Introduction to Biostatistics Descriptive Statistics and Sample Size Justification

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Introduction to BiostatisticsDescriptive
Statistics and Sample Size Justification

• Julie A. Stoner, PhD
• October 17, 2005

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Statistics Seminars

• Goal Interpret and critically evaluate
  biomedical literature
• Topics
• Sample size justification
• Exploratory data analysis
• Hypothesis testing

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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, 199912691-696

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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. 3121011-1015.1985.

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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

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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

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Why are descriptive statistics important?

• Identify signals/patterns from noise
• Understand relationships among variables
• Formal hypothesis testing should agree with
  descriptive results

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Outline

• Types of data
• Categorical data
• Numerical data
• Descriptive statistics
• Measures of location
• Measures of spread
• Descriptive plots

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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

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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

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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

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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

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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 (n1)(P/100)
• Median the 50th percentile, 50 of the data
  values lie below the median

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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 (71) (50/100) 4
• 4TH ordered observation is 148

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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

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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 1108/7158.29, Median 148
• The mean may be influenced by extreme data
  points.

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Skewed Distributions

• Data that is not symmetric and bell-shaped is
  skewed.
• Mean may not be a good measure of central
  tendency. Why?

Positive skew, or skewed to the right, mean gt
median
Negative skew, or skewed to the left, mean lt
median
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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.

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Descriptive Statistics Numerical Data

• Measures of spread
• Variance average squared deviation from the
  mean
• For n data points, x1, x2,, , xn the variance is
• Standard deviation square root of variance, in
  same units as original data

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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

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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?

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Descriptive Statistics Numerical Data

• The standard error of the mean is estimated as
• 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

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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

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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

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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

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Descriptive Statistics Categorical Data

• Measures of distribution
• Proportion
• Number of subjects with characteristics
• Total number subjects
• Percentage
• Proportion 100

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Descriptive Statistics Categorical Data

• Measures of distribution example
• What percentage of vaccinated individuals
  developed the flu?
• 198/400 0.495 49.5

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Example

• Consider the table of descriptive statistics for
  characteristics at baseline
• What do we conclude about comparability of the
  groups at baseline in terms of gender and age?

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Descriptive Plots

• Single variable
• Bar plot
• Histogram
• Box-plot
• Multiple variables
• Box-plot
• Scatter plot
• Kaplan-Meier survival plots

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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

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Barplot
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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

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Histogram
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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

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Boxplot
75th percentile
Median
25th percentile
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Box-plot
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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

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Scatter Plot
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Scatter Plot
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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

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Kaplan-Meier Survival Curves
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Kaplan-Meier Survival Curves
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Kaplan-Meier Survival Curves
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Descriptive Plots Guidelines

• Clearly label axes
• Indicate unit of measurement
• Note the scale when interpreting graphs

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Descriptive Statistics

• Exercises

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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. 11015-26

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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.

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• Proportion Male
• (336/654)100 51.4
• Proportion Smokers
• (65/654)100 9.9
• Proportion of Smokers who are Male
• (26/65)100 40

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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 3.75-30.75)
• compared to a median FEV of
• 2.5 liters/sec. (IQR 3-21) for
• non-smokers.

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Compare the age distribution between smokers and
non-smokers.

• Answer
• The smokers are
• older than the non-
• smokers in general.
• Specifically, the median
• age for the smokers is
• 13 years (IQR 15-123)
• compared to 9 years
• (IQR 11-83) for the
• non-smokers.

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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.

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Sample Size Justification
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Outline

• Statistical Concepts hypotheses and errors
• Effect size and variation
• Influence on sample size and power

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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. 199912691-696

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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 299690-694, 1978

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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

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Statistical ConceptsHypotheses

• 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

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Statistical ConceptsHypotheses

• 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.

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Statistical ConceptsErrors

• Errors associated with hypothesis testing

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Statistical ConceptsSignificance 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

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Statistical ConceptsPower

• 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

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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

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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.

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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

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Factors Influencing Power

• Assuming all other factors fixed,
• ? significance level ?
• ? power
• ? significant difference ?
• ? power
• ? variability in response ?
• ? power
• ? sample size ?
• ? power

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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?

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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.

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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.

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• 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

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• 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?

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• 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

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• 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

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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. 299690-694, 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. 293-113.
  1981.