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Biostatistics in Practice. Session 2: Summarization of Quantitative Information. Peter D. Christenson Biostatistician http://gcrc. /Biostat. Topics for this Session. Experimental Units Independence of Measurements Graphs: Summarizing Results Graphs: Aids for Analysis

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Biostatistics in practice
Biostatistics in Practice

Session 2:

Summarization of Quantitative Information

Peter D. Christenson


Topics for this session
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
Most Practical from this Session

Geometric Means

Confidence Intervals

Reference Ranges

Justify Methods from Graphs

Experimental units independence of measurements
Experimental Units_____Independence of Measurements

Statistical independence
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
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
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.

  • 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
Example, Continued

  • 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

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


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
Graphs:Summarizing Results

Common graphical summaries
Common Graphical Summaries

Graph NameY-axisX-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

Data graphical displays
Data Graphical Displays


Scatter plot

Raw Data


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

Data graphical displays1
Data Graphical Displays

Dot Plot

Box Plot

Raw Data


Data graphical displays2
Data Graphical Displays

Line or Profile Plot

Summarized - bars can represent various types of ranges

Data graphical displays3
Data Graphical Displays

Kaplan-Meier Plot

Probability of Surviving 5 years is 0.35

This is not necessarily 35% of subjects

Graphs aids for analysis
Graphs:Aids for Analysis

Graphical 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
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
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



Need to transform intensity to another scale, e.g. Log(intensity)

Need to summarize with percentiles, not mean.

Histograms: Not OK for Typical Analyses

Truncated Values


Undetectable in 28 samples (<LLOQ)


Need to use percentiles for most analyses.

Need to use median, not mean, and percentiles.

Look at the data for analysis requirements1
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.

Scatter Plot: Not OK for Typical Analyses

Gap and Outlier


Ott, Amer J Obstet Gyn 2005;192:1803-9.

Ferber et al, Amer J Obstet Gyn 2004;190:1473-5.

Should transform y-value to another scale, e.g. logarithm.

Consider analyzing subgroups.

Common summary measures
Common Summary Measures

Mean and SD or SEMGeometric MeanZ-ScoresCorrelationSurvival ProbabilityRisks, Odds, and Hazards

Summary statistics one variable
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
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



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.

Examples: Mean and SD



SD = 1.1 min.

Mean = 70.3

Mean = 1.0 min.

SD = 22.3

Summary statistics rule of thumb
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
    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]

  • Geometric means
    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(4.633-2*1.09)

    = 11.6

    upp2 = exp(4.633+2*1.09)

    = 909.6

    ≈ 909.6

    ≈ 102.8

    ≈ 11.6

    Confidence intervals
    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
    Z- Score = (Measure - Mean)/SD

    Mean = 60.6 min.SD = 9.6 min.

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

    Z-scores make different measures comparable.

    41 61 79

    Mean = 0SD = 1

    -2 0 2

    Z-Score = (Time-60.6)/9.6

    Outcome measure in case study
    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
    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: More Details

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

    95%, N=100: sample mean ± 1.98(SD/√N)

    95%, N= 30: sample mean ± 2.05(SD/√N)

    90%, N=100: sample mean ± 1.66(SD/√N)

    99%, N=100: 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
    Confidence Interval: Case Study

    Table 2

    Adjusted CI

    0.13 -0.12 -0.37

    Confidence Interval:

    -0.14 ±1.99(1.04/√73) =

    -0.14 ± 0.24 → -0.38 to 0.10

    close to

    Prediction Interval:

    -0.14 ±1.99(1.04) =

    -0.14 ± 2.07 → -2.21 to 1.93

    Ci for the antibody example
    CI for the Antibody Example

    GM = exp(4.633)

    = 102.8

    low2 = exp(4.633-2*1.09)

    = 11.6

    upp2 = exp(4.633+2*1.09)

    = 909.6

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

    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(4.633-2*1.09 /√394)

    = 92.1

    upp2 = exp(4.633+2*1.09 /√394)

    = 114.8

    Summary statistics two variables correlation
    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
    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.

    Correlation and measurement precision
    Correlation and Measurement Precision






    r=0 for s

    5 6


    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.

    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.

    Don’t know vital status of 16 subjects at 5 years.

    Summary statistics relative likelihood of an event
    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