1. Biostatistics in Practice Session 2: Summarization of Quantitative Information Peter D. Christenson Biostatistician http://research.LABioMed.org/Biostat

2. Topics for this Session Experimental Units Independence of Measurements Graphs: Summarizing Results Graphs: Aids for Analysis Summary Measures Confidence Intervals Prediction Intervals

3. Most Practical from this Session Geometric Means Confidence Intervals Reference Ranges Justify Analysis Methods from Graphs

4. Experimental Units _____ Independence of Measurements

5. Units and Independence Experiments may be designed such that each measurement does not give additional independent information. Many basic statistical methods require that measurements are “independent” for the analysis to be valid. Other methods can incorporate the lack of independence.

6. Example 1: Units and Independence Ten mice receive treatment A and a blood sample is obtained from each one. The same is done for 10 mice receiving treatment B. A protein concentration is measured in each of the 20 samples and an appropriate summary (average?, min?, %>10 nmol/ml?) is compared between groups A and B. The experimental unit is a mouse. Each of the 20 numbers are independent. A “basic” analysis requiring independence is valid.

7. Example 2: Units and Independence Ten mice receive treatment A, each is bled, and each blood sample is divided into 3 aliquots. The same is done for 10 mice receiving treatment B. A protein concentration is measured in each of the 60 aliquots. The experimental unit is a mouse. The 60 numbers are not independent. The 2 nd and 3 rd results for a sample are less informative than the 1 st. A “basic” analysis requiring independence is not valid unless a single number is used for each triplicate, giving 10+10 independent values.

8. Experimental Units in Case Study

9. A unit is a single child. Results from one child's three diets are not independent. The three results are probably clustered around a set-point for that child. The analysis must incorporate this possible correlated clustering. If the software is just given the 3x140 outcomes without distinguishing the individual children, the analysis would be wrong.

10. Modified Case Study Suppose an educational study used teaching method A in some schools and method B in others. The outcome is a test score later. The experimental unit is a school. Outcomes within a school are probably not independent. It would be wrong to use the method we will discuss in the next session (t- test) to compare the mean score among students given method A to those given B.

11. Another Example You apply treatment A to one pregnant mouse and measure a hormone in its offspring. Same for B. Suppose the results are: A Responses: 100, 98, 102, 99, 101 B Responses: 10, 8, 12, 9, 11 Can we conclude responses are greater under treatment A than under B?

12. Another Example You apply treatment A to one pregnant mouse and measure a hormone in its offspring. Same for B. Suppose the results are: A Responses: 100, 98, 102, 99, 101 B Responses: 10, 8, 12, 9, 11 No. The one mouse given A might have responded the same if given B. Same for the one mouse given B. Five offspring provide little independent information over 1 offspring. Each treatment was essentially only tested once.

13. Graphs: Summarizing Results

14. Common Graphical Summaries Graph NameY-axisX-axis HistogramCount or %Category ScatterplotContinuous Continuous Dot PlotContinuous Category Box PlotPercentiles Category Line PlotMean or value Category Kaplan-MeierProbabilityTime Many of the following examples are from StatisticalPractice.com

15. Data Graphical Displays HistogramScatter plot Raw Data Summarized* * Raw data version is a stem-leaf plot. We will see one later.

16. Data Graphical Displays Dot PlotBox Plot Raw Data Summarized

17. Data Graphical Displays Line or Profile Plot Summarized - bars can represent various types of ranges

18. Data Graphical Displays Kaplan-Meier Plot This is not necessarily 35% of subjects Probability of Surviving 5 years is 0.35

19. Graphs: Aids for Analysis

20. Graphical Aids for Analysis Most statistical analyses involve modeling. Parametric methods (t-test, ANOVA, Χ 2 ) have stronger requirements than non-parametric methods (rank -based). Every method is based on data satisfying certain requirements. Many of these requirements can be assessed with some useful common graphics.

21. Look at the Data for Analysis Requirements What do we look for? In Histograms (one variable): Ideal: Symmetric, bell-shaped. Potential Problems: Skewness. Multiple peaks. Many values at, say, 0, and bell-shaped otherwise. Outliers.

22. Example Histogram: OK for Typical* Analyses Symmetric. One peak. Roughly bell-shaped. No outliers. *Typical: mean, SD, confidence intervals, to be discussed in later slides.

23. Histograms: Not OK for Typical Analyses Skewed Need to transform intensity to another scale, e.g. Log(intensity) Multi-Peak Need to summarize with percentiles, not mean.

24. Histograms: Not OK for Typical Analyses Truncated Values Need to use percentiles for most analyses. Outliers Need to use median, not mean, and percentiles. LLOQ Undetectable in 28 samples (<LLOQ)

25. Look at the Data for Analysis Requirements What do we look for? In Scatter Plots (two variables): Ideal: Football-shaped; ellipse. Potential Problems: Outliers. Funnel-shaped. Gap with no values for one or both variables.

26. Example Scatter Plot: OK for Typical Analyses

27. Scatter Plot: Not OK for Typical Analyses Gap and Outlier Consider analyzing subgroups. Funnel-Shaped Should transform y-value to another scale, e.g. logarithm. Ott, Amer J Obstet Gyn 2005;192:1803-9. Ferber et al, Amer J Obstet Gyn 2004;190:1473-5.

28. Summary Measures

29. Common Summary Measures Mean and SD or SEM Geometric Mean Z-Scores Correlation Survival Probability Risks, Odds, and Hazards

30. Summary Statistics: One Variable Data Reduction to a few summary measures. Basic: Need Typical Value and Variability of Values Typical Values (“Location”): Mean for symmetric data. Median for skewed data. Geometric mean for some skewed data - details in later slides.

31. Summary Statistics: Variation in Values Standard Deviation, SD =~ 1.25 *(Average |deviation| of values from their mean). Standard, convention, non-intuitive values. SD of what? E.g., SD of individuals, or of group means. Fundamental, critical measure for most statistical methods.

32. Examples: Mean and SD Mean = 60.6 min. Note that the entire range of data in A is about 6SDs wide, and is the source of the “Six Sigma” process used in quality control and business. SD = 9.6 min.Mean = 15.1SD = 2.8 AB

33. Examples: Mean and SD SkewedMulti-Peak Mean = 1.0 min. SD = 1.1 min. Mean = 70.3 SD = 22.3

34. Summary Statistics: Rule of Thumb For bell-shaped distributions of data (“normally” distributed): ~ 68% of values are within mean ±1 SD ~ 95% of values are within mean ±2 SD “(Normal) Reference Range” ~ 99.7% of values are within mean ±3 SD

35. Summary Statistics: Geometric means Commonly used for skewed data. 1.Take logs of individual values. 2.Find, say, mean ±2 SD → mean and (low, up) of the logged values. 3.Find antilogs of mean, low, up. Call them GM, low2, up2 (back on original scale). 4.GM is the “geometric mean”. The interval (low2,up2) is skewed about GM (corresponds to graph). [See next slide]

36. Geometric Means These are flipped histograms rotated 90º, with box plots. Any log base can be used. ≈ 909.6 ≈ 11.6 GM = exp(4.633) = 102.8 low2 = exp(4.633-2*1.09) = 11.6 upp2 = exp(4.633+2*1.09) = 909.6 ≈ 102.8

37. Confidence Intervals Reference ranges - or Prediction Intervals -are for individuals. Contains values for 95% of individuals. _____________________________________ Confidence intervals (CI) are for a summary measure (parameter) for an entire population. Contains the (still unknown) summary measure for “everyone” with 95% certainty.

38. Z- Score = (Measure - Mean)/SD Standardizes a measure to have mean=0 and SD=1. Z-scores make different measures comparable. Mean = 60.6 min. Mean = 60.6 min. SD = 9.6 min. SD = 9.6 min. Z-Score = (Time-60.6)/9.6 -2 0 2 41 61 79 Mean = 0 SD = 1

39. Outcome Measure in Case Study GHA = Global Hyperactivity Aggregate For each child at each time: Z1 = Z-Score for ADHD from Teachers Z2 = Z-Score for WWP from Parents Z3 = Z-Score for ADHD in Classroom Z4 = Z-Score for Conner on Computer All have higher values ↔ more hyperactive. Z’s make each measure scaled similarly. GHA= Mean of Z1, Z2, Z3, Z4

40. Confidence Interval for Population Mean 95% Reference range - or Prediction Interval - or “Normal Range”, is sample mean ± 2(SD) _____________________________________ 95% Confidence interval (CI) for the (true, but unknown) mean for the entire population is sample mean ± 2(SD/√N) SD/√N is called “Std Error of the Mean” (SEM)

41. Confidence Interval: More Details Confidence interval (CI) for the (true, but unknown) mean for the entire population is 95%, N=100:sample mean ± 1.98(SD/√N) 95%, N= 30:sample mean ± 2.05(SD/√N) 90%, N=100:sample mean ± 1.66(SD/√N) 99%, N=100:sample mean ± 2.63(SD/√N) If N is small (N<30?), need normally, bell-shaped, data distribution. Otherwise, skewness is OK. This is not true for the PI, where percentiles are needed.

42. Confidence Interval: Case Study Confidence Interval: -0.14 ± 1.99(1.04/√73) = -0.14 ± 0.24 → -0.38 to 0.10 Table 2 Normal Range: -0.14 ± 1.99(1.04) = -0.14 ± 2.07 → -2.21 to 1.93 0.13 -0.12 -0.37 Adjusted CI close to

43. CI for the Antibody Example So, there is 95% assurance that an individual is between 11.6 and 909.6, the PI. So, there is 95% certainty that the population mean is between 92.1 and 114.8, the CI. GM = exp(4.633) = 102.8 low2 = exp(4.633-2*1.09) = 11.6 upp2 = exp(4.633+2*1.09) = 909.6 GM = exp(4.633) = 102.8 low2 = exp(4.633-2*1.09 /√394) = 92.1 upp2 = exp(4.633+2*1.09 /√394) = 114.8

44. Summary Statistics: Two Variables (Correlation) Always look at scatterplot. Correlation, r, ranges from -1 (perfect inverse relation) to +1 (perfect direct). Zero=no relation. Specific to the ranges of the two variables. Typically, cannot extrapolate to populations with other ranges. Measures association, not causation. We will examine details in Session 5.

45. Correlation Depends on Range of Data Graph B contains only the points from graph A that are in the ellipse. Correlation is reduced in graph B. Thus: correlation between two quantities may be quite different in different study populations. BA

46. Correlation and Measurement Precision A lack of correlation for the subpopulation with 5<x<6 may be due to inability to measure x and y well. Lack of evidence of association is not evidence of lack of association. B A r=0 for s B overall 5 6 12 10

47. Actually uses finer subdivisions than 0-2, 2-4, 4-5 years, with exact death times. Example: 100 subjects start a study. Nine subjects drop out at 2 years and 7 drop out at 4 yrs and 20, 20, and 17 died in the intervals 0-2, 2-4, 4-5 yrs. Then, the 0-2 yr interval has 80/100 surviving. The 2-4 interval has 51/71 surviving; 4-5 has 27/44 surviving. So, 5-yr survival prob is (80/100)(51/71)(27/44) = 0.35. Summary Statistics: Survival Probability Don’t know vital status of 16 subjects at 5 years.

48. Summary Statistics: Relative Likelihood of an Event Compare groups A and B on mortality. Relative Risk = Prob A [Death] / Prob B [Death] where Prob[Death] ≈ Deaths per 100 Persons Odds Ratio = Odds A [Death] / Odds B [Death] where Odds= Prob[Death] / Prob[Survival] Hazard Ratio ≈ I A [Death] / I B [Death] where I = Incidence = Deaths per 100 PersonDays