looking at data. clinical data example 1. kline et al. (2002) the researchers analyzed data from 934...
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Looking at Data
Clinical Data Example
1. Kline et al. (2002) The researchers analyzed data from 934
emergency room patients with suspected pulmonary embolism (PE). Only about 1 in 5 actually had PE. The researchers wanted to know what clinical factors predicted PE.
I will use four variables from their dataset today: Pulmonary embolism (yes/no) Age (years) Shock index = heart rate/systolic BP Shock index categories = take shock index and divide it
into 10 groups (lowest to highest shock index)
Descriptive Statistics
Types of Variables: Overview
Categorical Quantitative
continuousdiscreteordinalnominalbinary
2 categories +
more categories +
order matters +
numerical +
uninterrupted
Categorical Variables Also known as “qualitative.”
Categories.
treatment groups exposure groups disease status
Categorical Variables Dichotomous (binary) – two levels
Dead/alive Treatment/placebo Disease/no disease Exposed/Unexposed Heads/Tails Pulmonary Embolism (yes/no) Male/female
Categorical Variables
Nominal variables – Named categories Order doesn’t matter!
The blood type of a patient (O, A, B, AB) Marital status Occupation
Categorical Variables Ordinal variable – Ordered categories.
Order matters!
Staging in breast cancer as I, II, III, or IV Birth order—1st, 2nd, 3rd, etc. Letter grades (A, B, C, D, F) Ratings on a scale from 1-5 Ratings on: always; usually; many times; once in
a while; almost never; never Age in categories (10-20, 20-30, etc.) Shock index categories (Kline et al.)
Quantitative Variables Numerical variables; may be
arithmetically manipulated.
Counts Time Age Height
Quantitative Variables Discrete Numbers – a limited set of
distinct values, such as whole numbers.
Number of new AIDS cases in CA in a year (counts)
Years of school completed The number of children in the family (cannot have
a half a child!) The number of deaths in a defined time period
(cannot have a partial death!) Roll of a die
Quantitative Variables Continuous Variables - Can take on any
number within a defined range.
Time-to-event (survival time) Age Blood pressure Serum insulin Speed of a car Income Shock index (Kline et al.)
Looking at Data How are the data distributed?
Where is the center? What is the range? What’s the shape of the distribution (e.g.,
Gaussian, binomial, exponential, skewed)?
Are there “outliers”?
Are there data points that don’t make sense?
The first rule of statistics: USE COMMON SENSE!
90% of the information is contained in the graph.
Frequency Plots (univariate)
Categorical variables Bar Chart
Continuous variables Box Plot Histogram
Bar Chart Used for categorical variables to
show frequency or proportion in each category.
Translate the data from frequency tables into a pictorial representation…
Bar Chart: categorical variables
no
yes
Bar Chart for SI categories
Num
ber of Patients
Shock Index Category
0.0
16.7
33.3
50.0
66.7
83.3
100.0
116.7
133.3
150.0
166.7
183.3
200.0
1 2 3 4 5 6 7 8 9 10
Much easier to extract information from a bar chart than from a table!
Box plot and histograms: for continuous variables To show the distribution (shape,
center, range, variation) of continuous variables.
0.0
0.7
1.3
2.0
SI
Box Plot: Shock IndexS
ho
ck In
de
x U
nits
“whisker”
Q3 + 1.5IQR = .8+1.5(.25)=1.17575th percentile (0.8)
25th percentile (0.55)
maximum (1.7)
interquartile range(IQR) = .8-.55 = .25
minimum (or Q1-1.5IQR)
Outliers
median (.66)
Note the “right skew”
Bins of size 0.1
0.0
8.3
16.7
25.0
0.0 0.7 1.3 2.0
Histogram of SI
SI
Per
cent
0.0
2.0
4.0
6.0
0.0 0.7 1.3 2.0
Histogram
SI
Perc
ent
100 bins (too much detail)
0.0
66.7
133.3
200.0
0.0 0.7 1.3 2.0
Histogram
SI
Perc
ent
2 bins (too little detail)
0.0
0.7
1.3
2.0
SI
Box Plot: Shock IndexS
ho
ck In
de
x U
nits Also shows the
“right skew”
75th percentile
25th percentile
maximum
interquartile range
minimum
median
0.0
33.3
66.7
100.0
AGE
Box Plot: Age
Variables
Yea
rs
More symmetric
Histogram: Age
0.0
4.7
9.3
14.0
0.0 33.3 66.7 100.0
AGE (Years)
Pe
rce
nt
Not skewed, but not bell-shaped either…
Some histograms from your class (n=24)
Starting with politics…
Feelings about math and writing…
Optimism…
Diet…
Habits…
Measures of central tendency Mean Median Mode
Central Tendency Mean – the average; the balancing
point
calculation: the sum of values divided by the sample size
n
X++X+X=
n
x
=X n21
n
1=i ∑In math
shorthand:
Mean: exampleSome data: Age of participants: 17 19 21 22 23 23 23 38
25.23=8
38+23+23+23+22+21+19+17=
n
X
=X
n
1=ii∑
Mean of age in Kline’s data
Means Section of AGEGeometricHarmonic
Parameter Mean Median Mean Mean Sum ModeValue 50.19334 49 46.66865 43.00606 46730 49
556.9546
0.0
4.7
9.3
14.0
0.0 33.3 66.7 100.0
Pe
rce
nt
Mean of age in Kline’s data
The balancing point
0.0
4.7
9.3
14.0
0.0 33.3 66.7 100.0
Pe
rce
nt
Mean of Pulmonary Embolism? (Binary variable?)
0.0
33.3
66.7
100.0
0.0 0.3 0.7 1.0
Histogram
PE
Perc
ent
19.44% (181)
80.56%
(750)
1944.931
181
931
0*7501*1811
n
X
X
n
ii
Mean The mean is affected by extreme values
(outliers)
0 1 2 3 4 5 6 7 8 9 10
Mean = 3
0 1 2 3 4 5 6 7 8 9 10
Mean = 4
35
15
5
54321
4
5
20
5
104321
Slide from: Statistics for Managers Using Microsoft® Excel 4th Edition, 2004 Prentice-Hall
Central Tendency Median – the exact middle value
Calculation: If there are an odd number of
observations, find the middle value If there are an even number of
observations, find the middle two values and average them.
Median: exampleSome data: Age of participants: 17 19 21 22 23 23 23 38
Median = (22+23)/2 = 22.5
0.0
4.7
9.3
14.0
0.0 33.3 66.7 100.0AGE (Years)
Pe
rce
nt
Median of age in Kline’s data
Means Section of AGEGeometricHarmonic
Parameter Mean Median Mean Mean Sum ModeValue 50.19334 49 46.66865 43.00606 46730 49
0.0
4.7
9.3
14.0
0.0 33.3 66.7 100.0
Pe
rce
nt
Median of age in Kline’s data
50%
of mass
50%
of mass
Does PE have a median? Yes, if you line up the 0’s and 1’s,
the middle number is 0.
Median
The median is not affected by extreme values (outliers).
0 1 2 3 4 5 6 7 8 9 10
Median = 3
0 1 2 3 4 5 6 7 8 9 10
Median = 3
Central Tendency Mode – the value that occurs most
frequently
Mode: exampleSome data: Age of participants: 17 19 21 22 23 23 23 38
Mode = 23 (occurs 3 times)
Mode of age in Kline’s data
Means Section of AGEGeometricHarmonic
Parameter Mean Median Mean Mean SumModeValue 50.19334 49 46.66865 43.00606 46730 49
Mode of PE? 0 appears more than 1, so 0 is the
mode.
Measures of Variation/Dispersion Range Percentiles/quartiles Interquartile range Standard deviation/Variance
Range
Difference between the largest and the smallest observations.
0.0
4.7
9.3
14.0
0.0 33.3 66.7 100.0
Range of age: 94 years-15 years = 79 years
AGE (Years)
Pe
rce
nt
Range of PE? 1-0 = 1
Quartiles
25% 25% 25% 25%
The first quartile, Q1, is the value for which 25% of the observations are smaller and 75% are larger
Q2 is the same as the median (50% are smaller, 50% are larger)
Only 25% of the observations are greater than the third quartile
Q1 Q2 Q3
Interquartile Range
Interquartile range = 3rd quartile – 1st quartile = Q3 – Q1
Interquartile Range: age
Median(Q2) maximumminimum Q1 Q3
25% 25% 25% 25%
15 35 49 65 94
Interquartile range = 65 – 35 = 30
Average (roughly) of squared deviations of values from the mean
Variance
1
)( 2
2
n
XxS
n
ii
Why squared deviations? Adding deviations will yield a sum of
0. Absolute values are tricky! Squares eliminate the negatives.
Result: Increasing contribution to the variance
as you go farther from the mean.
Standard Deviation
Most commonly used measure of variation
Shows variation about the mean Has the same units as the original
data
1
)( 2
n
XxS
n
ii
Calculation Example:Sample Standard Deviation
Age data (n=8) : 17 19 21 22 23 23 23 38
n = 8 Mean = X = 23.25
3.67
280
18
)25.23(38)25.23(19)25.32(17 222
S
0.0
4.7
9.3
14.0
0.0 33.3 66.7 100.0
AGE (Years)
Pe
rce
nt
Std. dev is a measure of the “average” scatter around the mean.
Estimation method: if the distribution is bell shaped, the range is around 6 SD, so here rough guess for SD is 79/6 = 13
Std. Deviation age
Variation Section of AGE
Standard
Parameter Variance Deviation
Value 333.1884 18.25345
0.0
62.5
125.0
187.5
250.0
0.0 0.5 1.0 1.5 2.0
Std Dev of Shock Index
SI
Co
un
t
Std. dev is a measure of the “average” scatter around the mean.
Estimation method: if the distribution is bell shaped, the range is around 6 SD, so here rough guess for SD is 1.4/6 =.23
Std. Deviation SI
Variation Section of SI
Standard Std Error Interquartile
Parameter Variance Deviation of Mean Range Range
Value 4.155749E-02 0.2038566 6.681129E-03 0.24604321.430856
Std. dev is a measure of the “average” scatter around the mean.
Std. Dev of binary variable, PE
3959.930
8.145
1319
)1944.(0*750)1944.(1*181 22
S
19.44%
80.56%
Std. Deviation PE
Variation Section of PE
Standard
Parameter Variance Deviation
Value 0.156786 0.3959621
Comparing Standard Deviations
Mean = 15.5 S = 3.338 11 12 13 14 15 16 17 18 19 20 21
11 12 13 14 15 16 17 18 19 20 21
Data B
Data A
Mean = 15.5 S = 0.926
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5 S = 4.570
Data C
Regardless of how the data are distributed, a certain percentage of values must fall within K standard deviations from the mean:
Bienaymé-Chebyshev Rule
withinAt least
(1 - 1/12) = 0% …….….. k=1 (μ ± 1σ)
(1 - 1/22) = 75% …........ k=2 (μ ± 2σ)
(1 - 1/32) = 89% ………....k=3 (μ ± 3σ)
Note use of (sigma) to represent “standard deviation.”
Note use of (mu) to represent “mean”.
Symbol Clarification S = Sample standard deviation
(example of a “sample statistic”) = Standard deviation of the
entire population (example of a “population parameter”) or from a theoretical probability distribution
X = Sample mean µ = Population or theoretical mean
**The beauty of the normal curve:
No matter what and are, the area between - and + is about 68%; the area between -2 and +2 is about 95%; and the area between -3 and +3 is about 99.7%. Almost all values fall within 3 standard deviations.
68-95-99.7 Rule
68% of the data
95% of the data
99.7% of the data
Summary of Symbols
S2= Sample variance S = Sample standard dev 2 = Population (true or theoretical)
variance = Population standard dev. X = Sample mean µ = Population mean IQR = interquartile range (middle 50%)
Examples of bad graphics
What’s wrong with this graph?
from: ER Tufte. The Visual Display of Quantitative Information. Graphics Press, Cheshire, Connecticut, 1983, p.69
From: Visual Revelations: Graphical Tales of Fate and Deception from Napoleon Bonaparte to Ross Perot Wainer, H. 1997, p.29.
Notice the X-axis
Correctly scaled X-axis…
Report of the Presidential Commission on the Space Shuttle Challenger Accident, 1986 (vol 1, p. 145)
The graph excludes the observations where no O-rings failed.
http://www.math.yorku.ca/SCS/Gallery/
Smooth curve at least shows the trend toward failure at high and low temperatures…
Even better: graph all the data (including non-failures) using a logistic regression model
Tappin, L. (1994). "Analyzing data relating to the Challenger disaster". Mathematics Teacher, 87, 423-426
from: ER Tufte. The Visual Display of Quantitative Information. Graphics Press, Cheshire, Connecticut, 1983, p.74
What’s wrong with this graph?
Diagraphics II, 1994
What’s the message here?
Diagraphics II, 1994
From: Johnson R. Just the Essentials of Statistics. Duxbury Press, 1995.
From: Johnson R. Just the Essentials of Statistics. Duxbury Press, 1995.
From: Johnson R. Just the Essentials of Statistics. Duxbury Press, 1995.
From: Johnson R. Just the Essentials of Statistics. Duxbury Press, 1995.
For more examples… http://www.math.yorku.ca/SCS/Gallery/
“Lying” with statistics More accurately, misleading with
statistics…
Example 1: projected statistics
Lifetime risk of melanoma: 1935: 1/15001960: 1/6001985: 1/1502000: 1/742006: 1/60
http://www.melanoma.org/mrf_facts.pdf
Example 1: projected statistics
How do you think these statistics are calculated?
How do we know what the lifetime risk of a person born in 2006 will be?
Example 1: projected statistics
Interestingly, a clever clinical researcher recently went back and calculated (using SEER data) the actual lifetime risk (or risk up to 70 years) of melanoma for a person born in 1935.
The answer?Closer to 1/150 (one order of magnitude off)
(Martin Weinstock of Brown University, AAD conference 2006)
Example 2: propagation of statistics
In many papers and reviews of eating disorders in women athletes, authors cite the statistic that 15 to 62% of female athletes have disordered eating.
I’ve found that this statistic is attributed to about 50 different sources in the literature and cited all over the place with or without citations...
For example… In a recent review (Hobart and Smucker, The
Female Athlete Triad, American Family Physician, 2000):
“Although the exact prevalence of the female athlete triad is unknown, studies have reported disordered eating behavior in 15 to 62 percent of female college athletes.”
No citations given.
And… Fact Sheet on eating disorders: “Among female athletes, the
prevalence of eating disorders is reported to be between 15% and 62%.”Citation given: Costin, Carolyn. (1999) The Eating Disorder Source Book: A comprehensive guide to the causes, treatment, and prevention of eating disorders. 2nd edition. Lowell House: Los Angeles.
And… From a Fact Sheet on disordered
eating from a college website: “Eating disorders are significantly higher
(15 to 62 percent) in the athletic population than the general population.”
No citation given.
And… “Studies report between 15% and 62%
of college women engage in problematic weight control behaviors (Berry & Howe, 2000).” (in The Sport Journal, 2004)
Citation: Berry, T.R. & Howe, B.L. (2000, Sept). Risk factors for disordered eating in female university athletes. Journal of Sport Behavior, 23(3), 207-219.
And… 1999 NY Times article “But informal surveys suggest that
15 percent to 62 percent of female athletes are affected by disordered behavior that ranges from a preoccupation with losing weight to anorexia or bulimia.”
And “It has been estimated that the prevalence of
disordered eating in female athletes ranges from 15% to 62%.” ( in Journal of General Internal Medicine 15 (8), 577-590.)Citations:Steen SN. The competitive athlete. In: Rickert VI, ed. Adolescent Nutrition: Assessment and Management. New York, NY: Chapman and Hall; 1996:223 47. Tofler IR, Stryer BK, Micheli LJ. Physical and emotional problems of elite female gymnasts. N Engl J Med. 1996;335:281 3.
Where did the statistics come from?
The 15%: Dummer GM, Rosen LW, Heusner WW, Roberts PJ, and Counsilman JE. Pathogenic weight-control behaviors of young competitive swimmers. Physician Sportsmed 1987; 15: 75-84.
The “to”: Rosen LW, McKeag DB, O’Hough D, Curley VC. Pathogenic weight-control behaviors in female athletes. Physician Sportsmed. 1986; 14: 79-86.
The 62%:Rosen LW, Hough DO. Pathogenic weight-control behaviors of female college gymnasts. Physician Sportsmed 1988; 16:140-146.
Where did the statistics come from?
Study design? Control group? Cross-sectional survey (all) No non-athlete control groups
Population/sample size? Convenience samples Rosen et al. 1986: 182 varsity athletes from two
midwestern universities (basketball, field hockey, golf, running, swimming, gymnastics, volleyball, etc.)
Dummer et al. 1987: 486 9-18 year old swimmers at a swim camp
Rosen et al. 1988: 42 college gymnasts from 5 teams at an athletic conference
Where did the statistics come from? Measurement?
Instrument: Michigan State University Weight Control Survey
Disordered eating = at least one pathogenic weight control behavior:
Self-induced vomiting fasting Laxatives Diet pills Diuretics In the 1986 survey, they required use 1/month; in the
1988 survey, they required use twice-weekly In the 1988 survey, they added fluid restriction
Where did the statistics come from? Findings?
Rosen et al. 1986: 32% used at least one “pathogenic weight-control behavior” (ranges: 8% of 13 basketball players to 73.7% of 19 gymnasts)
Dummer et al. 1987: 15.4% of swimmers used at least one of these behaviors
Rosen et al. 1988: 62% of gymnasts used at least one of these behaviors
References
http://www.math.yorku.ca/SCS/Gallery/ Kline et al. Annals of Emergency Medicine 2002; 39: 144-152. Statistics for Managers Using Microsoft® Excel 4th Edition, 2004 Prentice-Hall Tappin, L. (1994). "Analyzing data relating to the Challenger disaster". Mathematics Teacher, 87, 423-
426 Tufte. The Visual Display of Quantitative Information. Graphics Press, Cheshire, Connecticut, 1983. Visual Revelations: Graphical Tales of Fate and Deception from Napoleon Bonaparte to Ross Perot
Wainer, H. 1997.