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Biostatistics in Practice
Session 2:
Summarization of Quantitative
Information
Peter D. Christenson
Biostatistician
http://research.LABioMed.org/Biostat
Topics for this Session
Experimental Units
Independence of Measurements
Graphs: Summarizing Results
Graphs: Aids for Analysis
Summary Measures
Confidence Intervals
Prediction Intervals
Most Practical from this Session
Geometric Means
Confidence Intervals
Reference Ranges
Justify Analysis Methods from Graphs
Experimental Units
_____
Independence of
Measurements
Units and Independence
Experiments 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.
Other methods can incorporate the lack of
independence.
Example 1: Units and Independence
Ten mice receive treatment A and a blood sample is
obtained from each one. The same is done for 10
mice receiving treatment B.
A protein concentration is measured in each of the 20
samples and an appropriate summary (average?,
min?, %>10 nmol/ml?) is compared between
groups A and B.
The experimental unit is a mouse.
Each of the 20 numbers are independent.
A “basic” analysis requiring independence is valid.
Example 2: Units and Independence
Ten mice receive treatment A, each is bled, and
each blood sample is divided into 3 aliquots. The
same is done for 10 mice receiving treatment B.
A protein concentration is measured in each of the 60
aliquots.
The experimental unit is a mouse.
The 60 numbers are not independent. The 2nd and 3rd
results for a sample are less informative than the 1st.
A “basic” analysis requiring independence is not valid
unless a single number is used for each triplicate,
giving 10+10 independent values.
Experimental Units in Case Study
Experimental Units in Case Study
A unit is a single child.
Results from one child's three diets are not
independent. The three results are probably
clustered around a set-point for that child.
The analysis must incorporate this possible
correlated clustering. If the software is just
given the 3x140 outcomes without
distinguishing the individual children, the
analysis would be wrong.
Modified Case Study
Suppose an educational study used teaching
method A in some schools and method B in
others. The outcome is a test score later.
The experimental unit is a school.
Outcomes within a school are probably not
independent. It would be wrong to use the
method we will discuss in the next session (ttest) to compare the mean score among
students given method A to those given B.
Another Example
You apply treatment A to one pregnant mouse and
measure a hormone in its offspring. Same for B.
Suppose the results are:
A Responses: 100, 98, 102, 99, 101
B Responses: 10, 8, 12, 9, 11
Can we conclude responses are greater under
treatment A than under B?
Another Example
You apply treatment A to one pregnant mouse and
measure a hormone in its offspring. Same for B.
Suppose the results are:
A Responses: 100, 98, 102, 99, 101
B Responses: 10, 8, 12, 9, 11
No. The one mouse given A might have responded the
same if given B. Same for the one mouse given B.
Five offspring provide little independent information
over 1 offspring. Each treatment was essentially only
tested once.
Graphs:
Summarizing Results
Common Graphical Summaries
Graph Name
Y-axis
X-axis
Histogram
Count or %
Category
Scatterplot
Continuous
Continuous
Dot Plot
Continuous
Category
Box Plot
Percentiles
Category
Line Plot
Mean or value
Category
Kaplan-Meier
Probability
Time
Many of the following examples are from StatisticalPractice.com
Data Graphical Displays
Histogram
Summarized*
Scatter plot
Raw Data
* Raw data version is a stem-leaf plot. We will see one later.
Data Graphical Displays
Dot Plot
Raw Data
Box Plot
Summarized
Data Graphical Displays
Line or Profile Plot
Summarized - bars can represent various types of ranges
Data Graphical Displays
Kaplan-Meier Plot
0.75
1.00
Kaplan-Meier survival estimate
0.50
Probability of Surviving 5
years is 0.35
0.00
0.25
This is not necessarily
35% of subjects
0
5
10
Years
15
20
Graphs:
Aids for Analysis
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.
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.
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.
Histograms: Not OK for Typical Analyses
Skewed
Multi-Peak
150
Frequency
20
100
50
0
10
0
0
1
2
3
4
5
6
7
8
Intensity
Need to transform
intensity to another scale,
e.g. Log(intensity)
20
70
120
Tumor Volume
Need to summarize
with percentiles, not
mean.
Histograms: Not OK for Typical Analyses
Truncated Values
Outliers
Undetectable in 28 samples (<LLOQ)
100
60
Frequency
50
40
30
20
50
10
0
0
0
LLOQ
5
10
Assay Result
Need to use percentiles
for most analyses.
0
4
8
Expression LogRatio
Need to use median,
not mean, and
percentiles.
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.
Example Scatter Plot: OK for Typical
Analyses
Scatter Plot: Not OK for Typical Analyses
Gap and Outlier
150
Funnel-Shaped
All Subjects:
r = 0.54 (95% CI: 0.27 to 0.73)
p = 0.0004
100
EPO > 300:
r = -0.04 (95% CI: -0.96 to 0.96)
p = 0.96
EPO < 150:
r = 0.23 (95% CI: -0.11 to 0.52)
p = 0.17
50
0
0
100
200
300
400
EPO
Ferber et al, Amer J Obstet
Gyn 2004;190:1473-5.
Consider analyzing
subgroups.
Ott, Amer J Obstet Gyn
2005;192:1803-9.
Should transform y-value
to another scale, e.g.
logarithm.
Summary Measures
Common Summary Measures
Mean and SD or SEM
Geometric Mean
Z-Scores
Correlation
Survival Probability
Risks, Odds, and Hazards
Summary Statistics: One Variable
Data Reduction to a few summary measures.
Basic: Need Typical Value and Variability of
Values
Typical Values (“Location”):
• Mean for symmetric data.
• Median for skewed data.
• Geometric mean for some skewed data
details in later slides.
-
Summary Statistics:
Variation in Values
• Standard Deviation, SD =~ 1.25 *(Average
|deviation| of values from their mean).
• Standard, convention, non-intuitive values.
• SD of what? E.g., SD of individuals, or of
group means.
• Fundamental, critical measure for most
statistical methods.
Examples: Mean and SD
A
B
15
25
Frequency
20
15
10
10
5
5
0
0
35
45
55
65
75
85
95
10
20
OD
Time
Mean = 60.6 min.
15
SD = 9.6 min.
Mean = 15.1
SD = 2.8
Note that the entire range of data in A is about 6SDs
wide, and is the source of the “Six Sigma” process
used in quality control and business.
Examples: Mean and SD
Skewed
Multi-Peak
150
Frequency
20
100
50
0
10
0
0
1
2
3
4
5
6
7
8
20
SD = 1.1 min.
120
Tumor Volume
Intensity
Mean = 1.0 min.
70
Mean = 70.3
SD = 22.3
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 SD
Summary Statistics:
Geometric means
Commonly used for skewed data.
1. Take logs of individual values.
2. Find, say, mean ±2 SD → mean and
(low, up) of the logged values.
3. Find antilogs of mean, low, up. Call
them GM, low2, up2 (back on original
scale).
4. GM is the “geometric mean”. The
interval (low2,up2) is skewed about
GM (corresponds to graph).
[See next slide]
Geometric Means
These are flipped
histograms rotated
90º, with box plots.
Any log base can
be used.
≈ 909.6
≈ 102.8
≈ 11.6
GM
= exp(4.633)
= 102.8
low2 = exp(4.633-2*1.09)
= 11.6
upp2 = exp(4.633+2*1.09)
= 909.6
Confidence Intervals
Reference ranges - or Prediction Intervals -are
for individuals.
Contains values for 95% of individuals.
_____________________________________
Confidence intervals (CI) are for a summary
measure (parameter) for an entire population.
Contains the (still unknown) summary measure
for “everyone” with 95% certainty.
Z- Score = (Measure - Mean)/SD
25
20
Frequency
Standardizes a
measure to
have
mean=0
and SD=1.
Mean = 60.6 min.
SD = 9.6 min.
15
10
5
0
35
45
41
65
Time
61
75
85
95
79
Mean = 0
SD = 1
25
20
Frequency
Z-scores make
different Mean = 60.6 min.
measures SD = 9.6 min.
comparable.
55
15
10
5
0
35
-245
55
0
65
2
75
85
95
Time
Z-Score = (Time-60.6)/9.6
Outcome Measure in Case Study
GHA = Global Hyperactivity Aggregate
For each child at each time:
Z1 = Z-Score for ADHD from Teachers
Z2 = Z-Score for WWP from Parents
Z3 = Z-Score for ADHD in Classroom
Z4 = Z-Score for Conner on Computer
All have higher values ↔ more hyperactive.
Z’s make each measure scaled similarly.
GHA= Mean of Z1, Z2, Z3, Z4
Confidence Interval for Population Mean
95% Reference range - or Prediction Interval - 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)
Confidence Interval: More Details
Confidence interval (CI) for the (true, but unknown)
mean for the entire population is
95%, N=100:
95%, N= 30:
90%, N=100:
99%, N=100:
sample mean ± 1.98(SD/√N)
sample mean ± 2.05(SD/√N)
sample mean ± 1.66(SD/√N)
sample mean ± 2.63(SD/√N)
If N is small (N<30?), need normally, bell-shaped, data
distribution. Otherwise, skewness is OK. This is not
true for the PI, where percentiles are needed.
Confidence Interval: Case Study
Table 2
Confidence Interval:
Adjusted
CI
0.13
-0.12
-0.37
-0.14 ± 1.99(1.04/√73) =
-0.14 ± 0.24 → -0.38 to 0.10
Normal Range:
-0.14 ± 1.99(1.04) =
-0.14 ± 2.07 → -2.21 to 1.93
close to
CI for the Antibody Example
GM
= exp(4.633)
= 102.8
low2 = exp(4.633-2*1.09)
= 11.6
So, there is 95%
assurance that an
individual is
between 11.6 and
909.6, the PI.
upp2 = exp(4.633+2*1.09)
= 909.6
GM
= exp(4.633)
= 102.8
low2 = exp(4.633-2*1.09 /√394)
= 92.1
upp2 = exp(4.633+2*1.09 /√394)
= 114.8
So, there is 95%
certainty that the
population mean
is between 92.1
and 114.8, the CI.
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.
Correlation Depends on Range of Data
A
B
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.
Correlation and Measurement Precision
A
B
overall
12
10
r=0 for
5
s
6
B
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.
Summary Statistics: Survival Probability
Example: 100 subjects start a study. Nine subjects
drop out at 2 years and 7 drop out at 4 yrs and 20, 20,
and 17 died in the intervals 0-2, 2-4, 4-5 yrs.
1.00
0.75
0.50
0.25
The 2-4 interval has 51/71
surviving; 4-5 has 27/44
surviving.
Actually uses finer
subdivisions than 0-2,
2-4, 4-5 years, with
exact death times.
0.00
Survival Probability
Then, the 0-2 yr interval
has 80/100 surviving.
Kaplan-Meier survival estimate
0
So, 5-yr survival prob is
(80/100)(51/71)(27/44) =
0.35.
5
10
Years
15
20
Don’t know vital
status of 16 subjects
at 5 years.
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 PersonDays