Lecture 2

Biostatistics
Overview/Teasers

First rules of statistics…
n Use common sense!
n Draw lots of pictures!

What’s wrong with this?
 Study with sample size of 10 (N=10)
 Results: “Objective scoring by blinded
investigators indicated that the treatment
resulted in improvement in all (100%) of the
subjects. Of patients showing overall
improvement, 78% were graded as having
either excellent or moderate improvement.”

Take-home message?
JAMA. 2010;303(12):1173-1179. doi:10.1001/jama.2010.312
Do the three groups differ
meaningfully in weight
change over time?

 How to think about, look at, and
describe data

Teaser 1,
 Hypothetical randomized trial comparing two diets:
 Those on diet 1 (n=10) lost an average of 34.5 lbs.
 Those on diet 2 (n=10) lost an average of 18.5 lbs.
 Conclusion: diet 1 is better?

Teaser 2,
 “400 shades of lipstick found to contain lead”,
FDA says” Washington Post, Feb. 14, 2012
 “What’s in Your Lipstick? FDA Finds Lead in
400 Shades,” Time.com February 15, 2012
 How worried should women who use lipstick
be?

Statistics in Medicine
Module 1:
Introduction to Data

Example Data
 Data compiled from previous Stanford students
(anonymous, non-identifiable)
 Sample size = 50

Each row stores
the data for 1
student (1
observation).

Each column stores the
values for 1 variable (e.g.,
ounces of coffee per day).

Types of data
 Quantitative
 Categorical (nominal or ordinal)
 Time-to-event

Quantitative variable
 Numerical data that you can add, subtract, multiply, and divide
 Examples:
 Age
 Blood pressure
 BMI
 Pulse
 Examples from our example data:
 Optimism on a 0 to 100 scale
 Exercise in hours per week
 Coffee drinking in ounces per day

Quantitative variable
 Continuous vs. Discrete
 Continuous: can theoretically take on any
value within a given range (e.g.,
height=68.99955… inches)
 Discrete: can only take on certain values
(e.g., count data)

Categorical Variables
 Binary = two categories
 Dead/alive
 Treatment/placebo
 Disease/no disease
 Exposed/Unexposed
 Heads/Tails
 Example data: played varsity sports in high school
(yes/no)

 Nominal = unordered categories
 The blood type of a patient (O, A, B, AB)
 Marital status
 Occupation

 Ordinal = Ordered categories
 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 Likert scale (e.g., strongly agree, agree,
neutral, disagree, strongly disagree)
 Age in categories (10-20, 20-30, etc.)
 Example data: non-drinker, light drinker, moderate
drinker, and heavy drinker of coffee

Coffee Drinking Categories (Ordinal)

Time-to-event variables
 The time it takes for an event to occur, if it occurs at all
 Hybrid variable—has a continuous part (time) and a binary part
(event: yes/no)
 Only encountered in studies that follow participants over time—
such as cohort studies and randomized trials
 Examples:
 Time to death
 Time to heart attack
 Time to chronic kidney disease

Always Plot Your Data!
 Are there “outliers”?
 Are there data points that don’t
make sense?
  How are the data distributed?

Are there points that don’t make
sense?
Oops!

How are the data distributed?
Categorical data:
 What are the N’s and percents in each
category?
Quantitative data:
 What’s the shape of the distribution (e.g., is it
normally distributed or skewed)?
 Where is the center of the data?
 What is the spread/variability of the data?

Frequency Plots (univariate)
Categorical variables
 Bar Chart
Quantiative/continuous variables
 Box Plot
 Histogram

Bar Chart
 Used for categorical variables to show
frequency or proportion in each
category.

Bar Chart: categorical
variables

Box plot and histograms: for
quantitative variables
 To show the distribution (shape, center,
range, variation) of quantitative
variables.

75th percentile (6)
25th percentile (2)
interquartile range
(IQR) = 6-2 = 4 median (3.25)
Boxplot of Exercise
maximum or
Q3 + 1.5 * IQR
minimum or
Q1 - 1.5 * IQR

Boxplot of Political Bent
(0=Most Conservative, 100=Most Liberal)
75th percentile (85)
maximum (100)
Q1 – 1.5 * IQR =
68 – 1.5 * 17 = 42.5
median (78)interquartile range
(IQR) = 85 – 68 = 17
“outliers”
minimum (27)

Bins of size = 2 hours/week
Histogram of ExerciseY-axis: The
percent of
observations
that fall
within each
bin.

42% of
students
(n=21)
exercise
between 2 and
3.999… hours
per week.
12% of students
(n=6) exercise
between 0 and
1.999… hours
per week.
Histogram of Exercise

2% of students
(n=1) exercise
≥ 12 hr/wk

Note the “right skew”

Bins of size = 0.2 hours/week
Too much detail!

Too little detail!

Histogram of Political Bent
Note the “left skew”
Also, could be described
as “bimodal” (two
peaks, two groups)

Shape of a Distribution
 Left-skewed/right-skewed/symmetric
Right-SkewedLeft-Skewed Symmetric

Shape of a Distribution
 Symmetric
 Bell curve (“normal distribution”)

Normal distribution (bell curve)
68% of
the data
95% of the data
99.7% of the data
Useful for many reasons:
-Has predictable behavior
-Many traits follow a
normal distribution in the
population
**Many statistics follow a
normal distribution (more
on this later!)**

Fruit and vegetable consumption
(servings/day)…

Feelings about math (0=lowest,
100=highest)
Closest to a normal
distribution!

Biostatistics
Describing Quantitative Data:
Where is the center?

Measures of “central tendency”
 Mean
 Median

Mean
 Mean – the average; the balancing
point
calculation: the sum of values divided by the
sample size
n
xxx
n
x
X n
n
i
i


 211
In math
shorthand:

Mean: example
Some data:
Age of participants: 17 19 21 22 23 23 23 38
25.23
8
38232323222119171




n
x
X
n
i
i

Mean of homework
The balancing point
Mean= 11.4 hours/week

The mean is affected by
extreme values…
Mean= 2.3 drinks/week
The balancing point

The mean is affected by extreme
values…
Mean= 2.9 drinks/week

Does a binary variable have a
mean?
Yes! If coded as a 0/1 variable…
Example: Played Varsity Sports in High School
(0=no, 1=yes)
60% (30)
40% (20)

mean?
60.
50
30
50
0*201*301




n
x
X
n
i
i
60% (30)
40% (20)

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: example
Some data:
Age of participants: 17 19 21 22 23 23 23 38
Median = (22+23)/2 = 22.5

Median of homework
50%
of
mass
50%
of mass
Median= 10
hours/week

Median of alcohol drinking
Median= 2.0 drinks/wk
50%
of
mass
50%
of mass

The median is NOT affected by
extreme values…

The median is NOT affected by
extreme values…
Median = 2.0
drinks/week
50%
of
mass
50%
of mass

Does Varsity Sports (binary
variable) have a median?
Yes, if you line up the 0’s and 1’s, the
middle number is 1.
60% (30)
40% (20)

 For skewed data, the median is preferred
because the mean can be highly
misleading…
Should I present means or
medians?

Hypothetical example: means
vs. medians…
10 dieters following diet 1 vs. 10 dieters following diet 2
Group 1 (n=10) loses an average of 34.5 lbs.
Group 2 (n=10) loses an average of 18.5 lbs.
Conclusion: diet 1 is better?

-30 -25 -20 -15 -10 -5 0 5 10 15 20
0
5
10
15
20
25
30
Percent
Weight change
Histogram, diet 2…
Mean=-18.5 pounds
Median=-19 pounds

-300 -280 -260 -240 -220 -200 -180 -160 -140 -120 -100 -80 -60 -40 -20 0 20
0
5
10
15
20
25
30
Percent
Weight Change
Histogram, diet 1…
Mean=-34.5 pounds
Median=-4.5 pounds

The data…
Diet 1, change in weight (lbs):
+4, +3, 0, -3, -4, -5, -11, -14, -15, -300
Diet 2, change in weight (lbs)
-8, -10, -12, -16, -18, -20, -21, -24, -26, -30

Compare medians via a “non-
parametric test”
We need to compare medians (ranked data) rather than means;
requires a “non-parametric test”
Apply the Wilcoxon rank-sum test (also known as the Mann-
Whitney U test) as follows…

Rank the data…
+4, +3, 0, -3, -4, -5, -11, -14, -15, -300
Ranks: 1 2 3 4 5 6 9 11 12 20
-8, -10, -12, -16, -18, -20, -21, -24, -26, -30
Ranks: 7 8 10 13 14 15 16 17 18 19

Sum the ranks…
+4, +3, 0, -3, -4, -5, -11, -14, -15, -300
Ranks: 1 2 3 4 5 6 9 11 12 20
Sum of the ranks: 1+2+3+4 +5 +6+9+11+12 +20 = 73
-8, -10, -12, -16, -18, -20, -21, -24, -26, -30
Ranks: 7 8 10 13 14 15 16 17 18 19
Sum of the ranks: 7+8+10+13 +14 +15+16+17+18 +19 = 137
Diet 2 is
superior to
Diet 1,
p=.018.

Biostatistics
Describing Quantitative Data:
What is the variability in the
data?

Measures of Variability
 Range
 Standard deviation/Variance
 Percentiles
 Inter-quartile range (IQR)

Range
 Difference between the largest and the
smallest observations.

Range of homework: 40 hours – 0 hours = 40 hours/wk

Standard deviation
 Challenge: devise a statistic that gives the
average distance from the mean.
 Distance from the mean:
?
)(
n
Xx
n
i
i 
 Average distance from the mean??:
Xxi 

Standard deviation
0
)(


n
Xx
n
i
i
 But this won’t work!

How can I get rid of negatives?
 Absolute values?
 Too messy mathematically!
 Squaring eliminates negatives!
n
Xx
S
n
i
i 

2
2
)(

 Average squared distance from the
mean:
Variance
1
)( 2
2




n
Xx
S
n
i
i
We lose a “degree of
freedom because we
have already
estimated the mean.

Standard Deviation
 Gets back to the units of the original data
 Roughly, the average spread around the
mean.
1
)( 2




n
Xx
S
n
i
i

The standard deviation is
affected by extreme values
 Because of the squaring, values farther
from the mean contribute more to the
standard deviation than values closer to the
mean:
25)510(
1)56(
5
2
2


X

Age data (n=8) : 17 19 21 22 23 23 23 38
Calculation Example:
Standard Deviation
n = 8 Mean = 23.25
3.6
7
280
18
)25.23(38)25.23(19)25.32(17 222





S

Homework (hours/week)
Mean = 11.4
Standard deviation = 10.5

100=highest)
Mean = 61
Standard deviation = 21

68-95-99.7 rule (for a perfect bell
curve)
68% of
the data
95% of the data
99.7% of the data

100=highest)
Mean +/- 1 std =
40 – 82
Percent
between 40
and 82 =
34/47 = 72%

100=highest)
Mean +/- 2 std =
19 – 100
Percent
between 19
and 100 =
46/47= 98%

100=highest)
100% of the
data!
Mean +/- 3 std =
0 – 100

standard deviation?
Yes! If coded as a 0/1 variable…
Example: Played Varsity Sports in High School
(0=no, 1=yes)
60% (30)
40% (20)

standard deviation?
49.
49
12
49
)36(.20)16(.30
150
)60.(0*200)6.(1*30 22





S
60% (30)
40% (20)

Understanding Standard Deviation:
Mean = 15
S = 0.9
Mean = 15
S = 5.1
Mean = 15
S = 3.7

Standard deviations vs. standard
errors
 Standard deviation measures the
variability of a trait.
 Standard error measures the variability
of a statistic, which is a theoretical
construct! (much more on this later!)

Percentiles
 Based on ranking the data
 The 90th percentile is the value for which
90% of observations are lower
 The 50th percentile is the median
 The 10th percentile is the value for which
10% of observations are lower
 Percentiles are not affected by extreme
values (unlike standard deviations)

Interquartile Range (IQR)
 Interquartile range = 3rd quartile – 1st
quartile
 The middle 50% of the data.
 Interquartile range is not affected by
outliers.

Boxplot of Political Bent
(0=Most Conservative, 100=Most Liberal)
interquartile range
(IQR) = 85 – 68 = 17

Symbols
 S2= Sample variance
 S = Sample standard deviation
 2 = Population (true or theoretical) variance
  = Population standard deviation
 X = Sample mean
 µ = Population mean
 IQR = interquartile range (middle 50%)

Biostatistics
Exploring real data: Lead in lipstick

2007 Headlines
 “Lipsticks Contain Excessive Lead, Tests
Reveal”
 “One third of lipsticks on the market
contain high lead”
Link to example news coverage:
http://www.reuters.com/article/2007/10/11/us-lipstick-lead-
idUSN1140964520071011

2007 report by a consumer
advocacy group…
 “One-third of the lipsticks tested
contained an amount of lead that
exceeded the U.S. Food and Drug
Administration’s 0.1 ppm limit for lead
in candy—a standard established to
protect children from ingesting lead.”

2007 report by a consumer
advocacy group…
 “One-third of the lipsticks tested
contained an amount of lead that
exceeded the U.S. Food and Drug
Administration’s 0.1 ppm limit for
lead in candy—a standard established
to protect children from ingesting lead.”
1 ppm = 1 part per million =1 microgram/gram

Recent Headlines
 “400 shades of lipstick found to contain
lead”, FDA says” Washington Post, Feb.
14, 2012
 “What’s in Your Lipstick? FDA Finds
Lead in 400 Shades,” Time February 15,
2012
Link to example news coverage:
http://healthland.time.com/2012/02/15/whats-in-your-lipstick-fda-finds-lead-in-400-shades/

How worried should women be?
 What is the dose of lead in lipstick?
 How much lipstick are women exposed
to?
 How much lipstick do women ingest?

Distribution of lead in lipstick
(FDA 2009, n=22)
Right-skewed!
Mean = 1.07
micrograms/gram
Median = 0.73
Std. Dev = 0.96
max = 3.06
99th percentile : 3.06
95th percentile: 3.05

(FDA 2012, n=400)
Right-skewed!
Mean = 1.11
micrograms/gram
Median = 0.89
Std. Dev = 0.97
max = 7.19

FDA 2009 (n=22) vs. FDA 2012
(n=400)
2012 (n=400)
Mean = 1.11
micrograms/gram
Median = 0.89
Std. Dev = 0.97
max = 7.19
2009 (n=22)
Mean = 1.07
micrograms/gram
Median = 0.73
Std. Dev = 0.96
max = 3.06

(n=400 samples, FDA 2012)
FDA data available at:
http://www.fda.gov/Cosmetics/ProductandIngredientSafety/ProductInforma
tion/ucm137224.htm#expanalyses

Percentiles
in mg/day
Hall B, Tozer S, Safford B, Coroama M, Steiling W, Leneveu-Duchemin MC, McNamara C, Gibney M. European consumer exposure to
cosmetic products, a framework for conducting population exposure assessments. Food and Chemical Toxicology 2007; 45: 2097 – 2108.
Data on lipstick exposure
Fig. 6 Lipstick
exposure for women
in grams/day.

Distribution of lipstick exposure:
Food and Chemical Toxicology 2007; 45: 2097 – 2108.
Percentiles
in mg/day

Highest use (1 in 30,000 women)
 1 in 30,000 women uses 218 milligrams of
lipstick per day.
 1 tube of lipstick contains 4000 milligrams.
 4000 mg/tube ÷ 218 mg/day = 18 days per
tube.
 The heaviest user goes through an entire
tube of lipstick in 18 days.

Exercise
Assuming that women ingest 50% of the
lipstick they apply daily, calculate:
1. What is the typical lead exposure to lipstick for
women, in micrograms (mcg) of lead (based on
medians)?
2. What is the highest daily lead exposure to lipstick
for women, in mcg of lead?
Lead in lipstick:
Median = 0.89
micrograms/gram
Maximum = 7.19 mcg/g
Daily lipstick usage:
Median = 17.11 milligrams
Maximum = 217.53 mg

Typical user
Daily exposure:
Daily ingestion:

Typical user
Daily exposure:
0.89 mcg/g x 17.11 mg x 1 g/1000 mg =
0.0152 mcg
Daily ingestion:
0.0152 mcg/2 = 0.0076 mcg

Highest user
Daily exposure:
7.19 mcg/g x 217.53 mg x 1 g/1000 mg =
1.56 mcg
Daily ingestion:
1.56 mcg/2 = 0.78 mcg
Frequency of usage this high:
1/30,000 * 1/400 =
1 woman in 12 million

To put these numbers in
perspective:
 “Provisional tolerable daily intake” for an adult
is 75 micrograms/day
 0.0076 mcg / 75 mcg = 0.02% of your PTDI
 0.78 mcg / 75 mcg = 1% of your PTDI (1 in 12
million women)
 Average American consumes 1 to 4 mcg of lead
per day from food alone.
US FDA report: Total Diet Study Statistics on Element Results. December 14, 2010.
http://www.fda.gov/downloads/Food/FoodSafety/FoodContaminantsAdulteration/TotalDietStudy/UC
M184301.pdf

Comparison with candy:
Median level of lead in milk chocolate = 0.016
mcg/g (FDA limit = 0.1 mcg/g)
Comparing concentrations of lead in lipstick
and chocolate:
0.016 mcg/g << 0.89 mcg/g << 7.19 mcg/g
US FDA report: Total Diet Study Statistics on Element Results. December 14, 2010.
http://www.fda.gov/downloads/Food/FoodSafety/FoodContaminantsAdulteration/TotalDietStudy/UC
M184301.pdf

Comparison with candy:
1 bar of chocolate has about 43 grams
Average American consumes 13.7 grams/day (11 pounds per
year)
Typical daily exposure from chocolate:
0.016 mcg/g x 13.7 g =0.22 mcg
Exposure from 1 chocolate bar:
0.016 mcg/g x 43 g =0.69 mcg

It all comes down to dose!
Typical daily exposure from chocolate (0.22
mcg) is 29 times the typical exposure from
lipstick (0.0076 mcg)
And extreme daily exposure to lead from
lipstick (0.78 mcg) is similar to exposure
from daily consumption of an average
chocolate bar (0.69 mcg)

Lecture 2

More Related Content

Similar to Lecture 2 (20)

Recently uploaded (20)

Lecture 2

Editor's Notes