Download Descriptive Statistics-II

Descriptive Statistics-II
Mahmoud Alhussami, MPH, DSc., PhD.
Shapes of Distribution


A third important property of data – after location
and dispersion - is its shape
Distributions of quantitative variables can be
described in terms of a number of features, many
of which are related to the distributions’ physical
appearance or shape when presented graphically.




modality
Symmetry and skewness
Degree of skewness
Kurtosis
Modality
The modality of a distribution concerns
how many peaks or high points there are.
 A distribution with a single peak, one
value a high frequency is a unimodal
distribution.

Modality

A distribution with two
or more peaks called
multimodal
distribution.
Symmetry and Skewness


A distribution is symmetric if the distribution could be split
down the middle to form two haves that are mirror images
of one another.
In asymmetric distributions, the peaks are off center, with a
bull of scores clustering at one end, and a tail trailing off at
the other end. Such distributions are often describes as
skewed.


When the longer tail trails off to the right this is a positively
skewed distribution. E.g. annual income.
When the longer tail trails off to the left this is called
negatively skewed distribution. E.g. age at death.
Symmetry and Skewness



Shape can be described by degree of asymmetry (i.e.,
skewness).
 mean > median
positive or right-skewness
 mean = median
symmetric or zero-skewness
 mean < median
negative or left-skewness
Positive skewness can arise when the mean is
increased by some unusually high values.
Negative skewness can arise when the mean is
decreased by some unusually low values.
 Left
skewed:
 Right
skewed:
 Symmetric:
7
Shapes of the Distribution

Three common shapes of frequency
distributions:
A
B
C
Symmetrical
and bell
shaped
Positively
skewed or
skewed to
the right
Negatively
skewed or
skewed to
the left
5 May 2017
8
Shapes of the Distribution

Three less common shapes of frequency
distributions:
A
Bimodal
5 May 2017
B
C
Reverse
J-shaped
Uniform
9

This guy
took a
VERY long
time!
10
Degree of Skewness




A skewness index can readily be calculated most
statistical computer program in conjunction with
frequency distributions
The index has a value of 0 for perfectly
symmetric distribution.
A positive value if there is a positive skew, and
negative value if there is a negative skew.
A skewness index that is more than twice the
value of its standard error can be interpreted as a
departure from symmetry.
Measures of Skewness or Symmetry

Pearson’s skewness coefficient



It is nonalgebraic and easily calculated. Also it
is useful for quick estimates of symmetry .
It is defined as:
skewness = mean-median/SD
Fisher’s measure of skewness.

It is based on deviations from the mean to the
third power.
Pearson’s skewness coefficient



For a perfectly symmetrical distribution, the mean will
equal the median, and the skewness coefficient will be
zero. If the distribution is positively skewed the mean
will be more than the median and the coefficient will be
the positive. If the coefficient is negative, the
distribution is negatively skewed and the mean less than
the median.
Skewness values will fall between -1 and +1 SD units.
Values falling outside this range indicate a substantially
skewed distribution.
Hildebrand (1986) states that skewness values above
0.2 or below -0.2 indicate severe skewness.
Fisher’s Measure of Skewness


The formula for Fisher’s skewness statistic is based on
deviations from the mean to the third power.
The measure of skewness can be interpreted in terms of
the normal curve




A symmetrical curve will result in a value of 0.
If the skewness value is positive, them the curve is skewed to
the right, and vice versa for a distribution skewed to the left.
A z-score is calculated by dividing the measure of skewness
by the standard error for skewness. Values above +1.96 or
below -1.96 are significant at the 0.05 level because 95%
of the scores in a normal deviation fall between +1.96 and
-1.96 from the mean.
E.g. if Fisher’s skewness= 0.195 and st.err. =0.197 the zscore = 0.195/0.197 = 0.99
Kurtosis
The distribution’s kurtosis is concerns how
pointed or flat its peak.
 Two types:



Leptokurtic distribution (mean thin).
Platykurtic distribution (means flat).
Kurtosis



There is a statistical index of kurtosis that can be
computed when computer programs are
instructed to produce a frequency distribution
For kurtosis index, a value of zero indicates a
shape that is neither flat nor pointed.
Positive values on the kurtosis statistics indicate
greater peakedness, and negative values indicate
greater flatness.
Fishers’ measure of Kurtosis
Fisher’s measure is based on deviation
from the mean to the fourth power.
 A z-score is calculated by dividing the
measure of kurtosis by the standard error
for kurtosis.

Normal Distribution
Also called belt shaped curve, normal
curve, or Gaussian distribution.
 A normal distribution is one that is
unimodal, symmetric, and not too peaked
or flat.
 Given its name by the French
mathematician Quetelet who, in the early
19th century noted that many human
attributes, e.g. height, weight, intelligence
appeared to be distributed normally.

Normal Distribution






The normal curve is unimodal and symmetric
about its mean ().
In this distribution the mean, median and mode
are all identical.
The standard deviation () specifies the amount
of dispersion around the mean.
The two parameters  and  completely define a
normal curve.
Also called a Probability density function. The
probability is interpreted as "area under the curve.“
The area under the whole curve = 1
5 May 2017
19
Sampling Distribution





A sample statistic is often unequal to the value of the
corresponding population parameter because of sampling
error.
Sampling error reflects the tendency for statistics to
fluctuate from one sample to another.
The amount of sampling error is the difference between the
obtained sample value and the population parameter.
Inferential statistics allow researchers to estimate how
close to the population value the calculated statistics is
likely to be.
The concept of sampling, which are actually probability
distributions, is central to estimates of sampling error.
Characteristics of Sampling
Distribution







Sampling error= sample mean-population mean.
Every sample size has a different sampling distribution of
the mean.
Sampling distributions are theoretical, because in practice,
no one draws an infinite number of samples from a
population.
Their characteristics can be modeled mathematically and
have determined by a formulation known as the central
limit theorem.
This theorem stipulates that the mean of the sampling
distribution is identical to the population mean.
A consequence of Central Limit Theorem is that if we
average measurements of a particular quantity, the
distribution of our average tends toward a normal one.
The average sampling error-the mean of the (meanμ)sample would always equal zero.
Standard Error of the Mean
The standard deviation of a sampling
distribution of the mean has a special
name: the standard error of the mean
(SEM).
 The smaller the SEM, the more accurate
are the sample means as estimates of the
population value.

Central Limit Theorem


describes the characteristics of the "population of the
means" which has been created from the means of an
infinite number of random population samples of size (N),
all of them drawn from a given "parent population".
It predicts that regardless of the distribution of the parent
population:



The mean of the population of means is always equal to the
mean of the parent population from which the population
samples were drawn.
The standard deviation of the population of means is always
equal to the standard deviation of the parent population
divided by the square root of the sample size (N).
The distribution of means will increasingly approximate a
normal distribution as the size N of samples increases.
Standard Normal Variable




It is customary to call a standard normal random
variable Z.
The outcomes of the random variable Z are
denoted by z.
The table in the coming slide give the area under
the curve (probabilities) between the mean and
z.
The probabilities in the table refer to the
likelihood that a randomly selected value Z is
equal to or less than a given value of z and
greater than 0 (the mean of the standard
normal).
5 May 2017
24
Source: Levine et al, Business Statistics, Pearson.
25
The 68-95-99.7 Rule for the Normal
Distribution
68% of the observations fall within one
standard deviation of the mean
 95% of the observations fall within two
standard deviations of the mean
 99.7% of the observations fall within three
standard deviations of the mean
 When applied to ‘real data’, these
estimates are considered approximate!

5 May 2017
26
Remember these probabilities (percentages):
# standard deviations
from the mean
Approx. area under the
normal curve
±1
.68
±1.645
.90
±1.96
.95
±2
.955
±2.575
.99
±3
.997
Practice: Find these values yourself using the Z
table.
Two Sample Z Test
27
Standard Normal Curve
5 May 2017
28
Standard Normal Distribution
50% of probability in
here –probability=0.5
5 May 2017
50% of probability in
here–probability=0.5
29
Standard Normal Distribution
95% of
probability
in here
2.5% of probability
in here
2.5% of
probability in here
Standard Normal
Distribution with 95% area
marked
5 May 2017
30
Calculating Probabilities
Probability calculations are always
concerned with finding the probability that
the variable assumes any value in an
interval between two specific points a and
b.
 The probability that a continuous variable
assumes the a value between a and b is
the area under the graph of the density
between a and b.

5 May 2017
31
If the weight of males is N.D.
with μ=150 and σ=10, what is
the probability that a
randomly selected male will
weigh between 140 lbs and
155 lbs?
Normal Distribution
32
Solution:
140
-1
150 155
0
0.5
X
Z
Z = (140 – 150)/ 10 = -1.00 s.d. from mean
Area under the curve = .3413 (from Z table)
Z = (155 – 150) / 10 =+.50 s.d. from mean
Area under the curve = .1915 (from Z table)
Answer: .3413 + .1915 = .5328
33
If IQ is ND with a mean of 100 and a S.D. of
10, what percentage of the population will
have
(a)IQs ranging from 90 to 110?
(b)IQs ranging from 80 to 120?
Solution:
Z = (90 – 100)/10 = -1.00
Z = (110 -100)/ 10 = +1.00
Area between 0 and 1.00 in the Z-table
is .3413; Area between 0 and -1.00 is also
.3413 (Z-distribution is symmetric).
Answer to part (a) is .3413 + .3413 = .6826.
34
 (b)
IQs ranging from 80 to 120?
Solution:
Z = (80 – 100)/10 = -2.00
Z = (120 -100)/ 10 = +2.00
Area between =0 and 2.00 in the Z-table
is .4772; Area between 0 and -2.00 is
also .4772 (Z-distribution is symmetric).
Answer is .4772 + .4772 = .9544.
35
Suppose that the average salary of college
graduates is N.D. with μ=$40,000 and
σ=$10,000.
(a)
(b)
(c)
(d)
(e)
What proportion of college graduates will earn
$24,800 or less?
What proportion of college graduates will earn
$53,500 or more?
What proportion of college graduates will earn
between $45,000 and $57,000?
Calculate the 80th percentile.
Calculate the 27th percentile.
36
(a) What proportion of college graduates
will earn $24,800 or less?
Solution:
Convert the $24,800 to a Z-score:
Z = ($24,800 - $40,000)/$10,000 = -1.52.
Always DRAW a picture of the distribution
to help you solve these problems.
37
.4357
$24,800
-1.52
$40,000
0
X
Z
First Find the area between 0 and -1.52 in the
Z-table. From the Z table, that area is .4357.
Then, the area from -1.52 to - ∞ is
.5000 - .4357 = .0643.
Answer: 6.43% of college graduates will earn
less than $24,800.
38
(b) What proportion of
college graduates will earn
.4115
$53,500 or more?
.0885
$40,000
$53,500
Solution:
+1.35
0
Z
Convert the $53,500 to a Z-score.
Z = ($53,500 - $40,000)/$10,000 = +1.35.
Find the area between 0 and +1.35 in the Ztable: .4115 is the table value.
When you DRAW A PICTURE (above) you see
that you need the area in the tail: .5 - .4115 .0885.
Answer: .0885. Thus, 8.85% of college
graduates will earn $53,500 or more.
39
.1915
(c) What proportion of college
graduates will earn between
$45,000 and $57,000?
$40k $45k
0
.5
$57k
1.7
Z
Z = $45,000 – $40,000 / $10,000 = .50
Z = $57,000 – $40,000 / $10,000 = 1.70
From the table, we can get the area under the
curve between the mean (0) and .5; we can get
the area between 0 and 1.7. From the picture
we see that neither one is what we need.
What do we do here? Subtract the small piece
from the big piece to get exactly what we need.
Answer: .4554 − .1915 = .2639
40
Parts (d) and (e) of this example ask you to
compute percentiles. Every Z-score is
associated with a percentile. A Z-score of 0
is the 50th percentile. This means that if
you take any test that is normally
distributed (e.g., the SAT exam), and your
Z-score on the test is 0, this means you
scored at the 50th percentile. In fact, your
score is the mean, median, and mode.
41
(d) Calculate the 80th percentile.
.5000
.3000
Solution:
$40,000
0
.84
First, what Z-score is associated
with the 80th percentile?
A Z-score of approximately +.84 will give you
about .3000 of the area under the curve. Also,
the area under the curve between -∞ and 0 is
.5000. Therefore, a Z-score of +.84 is
associated with the 80th percentile.
Z
ANSWER
Now to find the salary (X) at the 80th percentile:
Just solve for X: +.84 = (X−$40,000)/$10,000
42
(e) Calculate the 27th percentile.
.2300
.5000
.2700
Solution: First, what Z-score is associated
$40,000
with the 27th percentile? A Z-score
0
Z
-.61
of approximately -.61will give you
about .2300 of the area under the curve, with .2700 in
the tail. (The area under the curve between 0 and -.61
is .2291 which we are rounding to .2300). Also, the
area under the curve between 0 and ∞ is .5000.
Therefore, a Z-score of
-.61 is associated with the 27th percentile.
ANSWER
Now to find the salary (X) at the 27th percentile:
Just solve for X: -0.61 =(X−$40,000)/$10,000
X = $40,000 - $6,100 = $33,900
43
Graphical Methods

Frequency Distribution

Histogram

Frequency Polygon

Cumulative Frequency Graph

Pie Chart.
5 May 2017
44
Presenting Data

Table



Condenses data into a form that can make
them easier to understand;
Shows many details in summary fashion;
BUT
Since table shows only numbers, it may not be
readily understood without comparing it to
other values.
Principles of Table Construction
Don’t try to do too much in a table
 Us white space effectively to make table
layout pleasing to the eye.
 Make sure tables & test refer to each
other.
 Use some aspect of the table to order &
group rows & columns.

Principles of Table Construction
If appropriate, frame table with summary
statistics in rows & columns to provide a
standard of comparison.
 Round numbers in table to one or two
decimal places to make them easily
understood.
 When creating tables for publication in a
manuscript, double-space them unless
contraindicated by journal.

Frequency Distributions

A useful way to present data when you
have a large data set is the formation of a
frequency table or frequency distribution.

Frequency – the number of observations
that fall within a certain range of the data.
5 May 2017
48
Frequency Table
Age
Number of Deaths
<1
564
1-4
86
5-14
127
15-24
490
25-34
66
35-44
806
45-54
1,425
55-64
3,511
65-74
6,932
75-84
10,101
85+
9825
Total
34,524
49
Presenting Data
Chart
 Visual representation of a
frequency distribution that helps to gain
insight about what the data mean.
 Built with lines, area & text: bar
charts, pie chart
Bar Chart

Simplest form of chart
Used to display
nominal or ordinal
data
ETHICAL ISSUES SCALE
ITEM 8
60
50
PERCENT

40
30
20
10
0
Never
Seldom
Somet imes
Frequently
ACTING AGAINST YOUR OWN PERSONAL/RELIGIOUS VIEWS
Cluster Bar Chart
70
60
PERCENT
50
40
30
Employment
20
Full time RN
10
Part time RN
0
Self employed
Diploma
B achelor Degree
As sociate Degree
Post Bac
RN HIGHEST EDUCATION
Pie Chart


Alternative to bar
chart
Circle partitioned into
percentage
distributions of
qualitative variables
with total area of
100%
Doctorate NonNursing
Doctorate Nursing
MS NonNursing
Missing
MS Nursing
Juris Doctor
Diploma-Nursing
BS NonNursing
AD Nursing
BS Nursing
Histogram
Appropriate for interval, ratio and
sometimes ordinal data
 Similar to bar charts but bars are placed
side by side
 Often used to represent both frequencies
and percentages
 Most histograms have from 5 to 20 bars

Histogram
80
FREQUENCY
60
40
20
Std. Dev = 22.17
Mean = 61.6
N = 439.00
0
0.0
20.0
10.0
40.0
30.0
60.0
50.0
80.0
70.0
SF-36 VITALITY SCORES
100.0
90.0
Pictures of Data: Histograms

0
5
10
Number of Men
15
20
Blood pressure data on a sample of 113 men
80
100
120
140
160
Systolic BP (mmHg)
Histogram of the Systolic Blood Pressure for 113 men. Each bar
spans a width of 5 mmHg on the horizontal axis. The height of each
bar represents the number of individuals with SBP in that range.
5 May 2017
56
Frequency Polygon
Frequency Polygon
•First place a dot at the
midpoint of the upper base of
each rectangular bar.
20
18
16
14
•The points are connected with
straight lines.
Childrens w eights
12
•At the ends, the points are
connected to the midpoints of
the previous and succeeding
intervals (these intervals have
zero frequency).
10
8
6
4
2
0
4.5
14.5
5 May 2017
24.5
34.5
44.5
54.5
64.5
74.5
84.5
57
Hallmarks of a Good Chart
Simple & easy to read
 Placed correctly within text
 Use color only when it has a purpose, not
solely for decoration
 Make sure others can understand chart;
try it out on somebody first
 Remember: A poor chart is worse than no
chart at all.

Coefficient of Correlation
Measure of linear association between 2
continuous variables.
 Setting:



two measurements are made for each
observation.
Sample consists of pairs of values and you
want to determine the association between the
variables.
5 May 2017
59
Association Examples

Example 1: Association between a mother’s
weight and the birth weight of her child

2 measurements: mother’s weight and baby’s weight


Both continuous measures
Example 2: Association between a risk factor and
a disease

2 measurements: disease status and risk factor status

5 May 2017
Both dichotomous measurements
60
Correlation Analysis
When you have 2 continuous
measurements you use correlation
analysis to determine the relationship
between the variables.
 Through correlation analysis you can
calculate a number that relates to the
strength of the linear association.

5 May 2017
61
Scatter Plots and Association


You can plot the 2 variables in a scatter plot (one
of the types of charts in SPSS/Excel).
The pattern of the “dots” in the plot indicate the
statistical relationship between the variables (the
strength and the direction).



Positive relationship – pattern goes from lower left to
upper right.
Negative relationship – pattern goes from upper left to
lower right.
The more the dots cluster around a straight line the
stronger the linear relationship.
5 May 2017
62
Birth Weight Data
x (oz)
y(%)
112
111
107
119
92
80
81
84
118
106
103
94
63
66
72
52
75
118
120
114
42
72
90
91
x – birth weight in ounces
y – increase in weight between
70th and 100th days of life,
expressed as a percentage of
birth weight
63
Pearson Correlation Coefficient
Birth Weight Data
120
Increase in Birth Weight (%)
110
100
90
80
70
60
50
40
70
80
90
100
110
120
130
140
Birth Weight (in ounces)
5 May 2017
64
Calculations of Correlation
Coefficient

In SPSS:




Go to TOOLS menu and select DATA ANALYSIS.
Highlight CORRELATION and click “ok”
Enter INPUT RANGE (2 columns of data that
contain “x” and “y”)
Click “ok” (cells where you want the answer to
be placed.
5 May 2017
65
Pearson Correlation Results
x (oz)
x (oz)
y(%)
y(%)
1
-0.94629
1
Pearson Correlation Coefficient = -0.946
Interpretation:
- values near 1 indicate strong positive linear relationship
- values near –1 indicate strong negative linear relationship
- values near 0 indicate a weak linear association
5 May 2017
66
CAUTION!!!!
Interpreting the correlation coefficient
should be done cautiously!
 A result of 0 does not mean there is NO
relationship …. It means there is no
linear association.
 There may be a perfect non-linear
association.

5 May 2017
67
The Uses of Frequency Distributions


Becoming familiar with dataset.
Cleaning the data.



Outliers-values that lie outside the normal range of values for
other cases.
Inspecting the data for missing values.
Testing assumptions for statistical tests.


Assumption is a condition that is presumed to be true and
when ignored or violated can lead to misleading or invalid
results.
When DV is not normally distributed researchers have to
choose between three options:



Select a statistical test that does not assume a normal distribution.
Ignore the violation of the assumption.
Transform the variable to better approximate a distribution that is
normal. Please consult the various data transformation.
The Uses of Frequency Distributions
Obtaining information about sample
characteristics.
 Directing answering research questions.

Outliers



Are values that are extreme relative to the bulk
of scores in the distribution.
They appear to be inconsistent with the rest of
the data.
Advantages:


They may indicate characteristics of the population that
would not be known in the normal course of analysis.
Disadvantages:



They do not represent the population
Run counter to the objectives of the analysis
Can distort statistical tests.
Sources of Outliers
An error in the recording of the data.
 A failure of data collection, such as not
following sample criteria (e.g.
inadvertently admitting a disoriented
patient into a study), a subject not
following instructions on a questionnaire,
or equipment failure.
 An actual extreme value from an unusual
subjects.

Missing Data



Any systematic event external to the respondent (such as
data entry errors or data collection problems) or action on
the part of the respondent (such as refusal to answer) that
leads to missing data.
It means that analyses are based on fewer study
participants than were in the full study sample. This, in
turn, means less statistical power, which can undermine
statistical conclusion validity-the degree to which the
statistical results are accurate.
Missing data can also affect internal validity-the degree to
which inferences about the causal effect of the dependent
variable on the dependent variable are warranted, and also
affect the external validity-generalizability.
Strategies to avoid Missing Data
Persistent follow-up
 Flexibility in scheduling appointments
 Paying incentives.
 Using well-proven methods to track people
who have moved.
 Performing a thorough review of
completed data forms prior to excusing
participants.

Techniques for Handling Missing Data



Deletion techniques. Involve excluding subjects
with missing data from statistical calculation.
Imputation techniques. Involve calculating an
estimate of each missing value and replacing, or
imputing, each value by its respective estimate.
Note: techniques for handling missing data often
vary in the degree to which they affect the
amount of dispersion around true scores, and the
degree of bias in the final results. Therefore, the
selection of a data handling technique should be
carefully considered.
Deletion Techniques



Deletion methods involve removal of cases or variables with
missing data.
Listwise deletion. Also called complete case analysis. It is
simply the analysis of those cases for which there are no
missing data. It eliminates an entire case when any of its
items/variables has a missing data point, whether or not
that data point is part of the analysis. It is the default of
the SPSS.
Pairwise deletion. Called the available case analysis (unwise
deletion). Involves omitting cases from the analysis on a
variable-by-variable basis. It eliminates a case only when
that case has missing data for variables or items under
analysis.
Imputation Techniques
Imputation is the process of estimating
missing data based on valid values of
other variables or cases in the sample.
 The goal of imputation is to use known
relationship that can be identified in the
valid values of the sample to help estimate
the missing data

Types of Imputation Techniques
Using prior knowledge.
 Inserting mean values.
 Using regression

Prior Knowledge
Involves replacing a missing value with a
value based on an educational guess.
 It is a reasonable method if the researcher
has a good working knowledge of the
research domain, the sample is large, and
the number of missing values is small.

Mean Replacement
Also called median replacement for
skewed distribution.
 Involves calculating mean values from a
available data on that variable and using
them to replace missing values before
analysis.
 It is a conservative procedure because the
distribution mean as a whole does not
change and the researcher does not have
to guess at missing values.

Mean Replacement

Advantages:



Easily implemented and provides all cases with complete
data.
A compromise procedure is to insert a group mean for
the missing values.
Disadvantages:



It invalidates the variance estimates derived from the
standard variance formulas by understanding the data’s
true variance.
It distorts the actual distribution of values.
It depresses the observed correlation that this variable
will have with other variables because all missing data
have a single constant value, thus reducing the
variance.
Using Regression





Involves using other variables in the dataset as
independent variables to develop a regression
equation for the variable with missing data
serving as the dependent variable.
Cases with complete data are used to generate
the regression equation.
The equation is then used to predict missing
values for incomplete cases.
More regressions are computed, using the
predicted values from the previous regression to
develop the next equation, until the predicted
values from one step to the next are comparable.
Prediction from the last regression are the ones
used to replace missing values.
Using Regression

Advantages:


It is more objective than the researcher’s guess but not
as blind as simply using the overall mean.
Disadvantages:




It reinforces the relationships already in the data,
resulting in less generalizability.
The variance of the distribution is reduced because the
estimate is probably too close to the mean.
It assumes that the variable with missing data is
correlated substantially with missing data is correlated
substantially with the other variables in the dataset.
The regression procedure is not constrained in the
estimates it makes.
Categorical Data
Data that can be classified as belonging to a
distinct number of categories.



Binary – data can be classified into one of 2
possible categories (yes/no, positive/negative)
Ordinal – data that can be classified into
categories that have a natural ordering (i.e..
Levels of pain: none, moderate, intense)
Nominal- data can be classified into >2
categories (i.e.. Race: Arab, African, and
other)
5 May 2017
83
Proportions



Numbers by themselves may be misleading: they are on
different scales and need to be reduced to a standard basis
in order to compare them.
We most frequently use proportions: that is, the fraction of
items that satisfy some property, such as having a disease
or being exposed to a dangerous chemical.
"Proportions" are the same thing as fractions or
percentages. In every case you need to know what you are
taking a proportion of: that is, what is the DENOMINATOR
in the proportion.
x
p
n
5 May 2017
x
percent (100)  (100)
n
84
Proportions and Probabilities
We often interpret proportions as
probabilities. If the proportion with a
disease is 1/10 then we also say that the
probability of getting the disease is 1/10,
or 1 in 10.
 Proportions are usually quoted for samples.
 Probabilities are almost always quoted for
populations.

5 May 2017
85
Workers Example
Smoking
No
Yes

Cases Controls
Yes
11
35
No
50
203
Yes
84
45
No
313
270
For the cases:


Workers
Proportion of exposure=84/397=0.212 or 21.2%
For the controls:

Proportion of exposure=45/315=0.143 or 14.3%
5 May 2017
86
Prevalence
Disease Prevalence = the proportion of people with a given
disease at a given time.
disease prevalence =
Number of diseased persons at a given time
Total number of persons examined at that time
Prevalence is usually quoted as per 100,000 people
so the above proportion should be multiplied by
100,000.
5 May 2017
87
Interpretation
At time t
Cases (old  new)
Pr evalence 
Total
Problem of exposure, consequently
Not comparable measurement
Old = duration of the disease
New = speed of the disease
5 May 2017
88
Screening Tests
Through screening tests people are
classified as healthy or as falling into one
or more disease categories.
 These tests are not 100% accurate and
therefore misclassification is unavoidable.
 There are 2 proportions that are used to
evaluate these types of diagnostic
procedures.

5 May 2017
89
Sensitivity and Specificity

Sensitivity and specificity are terms used to
describe the effectiveness of screening tests. They
describe how good a test is in two ways - finding
false positives and finding false negatives

Sensitivity is the Proportion of diseased who
screen positive for the disease

Specificity is the Proportion of healthy who screen
healthy
5 May 2017
90
Sensitivity and Specificity
Condition Present
Condition Absent
……………………………………………………………………………………………
Test Positive
True Positive (TP)
False Positive (FP)
Test Negative
False Negative (FN)
True Negative (TN)
……………………………………………………………………………………………
 Test Sensitivity (Sn) is defined as the probability that the test is positive
when given to a group of patients who have the disease.



Sn= (TP/(TP+FN))x100.
It can be viewed as, 1-the false negative rate.
The Specificity (Sp) of a screening test is defined as the probability that
the test will be negative among patients who do not have the disease.


Sp = (TN/(TN+FP))X100.
It can be understood as 1-the false positive rate.
Positive & Negative Predictive Values
The positive predictive value (PPV) of a
test is the probability that a patient who
tested positive for the disease actually has
the disease. PPV = (TP/(TP+FP))X 100.
 The negative predictive value (NPV) of a
test is the probability that a patent who
tested negative for a disease will not have
the disease. NPV = (TN/(TN+FN))X100.

The Efficiency
The efficiency (EFF) of a test is the
probability that the test result and the
diagnosis agree.
 It is calculated as:
EFF = ((TP+TN)/(TP+TN+FP+FN)) X 100

Example

A cytological test was undertaken to screen
women for cervical cancer.
Test Positive
Test Negative
Total
Actually Positive
154 (TP)
225 (FP)
379
Actually Negative
362 (FN)
516 (TP+FN)
23,362 (TN)
23587(FP+TN)
23,724
Sensitivity =?
 Specificity = ?

5 May 2017
94
Relative Risk

Relative risks are the ratio of risks for two different
populations (ratio=a/b).
disease incidence in group 1
Relative Risk 
disease incidence in group 2

If the risk (or proportion) of having the outcome is 1/10 in
one population and 2/10 in a second population, then the
relative risk is: (2/10) / (1/10) = 2.0

A relative risk >1 indicates increased risk for the group in
the numerator and a relative risk <1 indicates decreased
risk for the group in the numerator.
5 May 2017
95
Relative Risk

Relative risk – the chance that a member of a group receiving
some exposure will develop a disease relative to the chance
that a member of an unexposed group will develop the same
disease.
P(disease | exposed)
RR 
P(disease | unexposed)

Recall: a RR of 1.0 indicates that the probabilities of disease
in the exposed and unexposed groups are identical – an
association between exposure and disease does not exist.
5 May 2017
96
Odds Ratio



Odds ratio (OR) is how strongly quantify the
presence or absence of property A associated
with the presence or absence of property B in a
given population. It is a measure of association
between an exposure and an outcome.
The OR represents the odds that an outcome will
occur given a particular exposure, compared to
the odds of the outcome occurring in the absence
of that exposure.
The odds ratio is the ratio of the odds of the
outcome in the two groups.



OR=1 Exposure does not affect odds of outcome
OR>1 Exposure associated with higher odds of outcome
OR<1 Exposure associated with lower odds of outcome
When is it used?
Odds ratios are used to compare the
relative odds of the occurrence of the
outcome of interest (disease or disorder),
given exposure to the variable of interest
(health characteristic). The odds ratio can
also be used to determine whether a
particular exposure is a risk factor for a
particular outcome, and to compare the
magnitude of various risk factors for that
outcome.
 Odd’s Ratio= A/B divided by C/D = AD/BC

Odd’s Ratio and Relative Risk

Odds ratios are better to use in casecontrol studies (cases and controls are
selected and level of exposure is
determined retrospectively)

Relative risks are better for cohort
studies (exposed and unexposed subjects
are chosen and are followed to determine
disease status - prospective)
5 May 2017
99
Odd’s Ratio and Relative Risk

When we have a two-way classification of
exposure and disease we can approximate the
relative risk by the odds ratio
Disease
Exposure


Yes
No
Yes
A
B
A+B
No
C
D
C+D
Relative Risk=A/(A+B) divided by C/(C+D)
Odd’s Ratio= A/B divided by C/D = AD/BC
5 May 2017
100
Case Control Study Example
Disease: Pancreatic Cancer
 Exposure: Cigarette Smoking

Disease
Exposure
5 May 2017
Yes
No
Yes
38
81
119
No
2
56
58
101
Example Continued

Relative risk for exposed vs. non-exposed



Numerator- proportion of exposed people that
have the disease
Denominator-proportion of non-exposed that
have the disease
Relative Risk= (38/119)/(2/58)=9.26
5 May 2017
102
Example Continued

Odd’s Ratio for exposed vs. non-exposed



Numerator- ratio of diseased vs. non-diseased
in the exposed group
Denominator- ratio of diseased vs. nondiseased in the non-exposed group
Odd’s Ratio= (38/81)/(2/56)=(38*56)/(2*81)
=13.14
5 May 2017
103