Biostatistics.school of public healthppt

School of Public Health
Name: Huruy Assefa
E-mail: Huruyame@yahoo.com
Mob.: 0914-728565
Chapter-1
Introduction to Biostatistics

Introduction
• What is statistics?
• Statistics: A field of study concerned with:
– collection, organization, analysis, summarization and
interpretation of numerical data, and
– the drawing of inferences about a body of data when only a
small part of the data is observed.
• Statistics helps us use numbers to communicate ideas
• Statisticians try to interpret and communicate the results
to others.
2

 Biostatistics: The application of statistical methods to
the fields of biological and medical sciences.
 Concerned with interpretation of biological data & the
communication of information derived from these data
 Has central role in medical investigations
Cont.
3

Uses of biostatistics
• Provide methods of organizing information
• Assessment of health status
• Health program evaluation
• Resource allocation
• Magnitude of association
– Strong vs weak association between exposure and
outcome
4

Cont.
• Assessing risk factors
– Cause & effect relationship
• Evaluation of a new vaccine or drug
– What can be concluded if the proportion of people
free from the disease is greater among the vaccinated
than the unvaccinated?
– How effective is the vaccine (drug)?
– Is the effect due to chance or some bias?
• Drawing of inferences
– Information from sample to population
5

What does biostatistics cover?
Research Planning
Design
Execution (Data collection)
Data Processing
Data Analysis
Presentation
Interpretation
Publication
The best way to
learn about
biostatistics is to
follow the flow of a
research from
inception to the
final publication
6

variable:
 It is a characteristic that takes on different
values in different persons, places, or things.
For example:
- heart rate,
- the heights of adult males,
- the weights of preschool children,
- the ages of patients seen in a dental clinic.
7

Types of variables
Quantitative Qualitative
Quantitative Variables
It can be measured in the
usual sense.
For example:
 the heights of adult males,
 the weights of preschool
children,
 the ages of patients seen in
a
 dental clinic.
Qualitative Variables
Many characteristics are not
capable of being measured. Some
of them can be ordered or ranked.
For example:
 classification of people into socio-
economic groups,
 social classes based on income,
education, etc.
8

Interval
Types of variables &
scale of measurement
Quantitative variables
(Numerical)
Qualitative variables
(Categorical)
Ratio
Nominal
Ordinal
9

Types of Statistics
1. Descriptive statistics:
• Ways of organizing and summarizing data
• Helps to identify the general features and trends in a
set of data and extracting useful information
• Also very important in conveying the final results of a
study
• Example: tables, graphs, numerical summary measures
10

Cont.
2. Inferential statistics:
• Methods used for drawing conclusions about a
population based on the information obtained
from a sample of observations drawn from that
population
• Example: Principles of probability, estimation,
confidence interval, comparison of two or more
means or proportions, hypothesis testing, etc.
11

Data
• Data are numbers which can be obtained by
measurement or counting
• The raw material for statistics
• Can be obtained from:
– Routinely kept records, literature
– Surveys
– Counting
– Experiments
– Reports
– Observation
– Etc
12

Types of Data
1. Primary data: collected from the items or
individual respondents directly by the researcher
for the purpose of a study.
2. Secondary data: which had been collected by
certain people or organization, & statistically
treated and the information contained in it is used
for other purpose by other people
13

Population and Sample
• Population:
– Refers to any collection of objects
• Target population:
– A collection of items that have something in common
for which we wish to draw conclusions at a particular
time.
• E.g., All hospitals in Ethiopia
– The whole group of interest
14

Cont.
Study (Sampled) Population:
• The subset of the target population that has at least
some chance of being sampled
• The specific population group from which samples are
drawn and data are collected
15

Cont.
Sample:
 A subset of a study population, about which
information is actually obtained.
 The individuals who are actually measured
and comprise the actual data.
16

Population
Sample
Information
• Role of statistics
in using information
from a sample to make
inferences about the
population
Cont.
17

Sample
Study Population
Target Population
E.g.: In a study of the prevalence
of HIV among adolescents in
Ethiopia, a random sample of
adolescents in Ayder of Mekelle
were included.
Target Population: All
adolescents in Ethiopia
Study population: All
adolescents in Mekelle
Sample: Adolescents in Ayder
sub-city who were included in
the study
Cont.
18

Generalizability
• Is a two-stage procedure:
• We need to be able to generalize from:
– the sample to the study population, &
– then from the study population to the target
population
• If the sample is not representative of the
population, the conclusions are restricted to
the sample & don’t have general applicability
19

Parameter and Statistic
• Parameter: A descriptive measure computed
from the data of a population.
– E.g., the mean (µ) age of the target population
• Statistic: A descriptive measure computed
from the data of a sample.
– E.g., sample mean age ( )
20

Descriptive Statistics:
Summarizing data
21

Measures of Central Tendency (MCT)
• On the scale of values of a variable there is a certain
stage at which the largest number of items tend to
cluster.
• Since this stage is usually in the centre of distribution,
the tendency of the statistical data to get concentrated
at a certain value is called “central tendency”
• The various methods of determining the point about
which the observations tend to concentrate are called
Measures of Central Tendency.
22

• The objective of calculating MCT is to determine a
single figure/value which may be used to
represent the whole data set.
• In that sense it is an even more compact
description of the statistical data than the
frequency distribution.
• Since a MCT represents the entire data, it
facilitates comparison within one group or
between groups of data.
Cont.
23

Characteristics of a good MCT
MCT is good or satisfactory if it possesses the following
characteristics.
1. It should be based on all the observations
2. It should not be affected by the extreme values
3. It should be as close to the maximum number of values as
possible
4. It should have a definite value
5. It should not be subjected to complicated and tedious
calculations (easy)
24

• The most common measures of central
tendency include:
– Arithmetic Mean
– Median
– Mode
– Others
Cont.
25

1. Arithmetic Mean
A. Ungrouped Data
• The arithmetic mean is the "average" of the data set
and by far the most widely used measure of central
location
• Is the sum of all the observations divided by the total
number of observations.
26

The heart rates for n=10 patients were as follows (beats
per minute):
167, 120, 150, 125, 150, 140, 40, 136, 120, 150
What is the arithmetic mean for the heart rate of these
patients?
Cont.
28

b) Grouped data
In calculating the mean from grouped data, we assume that all values falling into a
particular class interval are located at the mid-point of the interval. It is calculated as
follow:
x =
m f
f
i i
i=1
k
i
i=1
k


where,
k = the number of class intervals
mi = the mid-point of the ith
class interval
fi = the frequency of the ith
class interval
Cont.
29

Example. Compute the mean age of 169 subjects from the grouped data.
Mean = 5810.5/169 = 34.48 years
Class interval Mid-point (mi) Frequency (fi) mifi
10-19
20-29
30-39
40-49
50-59
60-69
14.5
24.5
34.5
44.5
54.5
64.5
4
66
47
36
12
4
58.0
1617.0
1621.5
1602.0
654.0
258.0
Total __ 169 5810.5
Cont.
30

When the data are skewed, the mean is “dragged” in the direction of
the skewness
• It is possible in extreme cases for all but one of the sample points to be on
one side of the arithmetic mean & in this case, the mean is a poor measure of
central location or does not reflect the center of the sample.
Cont.
31

Properties of the Arithmetic Mean
• For a given set of data there is one and only one
arithmetic mean (uniqueness)
• Easy to calculate and understand (simple)
• Influenced by each and every value in a data set
• Greatly affected by the extreme values
• In case of grouped data if any class interval is open,
arithmetic mean can not be calculated
32

Median
a) Ungrouped data
• The median is the value which divides the data set into two
equal parts.
• If the number of values is odd, the median will be the middle
value when all values are arranged in order of magnitude.
• When the number of observations is even, there is no single
middle value but two middle observations.
• In this case the median is the mean of these two middle
observations, when all observations have been arranged in the
order of their magnitude.
33

Cont.
35

Cont.
36

 The median is a better description (than the mean) of the
majority when the distribution is skewed
• Example:- Data: 14, 89, 93, 95, 96
– Skewness is reflected in the outlying low value of 14
– The sample mean is 77.4
– The median is 93
Cont.
37

b) Grouped data
• In calculating the median from grouped data, we
assume that the values within a class-interval are
evenly distributed through the interval.
• The first step is to locate the class interval in which
the median is located, using the following procedure.
• Find n/2 and see a class interval with a minimum
cumulative frequency which contains n/2.
• Then, use the following formal.
38

~
x = L
n
2
F
f
W
m
c
m












where,
Lm = lower true class boundary of the interval containing the median
Fc = cumulative frequency of the interval just above the median class
interval
fm = frequency of the interval containing the median
W= class interval width
n = total number of observations
Cont.
39

Example. Compute the median age of 169 subjects from
the grouped data.
n/2 = 169/2 = 84.5
Class interval Mid-point (mi) Frequency (fi) Cum. freq
10-19
20-29
30-39
40-49
50-59
60-69
14.5
24.5
34.5
44.5
54.5
64.5
4
66
47
36
12
4
4
70
117
153
165
169
Total 169
40

• n/2 = 84.5 = in the 3rd class interval
• Lower limit = 29.5, Upper limit = 39.5
• Frequency of the class = 47
• (n/2 – fc) = 84.5-70 = 14.5
• Median = 29.5 + (14.5/47)10 = 32.58 ≈ 33
Cont.
41

Properties of the median
• There is only one median for a given set of data (uniqueness)
• The median is easy to calculate
• Median is a positional average and hence it is insensitive to very
large or very small values (not affected by extreme values)
• Median can be calculated even in the case of open end intervals
• It is determined mainly by the middle points and less sensitive to
the remaining data points (weakness).
42

Mode
• The mode is the most frequently occurring value
among all the observations in a set of data.
• It is not influenced by extreme values.
• It is possible to have more than one mode or no
mode.
• It is not a good summary of the majority of the
data.
43

Mode
0
2
4
6
8
10
12
14
16
18
20
Mode
T. Ancelle, D. Coulombie
Mode
44

• It is a value which occurs most frequently in
a set of values.
• If all the values are different there is no
mode, on the other hand, a set of values
may have more than one mode.
a) Ungrouped data
45

• Example
• Data are: 1, 2, 3, 4, 4, 4, 4, 5, 5, 6
• Mode is 4 “Unimodal”
• Example
• Data are: 1, 2, 2, 2, 3, 4, 5, 5, 5, 6, 6, 8
• There are two modes - 2 & 5
• This distribution is said to be “bi-modal”
• Example
• Data are: 2.62, 2.75, 2.76, 2.86, 3.05, 3.12
• No mode, since all the values are different
Cont.
46

b) Grouped data
• To find the mode of grouped data, we
usually refer to the modal class, where the
modal class is the class interval with the
highest frequency.
• If a single value for the mode of grouped
data must be specified, it is taken as the
mid-point of the modal class interval.
47

Properties of mode
 It is not affected by extreme values
 It can be calculated for distributions with open
end classes
 Often its value is not unique
 The main drawback of mode is that often it
does not exist
49

Cont.
Which measure of central tendency is best with a
given set of data?
• Two factors are important in making this decisions:
– The scale of measurement (type of data)
– The shape of the distribution of the
observations
50

• The mean can be used for discrete and
continuous data.
• The median is appropriate for discrete and
continuous data as well, but can also be used for
ordinal data.
• The mode can be used for all types of data, but
may be especially useful for nominal and ordinal
measurements.
Cont.
51

(A) Symmetric and unimodal distribution —
Mean, median, and mode should all be
approximately the same
Mean, Median & Mode
Relationship between Mean, Median and Mode
52

(A) Bimodal — Mean and median should be
about the same, but may take a value that is
unlikely to occur; two modes might be best
Cont.
53

(C) Skewed to the right (positively skewed) —
Mean is sensitive to extreme values, so median
might be more appropriate
Mode
Median
Mean
Cont.
54

(D) Skewed to the left (negatively skewed) —
Same as (c)
Mode
Median
Mean
Cont.
55

Measures of Dispersion
56

Consider the following two sets of data:
A: 177 193 195 209 226 Mean = 200
B: 192 197 200 202 209 Mean = 200
Two or more sets may have the same mean and/or median but they may be
quite different.
57

These two distributions have the same mean,
median, and mode
58

• MCT are not enough to give a clear
understanding about the distribution of the
data.
• We need to know something about the
variability or spread of the values — whether
they tend to be clustered close together, or
spread out over a broad range
Cont.
59

• Measures that quantify the variation or dispersion of a
set of data from its central location
• Dispersion refers to the variety exhibited by the values
of the data.
• The amount may be small when the values are close
together.
• If all the values are the same, no dispersion
Measures of Dispersion
60

• Measures of dispersion include:
– Range
– Inter-quartile range
– Variance
– Standard deviation
– Coefficient of variation
– Standard error
– Others
Cont.
61

Range (R)
• The difference between the largest and smallest
observations in a sample.
• Range = Maximum value – Minimum value
• Example –
– Data values: 5, 9, 12, 16, 23, 34, 37, 42
– Range = 42-5 = 37
• Data set with higher range exhibit more variability
62

Properties of range
 It is the simplest crude measure and can be easily
understood
 It takes into account only two values which causes it to
be a poor measure of dispersion
 Very sensitive to extreme observations
 The larger the sample size, the larger the
range
63

Interquartile range (IQR)
• Indicates the spread of the middle 50% of the
observations, and used with median
IQR = Q3 - Q1
• Example: Suppose the first and third quartile for weights
of girls 12 months of age are 8.8 Kg and 10.2 Kg,
respectively.
IQR = 10.2 Kg – 8.8 Kg
i.e., 50% of the infant girls weigh between 8.8 and 10.2 Kg.
64

• It is a simple and versatile measure
• It encloses the central 50% of the observations
• It is not based on all observations but only on two
specific values
• It is important in selecting cut-off points in the
formulation of clinical standards
• Since it excludes the lowest and highest 25% values, it
is not affected by extreme values
• Less sensitive to the size of the sample
Properties of IQR:
65

• The variance is the average of the squares of the
deviations taken from the mean.
• It is squared because the sum of the deviations of the
individual observations of a sample about the sample
mean is always 0
0 = ( )
• The variance can be thought of as an average of
squared deviations
 -x
xi
Variance (2, s2)
66

• Variance is used to measure the dispersion of
values relative to the mean.
• When values are close to their mean (narrow
range) the dispersion is less than when there
is scattering over a wide range.
– Population variance = σ2
– Sample variance = S2
Cont.
67

a) Ungrouped data
 Let X1, X2, ..., XN be the measurement on N
population units, then:
mean.
population
the
is
N
X
=
where
N
)
(X
N
1
=
i
i
N
1
i
2
i
2








Cont.
68

A sample variance is calculated for a sample of individual values (X1, X2,
… Xn) and uses the sample mean (e.g:- ) rather than the population
mean µ.
Cont.
𝑿
69

Degrees of freedom
• In computing the variance there are (n-1) degrees of
freedom because only (n-1) of the deviations are
independent from each other
• The last one can always be calculated from the
others automatically.
• This is because the sum of the deviations from their
mean (Xi-Mean) must add to zero.
70

b) Grouped data
where
mi = the mid-point of the ith class interval
fi = the frequency of the ith class interval
= the sample mean
k = the number of class intervals
S
(m x) f
f - 1
2
i
2
i
i=1
k
i
i=1
k




x
71

 The main disadvantage of variance is that its
unit is the square of the unit of the original
measurement values.
 The variance gives more weight to the extreme
values as compared to those which are near to
mean value, because the difference is squared
in variance.
• The drawbacks of variance are overcome by the
standard deviation.
Properties of Variance:
72

Standard deviation (, s)
• It is the square root of the variance.
• This produces a measure having the same
scale as that of the individual values.
 
 2
and S = S2
73

• Following are the survival times of n=11
patients after heart transplant surgery.
• The survival time for the “ith” patient is
represented as Xi for i= 1, …, 11.
• Calculate the sample variance and SD?
Example:
74

Example. Compute the variance and SD of the age of 169 subjects from the
grouped data.
Mean = 5810.5/169 = 34.48 years
S2 = 20199.22/169-1 = 120.23
SD = √S2 = √120.23 = 10.96
Class
interval (mi) (fi) (mi-Mean) (mi-Mean)2 (mi-Mean)2 fi
10-19
20-29
30-39
40-49
50-59
60-69
14.5
24.5
34.5
44.5
54.5
64.5
4
66
47
36
12
4
-19.98
-9-98
0.02
10.02
20.02
30.02
399.20
99.60
0.0004
100.40
400.80
901.20
1596.80
6573.60
0.0188
3614.40
4809.60
3604.80
Total 169 1901.20 20199.22
Example:
76

Properties of SD
• The SD has the advantage of being expressed in the
same units of measurement as the mean
• SD is considered to be the best measure of dispersion
and is used widely because of the properties of the
theoretical normal curve.
• However, the drawback of SD is if the units of
measurements of variables of two data sets is not the
same.
77

Coefficient of variation (CV)
• When two data sets have different units of
measurements, or their means differ
sufficiently in size, the CV should be used as a
measure of dispersion.
• It is the best measure to compare the
variability of two series of sets of
observations.
• Data with less coefficient of variation is
considered more consistent.
78

100
x
SD
CV 

• “Cholesterol is more variable than systolic blood
pressure”
SD Mean CV (%)
SBP
Cholesterol
15mm
40mg/dl
130mm
200mg/dl
11.5
20.0
•CV is the ratio of the SD to the mean multiplied by 100.
Cont.
79

NOTE:
• The range often appears with the median as a
numerical summary measure
• The IQR is used with the median as well
• The SD is used with the mean
• For nominal and ordinal data, a table or graph is often
more effective than any numerical summary measure
80

Probability and Probability
Distributions
81

Probability
• Chance of observing a particular outcome.
• Likelihood of an event.
• Probability theory developed from the study of
games of chance like dice and cards.
• A process like flipping a coin, rolling a die or
drawing a card from a deck are probability
experiments.
82

Why Probability in Statistics?
• Results are not certain
• To evaluate how accurate our results are:
– Given how our data were collected, are our
results accurate ?
– Given the level of accuracy needed, how many
observations need to be collected ?
83

When can we talk about probability ?
When dealing with a process that has an uncertain
outcome
Experiment = any process with an uncertain outcome
• When an experiment is performed, one and only one
outcome is obtained
• Event = something that may happen or not when the
experiment is performed
84

Two Categories of Probability
• Objective and Subjective Probabilities.
• Objective probability
1) Classical probability and
2) Relative frequency probability.
85

Classical Probability
• Is based on gambling ideas
• Rolling a die -
– There are 6 possible outcomes:
– Total ways = {1, 2, 3, 4, 5, 6}.
• Each is equally likely
– P(i) = 1/6, i=1,2,...,6.
P(1) = 1/6
P(2) = 1/6
 …….
P(6) = 1/6
SUM = 1
86

• Definition: If an event can occur in N mutually exclusive
and equally likely ways, and if m of these posses a
characteristic, E, the probability of the occurrence of E =
m/N.
P(E)= the probability of E = P(E) = m/N
• If we toss a die, what is the probability of 4 coming up?
m = 1(which is 4) and N = 6
The probability of 4 coming up is 1/6.
Cont.
87

Relative Frequency Probability
• The proportion of times the event A occurs — in
a large number of trials repeated under
essentially identical conditions
• Definition: If a process is repeated a large number of
times (n), and if an event with the characteristic E occurs
m times, the relative frequency of E,
Probability of E = P(E) = m/n.
88

• If you toss a coin 100 times and head comes up 40 times,
P(H) = 40/100 = 0.4.
• If we toss a coin 10,000 times and the head comes up 5562,
P(H) = 0.5562.
• Therefore, the longer the series and the longer sample size, the closer the
estimate to the true value.
Example:
Of 158 people who attended a dinner party, 99 were ill.
P (Illness) = 99/158 = 0.63 = 63%.
Cont.
89

Subjective Probability
• Personalistic (represents one’s degree of belief in the
occurrence of an event).
• Personal assessment of which is more effective to
provide cure – traditional/modern
• Personal assessment of which sports team will win a
match.
• Also uses classical and relative frequency methods to
assess the likelihood of an event.
90

• E.g., If someone says that he is 95% certain that
a cure for EBOLA will be discovered within 5
years, then he means that:
P(discovery of cure for EBOLA within 5 years) = 95%
= 0.95
• Although the subjective view of probability has enjoyed
increased attention over the years, it has not fully accepted by
scientists.
Cont.
91

Mutually Exclusive Events
 Two events A and B are mutually exclusive if they
cannot both happen at the same time
P (A ∩ B) = 0
• Example:
– A coin toss cannot produce heads and tails
simultaneously.
– Weight of an individual can’t be classified simultaneously
as “underweight”, “normal”, “overweight”
92

Independent Events
• Two events A and B are independent if the
probability of the first one happening is the
same no matter how the second one turns
out. OR. The outcome of one event has no effect on the occurrence or
non-occurrence of the other.
P(A∩B) = P(A) x P(B) (Independent events)
P(A∩B) ≠ P(A) x P(B) (Dependent events)
Example:
– The outcomes on the first and second coin tosses
are independent
93

Intersection and union
• The intersection of two events A and B, A ∩ B, is the event that A and B
happen simultaneously
P ( A and B ) = P (A ∩ B )
• Let A represent the event that a randomly selected newborn is LBW, and B
the event that he or she is from a multiple birth
• The intersection of A and B is the event that the infant is both LBW and
from a multiple birth
• The union of A and B, A U B, is the event that either A happens or B
happens or they both happen simultaneously
P ( A or B ) = P ( A U B )
• In the example above, the union of A and B is the event that the newborn is
either LBW or from a multiple birth, or both
94

Properties of Probability
1. The numerical value of a probability always lies
between 0 and 1, inclusive.
 A value 0 means the event can not occur
 A value 1 means the event definitely will occur
 A value of 0.5 means that the probability that the event
will occur is the same as the probability that it will not
occur.
2. The sum of the probabilities of all mutually exclusive
outcomes is equal to 1.
P(E1) + P(E2 ) + .... + P(En ) = 1.
95

3. For two mutually exclusive events A and B,
P(A or B ) = P(AUB)= P(A) + P(B).
 If not mutually exclusive:
P(A or B) = P(A) + P(B) - P(A and B)
4. The complement of an event A, denoted by Ā or Ac, is the event that
A does not occur
• Consists of all the outcomes in which event A does NOT occur
P(Ā) = P(not A) = 1 – P(A)
• Ā occurs only when A does not occur.
• These are complementary events.
Cont.
96

Basic Probability Rules
1. Addition rule
 If events A and B are mutually exclusive:
P(A or B) = P(A) + P(B)
P(A and B) = 0
 More generally:
P(A or B) = P(A) + P(B) - P(A and B)
P(event A or event B occurs or they both occur)
97

Example:
The probabilities below represent years of
schooling completed by mothers of newborn infants
1. What is the probability that a mother has completed < 12
years of schooling?
2. What is the probability that a mother has completed 12 or more years
of schooling?
98

1. The probability that a mother has
completed < 12 years of schooling is:
P( 8 years) = 0.056 and
P(9-11 years) = 0.159
• Since these two events are mutually exclusive,
P( 8 or 9-11) = P( 8 U 9-11)
= P( 8) + P(9-11)
= 0.056+0.159
= 0.215
Cont.
99

2. The probability that a mother has completed 12
or more years of schooling is:
P(12) = P(12 or 13-15 or 16)
= P(12 U 13-15 U 16)
= P(12)+P(13-15)+P(16)
= 0.321+0.218+0.230
= 0.769
Cont.
100

– If A and B are independent events, then
P(A ∩ B) = P(A) × P(B)
– More generally,
P(A ∩ B) = P(A) P(B|A) = P(B) P(A|B)
P(A and B) denotes the probability that A and B
both occur at the same time.
2. Multiplication rule
101

Conditional Probability
• The conditional probability that event B has
occurred given that event A has already
occurred is denoted P(B|A) and is defined
provided that P(A) ≠ 0.
102

Example:
A study investigating the effect of prolonged exposure to
bright light on retina damage in premature infants.
Retinopathy
YES
Retinopathy
NO
TOTAL
Bright light
Reduced light
18
21
3
18
21
39
TOTAL 39 21 60
Cont.
103

• The probability of developing retinopathy is:
P (Retinopathy) = No. of infants with retinopathy
Total No. of infants
= (18+21)/(21+39)
= 0.65
Cont.
104

• We want to compare the probability of
retinopathy, given that the infant was exposed
to bright light, with that the infant was exposed
to reduced light.
• Exposure to bright light and exposure to
reduced light are conditioning events, events
we want to take into account when calculating
conditional probabilities.
Cont.
105

• The conditional probability of retinopathy, given
exposure to bright light, is:
• P(Retinopathy/exposure to bright light) =
No. of infants with retinopathy exposed to bright light
No. of infants exposed to bright light
= 18/21 = 0.86
Cont.
106

• P(Retinopathy/exposure to reduced light) =
# of infants with retinopathy exposed to reduced light
No. of infants exposed to reduced light
= 21/39 = 0.54
• The conditional probabilities suggest that premature infants
exposed to bright light have a higher risk of retinopathy than
premature infants exposed to reduced light.
Cont.
107

 For independent events A and B
P(A/B) = P(A).
 For non-independent events A and B
P(A and B) = P(A/B) P(B)
(General Multiplication Rule)
Cont.
108

Exercise:
Culture and Gonodectin (GD) test results for 240 Urethral
Discharge Specimens
GD Test
Result
Culture Result
yes
Gonorrhea
No
Gonorrhea
Total
Positive 175 9 184
Negative 8 48 56
Total 183 57 240
109

1. What is the probability that a man has
gonorrhea?
2. What is the probability that a man has a
positive GD test?
positive GD test and gonorrhea?
negative GD test and does not have gonorrhea
5. What is the probability that a man with
gonorrhea has a positive GD test?
Cont.
110

6. What is the probability that a man does not
have gonorrhea has a negative GD test?
7. What is the probability that a man does not
have gonorrhea has a positive GD test?
8. What is the probability that a man with positive
GD test has gonorrhea?
Cont.
111

Probability Distributions
• It is the way data are distributed, in order to draw
conclusions about a set of data
• Random Variable = Any quantity or characteristic that is
able to assume a number of different values such that any
particular outcome is determined by chance
• The probability distribution of a random variable is a
table, graph, or mathematical formula that gives the
probabilities with which the random variable takes
different values or ranges of values.
112

Discrete Probability Distributions
• For a discrete random variable, the probability
distribution specifies each of the possible outcomes
of the random variable along with the probability
that each will occur
• Examples can be:
– Frequency distribution
– Relative frequency distribution
– Cumulative frequency
113

• We represent a potential outcome of the
random variable X by x
 0 ≤ P(X = x) ≤ 1
 ∑ P(X = x) = 1
Cont.
114

The following data shows the number of diagnostic
services a patient receives
115

• What is the probability that a patient receives
exactly 3 diagnostic services?
P(X=3) = 0.031
• What is the probability that a patient receives at
most one diagnostic service?
P (X≤1) = P(X = 0) + P(X = 1)
= 0.671 + 0.229
= 0.900
Cont.
116

• What is the probability that a patient receives
at least four diagnostic services?
P (X≥4) = P(X = 4) + P(X = 5)
= 0.010 + 0.006
= 0.016
Cont.
117

Probability distributions can also
be displayed using a graph
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 1 2 3 4 5
No. of diagnostic services, x
Probability,
X=x
118

• Examples of discrete probability distributions
are the binomial distribution and the Poisson
distribution.
Cont.
119

Binomial Distribution
• Consider dichotomous (binary) random variable
• Is based on Bernoulli trial
– When a single trial of an experiment can result in only one of
two mutually exclusive outcomes (success or failure; dead or
alive; sick or well, male or female)
120

Example:
• We are interested in determining whether a newborn
infant will survive until his/her 70th birthday
• Let Y represent the survival status of the
child at age 70 years
• Y = 1 if the child survives and Y = 0 if he/she does not
• The outcomes are mutually exclusive and exhaustive
• Suppose that 72% of infants born survive to age 70 years
P(Y = 1) = p = 0.72
P(Y = 0) = 1 − p = 0.28
121

Characteristics of a Binomial Distribution
• The experiment consist of n identical trials[Fixed].
• Only two possible outcomes on each trial.
• The probability of A (success), denoted by p, remains the
same from trial to trial. The probability of B (failure),
denoted by q,
q = 1- p.
• The trials are independent.
• n and  are the parameters of the binomial distribution.
• The mean is n and the variance is n(1- )
122

• If an experiment is repeated n times and the
outcome is independent from one trial to
another, the probability that outcome A occurs
exactly x times is:
• P (X=x) = , x = 0, 1, 2, ..., n.
=
Cont.
123

• n denotes the number of fixed trials
• x denotes the number of successes in
the n trials
• p denotes the probability of success
• q denotes the probability of failure (1- p)
=
• Represents the number of ways of selecting x objects out of n where the
order of selection does not matter.
• where n!=n(n-1)(n-2)…(1) , and 0!=1
Cont.
124

Example:
• Suppose we know that 40% of a certain
population are cigarette smokers. If we take a
random sample of 10 people from this
population, what is the probability that we will
have exactly 4 smokers in our sample?
125

• If the probability that any individual in the
population is a smoker to be P=.40, then the
probability that x=4 smokers out of n=10 subjects
selected is:
P(X=4) =10C4(0.4)4
(1-0.4)10-4
= 10C4(0.4)4
(0.6)6
= 210(.0256)(.04666)
= 0.25
• The probability of obtaining exactly 4 smokers in the
sample is about 0.25.
Cont.
126

• We can compute the probability of observing zero
smokers out of 10 subjects selected at random, exactly
1 smoker, and so on, and display the results in a table,
as given, below.
• The third column, P(X ≤ x), gives the cumulative
probability. E.g. the probability of selecting 3 or fewer
smokers into the sample of 10 subjects is
P(X ≤ 3) =.3823, or about 38%.
Cont.
127

The probability in the above table can be
converted into the following graph
0
0.05
0.1
0.15
0.2
0.25
0.3
0 1 2 3 4 5 6 7 8 9 10
No. of Smokers
Probability
Cont.
129

Exercise
Each child born to a particular set of parents
has a probability of 0.25 of having blood type
O. If these parents have 5 children.
What is the probability that
a. Exactly two of them have blood type O
b. At most 2 have blood type O
c. At least 4 have blood type O
d. 2 do not have blood type O.
130

Solution for ‘a’
a.)
2637
.
0
)
75
.
0
(
)
25
.
0
(
2
5
=
2)
P(x 2
-
5
2








131

2. The Poisson Distribution
• Is a discrete probability distribution used to
model the number of occurrences of an event
that takes place infrequently in time or space
• Applicable for counts of events over a given
interval of time, for example:
– number of patients arriving at an emergency
department in a day
– number of new cases of HIV diagnosed at a clinic in
a month
132

• In such cases, we take a sample of days and observe the
number of patients arriving at the emergency department
on each day,
• We are observing a count or number of events, rather than a
yes/no or success/ failure outcome for each subject or trial,
as in the binomial.
• Suppose events happen randomly and independently in time
at a constant rate. If events happen with rate  events per
unit time, the probability of x events happening in unit time
is:
Cont.
133
P(x) =
e
x!
x
 


• where x = 0, 1, 2, . . .∞
• x is a potential outcome of X
• The constant λ (lambda) represents the rate at
which the event occurs, or the expected number
of events per unit time
• e = 2.71828
• It depends up on just one parameter, which is
the µ number of occurrences (λ).
Cont.
134

Example
• The daily number of new registrations of
cancer is 2.2 on average.
What is the probability of
a) Getting no new cases
b) Getting 1 case
c) Getting 2 cases
d) Getting 3 cases
e) Getting 4 cases
135

Solutions
a)
b) P(X=1) = 0.244
c) P(X=2) = 0.268
d) P(X=3) = 0.197
e) P(X=4) = 0.108
111
.
0
!
0
)
2
.
2
(
)
0
(
2
.
2
0




e
X
P
136

0 1 2 3 4 5 6 7
0.3
0.2
0.1
0.0
Probability
Poisson distribution with mean 2.2
137

Example:
• In a given geographical area, cases of tetanus are
reported at a rate of λ = 4.5/month
• What is the probability that 0 cases of tetanus will
be reported in a given month?
138

• What is the probability that 1 case of tetanus
will be reported?
Cont.
139

Continuous Probability Distributions
• A continuous random variable X can take on any value in
a specified interval or range
• The probability distribution of X is represented by a
smooth curve called a probability density function
• The area under the smooth curve is equal to 1
• The area under the curve between any two points x1 and
x2 is the probability that X takes a value between x1 and
x2
140

• The probability associated with any one particular value
is equal to 0
• Therefore, P(X=x) = 0
• Also, P(X ≥ x) = P(X > x)
• We calculate:
Pr [ a < X < b], the probability of an
interval of values of X.
Cont.
141

The Normal distribution
• Frequently called the “Gaussian distribution” or
bell-shape curve.
• Variables such as blood pressure, weight,
height, serum cholesterol level, and IQ score —
are approximately normally distributed
142

A random variable is said to have a normal distribution if it
has a probability distribution that is symmetric and bell-
shaped
Cont.
143

• A random variable X is said to follow ND, if and
only if, its probability density function is:
, - < x < .
f(x) =
1
2
e
x-
2
 









1
2
Cont.
144

π (pi) = 3.14159
e = 2.71828, x = Value of X
Range of possible values of X: -∞ to +∞
µ = Expected value of X (“the long run
average”)
σ2 = Variance of X.
µ and σ are the parameters of the normal
distribution — they completely define its
shape
Cont.
145

1. The mean µ tells you about location -
– Increase µ - Location shifts right
– Decrease µ – Location shifts left
– Shape is unchanged
2. The variance σ2 tells you about narrowness or
flatness of the bell -
– Increase σ2 - Bell flattens. Extreme values are more likely
– Decrease σ2 - Bell narrows. Extreme values are less likely
– Location is unchanged
Cont.
146

Properties of the Normal Distribution
1. It is symmetrical about its mean, .
2. The mean, the median and mode are almost equal. It is unimodal.
3. The total area under the curve about the x-axis is 1 square unit.
4. The curve never touches the x-axis.
5. As the value of  increases, the curve becomes more and more
flat and vice versa.
6. The distribution is completely determined by the parameters 
and .
148

• We cannot tabulate every possible
distribution
• Tabulated normal probability calculations are
available only for the ND with µ = 0 and σ2=1.
Cont.
150

Standard Normal Distribution
 It is a normal distribution that has a mean equal
to 0 and a SD equal to 1, and is denoted by N(0,
1).
 The main idea is to standardize all the data that is
given by using Z-scores.
 These Z-scores can then be used to find the area
(and thus the probability) under the normal
curve.
151

The standard normal distribution has
mean 0 and variance 1
• Approximately 68% of the area under the standard normal
curve lies between ±1, about 95% between ±2, and about 99%
between ±2.5
152

Z - Transformation
• If a random variable X~N(,) then we can
transform it to a SND with the help of Z-
transformation
Z = x - 

• Z represents the Z-score for a given x value
153

• Consider redefining the scale to be in terms of
how many SDs away from mean for normal
distribution, μ=110 and σ=15.
Value
X = 50 65 80 95 110 125 140 155 170
Z = -4 -3 -2 -1 0 1 2 3 4
SDs from mean using
Z=
𝑥−𝜇
𝜎
=
𝑥−110
15
Cont.
154

• This process is known as standardization and
gives the position on a normal curve with μ=0
and σ=1, i.e., the SND, Z.
• A Z-score is the number of standard deviations
that a given x value is above or below the
mean.
Cont.
155

Some Useful Tips
156

a) What is the probability that z < -1.96?
(1) Sketch a normal curve
(2) Draw a perpendicular line for z = -1.9
(3) Find the area in the table
(4) The answer is the area to the left of the line P(z < -
1.96) = 0.0250
157

b) What is the probability that -1.96 < z < 1.96?
The area between the values
P(-1.96 < z < 1.96) = 0.9750 - 0.0250 = 0.9500
158

c) What is the probability that z > 1.96?
• The answer is the area to the right of the line; found by
subtracting table value from 1.0000;
P(z > 1.96) =1.0000 - 0.9750 = 0.0250
159

Exercise
1. Compute P(-1 ≤ Z ≤ 1.5)
Ans: 0.7745
2. Find the area under the SND from 0 to 1.45
Ans: 0.4265
3. Compute P(-1.66 < Z < 2.85)
Ans: 0.9493
161

Applications of the Normal Distribution
Example:
• The diastolic blood pressures of males 35–44 years
of age are normally distributed with µ = 80 mm Hg
and σ2 = 144 mm Hg2
[σ = 12 mm Hg].
• Let individuals with BP above 95 mm Hg are
considered to be hypertensive
162

Cont.
163
a. What is the probability that a randomly selected
male has a BP above 95 mm Hg?
Approximately 10.6% of this population would be
classified as hypertensive.

b. What is the probability that a randomly
selected male has a DBP above 110 mm Hg?
Z = 110 – 80 = 2.50
12
P (Z > 2.50) = 0.0062
• Approximately 0.6% of the population has a
DBP above 110 mm Hg
Cont.
164

c. What is the probability that a randomly selected
male has a DBP below 60 mm Hg?
Z = 60 – 80 = -1.67
12
P (Z < -1.67) = 0.0475
• Approximately 4.8% of the population has a
DBP below 60 mm Hg
Cont.
165

Other Distributions
1. Student t-distribution
2. F- Distribution
3. 2 –Distribution
166

Sampling and Sampling
Distributions
167

• Researchers often use sample survey
methodology to obtain information about a
larger population by selecting and measuring a
sample from that population.
• Since population is too large, we rely on the
information collected from the sample.
• Inferences about the population are based on the
information from the sample drawn from that
population.
Cont.
168

• A sample is a collection of individuals selected
from a larger population.
• Sampling enables us to estimate the
characteristic of a population by directly
observing a portion of the population.
Cont.
169

Sample Information
Population
Cont.
170

Steps needed to select a sample and ensure that
this sample will fulfill its goals.
1. Establish the study's objectives
– The first step in planning a useful and efficient survey is
to specify the objectives with as much detail as
possible.
– Without objectives, the survey is unlikely to generate
valuable results.
– Clarifying the aims of the survey is critical to its
ultimate success.
– The initial users and uses of the data should be
identified at this stage.
171

2. Define the target population
– The target population is the total population for which
the information is required.
– Specifically, the target population is defined by the
following characteristics:
• Nature of data required
• Geographic location
• Reference period
• Other characteristics, such as socio-demographic characteristics
Cont.
172

3. Decide on the data to be collected
– The data requirements of the survey must be established.
– To ensure that the requirements are operationally sound, the
necessary data terms and definitions also need to be determined.
4. Set the level of precision
– There is a level of uncertainty associated with estimates
coming from a sample.
– The sample-to-sample variation is what causes the sampling
error.
– Researchers can estimate the sampling error associated
with a particular sampling plan, and try to minimize it.
Cont.
173

5. Decide on the methods on measurement
– Choose measuring instrument and method of approach to
the population
– Data about a person’s state of health may be obtained from
statements that he/she makes or from a medical
examination
– The survey may employ a self-administered questionnaire,
an interviewing
6. Preparing Frame
– List of all members of the population
– The elements must not overlap
Cont.
174

Sampling
• The process of selecting a portion of the
population to represent the entire population.
• A main concern in sampling:
– Ensure that the sample represents the population,
and
– The findings can be generalized.
175

Advantages of sampling:
• Feasibility: Sampling may be the only feasible method of
collecting information.
• Reduced cost: Sampling reduces demands on resource
such as finance, personnel, and material.
• Greater accuracy: Sampling may lead to better accuracy of
collecting data
• Sampling error: Precise allowance can be made for
sampling error
• Greater speed: Data can be collected and summarized
more quickly
176

Disadvantages of sampling:
• There is always a sampling error.
• Sampling may create a feeling of
discrimination within the population.
• Sampling may be inadvisable where every
unit in the population is legally
required to have a record.
177

Errors in sampling
1) Sampling error: Errors introduced due to errors
in the selection of a sample.
– They cannot be avoided or totally eliminated.
2) Non-sampling error:
- Observational error
- Respondent error
- Lack of preciseness of definition
- Errors in editing and tabulation of data
178

Sampling Methods
Two broad divisions:
A. Probability sampling methods
B. Non-probability sampling methods
179

Probability sampling
• Involves random selection of a sample
• A sample is obtained in a way that ensures every
member of the population to have a known, non
zero probability of being included in the sample.
• The method chosen depends on a number of
factors, such as
– the available sampling frame,
– how spread out the population is,
– how costly it is to survey members of the population
180

Most common probability
sampling methods
1. Simple random sampling
2. Systematic random sampling
3. Stratified random sampling
4. Cluster sampling
5. Multi-stage sampling
181

1. Simple random sampling
• Involves random selection
• Each member of a population has an equal
chance of being included in the sample.
182

• To use a SRS method:
– Make a numbered list of all the units in the
population
– Each unit should be numbered from 1 to N (where
N is the size of the population)
– Select the required number.
Cont.
183

• The randomness of the sample is ensured
by:
• use of “lottery’ methods
• a table of random numbers
Cont.
184

Example
• Suppose your school has 500 students and
you need to conduct a short survey on the
quality of the food served in the cafeteria.
• You decide that a sample of 10 students
should be sufficient for your purposes.
• In order to get your sample, you assign a
number from 1 to 500 to each student in
your school.
185

• To select the sample, you use a table of
randomly generated numbers.
• Pick a starting point in the table (a row and
column number) and look at the random
numbers that appear there. In this case, since
the data run into three digits, the random
numbers would need to contain three digits as
well.
Cont.
186

• Ignore all random numbers after 500 because they do
not correspond to any of the students in the school.
• Remember that the sample is without replacement,
so if a number recurs, skip over it and use the next
random number.
• The first 10 different numbers between 001 and 500
make up your sample.
Cont.
187

• SRS has certain limitations:
– Requires a sampling frame.
– Difficult if the reference population is dispersed.
– Minority subgroups of interest may not be
selected.
Cont.
188

2. Systematic random sampling
• Sometimes called interval sampling,
systematic sampling means that there is a
gap, or interval, between each selected unit in
the sample
• The selection is systematic rather than
randomly
189

• Important if the reference population is
arranged in some order:
– Order of registration of patients
– Numerical number of house numbers
– Student’s registration books
• Taking individuals at fixed intervals (every kth)
based on the sampling fraction, eg. if the
sample includes 20%, then every fifth.
Cont.
190

Steps in systematic random sampling
1. Number the units on your frame from 1 to N (where N is
the total population size).
2. Determine the sampling interval (K) by dividing the number
of units in the population by the desired sample size.
3. Select a number between one and K at random. This
number is called the random start and would be the first
number included in your sample.
4. Select every Kth unit after that first number
Note: Systematic sampling should not be used when a cyclic
repetition is inherent in the sampling frame.
191

Example
• To select a sample of 100 from a population of 400, you
would need a sampling interval of 400 ÷ 100 = 4.
• Therefore, K = 4.
• You will need to select one unit out of every four units
to end up with a total of 100 units in your sample.
• Select a number between 1 and 4 from a table of
random numbers.
192

• If you choose 3, the third unit on your frame
would be the first unit included in your
sample;
• The sample might consist of the following
units to make up a sample of 100: 3 , 7, 11, 15,
19...395, 399 (up to N, which is 400 in this
case).
Cont.
193

• Using the above example, you can see that
with a systematic sample approach there are
only four possible samples that can be
selected, corresponding to the four possible
random starts:
A. 1, 5, 9, 13...393, 397
B. 2, 6, 10, 14...394, 398
C. 3, 7, 11, 15...395, 399
D. 4, 8, 12, 16...396, 400
Cont.
194

3. Stratified random sampling
• It is done when the population is known to be have
heterogeneity with regard to some factors and those
factors are used for stratification
• Using stratified sampling, the population is divided into
homogeneous, mutually exclusive groups called strata,
and
• A population can be stratified by any variable that is
available for all units prior to sampling (e.g., age, sex,
province of residence, income, etc.).
• A separate sample is taken independently from each
stratum.
195

Why do we need to create strata?
• That it can make the sampling strategy more
efficient.
• A larger sample is required to get a more accurate
estimation if a characteristic varies greatly from
one unit to the other.
• For example, if every person in a population had
the same salary, then a sample of one individual
would be enough to get a precise estimate of the
average salary.
196

• Equal allocation:
– Allocate equal sample size to each stratum
• Proportionate allocation:
, j = 1, 2, ..., k where, k is
the number of strata and
– nj is sample size of the jth stratum
– Nj is population size of the jth stratum
– n = n1 + n2 + ...+ nk is the total sample size
– N = N1 + N2 + ...+ Nk is the total population
size
n
n
N
N
j j

Cont.
197

4. Cluster sampling
• Sometimes it is too expensive to spread a sample
across the population as a whole.
• Travel costs can become expensive if interviewers
have to survey people from one end of the country to
the other.
• To reduce costs, researchers may choose a cluster
sampling technique
• The clusters should be homogeneous, unlike stratified
sampling where by the strata are heterogeneous
198

Steps in cluster sampling
• Cluster sampling divides the population into groups or
clusters.
• A number of clusters are selected randomly to represent the
total population, and then all units within selected clusters
are included in the sample.
• No units from non-selected clusters are included in the
sample—they are represented by those from selected
clusters.
• This differs from stratified sampling, where some units are
selected from each group.
199

Example
• In a school based study, we assume students of
the same school are homogeneous.
• We can select randomly sections and include all
students of the selected sections only
200

• Sometimes a list of all units in the population is not
available, while a list of all clusters is either available or
easy to create.
• In most cases, the main drawback is a loss of efficiency
when compared with SRS.
• It is usually better to survey a large number of small
clusters instead of a small number of large clusters.
– This is because neighboring units tend to be more alike,
resulting in a sample that does not represent the whole
spectrum of opinions or situations present in the overall
population.
• Another drawback to cluster sampling is that you do not have
total control over the final sample size.
Cont.
201

5. Multi-stage sampling
• Similar to the cluster sampling.
• But it involves picking a sample from within each chosen
cluster, rather than including all units in the cluster.
• This type of sampling requires at least two stages.
• In the first stage, large groups or clusters are identified and
selected.
• In the second stage, population units are picked from
within the selected clusters (using any of the possible
probability sampling methods) for a final sample.
202

• If more than two stages are used, the process of
choosing population units within clusters continues until
there is a final sample.
• Also, you do not need to have a list of all of the units in
the population. All you need is a list of clusters and list of
the units in the selected clusters.
• Admittedly, more information is needed in this type of
sample than what is required in cluster sampling.
However, multi-stage sampling still saves a great amount
of time and effort by not having to create a list of all the
units in a population.
Cont.
203

B. Non-probability sampling
• The difference between probability and non-
probability sampling has to do with a basic
assumption about the nature of the population under
study.
• In probability sampling, every item has a known
chance of being selected.
• In non-probability sampling, there is an assumption
that there is an even distribution of a characteristic of
interest within the population.
204

• In non-probability sampling, since elements
are chosen arbitrarily, there is no way to
estimate the probability of any one element
being included in the sample.
• Also, no assurance is given that each item has
a chance of being included, making it
impossible either to estimate sampling
variability or to identify possible bias
Cont.
205

• Reliability cannot be measured in non-probability
sampling; the only way to address data quality is to
compare some of the survey results with available
information about the population.
• Still, there is no assurance that the estimates will meet
an acceptable level of error.
• Researchers are reluctant to use these methods
because there is no way to measure the precision of the
resulting sample.
Cont.
206

• Despite these drawbacks, non-probability
sampling methods can be useful when
descriptive comments about the sample itself
are desired.
• There are also other circumstances, such as
researches, when it is unfeasible or
impractical to conduct probability sampling.
Cont.
207

The most common types of non-
probability sampling
1. Convenience or haphazard sampling
2. Volunteer sampling
3. Judgment sampling
4. Quota sampling
5. Snowball sampling technique
208

1. Convenience or haphazard sampling
• Convenience sampling is sometimes referred
to as haphazard or accidental sampling.
• It is not normally representative of the target
population because sample units are only
selected if they can be accessed easily and
conveniently.
209

• The obvious advantage is that the method is
easy to use, but that advantage is greatly
offset by the presence of bias.
• Although useful applications of the technique
are limited, it can deliver accurate results
when the population is homogeneous.
Cont.
210

• For example, a scientist could use this method to
determine whether a lake is polluted or not.
• Assuming that the lake water is well-mixed, any
sample would yield similar information.
• A scientist could safely draw water anywhere on the
lake without bothering about whether or not the
sample is representative
Cont.
211

2. Volunteer sampling
• As the term implies, this type of sampling occurs
when people volunteer to be involved in the
study.
• In psychological experiments or pharmaceutical
trials (drug testing), for example, it would be
difficult and unethical to enlist random
participants from the general public.
• In these instances, the sample is taken from a
group of volunteers.
212

• Sometimes, the researcher offers payment to
attract respondents.
• In exchange, the volunteers accept the
possibility of a lengthy, demanding or
sometimes unpleasant process.
Cont.
213

• Sampling voluntary participants as opposed
to the general population may introduce
strong biases.
• Often in opinion polling, only the people
who care strongly enough about the subject
tend to respond.
• The silent majority does not typically
respond, resulting in large selection bias.
Cont.
214

3. Judgment sampling
• This approach is used when a sample is taken based
on certain judgments about the overall population.
• The underlying assumption is that the investigator
will select units that are characteristic of the
population.
• The critical issue here is objectivity: how much can
judgment be relied upon to arrive at a typical
sample?
215

• Judgment sampling is subject to the
researcher's biases and is perhaps even more
biased than haphazard sampling.
• Since any preconceptions the researcher may
have are reflected in the sample, large biases
can be introduced if these preconceptions are
inaccurate.
Cont.
216

• Researchers often use this method in
exploratory studies like pre-testing of
questionnaires and focus groups.
• They also prefer to use this method in
laboratory settings where the choice of
experimental subjects (i.e., animal, human)
reflects the investigator's pre-existing beliefs
about the population.
Cont.
217

• One advantage of judgment sampling is the
reduced cost and time involved in acquiring
the sample.
Cont.
218

4. Quota sampling
• This is one of the most common forms of non-
probability sampling.
• Sampling is done until a specific number of
units (quotas) for various sub-populations have
been selected.
219

• Since there are no rules as to how these
quotas are to be filled, quota sampling is
really a means for satisfying sample size
objectives for certain sub-populations.
Cont.
220

• As with all other non-probability sampling
methods, in order to make inferences about
the population, it is necessary to assume
that persons selected are similar to those not
selected.
• Such strong assumptions are rarely valid.
Cont.
221

• The main argument against quota sampling is
that it does not meet the basic requirement of
randomness.
• Some units may have no chance of selection
or the chance of selection may be unknown.
• Therefore, the sample may be biased.
Cont.
222

• Quota sampling is generally less expensive than
random sampling.
• It is also easy to administer, especially considering
the tasks of listing the whole population, randomly
selecting the sample and following-up on non-
respondents can be omitted from the procedure.
Cont.
223

• Quota sampling is an effective sampling
method when information is urgently required
and can be carried out sampling frames.
• In many cases where the population has no
suitable frame, quota sampling may be the
only appropriate sampling method.
Cont.
224

5. Snowball sampling
• A technique for selecting a research sample
where existing study subjects recruit future
subjects from among their acquaintances.
• Thus the sample group appears to grow like a
rolling snowball.
225

• This sampling technique is often used in hidden
populations which are difficult for researchers to
access; example populations would be drug users or
commercial sex workers.
• Because sample members are not selected from a
sampling frame, snowball samples are subject to
numerous biases. For example, people who have
many friends are more likely to be recruited into the
sample.
Cont.
226

Estimation

• Up until this point, we have assumed that the
values of the parameters of a probability
distribution are known.
• In the real world, the values of these population
parameters are usually not known
• Instead, we must try to say something about the
way in which a random variable is distributed
using the information contained in a sample of
observations
228

• The process of drawing conclusions about an
entire population based on the data in a sample is
known as statistical inference.
• Methods of inference usually fall into one of two
broad categories:
** Estimation or Hypothesis testing **
• For now, we will focus on using the observations in
a sample to estimate a population parameter
229

Estimation
• It is concerned with estimating the values of
specific population parameters based on
sample statistics.
• It is about using information in a sample to
make estimates of the characteristics
(parameters) of the source population.
230

Estimation, Estimator & Estimate
♣ Estimation is the computation of a statistic from sample
data, often yielding a value that is an approximation
(guess) of its target, an unknown true population
parameter value.
♣ The statistic itself is called an estimator and can be of two
types - point estimator or interval estimator.
♣ The value or values that the estimator assumes are called
estimates.
231

• Point estimation involves the calculation of a
single number to estimate the population
parameter
• Interval estimation specifies a range of
reasonable values for the parameter
Thus,
– A point estimate is of the form: [ Value ],
– Whereas, an interval estimate is of the form:
[ lower limit, upper limit ]
Point versus Interval Estimators
232

1. Point Estimate
• A single numerical value used to estimate the
corresponding population parameter.
Sample Statistics are Estimators of Population Parameters
Sample mean, 𝑿
Sample variance, S2
Sample proportion,
Sample Odds Ratio, OŔ
Sample Relative Risk, RŔ
Sample correlation coefficient, r
µ
2
P or π
OR
RR
ρ
233

2. Interval Estimation
• Interval estimation specifies a range of reasonable values for the
population parameter based on a point estimate.
• A confidence interval is a particular type of interval estimator and
Give a plausible range of values of the estimate likely to include
the “true” (population) value with a given confidence level.
Also give information about the precision of an estimate.
Wider CIs indicate less certainty.
CIs can also answer the question of whether or not an association
exists
234

Confidence Level
• Confidence Level
– Confidence in which the interval will contain the
unknown population parameter
• P (L, U) = (1 - α)
235

Estimation for Single Population
236

1. CI for a Single Population Mean
(normally distributed)
A. Known variance (large sample size)
• There are 3 elements to a CI:
1. Point estimate
2. SE of the point estimate
3. Confidence coefficient
• Consider the task of computing a CI estimate of μ for a
population distribution that is normal with σ known.
• Available are data from a random sample of size = n.
237

Assumptions
 Population standard deviation () is known
 Population is normally distributed
• A 100(1-)% C.I. for  is:
  is to be chosen by the researcher, most common values of  are 0.05,
0.01 and 0.1.
Cont.
238

Margin of Error
(Precision of the estimate)
239

Cont.
 As n increases, the CI decreases.
 As s increases, the length of CI increases.
 As the confidence level increases (α decreases),
the length of CI increases.
240

Example:
1. Waiting times (in hours) at a particular hospital are
believed to be approximately normally distributed with
a variance of 2.25 hr.
a. A sample of 20 outpatients revealed a mean waiting
time of 1.52 hours. Construct the 95% CI for the
estimate of the population mean.
b. Suppose that the mean of 1.52 hours had resulted from
a sample of 32 patients. Find the 95% CI.
c. What effect does larger sample size have on the CI?
241

a.
)
17
.
2
,
87
.
0
(
65
.
52
.
1
)
33
(.
96
.
1
52
.
1
20
25
.
2
96
.
1
52
.
1






• We are 95% confident that the true mean waiting
time is between 0.87 and 2.17 hrs.
 An incorrect interpretation is that there is 95%
probability that this interval contains the true
population mean.
Solution:
242

b.
c. The larger the sample size makes the CI
narrower (more precision).
)
.05
2
,
99
(.
53
.
52
.
1
)
27
(.
96
.
1
52
.
1
32
25
.
2
96
.
1
52
.
1






Cont.
243

Cont.
B. Unknown variance (small sample size, n ≤ 30)
• What if the  for the underlying population is
unknown and the sample size is small?
• As an alternative we use Student’s t
distribution.
244

Cont.
245

Example
• Standard error =
• t-value at 90% CL at 19 df =1.729
246

Cont.
247

Exercise
• Compute a 95% CI for the mean birth weight
based on n = 10, sample mean = 116.9 oz and
s =21.70.
• From the t Table, t9, 0.975 = 2.262
• Ans: (101.4, 132.4)
248

2. CIs for single population proportion, p
• Is based on three elements of CI
– Point estimate
– SE of point estimate
– Confidence coefficient
249

Cont.
250

Example 1
• A random sample of 100 people shows that 25
are left-handed. Form a 95% CI for the true
proportion of left-handers.
251

Interpretation
252

Example 2
• Suppose that among 10,000 female operating-room nurses,
60 women have developed breast cancer over five years. Find
the 95% for p based on point estimate.
• Point estimate = 60/10,000 = 0.006
• The 95% CI for p is given by the interval:
• The 95% CI for p is:
253

Hypothesis Testing

• The purpose of Hypothesis Testing is to aid the
researcher in reaching a decision (conclusion)
concerning a population by examining a
sample from that population.
255

Hypothesis
• Is a statement about one or more
• Is a claim (assumption) about a population
parameter
The purpose of Hypothesis Testing is to aid
the researcher in reaching a decision
(conclusion) concerning a population by
examining a sample from that population.
256

Examples of Research Hypotheses
Population Mean
• The average length of stay of patients
admitted to the hospital is five days
• The mean birth weight of babies delivered by
mothers with low SES is lower than those from
higher SES.
• Etc
257

Types of Hypothesis
1. The Null Hypothesis, H0
 Is a statement claiming that there is no difference between
the hypothesized value and the population value.
 (The effect of interest is zero = no difference)
 States the assumption (hypothesis) to be tested
 H0 is always about a population parameter, not about a
sample statistic
 Begin with the assumption that the Ho is true
– Similar to the notion of innocent until proven guilty
258

2. The Alternative Hypothesis, HA
• Is a statement of what we will believe is true if our
sample data causes us to reject Ho.
• Is generally the hypothesis that is believed (or needs to
be supported) by the researcher.
• Is a statement that disagrees (opposes) with Ho
(The effect of interest is not zero)
Cont.
259

Steps in Hypothesis Testing
1. Formulate the appropriate statistical hypotheses
clearly
• Specify HO and HA
H0:  = 0 H0:  ≤ 0 H0:  ≥ 0
H1:   0 H1:  > 0 H1:  < 0
two-tailed one-tailed one-tailed
2. State the assumptions necessary for computing
probabilities
• A distribution is approximately normal (Gaussian)
• Variance is known or unknown
260

3. Select a sample and collect data
• Categorical, continuous
4. Decide on the appropriate test statistic for
the hypothesis. E.g., One population
OR
Cont.
261

5. Specify the desired level of significance for
the statistical test (=0.05, 0.01, etc.)
6. Determine the critical value.
– A value the test statistic must attain to be
declared significant.
-1.96 1.96 1.645 -1.645
Cont.
262

7. Obtain sample evidence and compute the
test statistic
8. Reach a decision and draw the conclusion
• If Ho is rejected, we conclude that HA is true (or
accepted).
• If Ho is not rejected, we conclude that Ho may
be true.
263

Rules for Stating Statistical Hypotheses
1. One population
• Indication of equality (either =, ≤ or ≥) must appear
in Ho.
Ho: μ = μo, HA: μ ≠ μo
Ho: P = Po, HA: P ≠ Po
• Can we conclude that a certain population mean is
– not 50?
Ho: μ = 50 and HA: μ ≠ 50
– greater than 50?
Ho: μ ≤ 50 HA: μ > 50
264

• Can we conclude that the proportion of
patients with leukemia who survive more than
six years is not 60%?
Ho: P = 0.6 HA: P ≠ 0.6
Cont.
265

Statistical Decision
• Reject Ho if the value of the test statistic that
we compute from our sample is one of the
values in the rejection region
• Don’t reject Ho if the computed value of the
test statistic is one of the values in the non-
rejection region.
266

Another way to state conclusion
• Reject Ho if P-value < α
• Accept Ho if P-value ≥ α
 P-value is the probability of obtaining a test
statistic as extreme as or more extreme than the
actual test statistic obtained if the Ho is true
 The larger the test statistic, the smaller is the P-
value. OR, the smaller the P-value the stronger the
evidence against the Ho.
267

Types of Errors in Hypothesis Tests
• Whenever we reject or accept the Ho, we
commit errors.
• Two types of errors are committed.
– Type I Error
– Type II Error
268

Type I Error
• The probability of a type I error is the
probability of rejecting the Ho when it is true
• The probability of type I error is α
• Called level of significance of the test
• Set by researcher in advance
269

Type II Error
• The error committed when a false Ho is not
rejected
• The probability of Type II Error is 
• Usually unknown but larger than α
270

Action
(Conclusion)
Reality
Ho True Ho False
Do not
reject Ho
Correct action
(Prob. = 1-α)
Type II error (β)
(Prob. = β= 1-Power)
Reject Ho Type I error (α)
(Prob. = α = Sign. level)
Correct action
(Prob. = Power = 1-β)
Cont.
271

Type I & II Error Relationship
272

Hypothesis Testing of a Single Mean
(Normally Distributed)
273

Known Variance
274

Example: Two-Tailed Test
1. A simple random sample of 10 people from a certain population
has a mean age of 27. Can we conclude that the mean age of the
population is not 30? The variance is known to be 20. Let = .05.
 Answer, "Yes we can, if we can reject the Ho that it is 30."
A. Data
n = 10, sample mean = 27, 2 = 20, α = 0.05
B. Assumptions
Simple random sample
Normally distributed population
275

C. Hypotheses
Ho: µ = 30
HA: µ ≠ 30
D. Test statistic
As the population variance is known, we use Z as
the test statistic.
Cont.
276

E. Decision Rule
 Reject Ho if the Z value falls in the rejection region.
 Don’t reject Ho if the Z value falls in the non-rejection region.
 Because of the structure of Ho it is a two tail test. Therefore, reject Ho if
Z ≤ -1.96 or Z ≥ 1.96.
Cont.
277

F. Calculation of test statistic
G. Statistical decision
We reject the Ho because Z = -2.12 is in the rejection region. The
value is significant at 5% α.
H. Conclusion
We conclude that µ is not 30. P-value = 0.0340
A Z value of -2.12 corresponds to an area of 0.0170. Since there are two
parts to the rejection region in a two tail test, the P-value is twice this which
is .0340.
278

Hypothesis test using
confidence interval
• A problem like the above example can also be solved
using a confidence interval.
• A confidence interval will show that the calculated
value of Z does not fall within the boundaries of the
interval. However, it will not give a probability.
• Confidence interval
279

Example: One -Tailed Test
• A simple random sample of 10 people from a certain
population has a mean age of 27. Can we conclude that the
mean age of the population is less than 30? The variance is
known to be 20. Let α = 0.05.
• Data
n = 10, sample mean = 27, 2 = 20, α = 0.05
• Hypotheses
Ho: µ ≥ 30, HA: µ < 30
280

• Test statistic
• Rejection Region
• With α = 0.05 and the inequality, we have the entire rejection region at the
left. The critical value will be Z = -1.645. Reject Ho if Z < -1.645.
=
Lower tail test
281

• Statistical decision
– We reject the Ho because -2.12 < -1.645.
• Conclusion
– We conclude that µ < 30.
– p = .0170 this time because it is only a one tail test and not a two tail test.
Cont.
282

Unknown Variance
• In most practical applications the standard deviation of
the underlying population is not known
• In this case,  can be estimated by the sample standard
deviation s.
• If the underlying population is normally distributed,
then the test statistic is:
283

Example: Two-Tailed Test
• A simple random sample of 14 people from a certain population
gives a sample mean body mass index (BMI) of 30.5 and sd of
10.64. Can we conclude that the BMI is not 35 at α 5%?
• Ho: µ = 35, HA: µ ≠35
• Test statistic
• If the assumptions are correct and Ho is true, the test statistic
follows Student's t distribution with 13 degrees of freedom.
284

• Decision rule
– We have a two tailed test. With α = 0.05 it means that each tail is
0.025. The critical t values with 13 df are -2.1604 and 2.1604.
– We reject Ho if the t ≤ -2.1604 or t ≥ 2.1604.
• Do not reject Ho because -1.58 is not in the rejection region. Based
on the data of the sample, it is possible that µ = 35. P-value = 0.1375
Cont.
285

Sampling from a population that is not normally
distributed
• Here, we do not know if the population displays a
normal distribution.
• However, with a large sample size, we know from the
Central Limit Theorem that the sampling distribution
of the population is distributed normally.
286

• With a large sample, we can use Z as the test statistic
calculated using the sample sd.
Cont.
287

Hypothesis Tests for Proportions
• Involves categorical values
• Two possible outcomes
– “Success” (possesses a certain
characteristic)
– “Failure” (does not possesses that
characteristic)
• Fraction or proportion of population in the
“success” category is denoted by p
288

Proportions
289

(Normal Approximation to Binomial Distribution)
Hypothesis Testing about a Single Population
Proportion
290

Example
• We are interested in the probability of developing asthma
over a given one-year period for children 0 to 4 years of age
whose mothers smoke in the home. In the general population
of 0 to 4-year-olds, the annual incidence of asthma is 1.4%. If
10 cases of asthma are observed over a single year in a
sample of 500 children whose mothers smoke, can we
conclude that this is different from the underlying probability
of p0 = 0.014? Α = 5%
H0 : p = 0.014
HA: p ≠ 0.014
292

• The test statistic is given by:
Cont.
293

• The critical value of Zα/2 at α=5% is ±1.96.
• Don’t reject Ho since Z (=1.14) in the non-rejection
region between ±1.96.
• P-value = 0.2548
• We do not have sufficient evidence to conclude that
the probability of developing asthma for children
whose mothers smoke in the home is different from
the probability in the general population
Cont.
294

Sample size determination
295

Sample Size
• Sample Size: The number of study subjects
selected to represent a given study population.
• In estimating a certain characteristic of a
population, sample size calculations are
important to ensure that estimates are obtained
with required precision or confidence
• Should be sufficient to represent the
characteristics of interest of the study population.
296

Sample size determination depends on the:
– Objective of the study
– Design of the study
• Descriptive/Analytic
– Accuracy of the measurements to be made
– Degree of precision required for generalization
– Plan for statistical analysis
– Degree of confidence with which to conclude
Cont.
297

• The feasible sample size is also determined by
the availability of resources:
– time
– manpower
– transport
– available facility, and
– money
Cont.
298

Sample size for single sample
A. Sample size for estimating a single
population mean
B. Sample size to estimate a single population
proportion
299

A. Sample size for estimating a single
population mean
• where d = Margin of error =
= Absolute precision
= Half of the width (w) of CI
300
Where d = e in some text books

Examples:
1. Find the minimum sample size needed to estimate the drop
in heart rate (µ) for a new study using a higher dose of
propranolol than the standard one. We require that the
two-sided 95% CI for µ be no wider than 5 beats per minute
and the sample sd for change in heart rate equals 10 beats
per minute.
n = (1.96)2
102
/(2.5)2
= 62 patients
301

2. Suppose that for a certain group of cancer patients, we
are interested in estimating the mean age at diagnosis.
We would like a 95% CI of 5 years wide. If the population
SD is 12 years, how large should our sample be?
302

• Suppose d=1
• Then the sample size increases
Cont.
303

3. A hospital director wishes to estimate the
mean weight of babies born in the hospital.
How large a sample of birth records should be
taken if she/he wants a 95% CI of 0.5 wide?
Assume that a reasonable estimate of  is 2.
Ans: 246 birth records.
Cont.
304

But the population 2 is most of the
time unknown
As a result, it has to be estimated from:
• Pilot or preliminary sample:
– Select a pilot sample and estimate 2 with
the sample variance, s2
• Previous or similar studies
305

B. Sample size to estimate a single
population proportion
306

1. Suppose that you are interested to know the
proportion of infants who breastfed >18 months
of age in a rural area. Suppose that in a similar
area, the proportion (p) of breastfed infants was
found to be 0.20. What sample size is required to
estimate the true proportion within ±3% points
with 95% confidence. Let p=0.20, d=0.03, α=5%
Cont.
307

Sample Size: Two Samples
A. Estimation of the difference between two
population means
B. Estimation of the difference between two
population proportions
308

A. Sample size for estimating a difference in two
means
309

B. Sample size for estimating a difference in two
proportions
310

Data Screening
311

Data check entry
• One of the first steps to proper data screening is to
ensure the data is correct
– Check out each person’s entry individually
• Makes sense if small data set or proper data checking
procedure
• Can be too costly so…
– range of data should be checked
312

Normality
• All of the continuous data we are covering need
to follow a normal curve
• Skewness (univariate) – this represents the
spread of the data
313

Cont.
• skewness statistic is output by SPSS and SE
skewness is
3.2 violation of skewness assumption
Skewness
skewness
Skewness
skewness
S
Z
SE
Z


314

Cont.
• Kurtosis (univariate) – is how peaked the data is;
Kurtosis stat output by SPSS
• Kurtosis standard error
– for most statistics the skewness assumption is more
important that the kurtosis assumption
3.2 violation of kurtosis assumption
Kurtosis
kurtosis
Kurtosis
kurtosis
S
Z
SE
Z


315

Outliers
• technically it is a data point outside of you
distribution; so potentially detrimental
because may have undo effect on
distribution
316

Linearity
• relationships among variables are linear in
nature; assumption in most analyses
317

Homoscedasticity
• For grouped data this is the same as
homogeneity of variance
• For ungrouped data – variability for one
variables is the same at all levels of another
variable (no variance interaction)
318

Multicollinearity/Singularity
• If correlations between two variables are excessive
(e.g. 0.95) then this represents multicollinearity
• If correlation is 1 then you have singularity
• Often Multicollinearity/Singularity occurs in data
because one variable is a near duplicate of another
319

Biostatistics.school of public healthppt

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