Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Foundations of statistics wikipedia , lookup
Mean field particle methods wikipedia , lookup
Taylor's law wikipedia , lookup
Bootstrapping (statistics) wikipedia , lookup
Confidence interval wikipedia , lookup
History of statistics wikipedia , lookup
Categorical variable wikipedia , lookup
Regression toward the mean wikipedia , lookup
Biostatistics in Practice Session 2: Summarization of Quantitative Information Peter D. Christenson Biostatistician http://gcrc.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 Methods from Graphs Experimental Units _____ Independence of Measurements Statistical Independence Experimental units are the smallest independent entities for addressing a scientific question in an analysis of an experiment. “Independent” refers to the measurement that is made and the question, not the units. Definition: If knowledge of the value for a unit does not provide information about another unit’s value, given other factors (and the overall mean) in the analysis of the experiment, then the units are independent for this measurement. There may be a hierarchy of units. Importance of Independence Many basic statistical methods require that measurements are independent for the analysis to be valid. Other methods can incorporate the lack of independence. There can be some subjectivity regarding independence. Statistical methods use models. Models can be wrong. Example: Units and Independence Ten mice receive treatment A, each is bled, and blood samples are each divided into 3 aliquots. The same is done for 10 mice on treatment B. 1. A serum hormone is measured in the 60 aliquots and compared between A and B. The aliquots for a mouse are not independent. The unit is a mouse. A summary statistic from a mouse’s 3 aliquots (e.g., maximum or mean) are independent. N=10 and 10, not 30 and 30. Example, Continued 2. One of the 30 A aliquots is further divided into 25 parts and 5 different in vitro challenges are each made to a random set of 5 of the parts. The same is done for a single B aliquot. For this challenge experiment, each part is a unit, the values of challenge response are independent, and N=25+25. For comparing A and B, there are only N=1+1 experimental units, the two mice. Experimental Units in Case Study Experimental Units in Case Study There is a nested hierarchy of several "levels" of data: Schools, children within the schools, and diets received by every child. What would you use for the "N" for this study? Which outcomes do you intuitively think are correlated (in common language)? Results from one child's three diets? Results from children in the same school? Schools? Experimental Units in Case Study N = Number of children Results from one child's three diets cannot be modeled as independent. Results from children in the same school also could be “correlated” (dependent). They can be modeled as independent, if the effect of school is included in the analysis. Knowing one child’s score and the school mean gives no info on another child’s score. Units and Analysis in the Case Study N = Number of children Analysis: This method is a complex generalization of methods we discuss in Session 3. For any method, though, you need to inform the software of the correct experimental units. For some experiments, it is obvious and implicit. 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 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 Box Plot Raw Data 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 nonparametric 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 yvalue 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 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 Mean = 60.6 min. different SD = 9.6 min. measures comparabl e. 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”, if subjects normal, 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 Prediction Interval: -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