Biostatistics in Practice

1
Biostatistics in Practice
Session 2 Summarization of Quantitative
Information
Peter D. Christenson Biostatistician http//rese
arch.LABioMed.org/Biostat
2
Topics for this Session
Experimental Units Independence of Measurements
Graphs Summarizing Results Graphs Aids for
Analysis Summary Measures Confidence
Intervals Prediction Intervals
3
Most Practical from this Session
Geometric Means Confidence Intervals Reference
Ranges Justify Analysis Methods from Graphs
4
Experimental Units_____Independence of
Measurements
5
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.
6
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?, gt10 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.
7
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
1010 independent values.
8
Experimental Units in Case Study
9
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.
10
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 (t-test) to compare the mean score among
students given method A to those given B.
11
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?
12
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.
13
GraphsSummarizing Results
14
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
15
Data Graphical Displays
Histogram
Scatter plot
Raw Data
Summarized
Raw data version is a stem-leaf plot. We will
see one later.
16
Data Graphical Displays
Dot Plot
Box Plot
Raw Data
Summarized
17
Data Graphical Displays
Line or Profile Plot
Summarized - bars can represent various types of
ranges
18
Data Graphical Displays
Kaplan-Meier Plot
Probability of Surviving 5 years is 0.35
This is not necessarily 35 of subjects
19
GraphsAids for Analysis
20
Graphical Aids for Analysis
Most statistical analyses involve
modeling. Parametric methods (t-test, ANOVA, ?2)
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
Histograms Not OK for Typical Analyses
Skewed
Multi-Peak
Need to transform intensity to another scale,
e.g. Log(intensity)
Need to summarize with percentiles, not mean.
24
Histograms Not OK for Typical Analyses
Truncated Values
Outliers
Undetectable in 28 samples (ltLLOQ)
LLOQ
Need to use percentiles for most analyses.
Need to use median, not mean, and percentiles.
25
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.

26
Example Scatter Plot OK for Typical Analyses
27
Scatter Plot Not OK for Typical Analyses
Gap and Outlier
Funnel-Shaped
Ott, Amer J Obstet Gyn 20051921803-9.
Ferber et al, Amer J Obstet Gyn 20041901473-5.
Should transform y-value to another scale, e.g.
logarithm.
Consider analyzing subgroups.
28
Summary Measures
29
Common Summary Measures
Mean and SD or SEMGeometric MeanZ-ScoresCorr
elationSurvival ProbabilityRisks, Odds, and
Hazards
30
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.

31
Summary StatisticsVariation 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.

32
Examples Mean and SD
A
B
Mean 60.6 min.
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.
33
Examples Mean and SD
Skewed
Multi-Peak
SD 1.1 min.
Mean 70.3
Mean 1.0 min.
SD 22.3
34
Summary StatisticsRule 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

35
Summary Statistics Geometric means

• Commonly used for skewed data.
• Take logs of individual values.
• Find, say, mean 2 SD ? mean and (low, up) of the
  logged values.
• Find antilogs of mean, low, up. Call them GM,
  low2, up2 (back on original scale).
• GM is the geometric mean. The interval
  (low2,up2) is skewed about GM (corresponds to
  graph).
• See next slide

36
Geometric Means
These are flipped histograms rotated 90º, with
box plots. Any log base can be used.
GM exp(4.633) 102.8 low2
exp(4.633-21.09) 11.6 upp2
exp(4.63321.09) 909.6
909.6
102.8
11.6
37
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.
38
Z- Score (Measure - Mean)/SD
Standardizes a measure to have mean0 and SD1.
Z-scores make different measures comparable.
Mean 60.6 min.SD 9.6 min.
41 61 79
Mean 0SD 1
Mean 60.6 min.
SD 9.6 min.
-2 0 2
Z-Score (Time-60.6)/9.6
39
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. Zs make each measure scaled
similarly. GHA Mean of Z1, Z2, Z3, Z4
40
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/vN) SD/vN is called Std
Error of the Mean (SEM)
41
Confidence Interval More Details
Confidence interval (CI) for the (true, but
unknown) mean for the entire population is 95,
N100 sample mean 1.98(SD/vN) 95, N
30 sample mean 2.05(SD/vN) 90, N100 sample
mean 1.66(SD/vN) 99, N100 sample mean
2.63(SD/vN) If N is small (Nlt30?), need
normally, bell-shaped, data distribution.
Otherwise, skewness is OK. This is not true for
the PI, where percentiles are needed.
42
Confidence Interval Case Study
Table 2
Adjusted CI
0.13 -0.12 -0.37
Confidence Interval -0.14 1.99(1.04/v73)
-0.14 0.24 ? -0.38 to 0.10
close to
Normal Range -0.14 1.99(1.04) -0.14 2.07 ?
-2.21 to 1.93
43
CI for the Antibody Example
So, there is 95 assurance that an individual is
between 11.6 and 909.6, the PI.
GM exp(4.633) 102.8 low2
exp(4.633-21.09) 11.6 upp2
exp(4.63321.09) 909.6
So, there is 95 certainty that the population
mean is between 92.1 and 114.8, the CI.
GM exp(4.633) 102.8 low2
exp(4.633-21.09 /v394) 92.1 upp2
exp(4.63321.09 /v394) 114.8
44
Summary StatisticsTwo Variables (Correlation)

• Always look at scatterplot.
• Correlation, r, ranges from -1 (perfect inverse
  relation) to 1 (perfect direct). Zerono
  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.

45
Correlation Depends on Range of Data
B
A
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.
46
Correlation and Measurement Precision
A
B
overall
12 10
r0 for s
5 6
B
A lack of correlation for the subpopulation with
5ltxlt6 may be due to inability to measure x and y
well. Lack of evidence of association is not
evidence of lack of association.
47
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.
Then, the 0-2 yr interval has 80/100
surviving. The 2-4 interval has 51/71 surviving
4-5 has 27/44 surviving. So, 5-yr survival prob
is (80/100)(51/71)(27/44) 0.35.
Actually uses finer subdivisions than 0-2, 2-4,
4-5 years, with exact death times.
Dont know vital status of 16 subjects at 5 years.
48
Summary StatisticsRelative Likelihood of an
Event
Compare groups A and B on mortality. Relative
Risk ProbADeath / ProbBDeath where
ProbDeath Deaths per 100 Persons Odds Ratio
OddsADeath / OddsBDeath where Odds
ProbDeath / ProbSurvival Hazard Ratio
IADeath / IBDeath where I Incidence
Deaths per 100 PersonDays