Introduction to Biostatistics Descriptive Statistics and Sample Size Justification

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Introduction to Biostatistics

Descriptive Statistics and

Sample Size Justification

Julie A. Stoner, PhD

October 18, 2004

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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, 1999;12:691-696

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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. 312:1011-1015.1985.

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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

n

i

ixn

x1

1

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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 = (n+1)*(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 = (7+1) * (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 = 1108/7=158.29, 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 > median Negative skew, or skewed to the

left, mean < 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

2

1

2

1

1

n

ii xx

ns

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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

n

dsmes

.....

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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%

No Flu Flu

Vaccinated 202 198 400

Not Vaccinated

179 221 400

381 419 800

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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?

Parameter Enalapril+Felodipine ER

Enalapril

Number 109 108

Gender(% Male)

61% 54%

Age yearsMean (SD)

52(9) 53(11)

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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

Median

25th percentile

75th 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. 110:15-26

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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%

SMOKING STATUS

GENDER Non-smoker

Smoker Total

Female 279 39 318 Male 310 26 336 Total 589 65 654

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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 3.75-3=0.75) compared to a median FEV of 2.5 liters/sec. (IQR 3-2=1) 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 15-12=3) compared to 9 years (IQR 11-8=3) 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. 1999;12:691-696

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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 299:690-694, 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 testingTRUTH

Association NoAssociation

STUDYRejectNull

Correct Type I ErrorFalsepositive

Fail toRejectNull

Type IIErrorFalsenegative

Correct

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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. 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.