Biostatistics in Practice

Peter D. ChristensonBiostatistician

http://gcrc.LABioMed.org/Biostat

Session 2: Summarization of Quantitative

Information

Readings for Session 2from StatisticalPractice.com

Units: E.g., Mouse or litter, Not e.g., mg/ml.

RAW data, preferably.

Summary: Particular method depends on structure in the raw data.

Bell curve: often “natural”.

Want ranges (for what?).

• Units of Analysis“Experimental units”

• Look at the data

• Summary statistics• Typical values and their variability• Correlation

• Normal distribution

• Confidence intervals

Units of Analysis

Go over this entire reading at StatisticalPractice.com.

The author states that some students are “more similar” to each other than are other students, or some students are “independent”. What does this mean?

“Independent” really refers to the measurement that is made, not the units such as students or classes or schools.

If knowledge of the value for a student does not change the likelihood of another student’s value, given class means, then the students are independent for this measurement.

Would students from the same class likely be independent on height? How about on knowing some academic fact, such as what a case-control study is?

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 unit is a mouse, their means from 3 aliquots each are independent, N=10+10, and aliquots for a mouse are not independent.

2. One of the 30 A aliquots is further divided into 25 parts and 5 different in vitro challenges each made to 5 of the parts. The same is done for a single B aliquot. For the challenge experiment, each part is a unit, their values are independent, and N=25+25. For comparing A and B, there are only N=1+1 units, the two mice.

Look at the Data

Statistical methods depend on the “form” of a set of data, which can be assessed with some common useful graphics:

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

Examples on following slides are from StatisticalPractice.com

Data Graphical Displays

Histogram Scatter plot

Raw DataSummarized*

*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 - “antennae” can represent various ranges

Week

Look at the Data, Continued

What do we look for?

Histograms: Ideal: Symmetric, bell-shaped.Potential Problems:• Skewness.• Multiple peaks.• Many values at, say, 0, and bell-shaped otherwise• Outliers.

Scatter plots: Ideal: Football-shaped; ellipse.Potential Problems:• Outliers.• Funnel-shaped.• Gap with no values for one or both variables.

Example Histogram: OK for Default* Analyses

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

*software default, typical mean, SD, confidence intervals.

876543210

150

100

50

0

Intensity

Freq

uenc

yHistograms: Not OK for Default Analyses

Skewed

Need to transform intensity to another

scale, e.g. Log(intensity)

1207020

20

10

0

Tumor Volume

Freq

uenc

y

Multi-Peak

Need to summarize with percentiles, not mean.

Histograms: Not OK for Default Analyses

Truncated Values

Need to use percentiles for most analyses.

Outliers

Need to use median, not mean, and percentiles.

1050

60

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40

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Assay Result

Freq

uenc

y

LLOQ

Undetectable in 28 samples (<LLOQ)

840

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Expression LogRatio

Freq

uenc

y

Example Scatter Plot: OK for Typical Analyses

Scatter Plot: Not OK for Typical Analyses

Gap and Outlier

Consider analyzing subgroups.

Funnel-Shaped

Could transform y-value to another scale, e.g.

logarithm.

0 100 200 300 400

0

50

100

150

EPO

nRB

C C

ount

All Subjects:r = 0.54 (95% CI: 0.27 to 0.73)p = 0.0004

EPO < 150:r = 0.23 (95% CI: -0.11 to 0.52)p = 0.17

EPO > 300:r = -0.04 (95% CI: -0.96 to 0.96)p = 0.96

Ott, Amer J Obstet Gyn 2005;192:1803-9.Ferber et al, Amer J Obstet Gyn 2004;190:1473-5.

Summary Statistics: I

Typical Values (“Location”):• Mean for symmetric data.• Median for skewed data.• Geometric mean for some skewed data (see later slide).

Variation in Values (“Spread”. Standard deviation=SD):• Standard, convention, non-intuitive values.• SD=~ Avg. deviation of values from their mean.• SD of what? E.g., SD of individuals, or of group means.• Fundamental, critical measure for most statistical methods.

See graphs in reading for how mean and SD change if units of measurement change, e.g., nmoles to mg:

• Mean (a + b*X) = a + b*Mean(X)• SD (a + b*X) = b*SD(X)

Examples: Mean and SD

Mean = 60.6 min.

Note that the entire range of data in A is about 6SDs wide, and is the source of the “Six Sigma” process used in business quality control.

95857565554535

25

20

15

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Time

Freq

uenc

y

SD = 9.6 min.

201510

15

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OD

Freq

uenc

y

Mean = 15.1 min. SD = 2.8 min.

A B

876543210

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Intensity

Freq

uenc

yExamples: Mean and SD

Skewed

1207020

20

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Tumor Volume

Freq

uenc

y

Multi-Peak

Mean = 1.0 min.SD = 1.1 min. Mean = 70.3 min.

SD = 22.3 min.

Summary Statistics: IIRule 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• ~ 99.7% of values are within mean ±3 SD

Geometric means (see next slide):Used for some skewed data.1. Take logs of individual values.2. Find, say, mean ±2 SD → mean (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).

Geometric MeansThese are flipped histograms rotated 90º, and box plots.

Any base for the log transformation gives a symmetric distribution. [Ln used here; log10 gives same GM and bounds.]

=~ 909.6

=~ 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

=~ 102.8

Summary Statistics: III (Correlation)

We will examine calculation details later.

• With 2 continuous measures, always look at scatterplot.• See graphs in readings for values ranging from -1 (perfectly

inverse relation) to +1 (perfectly direct). Zero=no relation.• Measures linear association.• Very sensitive to outliers.• Specific to the ranges of the two variables.• Typically, cannot extrapolate to populations with other

ranges.• Subgroups may not have the same correlation; in fact, they

could have the opposite association (ecological fallacy).• Special correlations are used for non-symmetric data.• Measures association, not causation.

Correlation Depends on Ranges of X and Y

Graph B contains only the graph A points in the ellipse.

Correlation is reduced in graph B.

Thus: correlations for the same quantities X and Y may be quite different in different study populations.

BA

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.

Again, lack of evidence is not evidence of “lack” (of association in this setting).

BA

r=0 for s

B

overall

5 6

12

10

Confidence Intervals: I• See beginning of reading for the goal of confidence intervals.

• CIs are not about individuals, but rather about populations, i.e., groups of individuals.

• A mean from a sample estimates the mean of the entire population.

• 95% CI for the mean is a range of values we're 95% sure contains the unknown mean.

• Reading example: N=40 non-smokers. Vitamin C mean±2SD is 90±2*35 = 20 to 160 = “normal range”. Our estimate of the unknown mean for all non-smokers is 90, but how confident are we about that estimate? Need a ±range for it that we are 95% confident contains the unknown mean.

Confidence Intervals: II• Can calculate a CI for any unknown parameter.

• Typical 95% CI for a mean is roughly: mean ± 2SD/√N.• Larger SD → wider CI.• Larger N → narrower CI.• More confidence → wider CI.• For reading example, CI=~ 90 ± 2*35/√40 = 78 to 102.

• I am being sloppy with terminology. The underlined mean above is the always-to-be-unknown mean for the population (everyone). The other mean, before ±, is the mean that is calculated from the sample of N, denoted X-bar, and it estimates the unknown mean, denoted μ.

• Note explicit use of N; correct unit of analysis is critical. What if we measured vitamin C on 10 days for each subject?

Confidence vs. Prediction Intervals

• Typical 95% CI for a mean is roughly: mean ± 2SD/√N.• Recall that this CI is the range of values we're 95% sure

contains the unknown mean for “everyone”.

• What about (normal) ranges for individuals?• This is often called a prediction interval (PI) = “normal”

range = reference range.• 95% of individuals fall in a 95% PI.• 95% chance that an individual falls in a 95% PI.• Typical 95% PI for an individual is roughly: mean ± 2SD.

• With large N (how large? often N>30 is used), do not need bell-shaped data distribution for the CI, but that shape IS needed for the PI, regardless of N. Otherwise, we use percentiles for normal ranges.

CI and PI for the Antibody Example

=~ 909.6

=~ 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

=~ 102.8

So, there is 95% assurance that an individual is between 11.6 and 909.6, the PI.

So, there is 95% assurance that the pop- ulation mean is between 92.1 and 114.8, the CI.

GM = exp(4.633) = 102.8

lower = exp(4.633-2*1.09/√394) = 92.1

upper = exp(4.633+2*1.09/√394) = 114.8