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Introduction to Biostatistics Descriptive Statistics and Sample Size Justification Julie A. Stoner, PhD October 18, 2004 1 Statistics Seminars • Goal: Interpret and critically evaluate biomedical literature • Topics: – Sample size justification – Exploratory data analysis – Hypothesis testing 2 Example #1 • Aim: Compare two antihypertensive strategies for lowering blood pressure – Double-blind, randomized study – 5 mg Enalapril + 5 mg Felodipine ER to 10 mg Enalapril – 6-week treatment period – 217 patients – AJH, 1999;12:691-696 3 Example #2 • Aim: Demonstrate that D-penicillamine (DPA) is effective in prolonging the overall survival of patients with primary biliary cirrhosis of the liver (PBC) – – – – – Mayo Clinic Double-blind, placebo controlled, randomized trial 312 patients Collect clinical and biochemical data on patients Reference: NEJM. 312:1011-1015.1985. 4 Example #2 • Patients enrolled over 10 years, between January 1974 and May 1984 • Data were analyzed in July 1986 • Event: death (x) • Censoring: some patients are still alive at end of study (o) 1/1974 5/1984 6/1986 _____________________________X ___________________________o ________________________o 5 Statistical Inference • Goal: describe factors associated with particular outcomes in the population at large • Not feasible to study entire population • Samples of subjects drawn from population • Make inferences about population based on sample subset 6 Why are descriptive statistics important? • Identify signals/patterns from noise • Understand relationships among variables • Formal hypothesis testing should agree with descriptive results 7 Outline • Types of data – Categorical data – Numerical data • Descriptive statistics – Measures of location – Measures of spread • Descriptive plots 8 Types of Data • Categorical data: provides qualitative description – Dichotomous or binary data • Observations fall into 1 of 2 categories • Example: male/female, smoker/non-smoker – More than 2 categories • Nominal: no obvious ordering of the categories – Example: blood types A/B/AB/O • Ordinal: there is a natural ordering – Example: never-smoker/ex-smoker/light smoker/heavy smoker 9 Types of Data • Numerical data (interval/ratio data) – Provides quantitative description – Discrete data • Observations can only take certain numeric values • Often counts of events • Example: number of doctor visits in a year – Continuous data • Not restricted to take on certain values • Often measurements • Example: height, weight, age 10 Descriptive Statistics: Numerical Data • Measures of location – Mean: average value For n data points, x1, x2,, …, xn the mean is the sum of the observations divided by the number of observations 1 x n n x i i 1 11 Descriptive Statistics: Numerical Data • Measures of location – Mean: • Example: Find the mean triglyceride level (in mg/100 ml) of the following patients 159, 121, 130, 164, 148, 148, 152 Sum = 1022, Count = 7, Mean = 1022/7 = 146 12 Descriptive Statistics: Numerical Data • Measures of location – Percentile: value that is greater than a particular percentage of the data values • Order data • Pth percentile has rank r = (n+1)*(P/100) – Median: the 50th percentile, 50% of the data values lie below the median 13 Descriptive Statistics: Numerical Data • Measures of location – Median • Example: Find the median triglyceride level from the sample 159, 121, 130, 164, 148, 148, 152 Order: 121, 130, 148, 148, 152, 159, 164 Median: rank = (7+1) * (50/100) = 4 4TH ordered observation is 148 14 Descriptive Statistics: Numerical Data • Measures of location – Mode: most common element of a set – Example: Find the mode of the triglyceride values 159, 121, 130, 164, 148, 148, 152 Mode = 148 15 Descriptive Statistics: Numerical Data • Measures of location: comparison of mean and median – Example: Compare the mean and median from the sample of triglyceride levels 159, 141, 130, 230, 148, 148, 152 Mean = 1108/7=158.29, Median = 148 – The mean may be influenced by extreme data points. 16 Skewed Distributions • Data that is not symmetric and bell-shaped is skewed. Positive skew, or skewed to the right, mean > median Negative skew, or skewed to the left, mean < median • Mean may not be a good measure of central tendency. Why? 17 Motivation • Example: 1) 2 60 100 =54 2) 53 54 55 =54 • Both data sets have a mean of 54 but scores in set 1 have a larger range and variation than the scores in set 2. 18 Descriptive Statistics: Numerical Data • Measures of spread – Variance: average squared deviation from the mean For n data points, x1, x2,, …, xn the variance is n s 2 2 1 xi x n 1 i 1 – Standard deviation: square root of variance, in same units as original data 19 Descriptive Statistics: Numerical Data • Measures of spread – Standard Deviation: • Example: find the standard deviation of the triglyceride values 159, 121, 130, 164, 148, 148, 152 Distance from mean: 13, -25, -16, 18, 2, 2, 6 Sum of squared differences: 1418 Standard deviation: sqrt(1418/6)=15.37 20 Descriptive Statistics: Numerical Data • Standard deviation: How much variability can we expect among individual responses? • Standard error of the mean: How much variability can we expect in the mean response among various samples? 21 Descriptive Statistics: Numerical Data • The standard error of the mean is estimated as s.d . s.e.m. n where s.d. is the estimated standard deviation • Based on the formula, will the standard error of the mean will always be smaller or larger than the standard deviation of the data? – Answer: smaller 22 Descriptive Statistics: Numerical Data • Measures of spread – Minimum, maximum – Range: maximum-minimum – Interquartile range: difference between 25th and 75th percentile, values that encompass middle 50% of data 23 Descriptive Statistics: Numerical Data • Measures of spread – Example: find the range and the interquartile range for the triglyceride values 159, 121, 130, 164, 148, 148, 152 Range: 164 - 121 = 43 Interquartile Range: Order: 121, 130, 148, 148, 152, 159, 164 IQR: 159 - 130 = 29 24 Descriptive Statistics: Numerical Data • Helpful to describe both location and spread of data – Location: mean Spread: standard deviation – Location: median Spread: min, max, range interquartile range quartiles 25 Descriptive Statistics: Categorical Data • Measures of distribution – Proportion: Number of subjects with characteristics Total number subjects – Percentage: Proportion * 100% 26 Descriptive Statistics: Categorical Data • Measures of distribution: example No Flu Flu Vaccinated 202 198 400 Not Vaccinated 179 221 400 381 419 800 • What percentage of vaccinated individuals developed the flu? 198/400 = 0.495 49.5% 27 Example • Consider the table of descriptive statistics for characteristics at baseline Parameter Number Gender (% Male) Age years Mean (SD) Enalapril+ Felodipine ER 109 61% Enalapril 52(9) 53(11) 108 54% • What do we conclude about comparability of the groups at baseline in terms of gender and age? 28 Descriptive Plots: • Single variable – Bar plot – Histogram – Box-plot • Multiple variables – Box-plot – Scatter plot – Kaplan-Meier survival plots 29 Barplot • Goal: Describe the distribution of values for a categorical variable • Method: – Determine categories of response – For each category, draw a bar with height equal to the number or proportion of responses 30 Barplot 31 Histogram • Goal: Describe the distribution of values for a continuous variable • Method: – Determine intervals of response (bins) – For each interval, draw a bar with height equal to the number or proportion of responses 32 Histogram 33 Box-plot • Goal: Describe the distribution of values for a continuous variable • Method: – Determine 25th, 50th, and 75th percentiles of distribution – Determine outlying and extreme values – Draw a box with lower line at the 25th percentile, middle line at the median, and upper line at the 75th percentile – Draw whiskers to represent outlying and extreme values 34 Boxplot 75th percentile Median 25th percentile 35 Box-plot 36 Scatter Plot • Goal: Describe joint distribution of values from 2 continuous variables • Method: – Create a 2-dimensional grid (horizontal and vertical axis) – For each subject in the dataset, plot the pair of observations from the 2 variables on the grid 37 Scatter Plot 38 Scatter Plot 39 Kaplan-Meier Survival Curves • Goal: Summarize the distribution of times to an event • Method: – Estimate survival probabilities while accounting for censoring – Plot the survival probability corresponding to each time an event occurred 40 Kaplan-Meier Survival Curves 41 Kaplan-Meier Survival Curves 42 Kaplan-Meier Survival Curves 43 Descriptive Plots Guidelines • Clearly label axes • Indicate unit of measurement • Note the scale when interpreting graphs 44 Descriptive Statistics Exercises 45 Example • Below are some descriptive plots and statistics from a study designed to investigate the effect of smoking on the pulmonary function of children • Tager et al. (1979) American Journal of Epidemiology. 110:15-26 46 Example • The primary question, for this exercise, is whether or not smoking is associated with decreased pulmonary function in children, where pulmonary function is measured by forced expiratory volume (FEV) in liters per second. • The data consist of observations on 654 children aged 3 to 19. 47 GENDER Female Male Total SMOKING STATUS NonSmoker smoker 279 39 310 26 589 65 Total 318 336 654 • Proportion Male: – (336/654)100% = 51.4% • Proportion Smokers: – (65/654)100% = 9.9% • Proportion of Smokers who are Male: – (26/65)100% = 40% 48 Compare the FEV1 distribution between smokers and non-smokers • Answer – The smokers appear to have higher FEV values and therefore better lung function. Specifically, the median FEV for smokers is 3.2 liters/sec. (IQR 3.75-3=0.75) compared to a median FEV of 2.5 liters/sec. (IQR 3-2=1) for non-smokers. 49 Compare the age distribution between smokers and non-smokers. • Answer: – The smokers are older than the nonsmokers in general. Specifically, the median age for the smokers is 13 years (IQR 15-12=3) compared to 9 years (IQR 11-8=3) for the non-smokers. 50 Can you explain the apparent differences in pulmonary function between smokers and non-smokers displayed in Figure 1? • The relationship between FEV and smoking status is probably confounded by age (smokers are older and older children have better lung function). A comparison of FEV between smokers and non-smokers should account for age. 51 Sample Size Justification 52 Outline • Statistical Concepts: hypotheses and errors • Effect size and variation • Influence on sample size and power 53 Sample Size Justification • Example: Intensifying Antihypertensive Treatment – “A sample size calculation indicated that 114 patients per treatment group would be necessary for 90% power to detect a true mean difference in change from baseline of 3 mm Hg in sitting DBP between the two randomized treatment groups. This calculation assumed a two-sided test, =0.05, and standard deviation in sitting DBP of 7 mm Hg.” Source: AJH. 1999;12:691-696 54 Importance of Careful Study Design • Goal of sample size calculations: – Adequate sample size to detect clinically meaningful treatment differences – Ethical use of resources • Important to justify sample size early in planning stages • Examples of inadequate power: – NEJM 299:690-694, 1978 55 Type of Response • Sample size calculations depend on type of response variable and method of analysis – Continuous response • Example: cholesterol, weight, blood pressure – Dichotomous response • Example: yes/no, presence/absence, success/failure – Time to event • Example: survival time, time to adverse event 56 Statistical Concepts Hypotheses • Null hypothesis: H0 – Typically a statement of no treatment effect – Assumed true until evidence suggests otherwise – Example: H0: No difference in DBP between treatment groups • Alternative: HA – Reject null hypothesis in favor of alternative hypothesis – Often two-sided – Example: HA: DBP differs between treatment groups 57 Statistical Concepts Hypotheses • Alternative hypothesis may be one-sided or two-sided – Example: • Null hypothesis: Mean DBP is same in patients receiving different treatments • Alternative hypothesis: – One-sided: Mean DBP is lower in patients receiving treatment A – Two-sided: Mean DBP is different in patients receiving treatment A relative to treatment B • Choice of alternative does affect sample size calculations. Typically a two-sided test is recommended. 58 Statistical Concepts Errors • Errors associated with hypothesis testing TRUTH STUDY Association No Association Reject Null Correct Fail to Reject Null Type II Error False negative Type I Error False positive Correct 59 Statistical Concepts Significance Level • Significance level: – Probability of a Type I error – Probability of a false positive – Example: If the effect on DBP of the treatments do not differ, what is the probability of incorrectly concluding that there is a difference between the treatments? – When calculating sample size, we need to specify a significance level, meaning, the probability that we will detect a treatment effect purely by chance. – Typically chosen to be 5%, or 0.05 60 Statistical Concepts Power • Power: (1-) – Probability of detecting a true treatment effect – (1- probability of a false negative) = (1-probability of Type II error) = (1-) = probability of a true positive – Example: If the effects of the treatments do differ, what is the probability of detecting such a difference? – Typically chosen to be 80-99% 61 Treatment Effect • What is the minimal, clinically significant difference in treatments we would like to detect? • Pilot studies may indicate magnitude • Example: The authors felt that a 3 mm Hg difference in DBP between the treatment groups was clinically significant 62 Variability in Response • To estimate sample size, we need an estimate of the variability of the response in the population • Estimate variability from pilot or previous, related study • Example: The authors estimate that the standard deviation of DBP is 7 mm Hg. 63 Factors Influencing Sample Size Assuming all other factors fixed, • power sample size • significance level sample size • variability in response sample size • significant difference sample size 64 Factors Influencing Power Assuming all other factors fixed, • significance level power • significant difference power • variability in response power • sample size power 65 Summary • Sample size calculations are an important component of study design • Want sufficient statistical power to detect clinically significant differences between groups when such differences exist • Calculated sample sizes are estimates • Can manipulate sample size formulas to determine: – What is the power for detecting a particular difference given the sample size employed? – What difference can be detected with a certain amount of power given the sample size employed? 66 Factors Influencing Sample Size • A double-blind randomized trial was conducted to determine how inhaled corticosteroids compare with oral corticosteroids in the management of severe acute asthma in children. In the study, 100 children were randomized to receive one dose of either 2 mg of inhaled fluticasone or 2 mg of oral prednisone per kilogram of body weight. The primary outcome was forced expiratory volume (as a percentage of the predicted value) 4 hours after treatment administration. • Schuh et al., (2000) NEJM. 343(10)689-694. 67 Factors Influencing Sample Size • The null hypothesis is that the mean FEV, as a percentage of predicted value, is the same for both treatment groups. • The alternative hypothesis is that the mean FEV, as a percentage of predicted value, is different for the two treatment groups. 68 • What is a Type I Error in this example? – Incorrectly concluding that the treatments differ • What is a Type II Error in this example? – Failing to detect a true treatment difference 69 In the article the authors state “In order to allow detection of a 10 percentage point difference between the groups in the degree of improvement in FEV (as a percentage of the predicted value) from base line to 240 minutes and to maintain an error of 0.05 and a error of 0.10, the required size of the sample was 94 children.”. What is the power of the study and what does it mean? What is the significance level of the study and what does this level mean? 70 • Power: – The power is 90% – There is a 90% chance of detecting a treatment difference of 10 percentage points, given such a difference really exists • Significance Level: – The significance level is 0.05 – There is a 5% chance of concluding the treatments differ when in fact there is no difference 71 • Assuming a 5 percentage point difference between the groups, what happens to power? – The power of the study, as proposed, would be less than 90% • Assuming an 0.01 significance level what happens to power? – The power of the study, as proposed, would be less than 90% 72 References Descriptive Statistics • Altman, D.G., Practical Statistics for Medical Research. Chapman & Hall/CRC, 1991. Sample Size Justification • Freiman, J. A. et al. “The importance of beta, the type II error and sample size in the design and interpretation of the randomized control trial: Survey of 72 “negative” trials. N Engl J Med. 299:690-694, 1978. • Friedman, L. M., Furberg, C. D., DeMets, D. L., Fundamentals of Clinical Trials, Springer-Verlag, 1998, Chapter 7. • Lachin, J. M. “Introduction to sample size determination and power analysis for clinical trials”. Controlled Clinical Trials. 2:93-113. 1981. 73