Download Quantitative Analysis

DATA TYPES AND
QUANTITATIVE DATA ANALYSIS
PRESENTED TO
THIRD-TRIMESTER
YEAR 1
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DATA
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Information expressed qualitatively or
quantitatively
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Data are measurements of characteristics
Measurements are functions that assign
values in quantitative or quantitative form
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Characteristics are referred to as variables
Eg. Height, weight, sex, tribe, etc
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VARIABLES AND DATA TYPES
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Variable as characterization of event
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Classification of Variables
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Qualitative: usually categorical; values/members fall
into one of a set of mutually exclusive & collectively
exhaustive classes. eg. Sex, crop variety, animal
breed, source of water, type of house
Quantitative: numeric values possessing an inherent
order.
 Discrete: eg. # of children/farmers/animals, etc
 Continuous: height, weight, distance, etc
Random and Fixed
Data Types
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Scales of measurements
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Nominal
Ordinal
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Interval
Ratio
Levels of measurement distinguished on the basis of the following
criteria:
 Magnitude or size; Direction
 Distance or interval; Origin
 Equality of points; Ratios of intervals; Ratio of points
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NOMINAL DATA
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Example: Sex (Gender) coded M,F or 0,1
‘Numbers’ simply identify, classify, categorize or
distinguish.
The score has no size or magnitude
Score has equality because two subjects are similar
(equal) if they have same number
Weakest level of measurement; poor
Arithmetic operations CANNOT be performed on
nominal data types
ORDINAL DATA
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Associated with qualitative random variables
Generated from ranked responses (or from a
counting process).
Have properties of nominal-data, in addition to
DIRECTION
Numeric or non-numeric
Next to nominal in terms of weakness
Arithmetic operations must be avoided
Egs: knowledge (low, average, high), socioeconomic status, attitude, opinion (like, dislike,
strongly dislike), etc.
INTERVAL and RATIO
INTERVAL
– Numeric, have magnitude or size, direction, distance or interval,
and origin
– Interval scale has no absolute 0 that is NOT independent of
system of measurement [0oC not same temperature as 0oF]
– Eg. Temperature in degrees Fahrenheit or Celsius
RATIO
• Weight of cassava in kilogram or pounds weight
– Numeric, have magnitude or size, direction, distance or interval,
and origin
– Absolute origin exists and not system dependent
All arithmetic operations can be performed on such data types
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DATA COLLECTION PROCESSES
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Processes include (not mutually exclusive)
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Routine Records;
Survey Data;
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Experimental data;
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ROUTINE (MONITORING) DATA
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Data periodically recorded essentially for
administrative use of the establishment and for
studying trends or patterns.
Examples – medical records, meteorological data
Some statistical analysis of data possible on
description and prescription
Cheap data, and planning could be haphazard
EXPERIMENTAL DATA
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Treatments are the investigated factors of variation
Treatments are controlled by the designer
Treatment levels may be fixed, random, qualitative,
quantitative
Comparative experimental data require inductive
analysis
Emphasis on inference including estimation of
effects and test of hypotheses.
SURVEY DATA COLLECTION
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Information on characteristics, opinions, attitudes,
tendencies, activities or operations of the individual
units of the population
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Based on a small set of the population
Can be planned; preference for random surveys
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Researcher or investigator has no (or must not
exercise) control over the respondent or data
Which procedure to use?
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Depends on study objectives
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All 3 procedures are possible while in the
community
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Monitoring and Survey procedures will be
most used during the first year.
We discuss SURVEY further
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SAMPLING (SURVEY) METHODS
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Ensure units of population have same chance of
being in the sample.
Sampling Types
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Probability sampling - the selection of sampling units
is according to a probability (random & non-random)
scheme.
Non-probability sampling - selection of samples not
objectively made, but influenced a great deal by the
sampler. Example – haphazard and use of
volunteers
Preference is for probability sampling, but situation
may determine otherwise
SYSTEMATIC SAMPLING Procedure
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Sampling units are selected according to a
pre-determined pattern.
For instance, given a sampling intensity of
10% from a population of 100 numbered
trees or units (strips etc) might require your
observing every 1 out of 10 trees (units,
strips) in an ordered manner or sequence
Selection in Systematic Procedure
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E.g. if by some process, random or non-random, the
3rd tree (unit or strip) is selected first, then the 13th,
23rd, 33rd, 43rd,..., 93rd trees (unit, strips) will
accordingly be selected. Strictly, this type of
selection as illustrated with the population of 100
trees (units) involves only one sample.
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Improve by selecting 1st unit randomly from 1 to 10,
or 1 to 100, and by MULTIPLE random starts
Applications of Systematic Sampling
_ Population is unknown
_ Baseline studies on spatial distribution
patterns of population
_ Baseline studies on extent/distribution of
pests, pathogens, etc.
_ Mapping purposes
_ Regeneration studies
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Advantages of Systematic Sampling
_ Easy to set-up
_ Relative speed in data collection
_ Total coverage of population assured
_ Good base for future designs, as position
of characters can easily be mapped (with
known coordinates)
_ Demarcation of units not necessary, as
sampling units are defined by first unit.
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Disadvantages of Systematic Sampling
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With only one random observation,
sampling error not valid
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Unknown trend(s) in population can
influence results adversely [Examples:
topography, season of sampling interval]
Avoiding the disadvantages
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The first major disadvantage on sampling
error can be rectified by introducing several
multiple random starts through stratification
of the population
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The second problem of trend is more
difficult but simply relates to the choice of the
sampling interval.
Simple/Unrestricted Random Sampling
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Unlike the systematic sampling, sampling
units need not be equally spaced.
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We shall define this as that sampling
procedure which ensures equal probability
for all samples of the same size (without any
restriction imposed on the selection process).
Illustration of SRS
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Given a pop. Size of N from which a sample of size n will be
drawn, the number of possible ways of obtaining the sample is
 {( N Nn!)!n!}
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Supposing a population is known to have 5 units, and a sample
size of 3 is required.
From this population of 5 units, there are 10 possible ways of
obtaining a sample of size 3. [The formula is 5C3= 5!/{(5-3)!
3!} = 10].
Each of these combinations is unique and has the same
chance (1/10) of being selected.
Thus SRS is a random sampling procedure where each sample
of size n has the same probability of selection.
SRS selection process
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(i) Select randomly one 'sample
combination' from the number 1 to 10 (as
there are 10 possible combinations).
(ii) Use the table of random numbers to
select 3 numbers from 1 to 5 or select three
numbers from a 'hat' containing all the five
numbers. This option seems easier and more
practicable than (i).
Summary - SRS
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Application: Applied when the population is
known to be homogeneous. Procedure is
suitable for units defined by plot sizes.
Advantage: Easy to apply, though not as
easy as the systematic procedure.
Disadvantage: Requires knowledge of all
the units in the population (construction of
the frame is necessary)
STRATIFIED RANDOM SAMPLING
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Requires dividing the population into non-overlapping
homogeneous units, which we are called STRATA.
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SRS is then applied to each stratum, hence stratified random
sampling (STRS).
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Examples of strata types or criteria are ages of plantation,
species types, aspect, topography/ altitude, farm types, habitat
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Dividing the population into such homogeneous units usually
leads to better estimates of the desired population parameters.
Where/when to apply Stratified RS
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Very suitable for heterogeneous areas (or
units) that can be identified and classified
into homogeneous entities.
Supplementary information, e.g. rem sensing
aerial photographs, useful for stratification.
Choice of strata should ensure variation
between units within strata is less than the
variation between strata.
Advantages/Disadvantages of STRS
Advantages
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Estimates are more precise
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Separate estimates and inferences for strata are possible
Disadvantages
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Sample size depends on type of allocation to be used
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Sampling likely to be efficient in some strata than others
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Errors in strata classification affect overall estimate
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Frame construction for each stratum is required.
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Allocation of units (n) to strata
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Equal allocation - Equal (same) number of
units are collected from each stratum.
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Proportional allocation - The number of units
per strata is proportional to the size of the
strata.
ANALYSING QUALITATIVE DATA
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Qualitative data are essentially labels of a
categorical variable
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Statistical Analyses involve totals,
percentages and conversion to pie-charts
and bar charts (bar-graphs).
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Sophisticated analyses include categorical
modelling
EXAMPLE
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Chart of A,B,C
35
30
25
20
Series1
1
15
2
10
3
5
0
1
29
2
3
Hse
Freque
ncy
Percen
t
Degree
of 360
A=1
36
72%
260
B=2
10
20%
72
C=3
4
8%
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You can have multiple bar graphs (i.e, can have
more than one variable illustrated on a bar chart.
Example is given below:
40
80
35
70
30
60
25
50
Male
20
Female
15
30
10
20
5
10
0
Male
0
1
30
Female
40
2
3
1
2
3
Contingency Table
This involves count summaries for 2 or more
categories placed in row-column format:
Example of a 2 by 3 contingency table:
Gender
Male
Female
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Group
A
B
C
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34
10
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4
2
Assess association
between Gender & Group
ANALYSING QUANTITATIVE DATA
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Basic analyses involve determining the
CENTRE and SPREAD of data.
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Inferential, probability and non-probability
based
Measuring Centre
Statistics include
–
–
–
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MODE (most frequently occurring observation)
MEDIAN (observation lying at the centre of an
ordered data) – best for INCOME data
MEAN (a sufficient, consistent, unbiased statistic,
utilising ALL observations)
EXAMPLE
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Consider that we selected RANDOMLY 10
houses out of 50, and observed the number
of school-aged children who do not go to
school as follows:
1
2 4
4
1
1
6
Find MEDIAN, MODE, MEAN
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0
5
2
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MODE: 1 as it appeared most often (most households have at
least 1 child of school-going age not in school)
MEDIAN: Centremost observation after ordering data lies between
the 4th and 5th data, i.e., between 2 and 2 (= 352)
0
1
1
1
2
2
4
4
5
6
Interpretation: 50% of the sampled population have up to 2 children of
school-going age not in school)
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MEAN: We use the arithmetic mean = sum of data divided by no.
of observations, = (0+1+1+1+ 2+2+4+4+5+6)/10=2.6
Measuring Spread
Statistics include
–
–
–
–
–
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MINIMUM, MAXIMUM (ie EXTREME data)
RANGE (a single statistic calculated as
MAXIMUM minus MINIMUM value)
MEAN of the sum of the ABSOLUTE DEVIATION
STANDARD DEVIATION (SD, but use the divisor
n-1, not n as in most calculators).
STANDARD ERROR
EXAMPLE
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Consider that we selected RANDOMLY 10
houses out of 50, and observed the number
of school-aged children who do not go to
school as follows:
1
2 4
4
1
1
6
0
5
2
Find STANDARD DEVIATION, STANDARD
ERROR and CONFIDENCE LIMITS
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CALCULATING SPREAD: STANDARD DEVIATION
Standard Deviation:
Square
Dev
X
Deviation
1
-1.6
2.56
1
-1.6
2.56
1
-1.6
2.56
0
-2.6
6.76
2
-0.6
0.36
2
-0.6
0.36
4
1.4
1.96
4
1.4
1.96
5
2.4
5.76
6
3.4
11.56
n
SD 
(X
38
36.4
n 1
36.4
9
SD 
Approximate SD = ASD 
Range
4
SD 
X
 X )2
i
n
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i
2
i

= 2.01
( X i ) 2
i
n
n 1
= (6-0)/4 = 1.5 (valid if sample is large and distribution is normal)
Sampling fraction (f) and Finite
Population Correction Factor (fpc)
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Sampling fraction= f = n/N = 10/50 = 0.20
(represents the proportion of the population
that is sampled, i.e. observed)
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If f < 0.05, fpc is ignored. In our case, f > 0.5
(indeed equals 0.20), fpc must be calculated
and used for the sampling error computation
fpc = (N-n)/N = 1– n/N = 1- 0.20 = 0.80
CALCULATING SPREAD: STANDARD ERROR
n
(X
i
 X )2
i
n 1
n
SE 
SE 
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SD
n
fpc 
(1 
2.01
10
n
)
N
x 0.80
= 0.57
Confidence (Fiducial) Limits
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Given a level of significance, 5%, can obtain a 95% confidence limit on
the mean number of non-school going children by multiplying SE by
1.96, that is:
P(2.6-1.96*0.57 < true number < 2.6+1.96*0.57) =1-0.05= 0.95
P(1.5 < true number per household < 3.7) = 0.95
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Interpretation: 95% certain that true number of children in community
who are of school-age but at home is between 1.5 (1) and 3.7 (4).
OR can conclude (after multiplying by the total 50 households
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75 to 185 school-aged children in the community are not in school
Combining Spread and Centre
BOX PLOT
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HISTOGRAM
Further Analysis of Quantitative Data
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Histograms give idea of the distribution of the data;
very useful for quantitative data
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An excellent alternative to histogram is the stem-leaf
diagram.
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Measures of association – correlation analysis,
dependence (cause-effect) relations (regression
procedures) – 2006/2007
DATA ANALYSIS IS ENDLESS!!!
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ENJOY YOUR TIME DURING TTFPP

END
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KS Nokoe, PT Birteeb, IK Addai, M Agbolosu, L Kyei,