Basic Statistical Concepts

1. Basic Statistical Concepts Donald E. Mercante, Ph.D. Biostatistics School of Public Health L S U - H S C

2. Sample Population Parameters Statistics

3. Two Broad Areas of Statistics Descriptive Statistics - Numerical descriptors - Graphical devices - Tabular displays Inferential Statistics - Hypothesis testing - Confidence intervals - Model building/selection

4. Descriptive Statistics When computed for a population of values, numerical descriptors are called Parameters When computed for a sample of values, numerical descriptors are called Statistics

5. Descriptive Statistics Two important aspects of any population Magnitude of the responses Spread among population members

6. Descriptive Statistics Measures of Central Tendency (magnitude) Mean - most widely used - uses all the data - best statistical properties - susceptible to outliers Median - does not use all the data - resistant to outliers

7. Descriptive Statistics Measures of Spread (variability) range - simple to compute - does not use all the data variance - uses all the data - best statistical properties - measures average distance of values from a reference point

8. Properties of Statistics • Unbiasedness - On target • Minimum variance - Most reliable • If an estimator possesses both properties then it is a MINVUE = MINimum Variance Unbiased Estimator • Sample Mean and Variance are UMVUE = Uniformly MINimum Variance Unbiased Estimator

9. Inferential Statistics - Hypothesis Testing - Interval Estimation

10. Hypothesis Testing Specifying hypotheses: H0: “null” or no effect hypothesis H1: research or alternative hypothesis Note:Only H0 (null) is tested.

11. Errors in Hypothesis Testing

12. Hypothesis Testing In parametric tests, actual parameter values are specified for H0and H1. H0: µ < 120 H1: µ > 120

13. Hypothesis Testing Another example of explicitly specifying H0and H1. H0:  = 0 H1:  0

14. Hypothesis Testing General framework: Specify null & alternative hypotheses Specify test statistic State rejection rule (RR) Compute test statistic and compare to RR State conclusion

15. Common Statistical Tests

16. Common Statistical Tests (cont.)

17. Advanced Topics

18. P-Values p = Probability of obtaining a result at least this extreme given the null is true. P-values are probabilities 0 < p < 1 Computed from distribution of the test statistic

19. Epidemiological Concepts Ratea proportion, specifically a fraction, where The numerator, c, is included in the denominator: Useful for comparing groups of unequal size Example:

20. Epidemiological Concepts Measures of Morbidity: Incidence Rate: # new cases occurring during a given time interval divided by population at risk at the beginning of that period. Prevalence Rate: total # cases at a given time divided by population at risk at that time.

21. Epidemiological Concepts Most people think in terms of probability (p) of an event as a natural way to quantify the chance an event will occur => 0<=p<=1 0 = event will certainly not occur 1 = event certain to occur But there are other ways of quantifying the chances that an event will occur….

22. Epidemiological Concepts Odds and Odds Ratio: For example, O = 4 means we expect 4 times as many occurrences as non-occurrences of an event. In gambling, we say, the odds are 5 to 2. This corresponds to the single number 5/2 = Odds.

23. Epidemiological Concepts The relationship between probability & odds

24. Epidemiological Concepts Odds<1 correspond To probabilities<0.5 0<Odds<

25. Example 1: Odds Ratio Death sentence by race of defendant in 147 trials

26. Example 2: Odds Ratio Odds of death sentence = 50/97 = 0.52 For Blacks: O = 28/45 = 0.62 For Nonblacks: O = 22/52 = 0.42 Ratio of Black Odds to Nonblack Odds = 1.47 This is called the Odds Ratio

27. Logistic Regression Odds ratios are directly related to the parameters of the logit (logistic regression) model. Logistic Regression is a statistical method that models binary (e.g., Yes/No; T/F; Success/Failure) data as a function of one or more explanatory variables. We would like a model that predicts the probability of a success, ie, P(Y=1) using a linear function.

28. Logistic Regression Problem: Probabilities are bounded by 0 and 1. But linear functions are inherently unbounded. Solution: Transform P(Y=1) = p to an odds. If we take the log of the odds the lower bound is also removed. Setting this result equal to a linear function of the explanatory variables gives us the logit model.

29. Logistic Regression Logit or Logistic Regression Model Where pi is the probability that yi = 1. The expression on the left is called the logit or log odds.

30. Logistic Regression Probability of success: Odds Ratio for Each Explanatory Variable:

31. Screening Tests

32. Screening Tests How do we evaluate the usefulness of such a test? Diagnostics: sensitivity specificity False positive rate False negative rate predictive value positive predictive value negative

33. Screening Tests

34. Screening Tests

35. Screening Tests

36. Interval Estimation Statistics such as the sample mean, median, variance, etc., arecalled point estimates -vary from sample to sample -do not incorporate precision

37. Interval Estimation Estimates Take as an example the sample mean: X ——————>  (popn mean) Or the sample variance: S2 ——————> 2 (popn variance)

38. Interval Estimation Recall Example 1, a one-sample t-test on the population mean. The test statistic was This can be rewritten to yield:

39. Interval Estimation Which can be rearranged to give a (1-)100% Confidence Interval for : Form: Estimate ± Multiple of Std Error of the Est.

40. Interval Estimation Example 1: Standing SBP Mean = 140.8, s.d. = 9.5, N = 12 95% CI for : 140.8 ± 2.201 (9.5/sqrt(12)) 140.8 ± 6.036 (134.8, 146.8)