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Introduction to Biostatistics Descriptive

Statistics and Sample Size Justification

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

Statistics Seminars

- Goal Interpret and critically evaluate

biomedical literature - Topics
- Sample size justification
- Exploratory data analysis
- Hypothesis testing

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

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.

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

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

Why are descriptive statistics important?

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

descriptive results

Outline

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

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

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

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

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

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

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

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

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.

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

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.

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

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

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?

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

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

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

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

Descriptive Statistics Categorical Data

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

Descriptive Statistics Categorical Data

- Measures of distribution example
- What percentage of vaccinated individuals

developed the flu? - 198/400 0.495 49.5

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?

Descriptive Plots

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

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

Barplot

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

Histogram

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

Boxplot

75th percentile

Median

25th percentile

Box-plot

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

Scatter Plot

Scatter Plot

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

Kaplan-Meier Survival Curves

Kaplan-Meier Survival Curves

Kaplan-Meier Survival Curves

Descriptive Plots Guidelines

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

Descriptive Statistics

- Exercises

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

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.

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

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.

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.

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.

Sample Size Justification

Outline

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

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

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

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

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

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.

Statistical Concepts Errors

- Errors associated with hypothesis testing

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

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

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

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.

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

Factors Influencing Power

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

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?

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.

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.

- 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

- 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?

- 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

- 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

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.