Types of Statistics. Censored data. Files for today (June 27) Lecture and Homework INTRODUCTION TO BIOSTATISTICS. Today s Outline

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1 INTRODUCTION TO BIOSTATISTICS FOR GRADUATE AND MEDICAL STUDENTS Files for today (June 27) Lecture and Homework Descriptive Statistics and Graphically Visualizing Data Lecture #2 (1 file) PPT presentation Homework #1 (1 file) Biostat_HW_Huet_assigned62713.docx Beverley Adams Huet, MS Assistant Professor Department of Clinical Sciences, Division of Biostatistics June 27, Medical Student Research Page -- Summer II June 27, Today s Outline Describing data Descriptive statistics Measures of central tendency Measures of dispersion Other statistics Coefficient of variation Standard error of the mean Transformations Histograms and other graphs Types of Statistics Descriptive statistics Which summary statistics to use to organize and describe the data? Proportion, mean, median, SD, percentiles Inferential statistics Generalizing from the sample. Which test? T-test, Fisher s Exact, ANOVA, survival analysis Bayesian approaches June 27, June 27, Summary of commonly used statistical tests Type of Outcome Variable: Continuous Rank, Score, or Goal: Binomial Survival measurement Measurement (from a normal (from non-normal (e.g. heads or tails) (Time to event) distribution) distribution) Median, interquartile Kaplan-Meier survival Describe one group: Mean, SD Proportion range curve Censored data Cannot be measured beyond some limit Compare one group to a hypothetical value: One-sample t test Wilcoxon Signed-Rank test Chi-square or binomial test Compare two unpaired (independent) groups: Compare two paired groups: Compare three or more unmatched groups: Compare three or more matched groups: Quantify association between two variables: Mann-Whitney Fisher's exact test Two sample (unpaired) Log-rank test or (Wilcoxon Rank Sum) (or chi-square for large t test Mantel-Haenszel test samples) Wilcoxon Signed-Rank Conditional proportional Paired t test McNemar test test hazards Cox proportional hazard One-way ANOVA Kruskal-Wallis test Chi-square test Repeated-measures Conditional proportional Friedman test Cochrane Q test analysis hazards Pearson correlation Spearman correlation Contingency coefficients Left censoring Right censoring Predict value from another measured variable: Simple linear Nonparametric Simple logistic Cox proportional hazard Predict value from several Multiple linear Multiple logistic Cox proportional hazard measured or binomial, ANCOVA variables: June 27, June 27,

2 Left Censored data Cannot be measured beyond some limit Lab data undetectable, below lower limit Example CRP <.2 mg/dl Censored at the limit of detectability Subject CRP < Right Censored data Cannot be measured beyond some limit Right censoring - Survival data the period of observation was cut off before the event of interest occurred. Note an event in a survival analysis may be infection, fracture, transplant, metastasis June 27, June 27, Right censored survival data Survival time known Censored 1 Survival Survival Analysis Right censored survival data 9 8 Event at 3 months.2 Subject Lost to follow-up at 9 months Time Step function Survival time known Censored 2 1 Subject Study time, months June 27, June 27, Study time, months 1 Descriptive statistics Measures of Central Tendency Measures of Dispersion Measures of Central Tendency* *or Measures of Location Mean Median Geometric mean Mode June 27,

3 Measures of Central Tendency* Mean Arithmetic average or balance point Discrete/continuous data; symmetric distribution May be sensitive to outliers Sample mean symbol is denoted as x-bar X X N *or Measures of Location Fasting plasma glucose, n=6 SubjectID Glucose mg/dl Mean 152 June 27, Fasting plasma glucose, n=6 Glucose mg/dl X Mean SubjectID Glucose mg/dl Mean 152 Median 14 Glucose, mg/dl June 27, Fasting Plasma Glucose What about other measures of central tendency? Measures of Central Tendency* *or Measures of Location Measures of Central Tendency In a symmetric distribution, the median, mode and mean will have the same value. Median In a non-symmetric (skewed) distribution, the median, mode and mean may not have the same value. Middle value when the data are ranked in order (if the sample size is an even number then the median is the average of the two middle values) 5 th percentile Ordinal/discrete/continuous data Useful with highly skewed discrete or continuous data Relatively insensitive to outliers June 27, June 27, Measures of Central Tendency The median of 13, 11, 17 is 13 The median of 13, 11, 568 is 13 The median of 14, 12, 11, 568 is 13 Measures of Central Tendency SubjectID Glucose mg/dl Mean 152 Median 14 Order the glucose values from smallest to largest Glucose SubjectID mg/dl June 27, June 27,

4 The median is often better than the mean for describing the center of the data Gonick & Smith (1993) The Cartoon Guide to Statistics. June 27, Geometric mean Usually smaller than the arithmetic mean. Can be used instead of arithmetic mean when data have a skewed or log-normal distribution Find the mean on the log scale, taking the antilog of this mean yields the geometric mean. Report for log transformed data G n x x3 *... x June 27, * x 2 * n Geometric mean Creatinine Log 1 (Creatinine) Sometimes we can transform our data Histograms Log transformed data June 27, Geometric mean Log e transformed data SubjectID Glucose mg/dl ln(glucose) Mean SD Median Geometric mean Take the antilog of the mean exp( ) = Geometric mean: Back-transform (antilog) the mean of the log transformed data June 27, Measures of Central Tendency Mode Most frequently occurring value in the distribution Nominal/ordinal/discrete/continuous data Measures of Dispersion The mode of 13, 11, 22, 11, 17 = 11 Gonick & Smith (1993) The Cartoon Guide to Statistics. June 27, June 27,

5 Measures of Dispersion Also known as Measures of Spread Measures of Variability Commonly used measures of variability Standard deviation Range Percentiles June 27, Measures of Dispersion Standard Deviation Square root of average of the squared deviations Has the same units as the original observation NEVER negative Lower case sigma s x X 2 x Population standard deviation (Greek letters) n X X 2 n 1 Sample standard deviation (Roman letters) [n-1 in the denominator corrects for bias] June 27, Standard deviation calculation Deviation from mean Glucose Glucose minus Squared SubjectID mg/dl mean of 152 deviation Standard deviation and the Normal Distribution Mean 152 Square root of variance Median 14 Divide by n-1 Sum of squares n 6 Variance SD June 27, Approximately 68% of the observations fall within 1 SD of the mean Approximately 95% of the observations fall within 2 SDs of the mean Approximately 99.7% of the observations fall within 3 SDs of the mean 28 Percentiles From Primer of Biostatistics by Stanton A Glantz June 27, Percentiles The value below which a given percentage of the values occur The 5 th percentile is the median Quartiles The 25 th percentile is first quartile (Q1) The 75 th percentile is third quartile (Q3) Interquartile range is the difference between 25 th and 75 th percentiles Other commonly reported percentiles 1 th and 9 th percentiles 5 th and 95 th percentiles June 27,

6 Special Percentiles Quartiles Percentiles 25% 25% 25% 25% Q1 Q2 Q3 From Primer of Biostatistics by Stanton A Glantz IQR June 27, June 27, Summarizing data with medians and percentiles Box and Whisker Plot From Lang and Secic, How to Report Statistics in Medicine June 27, June 27, Box and Whisker plots Descriptive statistics with percentiles FIG. 1. Serum levels of OPG (A) and srankl (B) in CS patients and controls. Values are median, 25th and 75th percentile, and range. *, P <.1 by Mann- Whitney U test. Dovio, A. et al. J Clin Endocrinol Metab 27;92: proc means n mean std median min p25 p75 max maxdec=5 ;; title3 'Descriptive statistics'; class group; var hdl; run; June 27, June 27, Copyright 27 The Endocrine Society 6

7 Measures of shape Skewness measures symmetry Characteristics of the Insulin Resistance Syndrome in a Japanese Population The Jichi Medical School Cohort Study Arteriosclerosis, Thrombosis, and Vascular Biology. 1996;16: Skewed left or right?? Skewed left a long left tail Skewed right a long right tail Figure 1. Histograms of the fasting insulin levels in Japanese men and women. June 27, June 27, Examples of variables having a skewed distribution Skewness measures symmetry Triglycerides, insulin, HOMA Bilirubin, leptin, CRP, viral load counts Urine albumin Income Health care costs Hospital length of stay Other statistics Coefficient of variation Standard error of the mean June 27, June 27, Coefficient of variation (CV) The standard deviation expressed as a proportion of the mean: CV SD Mean Often expressed as a percent: CV % 1 SD Mean Coefficient of variation (CV) The magnitude of total intra-individual variability based on coefficient of variation (CV) for these lipids in premenopausal women (CV, 4% to 8.1%) was similar to that found for men (CV, 4.3% to 9.1%) and for postmenopausal women (CV, 3.7% to 6.7%). Metabolism. 2 Sep;49(9): June 27, June 27,

8 Coefficient of variation (CV) Coefficient of variation (CV) Femoral Neck Hip Bone Density ID1 ID2 ID3 ID4 Scan Scan Scan Mean SD CV CV%.36%.3% 2.1% 2.99% CV SD Mean Subject A CO2 AST ALT Day Day Day Mean SD CV CV% 5.21% 11.62% 2.11% CV % 1 SD Mean June 27, A measure of spread A unitless fraction (SD/mean) Used for comparing the relative variability of variables measured in different units e.g., HDL-cholesterol (4.6%) and triglycerides (2.7%) Used for comparing the relative variability of variables measured in same units e.g., inter-assay versus intra-assay variability June 27, Standard error of the mean Is NOT a descriptive statistic The standard error of the mean is useful in the calculations of confidence intervals and significance tests Do not summarize continuous data with the mean and the standard error of the mean. Lang and Secic (26) How to Report Statistics in Medicine June 27, Standard error of the mean (also called SEM, SE, standard error) SEM = Which is smaller - SD or SEM? Standard deviation Square root of the sample size or SEM SD n June 27, Standard error (SE, SEM, standard error of the mean) Why is the standard error commonly used as descriptive statistic and graphed as error bars? It is smaller than the standard deviation Looks better? The only role of the standard error is to distort and conceal the data. Feinstein, Clinical Biostatistics June 27, Standard deviation (SD) and Standard error of the mean (SEM) Can convert one to the other if the sample size (n) is known SEM SD n SD SEM n These formulas do need to be memorized! June 27,

9 Adding a constant Standard deviation (SD) and Standard error of the mean (SEM) Can convert one to the other if the sample size (n) is known Making friends with your data Don t run away! SD = 4, n = 64, SEM =? SEM = 4/8 = 5 SEM = 12, n = 81, SD =? SD = 12 * 9 =18 June 27, Look at your data! Make friends with your data! June 27, Transformations Transformations Plasma Glucose Why transform data? Many statistical analyses include assumptions Normality (Normally distributed) The different groups have the same standard deviation Linearity for correlation or modeling Linear Nonlinear mmol/l mg/dl June 27, June 27, Linear Transformations Add, subtract, multiply, divide A straight line is obtained when plotting the new values against the original values Mean and standard deviation of the transformed values are easily obtained Linear Transformations Adding/subtracting a constant The mean increases/decreases by the same amount as the constant The standard deviation is unaffected X X Obs X X+2 a b c d e f Mean SD Median June 27, June 27,

10 Linear Transformations - Multiplying by a constant mmol/l Conversion from conventional units to Standard International (SI) units Plasma Glucose SubjectID Glucose mg/dl Convert to SI units Glucose mmol/l Mean SD Median Nonlinear transformation mg/dl Both the mean and SD are multiplied by.555 Multiply by.555 to convert to mg/dl to mmol/l Mean *.555= 8.44 SD *.555= 3.2 Median 14 14*.555= 7.77 June 27, June 27, Fig 1--Serum triglyceride and log1 serum triglyceride concentrations in cord blood for 282 babies, with best fitting normal distribution Transformations Common non-linear transformations Log transformation (log 1 or log e ) Square-root transformation Less dramatic than the log transformation Reciprocal transformation More drastic than the log transformation Use for extremely skewed data distributions Bland, J M. et al. BMJ 1996;312:179 June 27, June 27, Copyright 1996 BMJ Publishing Group Ltd. Transformations For assessing linear association Log transformation % Liver Fat Untransformed % Total Body Fat X Pearson Correlation coefficient =.28 % Liver Fat loge transformed % Total Body Fat Pearson Correlation coefficient =.38 When might a log transformation be useful? Remove positive (right) skewness The standard deviation is greater than half of the mean (if the measure cannot be negative) The mean is larger than the median Mean liver fat = 6.2% SD=6.3% (median=4.2) The mean is proportional to the standard deviation When comparing several groups the variances or standard deviations are not equal June 27, June 27,

11 Log Transformation Serum Creatinine Leptin (ng/ml) Group A Group B n Mean SD Median These data might need a log transformation why? June 27, Original units Log transformed 62 Histogram with normal distribution overlay Statistics and graph software SigmaPlot and GraphPad Prism can be downloaded from the UTSW Information Resources INTRAnet June 27, Histogram Group the data into intervals (x-axis) Height of the bar indicates the frequency (y-axis) Each bar begins/ends at the true limits of the interval. Bars are presented next to each other (unless the data in the next interval has a frequency of ). Bars are usually the same width Frequencies correspond to the area of each bar June 27, Histograms for continuous/discrete data Absolute frequency histogram Relative frequency histogram Spaghetti plot for Repeated Measurements Each subject is observed under multiple conditions or on multiple occasions Spaghetti plot Absolute Relative Multiple time points June 27,

12 Forest plot for Meta-analysis Look at your data! A difference must be a difference to make a difference Do not rely on p-values. The non-significant results might be just as interesting or enlightening. Descriptive statistics Forest plot June 27, June 27,