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Biostatistics in Practice Session 2: Summarization of Quantitative Information Peter D. Christenson Biostatistician http://research.LABioMed.org/Biostat Topics for this Session Experimental Units Independence of Measurements Graphs: Summarizing Results Graphs: Aids for Analysis Summary Measures Confidence Intervals Prediction Intervals Most Practical from this Session Geometric Means Confidence Intervals Reference Ranges Justify Analysis Methods from Graphs Experimental Units _____ Independence of Measurements 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. 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. 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 2nd and 3rd results for a sample are less informative than the 1st. A “basic” analysis requiring independence is not valid unless a single number is used for each triplicate, giving 10+10 independent values. Experimental Units in Case Study Experimental Units in Case Study 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. 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 (ttest) to compare the mean score among students given method A to those given B. 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? 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. Graphs: Summarizing Results Common Graphical Summaries Graph Name Y-axis X-axis Histogram Count or % Category Scatterplot Continuous Continuous Dot Plot Continuous Category Box Plot Percentiles Category Line Plot Mean or value Category Kaplan-Meier Probability Time Many of the following examples are from StatisticalPractice.com Data Graphical Displays Histogram Summarized* Scatter plot Raw Data * Raw data version is a stem-leaf plot. We will see one later. Data Graphical Displays Dot Plot Raw Data Box Plot Summarized Data Graphical Displays Line or Profile Plot Summarized - bars can represent various types of ranges Data Graphical Displays Kaplan-Meier Plot 0.75 1.00 Kaplan-Meier survival estimate 0.50 Probability of Surviving 5 years is 0.35 0.00 0.25 This is not necessarily 35% of subjects 0 5 10 Years 15 20 Graphs: Aids for Analysis Graphical Aids for Analysis Most statistical analyses involve modeling. Parametric methods (t-test, ANOVA, Χ2) 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. 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. 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. Histograms: Not OK for Typical Analyses Skewed Multi-Peak 150 Frequency 20 100 50 0 10 0 0 1 2 3 4 5 6 7 8 Intensity Need to transform intensity to another scale, e.g. Log(intensity) 20 70 120 Tumor Volume Need to summarize with percentiles, not mean. Histograms: Not OK for Typical Analyses Truncated Values Outliers Undetectable in 28 samples (<LLOQ) 100 60 Frequency 50 40 30 20 50 10 0 0 0 LLOQ 5 10 Assay Result Need to use percentiles for most analyses. 0 4 8 Expression LogRatio Need to use median, not mean, and percentiles. 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. Example Scatter Plot: OK for Typical Analyses Scatter Plot: Not OK for Typical Analyses Gap and Outlier 150 Funnel-Shaped All Subjects: r = 0.54 (95% CI: 0.27 to 0.73) p = 0.0004 100 EPO > 300: r = -0.04 (95% CI: -0.96 to 0.96) p = 0.96 EPO < 150: r = 0.23 (95% CI: -0.11 to 0.52) p = 0.17 50 0 0 100 200 300 400 EPO Ferber et al, Amer J Obstet Gyn 2004;190:1473-5. Consider analyzing subgroups. Ott, Amer J Obstet Gyn 2005;192:1803-9. Should transform y-value to another scale, e.g. logarithm. Summary Measures Common Summary Measures Mean and SD or SEM Geometric Mean Z-Scores Correlation Survival Probability Risks, Odds, and Hazards 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. - 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. Examples: Mean and SD A B 15 25 Frequency 20 15 10 10 5 5 0 0 35 45 55 65 75 85 95 10 20 OD Time Mean = 60.6 min. 15 SD = 9.6 min. Mean = 15.1 SD = 2.8 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. Examples: Mean and SD Skewed Multi-Peak 150 Frequency 20 100 50 0 10 0 0 1 2 3 4 5 6 7 8 20 SD = 1.1 min. 120 Tumor Volume Intensity Mean = 1.0 min. 70 Mean = 70.3 SD = 22.3 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 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] Geometric Means These are flipped histograms rotated 90º, with box plots. Any log base can be used. ≈ 909.6 ≈ 102.8 ≈ 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 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. Z- Score = (Measure - Mean)/SD 25 20 Frequency Standardizes a measure to have mean=0 and SD=1. Mean = 60.6 min. SD = 9.6 min. 15 10 5 0 35 45 41 65 Time 61 75 85 95 79 Mean = 0 SD = 1 25 20 Frequency Z-scores make different Mean = 60.6 min. measures SD = 9.6 min. comparable. 55 15 10 5 0 35 -245 55 0 65 2 75 85 95 Time Z-Score = (Time-60.6)/9.6 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 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) Confidence Interval: More Details Confidence interval (CI) for the (true, but unknown) mean for the entire population is 95%, N=100: 95%, N= 30: 90%, N=100: 99%, N=100: sample mean ± 1.98(SD/√N) sample mean ± 2.05(SD/√N) sample mean ± 1.66(SD/√N) 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. Confidence Interval: Case Study Table 2 Confidence Interval: Adjusted CI 0.13 -0.12 -0.37 -0.14 ± 1.99(1.04/√73) = -0.14 ± 0.24 → -0.38 to 0.10 Normal Range: -0.14 ± 1.99(1.04) = -0.14 ± 2.07 → -2.21 to 1.93 close to CI for the Antibody Example GM = exp(4.633) = 102.8 low2 = exp(4.633-2*1.09) = 11.6 So, there is 95% assurance that an individual is between 11.6 and 909.6, the PI. 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 So, there is 95% certainty that the population mean is between 92.1 and 114.8, the CI. 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. Correlation Depends on Range of Data A B 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. Correlation and Measurement Precision A B overall 12 10 r=0 for 5 s 6 B 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. Summary Statistics: Survival Probability 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. 1.00 0.75 0.50 0.25 The 2-4 interval has 51/71 surviving; 4-5 has 27/44 surviving. Actually uses finer subdivisions than 0-2, 2-4, 4-5 years, with exact death times. 0.00 Survival Probability Then, the 0-2 yr interval has 80/100 surviving. Kaplan-Meier survival estimate 0 So, 5-yr survival prob is (80/100)(51/71)(27/44) = 0.35. 5 10 Years 15 20 Don’t know vital status of 16 subjects at 5 years. Summary Statistics: Relative Likelihood of an Event Compare groups A and B on mortality. Relative Risk = ProbA[Death] / ProbB[Death] where Prob[Death] ≈ Deaths per 100 Persons Odds Ratio = OddsA[Death] / OddsB[Death] where Odds= Prob[Death] / Prob[Survival] Hazard Ratio ≈ IA[Death] / IB[Death] where I = Incidence = Deaths per 100 PersonDays