effect size (cohen’s d): estimated standardized difference

    • cohen’s d: $\frac{\mu_{A}-\mu_0}{\sigma}$ » $\frac{\delta}{\sigma}$

    • <0.2: negligble

    • 0.2: small

    • 0.5: medium

    • >0.8: large

  • alpha: 5%