Transcript Session Slides/Handout

Biostatistics in Practice
Session 2:
Summarization of Quantitative
Information
Peter D. Christenson
Biostatistician
http://gcrc.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 Methods from Graphs
Experimental Units
_____
Independence of
Measurements
Statistical Independence
Experimental units are the smallest independent
entities for addressing a scientific question in an
analysis of an experiment.
“Independent” refers to the measurement that is
made and the question, not the units.
Definition: If knowledge of the value for a unit
does not provide information about another
unit’s value, given other factors (and the overall
mean) in the analysis of the experiment, then the
units are independent for this measurement.
There may be a hierarchy of units.
Importance of Independence
Many basic statistical methods require that
measurements are independent for the
analysis to be valid.
Other methods can incorporate the lack of
independence.
There can be some subjectivity regarding
independence. Statistical methods use
models. Models can be wrong.
Example: Units and Independence
Ten mice receive treatment A, each is bled, and
blood samples are each divided into 3 aliquots.
The same is done for 10 mice on treatment B.
1. A serum hormone is measured in the 60
aliquots and compared between A and B.
The aliquots for a mouse are not independent.
The unit is a mouse.
A summary statistic from a mouse’s 3 aliquots
(e.g., maximum or mean) are independent.
N=10 and 10, not 30 and 30.
Example, Continued
2. One of the 30 A aliquots is further divided into
25 parts and 5 different in vitro challenges are
each made to a random set of 5 of the parts.
The same is done for a single B aliquot.
For this challenge experiment, each part is a
unit, the values of challenge response are
independent, and N=25+25.
For comparing A and B, there are only N=1+1
experimental units, the two mice.
Experimental Units in Case Study
Experimental Units in Case Study
There is a nested hierarchy of several
"levels" of data: Schools, children within
the schools, and diets received by every
child. What would you use for the "N"
for this study?
Which outcomes do you intuitively think
are correlated (in common language)?
Results from one child's three diets?
Results from children in the same
school? Schools?
Experimental Units in Case Study
N = Number of children
Results from one child's three diets cannot
be modeled as independent.
Results from children in the same school
also could be “correlated” (dependent).
They can be modeled as independent, if the
effect of school is included in the analysis.
Knowing one child’s score and the school
mean gives no info on another child’s
score.
Units and Analysis in the Case Study
N = Number of children
Analysis:
This method is a complex generalization of
methods we discuss in Session 3.
For any method, though, you need to inform
the software of the correct experimental
units. For some experiments, it is obvious
and implicit.
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 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
Box Plot
Raw Data
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 nonparametric 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 yvalue 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
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
Mean = 60.6 min.
different SD = 9.6 min.
measures
comparabl
e.
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”, if subjects normal, 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
Prediction Interval:
-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