Biostatistics in Practice

Session 2: Summarization of Quantitative

Information

Youngju PakBiostatistician

http://research.LABioMed.org/Biostat 1

Topics for this Session

Experimental Units

Independence of Measurements

Graphs: Summarizing Results

Graphs: Aids for Analysis

Summary Measures

Confidence Intervals

Prediction Intervals 2

Experimental Units_____

Independence of Measurements

3

Units and IndependenceExperiments may be designed such that each measurement does not give additional independent information.

Many basic statistical methods require that measurements are “independent” for the analysis to be valid.

In mathematics, two events are independent if and only if the occurrence of one event makes it neither more nor less probable that the other occurs. 4

Population

Sample

Sample estimate of population parameter

Population parameter

Sampling mechanism: random sample or convenience sample

Confidence Interval

for population parameter

5

Summarizing the Data with Descriptive Statistics

6

Experimental Units in Case Study

What is the experimental unit in this study? 1. School 2. Child 3. Parent 4. GHA score (results from three diets)

Are all GHA scores(eg. 153 x 3 groups=459 GHA scores for 3-4 years old children)

The analysis MUST incorporate this possible correlation (clustering) if there exists.

7

Common Descriptive Statistics used

Sample Mean and Standard Deviation (SD)

Sample Median and Inter-Quartile Range (IRQ)

Sample Correlation

Sample Survival Probability

Sample Risks & Odds 8

Mean vs. Median(measure the central tendency)

• Mean – What most people

think of as “average”– Easy to calculate– Easily distorted– Be cautious with

SKEWED data– Calculate:

sum of data / number of data points

• Median– Relatively easy to

obtain– Not affected by

extreme values so it is considered a “ROBUST” statistic

– Calculate: • Sort data • If odd number points,

the middle is the median

• Otherwise, the median is the average of the middle two numbers

9

Standard Deviation (SD) &Inter-Quartile Range(IRQ)(measuring the variability or scatterness of the data )

• Inter-Quartile Range (IQR)=

75th percentile (Q3) - 25th percentile(Q1)

, where 25% of the data <Q1 , 75% of the data < Q3

• SD is usually used for the normally distributed data (bellshape, symmetric around the mean)

• IQR is usually used when the data distribution is skewed.• Range = Max -Min

10

Summarization of the Case Study

How are the outcome measures summarized? e.g., Table 2:

11

Summary Statistics:Relative Likelihood of an Event

Compare groups A and B on mortality.

Relative Risk = ProbA[Death] / ProbB[Death]where Prob[Death] ≈ Deaths per 100 Persons

Odds Ratio = OddsA[Death] / OddsB[Death] where Odds= Prob[Death] / Prob[Survival]

Hazard Ratio ≈ IA[Death] / IB[Death]where I = Incidence

= Deaths per 100 PersonDays12

Summarizing the Data with Graphs

13

Data Graphical DisplaysMany of the following examples are from StatisticalPractice.com

Histogram Scatter plot

Raw DataSummarized*

* Raw data version is a stem-leaf plot. We will see one later.14

Data Graphical Displays

Dot Plot Box Plot

Raw Data Summarized

15

Bar Charts

16

Pie Charts

17

Data Graphical DisplaysLine or Profile Plot

Summarized - bars can represent various types of ranges18

Data Graphical Displays

Kaplan-Meier Plot

Interval (Start-End)

# At Risk at Start of Interval

# Censored During Interval

# At Risk at End of Interval

# Who Died at End of Interval

Proportion Surviving This Interval

Cumulative Survival at End of Interval

0-1 7 0 7 16/7 = 0.86

0.86

1-4 6 2 4 13/4 = 0.75

0.86 * 0.75 = 0.64

4-10 3 1 2 11/2 = 0.5

0.86 * 0.75 * 0.5 = 0.31

10-12 1 0 1 01/1 = 1.0

0.86 * 0.75 * 0.5 * 1.0 = 0.31

(Source: www.cancerguide.org) 19

Graphs:

Aids for Analysis

20

Graphical Aids for Analysis

Most statistical analyses involve modeling.

Parametric methods (t-test, ANOVA, Χ2) have stronger requirements than non-parametric methods (rank -based).

Every method is based on data satisfying certain requirements.

Many of these requirements can be assessed with some useful common graphics.

21

Look at the Data for Analysis Requirements

What do we look for?

In Histograms (one variable):Ideal: Symmetric, bell-shaped.

Potential Problems:• Skewness.• Multiple peaks.• Many values at, say, 0, and bell-shaped

otherwise.• Outliers. 22

Example Histogram: OK for Typical* Analyses

• Symmetric.• One peak.• Roughly bell-shaped.• No outliers.

*Typical: mean, SD, confidence intervals, to be discussed in later slides. 23

Z- Score = (Measure - Mean)/SD

35 45 55 65 75 85 95

0

5

10

15

20

25

Time

Fre

qu

ency

Standardizes a measure to have mean=0 and SD=1.

Z-scores make different measures comparable.

35 45 55 65 75 85 95

0

5

10

15

20

25

Time

Fre

qu

ency

Mean = 60.6 min.

Mean = 60.6 min.SD = 9.6 min.

SD = 9.6 min.

Z-Score = (Time-60.6)/9.6

-2 0 2

41 61 79

Mean = 0SD = 1

24

Outcome Measure in Case StudyGHA = Global Hyperactivity Aggregate

For each child at each time:Z1 = Z-Score for ADHD from TeachersZ2 = Z-Score for WWP from ParentsZ3 = Z-Score for ADHD in ClassroomZ4 = Z-Score for Conner on Computer, where weekly score=changes from T0All have higher values ↔ more hyperactive.Z’s make each measure scaled similarly.

GHA= Mean of Z1, Z2, Z3, Z4 25

Summary Statistics:Rule of Thumb

For bell-shaped distributions of data (“normally” distributed):

• ~ 68% of values are within mean ±1 SD

• ~ 95% of values are within mean ±2 SD “(Normal) Reference Range”

• ~ 99.7% of values are within mean ±3 SD26

876543210

150

100

50

0

Intensity

Fre

qu

en

cyHistograms: Not OK for Typical Analyses

Skewed

Need to transform intensity to another scale,

e.g. Log(intensity)

1207020

20

10

0

Tumor Volume

Fre

quen

cy

Multi-Peak

Need to summarize with percentiles, not

mean.27

Look at the Data for Analysis Requirements

What do we look for?

In Scatter Plots (two variables): Ideal: Football-shaped; ellipse.

Potential Problems:• Outliers.• Funnel-shaped.• Gap with no values for one or both variables. 28

Example Scatter Plot: OK for Typical Correlation Analyses

29

Summary Statistics:Two Variables (Correlation)

• Always look at scatterplot.• Correlation, r, ranges from -1 (perfect

inverse relation) to +1 (perfect direct). Zero=no relation.

• Specific to the ranges of the two variables.• Typically, cannot extrapolate to populations

with other ranges.• Measures association, not causation.

We will examine details in Session 5.30

Correlation Depends on Range of Data

Graph B contains only the points from graph A that are in the ellipse.

Correlation is reduced in graph B.

Thus: correlation between two quantities may be quite different in different study populations.

BA

31

Correlation and Measurement Precision

A lack of correlation for the subpopulation with 5<x<6 may be due to inability to measure x and y well.

Lack of evidence of association is not evidence of lack of association.

B

A

r=0 for s

Boverall

5 6

12

10

32

Confidence Interval (CI)• How well your sample mean(m) reflects

the true( or population) mean How confident? 95%?

• A confidence interval (CI) is one of inferential statistics that estimate the true unknown parameter using interval scales.

33

Confidence Interval for Population Mean

95% Reference range or “Normal Range”, is

sample mean ± 2(SD) _____________________________________

95% Confidence interval (CI) for the (true, but unknown) mean for the entire population is

sample mean ± 2(SD/√N)

SD/√N is called “Std Error of the Mean” (SEM)34

Confidence Interval: Case Study

Confidence Interval:

-0.14 ± 1.99(1.04/√73) =

-0.14 ± 0.24 → -0.38 to 0.10

Table 2

Normal Range:

-0.14 ± 1.99(1.04) =

-0.14 ± 2.07 → -2.21 to 1.93

0.13 -0.12 -0.37

Adjusted CI

close to

35