1. 1 Point and Interval Estimates Examples with z and t distributions Single sample; two samples Result: Sums (and differences) of normally distributed RV are normally distributed. Determining the variance of the difference between means for two independent samples Pooled estimates of the variance (when two independent estimates are available) Degrees of freedom for the variance of the difference between the means of two independent samples (equal/not equal variances) Estimating the variance for use with proportions, and CI with proportions: Bayesian Credibility Intervals –Prior Distribution –Joint Distribution of prior and data –Posterior Distribution

2. 2 Introduction to Biostatistics (PUBHLTH 540) Examples of Point and Interval Estimates + Credibility Intervals Examples from Seasons Study Assumptions: Subjects are SRS from population. Assume different groups are independent SRS from different stratum (ie. gender) Details: Use t-distribution for interval estimates when sample sizes are small (unless estimate is of a proportion) –requires an assumption that the underlying random variable is normally distributed When response is binary (yes/no), we estimate the population mean by the sample mean (equal to the sample proportion ), and the sample variance by

3. 3 Examples: Point and Interval Estimate of Wt Examples from Seasons Study (see ejs09b540p34.sas). What is a 95% Confidence Interval for Weight? (see: http://dostat.stat.sc.edu/prototype/calculators/index.php3 )?dist=T to get t-percentiles)http://dostat.stat.sc.edu/prototype/calculators/index.php3 Weight n291Lower 95Upper 95 Mean77.6275.679.7 Std17.79 df290 statist1.968 The mean weight is estimated as 77.6 kg, with a 95% CI of (75.6, 79.7) Use applets to get t value

4. 4 Examples: Point and Interval Estimate of Wt Answer: Same as before--The mean weight is estimated as 77.6 kg, with a 95% CI of (75.6, 79.7) Suppose we assume the Seasons study subjects were a SRS from people in the US. What is a point and interval estimate of weight for the US population?

5. 5 Examples: Point and Interval Estimate of Wt- separately for men and women Examples from Seasons Study ejs09b540p34.sas (see: http://dostat.stat.sc.edu/prototype/calculators/index.php3?dist=T to get t-percentiles) For men, the mean weight is estimated as 85.9 kg (95% CI (83.3,88.5) while for women, mean wt is 69.7 kg (95% CI (67.2, 72.3) Table 3. Description of weight by gender Male(0) Analysis Variable : wt Wt (kg) (formerly cc5a) N Mean Std Dev Variance Std Error ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ 142 85.90 15.82 250.32 1.33 ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Female(1) Analysis Variable : wt Wt (kg) (formerly cc5a) N Mean Std Dev Variance Std Error ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ 149 69.73 15.92 253.42 1.30 ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Source: ejs09b540p34.sas 10/20/2009 by ejs Use Applet-- men Use Applet-- women

6. 6 Examples: Point and Interval Estimate of Wt- adjusting for gender in US population Suppose we assume the Seasons study male subjects were a SRS from males in the US, and similarly, and female subjects were an independent SRS from females in the US. In 2000, there were 138.05 million males, and 143.37 million females in the U.S.. Using the Seasons study estimates, what is a point and interval estimate of weight for the US population? Males Females

7. 7 Example: Linear Combinations of Random variables Estimate:

8. 8 Example: Linear Combinations of Random variables What are the DF for the t-dist? If variances are equal, use df=n1+n2-2, and replace individual variance estimates by a pooled variance. If variances are not equal, see p270- 271 in text for df approximation.

9. 9 Note: Common estimate of a variance- Pooled Estimate If we assume the population variance in weight is equal for males and females, we can estimate a pooled (common) variance (see p267 in text): More generally: for Wt:

10. 10 Example: Linear Combinations of Random variables Assuming not equal: from p270- 271 in text,

11. 11 Example: Linear Combinations of Random variables Wt MeanconstantsEstimateVar(Mean) Male0.4985.91.76 Female0.5169.71.70 Wt Average 77.60.865 df289 Approx df288.5 95% CI (75.8, 79.5) t-0.9751.968 Wt is estimated as 77.6 kg with a 95% CI of (75.8,79.5)

12. 12 Examples: Proportion of Subjects who are obese (BMI>30) (see p327 text) Estimate the proportion of subjects obese, and a 95% CI Create 0/1 variable 1=obese 0=normal wt Use Z-dist for CI (since np>5) Variance estimate: Obese No221 Yes70Lower 95 Total2910.1914 P_hat0.2405498280.2897 var (phat)0.000627786Upper95 See:ejs09b540p34.sas

13. 13 Examples: Proportion of Subjects who are obese (BMI>30) (see p327 text) Single random variable (0/1) is called a Bernoulli random variable. Variance is estimated using maximum likelihood estimator (biased): Usual estimate of the variance (used in other settings) is: Normal Approximation is used commonly when nP>5 and n(1-P)>5 (NOT t-dist) Example: Sample finds 4 of 10 subjects obese Note: nP is not large enough here for the normal approximation to be “good”. 95% CI

14. 14 Examples: Credibility Intervals Bayesian Approach Recall that we could estimate the mean using Maximum Likelihood Example: We select a srs with replacement of n=10 and observe x=4. What is p? Solution 1: Use the sample mean: Solution 2: Use value of the parameter p that maximizes the likelihood, given the data. Likelihood: The likelihood is a function of p. We can think of a set of possible values, i.e. 0, 0.1, 0.2, …, 0.8, 0.9, 1 of p. The maximum likelihood estimate is the value of p where the likelihood is largest.

15. 15 Binomial Distribution Likelihood We select a srs with replacement of n=10 and observe x=4. What is p? Parameter p L(p)Parameter p L(p) 0.050.0010.550.1596 0.100.01120.600.1115 0.150.04010.650.0689 0.200.08810.700.0368 0.250.14600.750.0162 0.300.20010.800.0055 0.350.23770.850.0012 0.400.25080.900.0001 0.450.23840.950.0000 0.500.20511.000.0000

16. 16 Binomial Distribution Maximum Likelihood Likelihood: pL(p)p 0.050.0010.400.2508 0.100.01120.450.2384 0.150.04010.500.2051 0.200.08810.550.1596 0.250.14600.600.1115 0.300.20010.650.0689 0.350.2377etc 0.05 0.1 0.2 0.30.40.5 Maximum Likelihood 0.60.70.9

17. 17 Examples: Credibility Intervals Bayesian Approach-Prior Suppose we assume each parameter is equally likely. This is called a uniform prior distribution Parameter p Prior Prob. Parameter p Prior Prob. 0.05 0.550.05 0.100.050.600.05 0.150.050.650.05 0.200.050.700.05 0.250.050.750.05 0.300.050.800.05 0.350.050.850.05 0.400.050.900.05 0.450.050.950.05 0.500.051.000.05 Prior distribution

18. 18 Examples: Credibility Intervals Bayesian Approach-Data|p We select a srs with replacement of n=10 and observe x=4. The likelihood is the Pr(Data|p) Parameter p L(p|x)Parameter p L(p|x) 0.050.0010.550.1596 0.100.01120.600.1115 0.150.04010.650.0689 0.200.08810.700.0368 0.250.14600.750.0162 0.300.20010.800.0055 0.350.23770.850.0012 0.400.25080.900.0001 0.450.23840.950.0000 0.500.20511.000.0000

19. 19 Examples: Credibility Intervals Bayesian Approach-Posterior Combining the Likelihood and the prior, we have the joint probabilities We sum these probabilities over all possible possible values of p, and divide by this sum to form posterior probabilities:

20. 20 Examples: Credibility Intervals Bayesian Approach-Posterior Credibility Intervals are like Confidence Intervals for parameters in the Posterior Distribution (Uniform Prior)

21. 21 Examples: Credibility Intervals Bayesian Approach-Posterior Credibility Intervals are like Confidence Intervals for parameters in the Posterior Distribution (Symmetric Prior)

22. 22 Examples: Credibility Intervals Bayesian Approach-Posterior Credibility Intervals are like Confidence Intervals for parameters in the Posterior Distribution (Tiered Prior)

23. 23 Examples: Credibility Intervals Bayesian Approach-Conclusions Credibility Intervals (for the same data) depend on the Prior Distribution PriorCredibility IntervalConfidence Uniform(0.15, 0.70)0.96 Symmetric(0.15, 0.35)0.91 Tiered(0.15, 0.55)0.89 Frequentist 95% Confidence Intervals based on Normal Approximation (0.10, 0.70) Credibility Interval- Intuitive Interpretation- prob parameter is in interval is confidence Frequentist Confidence Interval- awkward interpretation- includes parameter for 95% of samples, if repeated