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# BIOSTATISTICS

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### BIOSTATISTICS

1. 1. BIOSTATISTICS II Capita Selecta, 2009 Part I Analysis of Variance Part II Generalized Linear Models Part III Multiple regression and model building Part IV Sample size calculations Part V Measuring agreement Part VI Systematic review and meta-analysis Søren Lundbye Christensen Johannes J. Struijk
2. 2. Part III Multiple regression & model building Literature: any serious book on statistics Martin Bland, ”Introduction to medical statistics” Oxford Univ. Press, 2000,chapter 17.
3. 3. Multiple regression & model building Basic model: Best (minimum mean square error) estimator: Solution for b: We immediately see a problem: if some of the independent variables are linearly related then the inverse of the covariance matrix doesn’t exist. exbxbxbbY kk  22110 eXbY  bXY ˆˆ    xyxx xyxx SSb SbS bXXYX 1 TT ˆ ˆ ˆ    
4. 4. Multiple regression & model building Maximum voluntary contraction (MVC) of the quadriceps muscle as function of age and height of 41 alcoholics.
5. 5. Multiple regression & model building
6. 6. Multiple regression & model building Model: MVC = b0 + b1xHeight + b2xAge Multiple correlation coefficient R2 = SSReg / SST (proportion of variability accounted for)
7. 7. Multiple regression & model building
8. 8. Multiple regression & model building Interaction: MVC = b0 + b1xHeight + b2xAge + b3xHeightxAge Note: adjusted Ra 2 = 1- (1-R2)(n-1)/(n-p-1)
9. 9. Multiple regression & model building Polynomial regression: MVC = b0 + b1xHeight + b2xHeight2
10. 10. Multiple regression & model building
11. 11. Multiple regression & model building Dichotomous variables Examples Sex: man / woman Liver disease: yes / no Assign 0’s and 1’s to those variables and use the standard techniques
12. 12. Multiple regression & model building Variance inflation factor: VIF = 1/(1-Ri 2) VIF>10 is real problem (Ri 2 >90%: 90 of influence of xi is explained by other x’s) Leverage: Cook’s distance (influential points)
13. 13. Multiple regression & model building Many variables? Step-up (forward) Step-down (backward) Forward-backward Best subset F1,n-q=(SSE(q)-SSE(q+1)) / (SSE(q+1)/(n-q))
14. 14. Part IV Sample Size Calculation Literature: Machin et al., (1997), ”Sample size tables for clinical studies”, Blackwell, Oxford Altman (1982), ”How large a sample?” In: Statistics in Practice (Eds. Gore & Altman), Blackwell Publishing Ltd., London Lehr (1992), ”Sixteen s squared over d squared: a relation for crude sample size estimates”, Stat. in Med., 11:1099-1102 Martin Bland, ”Introduction to medical statistics” Oxford Univ. Press, 2000,chapter 18.
15. 15. Sample Size Calculation Importance of sample size Common error to have a sample that is too small: low power, Type II error: no rejection of the null hyptoheses.
16. 16. Sample Size Calculation A little taxonomy of sample size calculations Power – chance of rejecting the null-hypothesis if it is false Significance level – cutt-off level of the p-value below which we reject the null-hypothesis Variability - e.g., standard deviation for numerical data Smallest effect of interest – magnitude of the effect that we want to be able to detect as being statistically significant
17. 17. Sample Size Calculations Sample size calculations are important for: - Estimation: effect on confidence intervals - Examples: estimation of population mean estimation of correlation coefficient - Tests: effect on confidence level and power - Example: 1-sample test Literature: Martin Bland, ”Introduction to medical statistics” Oxford Univ. Press, 2000,chapter 18.
18. 18. Sample Size Calculations Methods of Sample Size Calculations - Do the math - Special tables - Nomograms - Simulation - Computer software
19. 19. Sample Size Calculations Estimation of population mean μ. Assume sample size = n. Estimated mean: Estimated variance: Estimated standard error: Confidence interval:   n i in XM 1 1      n i in MXS 1 2 1 12 nSMSE /)(   )(,)( 2/2/ MSEzMMSEzM  
20. 20. Sample Size Calculations Estimation of population mean μ. Width of the confidence interval: For a desired width, Wd, of the CI we can thus calculate n: Thus, n depends on • confidence level, • desired width of the confidence interval, • variance, • distribution of the data. n S zWidth 2/2  2 2/2         dW Sz n
21. 21. Sample Size Calculations Estimation of correlation coefficient ρ. Assume sample size = n Estimated correlation = r (has a very nasty distribution) Fisher’s z transformation: has a normal distribution with Mean: SE:          r r z 1 1 ln 2 1                       1 1 ln 2 1 121 1 ln 2 1 n z  31)(S  nzE
22. 22. Sample Size Calculations Estimation of correlation coefficient ρ. Confidence interval: Example: expected r = 0.5; desired 95% CI = [0.4, 0.6] z0.4=0.424; z0.5=0.549; z0.6=0.693 z0.6-z0.5=0.144; z0.5-z0.4=0.126     31,31 2/2/   nzznzz   246126.03196.1  nn
23. 23. Sample Size Calculations Paired-sample test. Test statistic: )(dse d z d  zα -zβ+μd/se(d) μd/se(d) 0 β H0
24. 24. Sample Size Calculations Altman’s nomogram Altman (1982), ”How large a sample?”, in Statistics in practice, eds. Gore & Altman, BMA London. Example: difference of capillary density (per mm2) in the feet of ulcerated patients (better foot minus worse foot): Min. diff. to be detected 4 mm-2 SD(difference) = 6.1 Standardized difference = 2 x (4/6.1)= 1.31 Required Power = 0.80 Significance level = 0.05
25. 25. Sample Size Calculations Using the formula: zα = 1.96 (α = 0.05) zβ = 0.86 (Power = 80%) Min. μd = 4.0 VAR(d) = 6.12 =37.21 n = 18 zα -zβ+μd/se(d) μd/se(d) 0 β H0
26. 26. Part V Measuring agreement Literature: Bland, Altman, (1999), ”Measuring agreement in method comparison studies”, Stat Meth Med Res, 8:135-160 Landis, Koch, (1977), ”The measurement of observer agreement for categorical data”, Biometrics, 33:159-174
27. 27. Measuring agreement Methods used in the literature: Data Method Ordinal Cohen’s kappa Spearman’s rank-order correlation coefficient Kendall’s tau Kendall’s coefficient of concordance Interval/ratio Pearson’s correlation coefficient Intraclass correlation coefficient Tukey’s mean-difference plot (Bland-Altman plot)
28. 28. Measuring agreement
29. 29. Measuring agreement Cohen’s kappa (Ordinal data) Doctor 1 Doctor 2 Schizo- Bipolar Other Row sum Schizo- 31 4 2 37 Bipolar 6 29 8 43 Other 10 7 3 20 Column sum 10 7 13 100 agreement rate = 0.63 κ = 0.41 σκ= 0.077
30. 30. Measuring agreement More than two judges (Ordinal data) For example: Kendall’s coefficient of concordance (related to Friedman’s two-way ANOVA on ranks) MGP 2009 - Song District 1 2 3 4 5 Totals NJutl 1 2 3 5 4 MJutl 1 2 4 3 5 SJutl 1 2 3 5 4 Sjæll 1 2 3 4 5 Cophn 1 2 4 3 5 Sum 5 10 17 20 23 T=75 Sumsq 25 100 289 400 529 U=1343
31. 31. Measuring agreement Kendall’s coefficient of concordance, W m = number of raters n = number of classes W = 218 / 250 = 0.872
32. 32. Measuring agreement NUMERICAL VARIABLES Correlation coefficient Intraclass correlation coefficient Bland-Altman plot (Tukey plot) Manual Automated
33. 33. Measuring agreement Pearson’s product-moment correlation coefficient Ignores bias and gain! Only for two raters.
34. 34. Measuring agreement Intraclass correlation coefficient (also for multiple raters) = Between pairs variance / Total variance. k = number of subjects (or measured objects) n = number of raters (or methods) This takes into account the systematic difference!
35. 35. Measuring agreement Bland-Altman plot Tukey mean-difference plot
36. 36. Measuring agreement Bias Proportional errorHeterogeneous variance
37. 37. Part VI Systematic review and meta-analysis Literature: Chalmers, Altman, (eds), (1995), ”Systematic reviews”, Br. Med. J. Publ. Group, London Higgins et al., (2003), ”Measuring inconsistency in meta- analysis”, Br. Med. J., 237:557-560 Cochrane Handbook: at http://www.cochrane.org
38. 38. Systematic review and meta-analysis Systematic review = Formalized and stringent process of combining the information from all (published and unpublished) of the same health condition.
39. 39. Systematic review and meta-analysis Why systematic reviews? Reduction of information Generalization to a wider population Consistency by comparing different studies Reliability of recommendations Power and precision increases
40. 40. Systematic review and meta-analysis Meta-analysis = Systematic review with focus on numerical results To combine results f rom individual studies to estimate an overall / average effect of interest (example: the relative risk of getting cancer because of using mobile phones)
41. 41. Systematic review and meta-analysis Meta-analysis From a statistical angle, meta-analysis is an application of multifactorial methods: Multiple studies of the same thing. Combine the results of the studies: - Treatment / risk factor is one independent factor - Study is a second independent factor
42. 42. Systematic review and meta-analysis Meta-analysis Clear definition of the question / effect of interest. Example: - Does lowering serum cholesterol reduce risk of dying from coronary artery disease? - Does a diet to lower serum cholesterol reduce risk of dying from coronary artery disease? Study where attempt to lower cholesterol failed should be included?
43. 43. Systematic review and meta-analysis Meta-analysis – PUBLICATION BIAS Simple literature search is not good enough! - Bias towards positive results (sometimes to negative results) - More positive results in English literature? - Unpublished studies are important.
44. 44. Systematic review and meta-analysis Meta-analysis – Example from M. Bland, ch. 17
45. 45. Systematic review and meta-analysis Meta-analysis – Example from M. Bland, ch. 17
46. 46. Systematic review and meta-analysis Meta-analysis – Example from M. Bland, ch. 17 ln(o) = b0+b1T+b2S1+ ... +b5S4+b6S5+b7TS1+ ... +b11TS5
47. 47. Systematic review Example (Mailis-Gagnon et al., (2004), ”Spinal cord stimulation for chronic pain”, The Cochrane Library, issue 3) 1692 papers : only 2 admitted to the review Result: further study needed(!) http://thecochranelibrary.com