Lecture 1: Measurements, Statistics,
Probability, and Data Display
Karen Bandeen-Roche, PhD
Department of Biostatistics
Johns Hopkins University
July 11, 2011
Introduction to Statistical
Measurement and Modeling
What is statistics?
The study of …
(i.) … populations
(ii.) …variation
(iii.) … methods of the reduction of data.
“The original meaning of the word … suggests
that it was the study of populations of human
beings living in political union.”
Sir R. A. Fisher
What is statistics?
 “… Statistical Science [is] the particular aspect
of human progress which gives the 20th century
its special character…. It is to the statistician
that the present age turns for what is most
essential in all its more important activities.”
Sir R. A. Fisher
What is statistics?
Less complimentary views
 “Science is difficult. You need mathematics and
statistics, which is dull like learning a language.”
Richard Gregory
 “There are three kinds of lies: lies, damned lies
and statistics.”
Mark Twain, quoting Disraeli
What is statistics?
 Statistics in concerned with METHODS for
COLLECTING & DESCRIBING DATA and
then for ASSESSING STRENGTH OF
EVIDENCE in DATA FOR/AGAINST
SCIENTIFIC IDEAS!”
Scott L. Zeger
What is statistics?
 the art and science of gathering, analyzing,
and making inferences from data.”
Encyclopaedia Britannica
Poetry
Music
Statistics
Mathematics
Physics
What is biostatistics?
 The science of learning from biomedical data
involving appreciable variability or
uncertainty.
Amalgam
Data examples
 Osteoporosis screening
 Importance: Osteoporosis afflicts millions of older
adults (particularly women) worldwide
 Lowers quality of life, heightens risk of falls etc.
 Scientific question: Can we detect osteoporosis earlier
and more safely?
 Method: ultrasound versus dual photon
absorptiometry (DPA) tried out on 42 older women
 Implications: Treatment to slow / prevent onset
Osteoporosis data
DPA scores by osteoporosis groups
0.6
1600
0.7
1700
0.8
1800
0.9
1.0
1900
1.1
2000
1.2
Ultrasound scores by osteoporosis groups
control
case
control
case
Data examples
 Temperature modeling
 Importance: Climate change is suspected. Heat waves,
increased particle pollution, etc. may harm health.
 Scientific question: Can we accurately and precisely
model geographic variation in temperature?
 Method: Maximum January-average temperature over
30 years in 62 United States cites
 Implications: Valid temperature models can support
future policy planning
United States temperature map
http://green-enb150.blogspot.com/2011/01/isorhythmic-map-united-states-weather.html
Modeling geographical variation:
Latitude and Longitude
http://www.enchantedlearning.com/usa/activity/latlong/
Temperature data
100
120
140
160
60
70
80
80
120
140
160
20
30
40
50
temp
40
50
80
100
longtude
20
30
latitude
20
30
40
50
60
70
80
20
30
40
50
Data examples
 Boxing and neurological injury
 Importance: (1) Boxing and sources of brain jarring
may cause neurological harm. (2) In ~1986 the IOC
considered replacing Olympic boxing with golf.
 Scientific question: Does amateur boxing lead to
decline in neurological performance?
 Method: “Longitudinal” study of 593 amateur boxers
 Implications: Prevention for brain injury from
subconcussive blows.
Boxing data
-20
-10
0
blkdiff
10
20
Lowess smoother
0
bandw idth = .8
100
200
blbouts
300
400
Data examples
 Temperature modeling
 Importance: Climate change is suspected. Heat waves,
increased particle pollution, etc. may harm health.
 Scientific question: Can we accurately and precisely
model geographic variation in temperature?
 Implications: Valid temperature models can support
future policy planning
Course objectives
 Demonstrate familiarity with statistical tools for
characterizing population measurement properties
 Distinguish procedures for deriving estimates from
data and making associated scientific inferences
 Describe “association” and describe its importance in
scientific discovery
 Understand, apply and interpret findings from
 methods of data display
 standard statistical regression models
 standard statistical measurement models
 Appreciate roles of statistics in health science
Basic paradigm of statistics
 We wish to learn about populations
 All about which we wish to make an inference
 “True” experimental outcomes and their mechanisms
 We do this by studying samples
 A subset of a given population
 “Represents” the population
 Sample features are used to infer population features
 Method of obtaining the sample is important
 Simple random sample: All population elements /
outcomes have equal probability of inclusion
Basic paradigm of statistics
Probability
Truth for
Population
Observed Value for a
Representative Sample
Statistical inference
Tools for description
 Populations
 Samples
 Probability
 Probability
 Parameters
 Statistics / Estimates
 Values, distributions
 Data displays
 Hypotheses
 Statistical tests
 Models
 Analyses
Probability
 Way for characterizing random experiments
 Experiments whose outcome is not determined
beforehand
 Sample space: Ω := {all possible outcomes}
 Event = A ⊆ Ω := collection of some outcomes
 Probability = “measure” on Ω
 Our course: measure of relative frequency of occurrence
 “Bayesian”: measure of relative belief in occurrence
Probability measures
 Satisfy following axioms:
i) P{Ω} = 1: reads "probability of Ω"
ii) 0 ≤ P{A} ≤ 1 for each A
> 0 = “can’t happen”; 1 = “must happen”
iii) Given disjoint events {Ak}, P{  kK1 Ak } = Σ P{Ak}
> “disjoint” = “mutually exclusive”; no two can
happen at the same time
Random variable (RV)
 A function which assigns numbers to outcomes of a
random experiment - X:Ω → ℝ
 Measurements
 Support:= SX = range of RV X
 Two fundamental types of measurements
 Discrete: SX is countable (“gaps” in possible values)
 Binary: Two possible outcomes
 Continuous: SX is an interval in ℝ
 “No gaps” in values
Random variable (RV)
 Example 1: X = number of heads in two fair coin tosses
 SX = {0,1,2}
 Example 2: Draw one of your names out of a hat.
X=age (in years) of the person whose name I draw.
 SX =
 Mass function:
Probability distributions
 Heuristic: Summarizes possible values of a random
variable and the probabilities with which each occurs
 Discrete X: Probability mass function = list exactly as
the heuristic: p:x → P(X=x)
 Example = 2 fair coin tosses:
 P{HH} = P{HT} = P{TH} = P{TT} = ¼
 Mass function:
x
p(x) = P(X=x)
0
¼
1
½
2
¼
y  {0,1,2}
0
Cumulative probability distributions
 F: x → P(X ≤ x) = cumulative distribution function CDF
 Discrete X: Probability mass function = list exactly as
the heuristic
 Example = 2 fair coin tosses:
x (-,0)
0
x (0,1)
1
x (1,2)
2
x (2,)
F(x)
0
1/4
1/4
3/4
3/4
1
1
> draw picture of p, F
p(x)
0
1/4
0
1/2
0
1/4
0
Cumulative probability distributions
 Example = 2 fair coin tosses:
 Notice: p(x) recovered as differences in values of F(x)
 Suppose x1≤ x2≤ … and SX = {x1, x2, …}
 p(xi) = F(xi) - F(xi-1), each i (define x0= -∞ and F(x0)=0)
Cumulative probability distributions
 Draw one of your names out of a hat. X=age (in
years) of the person whose name I draw
What about continuous RVs?
 Can we list the possible values of a random variable
and the probabilities with which each occurs?
 NO. If SX is uncountable, we can’t list the values!
 The CDF is the fundamental distributional quantity
 F(x) = P{X≤x}, with F(x) satisfying
i) a ≤ b ⇒ F(a) ≤ F(b);
ii) lim (b→∞) F(b) = 1;
iii) lim (b→-∞) F(b) = 0;
iv) lim (bn ↓ b) F(bn) = b
v) P{a<X≤b} = F(b) - F(a)
Two continuous CDFs
 “Exponential”
0.8
0.6
0.2
0.4
P(US<=score)
0.6
0.4
0.2
0.0
0.0
P(US<=score)
0.8
1.0
1.0
 “Normal”
1400
1600
1800
ultrasound scores
2000
2200
0
2000
4000
6000
ultrasound scores
8000
10000
Mass function analog: Density
 Defined when F is differentiable everywhere
(“absolutely continuous”)
 The density f(x) is defined as
lim(ε↓0) P{X є [x-ε/2,x+ε/2]}/ε
= lim(ε↓0) [F(x+ε/2)-F(x-ε/2)]/ε
= d/dy F(y) |y=x
 Properties
i) f ≥ 0
ii) P{a≤X≤b} = ∫ab f(x)dx
iii) P{XεA} = ∫A f(x)dx
iv) ∫-∞∞ f(x)dx = 1
Two densities
 “Exponential
4e-04
2e-04
3e-04
density
0.0020
0.0015
1e-04
0.0010
0.0005
0e+00
0.0000
density
0.0025
5e-04
0.0030
 “Normal”
1400
1600
1800
ultrasound score
2000
2200
0
2000
4000
6000
ultrasound score
8000
10000
Probability model parameters
 Fundamental distributional quantities:
 Location: ‘central’ value(s)
 Spread: variability
 Shape: symmetric versus skewed, etc.
Location and spread
(Different Locations)
(Different Spreads)
Probability model parameters
 Location
 Mean: E[X] = ∫ xdF(x) = µ
 Discrete FV: E[X] = ΣxεSX xp(x)
 Continuous case: E[X] = ∫ xf (x)dx
 Linearity property: E[a+bX] = a + bE[X]
 Physical interpretation: Center of mass
Probability model parameters
 Location
 Median
 Heuristic: Value so that ½ of probability weight above, ½
below

Definition: median is m such that F(m) ≥ 1/2, P{X≥m} ≥ ½
 Quantile ("more generally"...)

Definition: Q(p) = q: FX(q) ≥ p, P{X≥q} ≥ 1-p

Median = Q(1/2)
Probability model parameters
 Spread
 Variance: Var[X] = ∫(x-E[X])2dF(x) = σ2
 Shortcut formula:
E[X2]-(E[X])2
 Var[a+bX] = b2Var[X]
 Physical interpretation: Moment of inertia
 Standard deviation: SD[X] = σ
= √(Var[X])
 Interquartile range (IQR) = Q(.75) - Q(.25)
Pause / Recapitulation
 We learn about populations through representative
samples
 Probability provides a way to characterize populations
 Possibly unseen (models, hypotheses)
 Random experiment mechanisms
 We will now turn to the characterization of samples
 Formal: probability
 Informal: exploratory data analysis (EDA)
Describing samples
 Empirical CDF
 Given data X1,...,Xn, Fn(x) = {#Xi's ≤ x}/n
 Define indicator 1{A}:= 1 if A true
= 0 if A false
 ECDF = Fn = (1/n)Σ 1{Xi≤x}
= probability (proportion) of values ≤ x in sample
 Notice is real CDF with correct properties
 Mass function px = 1/n if x ε {X1,...,Xn};
= 0 otherwise.
Sample statistics
 Statistic = Function of data
 As defined in probability section, with F=Fn
 Mean = X n = ∫ xdFn(x)
= (1/n) Σ Xi.

1 n
 Variance = s2 = n  1  1 X i  X
i 1
 Standard deviation = s

2
Sample statistics - Percentiles
 “Order statistics” (sorted values):
 X(1) = min(X1,...,Xn)
 X(n) = max(X1,...,Xn)
 X(j) = jth largest value, etc.
 Median = mn = {x:Fn(x)≥1/2} and {x:PFn{X≥x}≥1/2
= X((n+1)/2) = middle if n odd;
= [X(n/2)+X(n/2+1)]/2 = mean of middle two if n even
 Quantile Qn(p) = {x:Fn(x)≥p} and {x:PFn{X≥x}≥1-p}
 Outlier = data value "far" from bulk of data
Describing samples - Plots
 Stem and leaf plot: Easy “density” display

Steps
 Split into leading digits, trailing digits
 Stems: Write down all possible leading digits in
order, including “might have occurred's”
 Leaves: For each data value, write down first trailing
digit by appropriate value (one leaf per datum).

Issue: # stems
 Chiefly science
 Rules of thumb: root-n, 1+3.2log10n
Describing samples - Plots
 Boxplot
 Draw box whose "ends" are Q(1/4) and Q(3/4)
 Draw line through box at median
 Boxplot criterion for "outlier": beyond "inner fences"
= hinges +/- 1.5*IQR
 Draw lines ("Whiskers") from ends of box to last
points inside inner fences
 Show all outliers individually
 Note: perhaps greatest use = with multiple batches
Osteoporosis data
Osteo
Age
US Score
DPA
0
58
1606
0.837
0
68
1650.25
0.841
0
53
1659.75
0.917
0
68
1662
0.975
0
54
1760.75
0.722
0
56
1770.25
0.801
0
77
1773.5
1.213
0
54
1789
1.027
0
62
1808.25
1.045
0
59
1812.5
0.988
0
72
1822.38
0.907
0
53
1826
0.971
0
61
1828
0.88
0
51
1868.5
0.898
0
61
1898.25
0.806
0
52
1908.88
0.994
0
66
1911.75
1.045
0
53
1935.75
0.869
0
62
1937.75
0.968
0
59
1946
0.957
0
50
2004.5
0.954
0
61
2043.08
1.072
Osteoporosis data
Osteo
Age
US Score
DPA
1
73
1588.66
0.785
1
63
1596.83
0.839
1
61
1608.16
0.786
1
75
1610.5
0.825
1
25
1617.75
0.916
1
64
1626.5
0.839
1
69
1658.33
1.191
1
62
1663.88
0.648
1
68
1674.8
0.906
1
58
1690.5
0.688
1
57
1695.15
0.834
1
62
1703.88
0.6
1
64
1704
0.762
1
66
1704.8
0.977
1
58
1715.75
0.704
1
70
1716.33
0.916
1
62
1739.41
0.86
1
67
1756.75
0.776
1
70
1800.75
0.799
1
42
1884.13
0.879
Introduction: Statistical Modeling
 Statistical models: systematic + random
 Probability modeling involves random part
 Often a few parameters “Θ” left to be estimated by data
 Scientific questions are expressed in terms of Θ
 Model is tool / lens / function for investigating scientific
questions
 "Right" versus "wrong" misguided
 Better: “effective” versus “not effective”
Modeling: Parametric Distributions
 Exponential distribution
F(x)
= 1-e-λx
if x ≥ 0
= 0 otherwise
 Model parameter: λ = rate
 E[X] = 1/λ
 Var[X] = 1/λ2
 Uses
 Time-to-event data
 “Memoryless”
Modeling: Parametric Distributions
 Normal distribution
f(x) =
on support SX = (-∞, ∞).
 Distribution function has no closed form:
 F(x) := ∫-∞x f (t)dt, f given above
 F(x) tabulated, available from software packages
 Model parameters: μ=mean; σ2=variance
Normal distribution
 Characteristics
a) f(x) is symmetric about μ
b) P{μ-σ≤X≤μ+σ} ≈ .68
c) P{μ-2σ≤X≤μ+2σ} ≈ .95
 Why is the normal distribution so popular?
a) If X distributed as (“~”) Normal with parameters
(μ,σ) then (X-μ)/σ = “Z” ~ Normal (μ=0,σ=1)
b) Central limit theorem: Distributions of sample means
converge to normal as n →∞
Normal distributions
Application
 Question: Is the normal distribution or exponential
distribution a good model for ultrasound measurements
in older women?
 If so, then comparisons between cases, controls reduce to
comparisons of mean, variance
 Method
 Each model predicts the distribution of measurements
 ECDF Fn characterizes the distribution in our sample
 Compare Fn to
 Normal CDF with mean= 1761.43, SD=120.31
 Exponential CDF with rate = 1/1761.43
Aside
 When is the proposed method a good idea?
 Need Fn to well approximate F if the sample is
representative of a population distributed as F
 Glivenko-Cantelli theorem: Let X1, . . . ,Xn be a
sequence of random variables obtained through
simple random sampling from a population distributed
as F. Then P(lim supx(|Fn(x) − F(x)|) = 0) = 1.
Application – Two models
 “Exponential”
0.8
0.6
0.2
0.4
P(US<=score)
0.6
0.4
0.2
0.0
0.0
P(US<=score)
0.8
1.0
1.0
 “Normal”
1400
1600
1800
ultrasound scores
2000
2200
0
2000
4000
6000
ultrasound scores
8000
10000
Application – Two models
 “Exponential”
0.6
0.2
0.4
P(US<=score)
0.6
0.4
0.0
0.2
0.0
P(US<=score)
0.8
0.8
1.0
1.0
 “Normal”
1400
1600
1800
Ultrasound scores
2000
2200
0
2000
4000
6000
Ultrasound scores
8000
10000
12000
Main points
 The goal of biostatistics is to learn from biomedical
data involving appreciable variability or uncertainty
 We do this by inferring features of populations from
representative samples of them
 Probability is a tool for characterizing populations,
samples and the uncertainty of our inferences from
samples to populations
 Definitions
 Random variables
 Distributions
 Parameters: Location, spread, other
Main points
 Describing sample distributions is a key step to
making inferences about populations
 If the sample “is” the population: The only step
 ECDF, Summary statistics, data displays
 Models are lenses to focus questions for statistical
analysis
 Parametric distributions
 Normal distribution