2. Topics to covered
1. Introduction
2. Population and sample
3. Descriptive vs inferential statistics
4. Data types: experimental vs observational; primary vs
secondary; quantitative vs qualitative..
5. Research design: sampling methods and sample size
determination
6. Biological data analysis: mean comparisn, ANOVA,
regression….
7. Experimental designs : univactorial and
multifactorial….
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3. Introduction
• Statistics — the science of collecting, describing, analyzing,
and interpreting data, so that inferences (conclusions
about a population based on data from merely a sample)
can be made with quantifiable certainty.
Statistics requires
• defining a population of interest,
• drawing a sample from that population,
• measuring something on each member of that sample,
• making a conclusion about the population based on the
measured quantities and,
• finally, using probability to say something about how sure
you are of your conclusion.
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4. Int…
• STATIC or DATUM means a measured or
counted fact or piece of information stated as a
figure such as height of one person, birth of a
baby, etc.
• They are collected from records, experiments
and surveys, in all walks of life: sciences,
economics, politics, education, industry,
business, administration etc.
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5. Introduction
• The field of statistics exists because it is usually impossible to
collect data from all individuals of interest (population).
• Our only solution is to collect data from a subset (sample) of
the individuals of interest, but our real desire is to know the
“truth” about the population.
• Quantities such as means, standard deviations and
proportions are all important values and are called
“parameters” when we are talking about a population. Since
we usually cannot get data from the whole population.
• When they are calculated from sample data, these quantities
are called “statistics.”
• A statistic estimates a parameter.
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6. Population and sample
In statistics, the collection of all individuals or
items under consideration is called
population.
Sample is that part of the population from
which information is collected. (Weiss, 1999)
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13. Biostatistics….
Biostatistics:
• It is the branch of statistics concerned with mathematical facts
and data related to biological events.
• It is the science that helps in managing medical uncertainties.
Biostatistics covers applications and contributions not only from
health, medicines and, nutrition but also from fields such as
genetics, biology, epidemiology, and many others.
• It is mainly consists of various steps like generation of
hypothesis, collection of data, and application of statistical
analysis.
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17. The three main aspects of statistics:
1)Design: designing the process of data
collection (Identify population, what kind and
how much data needed, how to collect a
sample)
2)Description: the methods of
summarizing/describing data.
3) Inference: infer “general rules” about a
population from a sample.
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19. Major branches of statistics
1. Descriptive statistics or exploratory data analysis: defined as the
branch of statistics that describes the contents of data or makes a
picture based on the data.
• summarize and portray the characteristics of the contents of a data set
or to identify patterns in a data set.
• • Descriptive Statistics Descriptive: statistics consists of methods for
organizing, displaying, and describing data by using tables, graphs, and
summary measures.
• Descriptive statistics do not generalize beyond the available data
• Descriptive statistics consist of methods for organizing and
summarizing information (Weiss, 1999).
• Descriptive statistics includes the construction of graphs, charts, and
tables, and the calculation of various descriptive measures such as
averages, measures of variation, and percentiles.
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20. Major Types of Descriptive Statistics
1. Measures of Frequency: Count, Percent,
Frequency.
2. Measures of Central Tendency : Mean, Median,
and Mode, SD.
3. Measures of Dispersion or Variation :Range,
Variance, Standard Deviation.
• 4. Measures of Position : Percentile ,Quartile,
Decils
Descriptive statistics do not generalize beyond the
available data
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21. • A data set in its original form is usually very large.
Consequently, such a data set is not very helpful
in drawing conclusions or making decisions. It is
easier to draw conclusions from summary tables
and diagrams than from the original version of a
data set. So, we reduce data to a manageable size
by constructing tables, drawing graphs, or
calculating summary measures such as averages.
The portion of statistics that helps us do this type
of statistical analysis is called descriptive
statistics.
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22. 2. Inferential statistics or confirmatory data analysis: researchers use
statistics used to draw conclusions about the world or to test formal
hypotheses.
• The field of statistics, which is relatively young, traces its origins to
questions about games of chance.
• The foundation of statistics rests on the theory of probability, a
subject with origins many centuries ago in the mathematics of
gambling.
• Inferential statistics includes methods like point estimation, interval
estimation and hypothesis testing which are all based on probability
theory.
• Inferential statistics generalizes from the sample.
• Hypothesis testing, confidence intervals: t-test, Fisher’s Exact, ANOVA,
survival analysis, – Bayesian approaches
• Making decisions in the face of uncertainty.
• Inferential statistics consist of methods for drawing and measuring
the reliability of conclusions about population based on information
obtained from a sample of the population. (Weiss, 1999)
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23. • •A major portion of statistics deals with making decisions,
inferences, predictions, and forecasts about populations
based on results obtained from samples.
• For example, we may make some decisions about the
political views of all college and university students based
on the political views of 1000 students selected from a few
colleges and universities. The area of statistics that deals
with such decision-making procedures is referred to as
inferential statistics. This branch of statistics is also called
inductive reasoning or inductive statistics.
• Inferential Statistics Inferential statistics consists of
methods that use sample results to help make decisions or
predictions about a population.
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24. • Deductive Statistics
• – Deduces properties of samples from a
complete knowledge about population
characteristics
• • Inductive Statistics
• – Concerned with using known sample
information to draw conclusions, or make
inferences regarding the unknown population
• – Same as inferential statistics
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25. Inferential statistics
• Generalize from the sample.
• Hypothesis testing, confidence intervals
• – t-test, Fisher’s Exact, ANOVA, survival
analysis, bayesian approaches
• Making decisions in the face of uncertainty
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26. • Probability, which gives a measurement of the
likelihood that a certain outcome will occur,
acts as a link between descriptive and
inferential statistics.
Probability is used to make statements about
the occurrence or nonoccurrence of an event
under uncertain conditions.
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27. Statistical Parameters
• Parameter is an unknown numerical summary of the
population.
• A statistic is a known numerical summary of the sample which
can be used to make inference about parameters (Agresti &
Finlay, 1997).
• Usually the features of the population under investigation can
be summarized by numerical parameters.
• Hence the research problem usually becomes as an
investigation of the values of parameters.
• These population parameters are unknown and sample
statistics are used to make inference about them.
• That is, a statistic describes a characteristic of the sample
which can then be used to make inference about unknown
parameters.
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28. Data
• A collective recording of observations either numerical or
otherwise.
• A collection of facts from which conclusions may be drawn.
Classification of data
• Depending on the nature of the variable, data
is classified into 2 broad categories :-
A. Qualitative data
B. Quantitative data
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31. Quantitative data
• DISCRETE DATA: when the variable under observation takes
only fixed values like whole numbers, the data is discrete.
DMFT, no. of children ( counted items).
• CONTINOUS DATA : If the variable can take any value in a given
range, decimal or fractional, it is called as continuous data like
arch length, mesiodistal width of the erupted teeth. ( measured
characteristics )
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38. • However, in most cases qualitative data are
converted to quanitative data
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39. Qualitative Data
• NOMINAL DATA: consists of named categories,
with no implied order among the categories.
• ORDINAL DATA: consists of ordered categories,
where differences between categories cannot
be considered to be equal. Ex- Likert’s scale.
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40. Data source: Data can be collected through
1. PRIMARY SOURCE: the data generated by the
investigator himself. This is first hand
information.
1. SECONDARY SOURCE: the data is already
recorded is utilized to serve the purpose of
the objective of the study. Exa – records of
the OPD of dental clinics.
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41. Collection of primary data
1) DIRECT PERSONAL INTERVIEWS
2) ORAL HEALTH EXAMINATION
3) QUESTIONNAIRE BASED
4. Lab experiment
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45. Data Sources
• Data arise from experimental or observational studies and it is
important to distinguish the two.
• In an experiment the researcher deliberately imposes a treatment on
one or more subjects or experimental units (not necessarily human).
• The experimenter then measures or observes the subjects’ response
to the treatment.
• Example: To assess whether or not saccharine is carcinogenic, a
researcher feeds 25 mice daily doses of saccharine, After 2 months
10 of the mice have developed tumors.
• By definition, this is an experiment, but not a very good one.
• In the saccharine example: we don’t know whether 10/25 with
tumors is high because there is no control group to which comparison
can be made
• Solution: Select 25 more mice and treat them exactly the same but
give them daily doses of an inert substance (a placebo)
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52. Parametric and Nonparametric statistics
• Parametric statistical procedures rely on
assumptions about the shape of the distribution
(i.e., assume a normal distribution) in the
underlying population and about the form or
parameters (i.e., means and standard deviations)
of the assumed distribution.
• Nonparametric statistical procedures rely on no
or few assumptions about the shape or
parameters of the population distribution from
which the sample was drawn.
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54. Parametric and Nonparametric statistics….
• Nonparametric tests do not rely on assumptions about the shape or
parameters of the underlying population distribution.
• If the data deviate strongly from the assumptions of a parametric
procedure, using the parametric procedure could lead to incorrect
conclusions.
• If you determine that the assumptions of the parametric procedure
are not valid, use an analogous nonparametric procedure instead.
• The parametric assumption of normality is particularly worrisome for
small sample sizes (n < 30). Nonparametric tests are often a good
option for these data.
• It can be difficult to decide whether to use a parametric or
nonparametric procedure in some cases.
• Nonparametric procedures generally have less power for the same
sample size than the corresponding parametric procedure if the data
truly are normal.
• Interpretation of nonparametric procedures can also be more difficult
than for parametric procedures.
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55. Disadvantages of nonparametric test
Two main draw backs:
1. are less statistically powerful than the analogous parametric procedure when
the data truly are approximately normal.
“Less powerful” means that there is a smaller probability that the procedure will
tell us that two variables are associated with each other when they in fact
truly are associated.
If you are planning a study and trying to determine how many patients to include,
a nonparametric test will require a slightly larger sample size to have the same
power as the corresponding parametric test.
2. The second drawback associated with nonparametric tests is that their results
are often less easy to interpret than the results of parametric tests.
Many nonparametric tests use rankings of the values in the data rather than
using the actual data.
Knowing that the difference in mean ranks between two groups is five does not
really help our intuitive understanding of the data.
On the other hand, knowing that the mean systolic blood pressure of patients
taking the new drug was five mmHg lower than the mean systolic blood
pressure of patients on the standard treatment is both intuitive and useful.
• In short, nonparametric procedures are useful in many cases and necessary in
some, but they are not a perfect solution.
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56. Research
• Research: the systematic investigation into and study of
materials and sources in order to establish facts and
reach new conclusions
•In the broadest sense of the word, the research includes
gathering of data, information and facts for the
advancement of knowledge.
• Research must be systematic and follow a series of steps
and a standard protocol.
•These rules are broadly similar but may vary slightly
between the different fields of science.
•Scientific research must be organized and undergo
planning, including performing literature reviews of past
research and evaluating what questions need to be
answered.
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61. Variables
• A characteristic that varies from one person or thing to
another is called a variable, i.e. a variable is any
characteristic that varies from one individual member
of the population to another.
• Types of Variable:
1. Quantitative (or numerical) variables
2. Qualitative (or non numerical) variables
• Example: height, weight, number of siblings, sex,
marital status, and eye color.
• The first three of these variables yield numerical
information last three yield non-numerical information
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71. Quantitative variables
• Quantitative variables can be classified as
either discrete or continuous.
• Discrete variables: a variable is discrete if it can
assume only a finite numbers of values or as
many values as there are integers.
• Continuous variables: quantities such as length,
weight, or temperature can in principle be
measured arbitrarily accurately. There is no
indivisible unit.
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72. Continuous & Discrete Variables
Continuous (quantitative)
Variable/data
• A variable can take on
any value between two
specified values.
• An infinite number of
values.
• Also known as
quantitative variable
E.g. Income & age;
Scale: Interval & Ratio
Discrete Variable
• A variable whose attribute
are separate from one
another.
• Also known as qualitative
variable
E.g. Marital status, gender &
nationality.
Scale: Nominal & Ordinal
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73. Continuous and Discrete data
• A continuous variable can take any value within the
measured Range. For example, if we measure fish
length, the variable can be an infinite number of
lengths between any two integers (thus, we are only
limited by the sensitivity of our measurement devices)
• A discrete variable can generally only take on values
that are consecutive integers (no fractional values are
possible)
• For example, if we count the number of ants in a
colony there can be 221 ants or 222 ants, but not 221.5
ants
• Nominal scale data are always discrete; other data
types can be either continuous or discrete
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74. Continuous variables can be summarize with
• –Means, medians, ranges, percentiles,
standard deviation
• • Numerous graphical approaches
• – Scatterplots, dot plots, box and whisker
plots
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81. Levels of measurement scales
• The level of measurement refers to the relationship
among the values that are assigned to the attributes,
feelings or opinions for a variable.
• Typically, there are four levels of measurement scales
or methods of assigning numbers:
(a) Nominal scale,
(b) Ordinal scale,
(c) Interval scale, and
(d) Ratio scale.
Note that, there is no unique way that you can use to
select a particular scaling technique for your research
study.
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82. 1. Nominal/ categorical scale data
• Nominal Scale is the crudest among all measurement scales but it is
also the simplest scale.
• In this scale the different scores on a measurement simply indicate
different categories.
• The nominal scale does not express any values or relationships between
variables.
• The assigned numbers have no arithmetic properties and act only as
labels.
• The only statistical operation that can be performed on nominal scales
is a frequency count. We cannot determine an average except mode.
• Data doesn’t have a numerical measurement
• Eye color, sex, with or without some attribute
• For example: labeling men as ‘1’ and women as ‘2’ which is the most
common way of labeling gender for data recording purpose does not
mean women are ‘twice something or other’ than men. Nor it suggests
that men are somehow ‘better’ than women.
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83. 2. Ordinal scale data
• Ordinal Scale involves the ranking of items along the continuum of the characteristic
being scaled.
• In this scale, the items are classified according to whether they have more or less of a
characteristic.
• The main characteristic of the ordinal scale is that the categories have a logical or
ordered relationship. This type of scale permits the measurement of degrees of
difference, (i.e. ‘more’ or ‘less’) but not the specific amount of differences (i.e. how
much ‘more’ or ‘less’). This scale is very common in marketing, satisfaction and
attitudinal research.
• Data consist of an ordering or ranking of measurements only
• • Exact measurement data unknown or not taken (e.g., we may only know
larger/smaller, lighter/darker, etc.)
• • Often ratio or interval data is converted to ordinal data to aid interpretation (i.e.,
exact measurements assigned ranks) and statistical analysis (e.g., grades)
• Using ordinal scale data, we can perform statistical analysis like Median and Mode, but
not the Mean.
• For example, a fast food home delivery shop may wish to ask its customers:
• How would you rate the service of our staff? (1) Excellent • (2) Very Good • (3) Good •
(4) Poor • (5) Worst •
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84. 3. Interval scale data
• Interval Scale is a scale in which the numbers are used to rank
attributes such that numerically equal distances on the scale represent
equal distance in the characteristic being measured.
• An interval scale contains all the information of an ordinal scale, but it
also one allows to compare the difference/distance between
attributes. Interval scales may be either in numeric or semantic
formats.
• The interval scales allow the calculation of averages like Mean, Median
and Mode and dispersion like Range and Standard Deviation.
• For example, the difference between ‘1’ and ‘2’ is equal to the
difference between ‘3’ and ‘4’. Further, the difference between ‘2’ and
‘4’ is twice the difference between ‘1’ and ‘2’.
• Constant interval, but no true zero, so can’t express in terms of ratios
• • Temperature scale is a good example (zero point is arbitrary; can’t say
40º is twice as hot as 20º)
• Other biological examples could be time of day and lat/long
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86. 4. Ratio Scale
Ratio Scale data: the highest level of measurement scales:
• Constant size interval between adjacent units on the measurement scale
• There exists a zero point on the measurement scale, which allows us to talk in
terms of the ratios of measurements (e.g., x is twice as large as y)
• Most data on a ratio scale (examples include lengths, weights, numbers of
items, volume, rates, lengths of time)
• This has the properties of an interval scale together with a fixed (absolute)
zero point. The absolute zero point allows us to construct a meaningful ratio.
• Ratio scales permit the researcher to compare both differences in scores and
relative magnitude of scores.
• Examples of ratio scales include weights, lengths and times.
• Example 1., the number of customers of a bank’s ATM in the last three
months is a ratio scale. This is because you can compare this with previous
three months.
• Example 2., the difference between 10 and 15 minutes is the same as the
difference between 25 and 30 minutes and 30 minutes is twice as long as 15
minutes.
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88. 5. Comparative scales
• In comparative scaling, the respondent is asked
to compare one object with another.
• The comparative scales can further be divided
into the following four types of scaling
techniques:
• (a) Paired Comparison Scale,
• (b) Rank Order Scale,
• (c) Constant Sum Scale, and
• (d) Q-sort Scale.
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89. a. Paired Comparison Scale:
This is a comparative scaling technique in which a respondent is presented with
two objects at a time and asked to select one object according to some criterion.
The data obtained are ordinal in nature.
For example, there are four types of cold drinks -Coke, Pepsi, Sprite, and Limca.
The respondents can prefer Pepsi to Coke or Coke to Sprite, etc.
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90. b. Rank Order Scale:
• This is another type of comparative scaling
technique in which respondents are presented
with several items simultaneously and asked to
rank them in the order of priority.
• This is an ordinal scale that describes the favoured
and unfavoured objects, but does not reveal the
distance between the objects.
• The resultant data in rank order is ordinal data.
This yields better results when direct comparison
are required between the given objects.
• The major disadvantage of this technique is that
only ordinal data can be generated.
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92. c. constant Sum Scale:
• In this scale, the respondents are asked to allocate a
constant sum of units such as points, rupees, or chips among
a set of stimulus objects with respect to some criterion.
• For example, you may wish to determine how important the
attributes of price, fragrance, packaging, cleaning power, and
lather of a detergent are to consumers.
• Respondents might be asked to divide a constant sum to
indicate the relative importance of the attributes. The
advantage of this technique is saving time.
• However, main disadvantages are the respondents may
allocate more or fewer points than those specified. The
second problem is respondents might be confused.
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94. d. Q-sort scale
• This is a comparative scale that uses a rank order
procedure to sort objects based on similarity with
respect to some criterion.
• The important characteristic of this methodology is that
it is more important to make comparisons among
different responses of a respondent than the responses
between different respondents.
• Therefore, it is a comparative method of scaling rather
than an absolute rating scale.
• In this method the respondent is given statements in a
large number for describing the characteristics of a
product or a large number of brands of a product.
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96. 6. Noncomparative scales
• In non-comparative scaling respondents need
only evaluate a single object. Their evaluation is
independent of the other object which the
researcher is studying.
• The non-comparative scaling techniques can be
further divided into:
• (a)Continuous Rating Scale, and
• (b)Itemized Rating Scale.
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97. a. Continuous Rating Scales:
• It is very simple and highly useful.
• In continuous rating scale, the respondent’s
rate the objects by placing a mark at the
appropriate position on a continuous line that
runs from one extreme of the criterion variable
to the other.
• Example : Question: How would you rate the
TV advertisement as a guide for buying?
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99. b. Itemized Rating Scales:
• Itemized rating scale is a scale having numbers or brief
descriptions associated with each category.
• The categories are ordered in terms of scale position
and the respondents are required to select one of the
limited number of categories that best describes the
product, brand, company, or product attribute being
rated.
• rating scales are widely used in marketing research.
• Itemised rating scales is further divided into three parts,
namely
(a) Likert scale,
(b) Semantic Differential Scale, and
(c) Stapel Scale.
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100. The itemised rating scales can be in the form of : (a) graphic, (b)
verbal, or (c) numeric as shown below :
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103. 7. Likert Scale:
• Likert, is extremely popular for measuring attitudes, because, the
method is simple to administer.
• With the Likert scale, the respondents indicate their own attitudes by
checking how strongly they agree or disagree with carefully worded
statements that range from very positive to very negative towards the
attitudinal object.
• Respondents generally choose from five alternatives (say strongly
agree, agree, neither agree nor disagree, disagree, strongly disagree).
• A Likert scale may include a number of items or statements.
• Disadvantage of Likert Scale is that it takes longer time to complete
than other itemised rating scales because respondents have to read
each statement.
• Despite the above disadvantages, this scale has several advantages.
• It is easy to construct, administer and use.
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105. Semantic Differential Scale:
This is a seven point rating scale with end points associated with bipolar labels (such as
good and bad, complex & simple) that have semantic meaning.
It can be used to find whether a respondent has a positive or negative attitude towards
an object.
It has been widely used in comparing brands, products and company images.
It has also been used to develop advertising and promotion strategies and in a new
product development study.
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106. 8. Staple Scale:
• The Stapel scale was originally developed to
measure the direction and intensity of an
attitude simultaneously.
• Modern versions of the Stapel scale place a
single adjective as a substitute for the
Semantic differential when it is difficult to
create pairs of bipolar adjectives.
• The modified Stapel scale places a single
adjective in the centre of an even number of
numerical Values.
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109. Selection of an appropriate scaling technique
• A number of issues decide the choice of
scaling technique. Some significant issues are:
1) Problem Definition and Statistical Analysis,
2) The Choice between Comparative and Non-
comparative Scales,
3) Type of Category Labels,
4) Number of Categories,
5) Balanced versus Unbalanced Scale, and
6) Forced versus Non-forced Categories
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112. Accuracy and Precision
• Accuracy = closeness of a measured value to its
true value (Bias = inaccuracy)
• Precision=closeness of repeated measurements
of the same quantity (Variation or variability =
imprecision).
• Many fields within biology differ in their ability
to measure variables accurately and precisely
• Most continuous variables are approximate,
while discrete are exact.
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113. Significant Figures
• The last digit of measurement implies
precision= limits of measurement scale
between which the true measurement lies
• A length measurement of 14.8 mm implies
that the true value lies between 14.75 and
14.85
• ***The limit always carries one figure past the
last significant digit measured by the
investigator
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114. Rule of thumb for significant figures (Sokal and Rohlf)
• The number of unit steps from the smallest to the largest measurement
in an array should usually be between 30 and 300.
• Example: If we were measuring the diameter of rocks to the nearest
mm and the range is from 5-9mm, that is only four unit steps from
smallest to largest and we should measure an additional significant
figure (e.g., 5.3 – 9.2 mm, with 3.9 unit steps).
• In contrast if we were measuring the length of bobcat whiskers within
the range of 10-150mm, there would be no need to measure to another
significant figure (we already have 140 unit steps) Reasoning: The
greater the number of unit steps, the less relative error for each
mistake of one measurement unit.
• Also, the proportional error reduction decreases quickly above high
numbers of unit steps (300), making measurement to this level of
precision not worthwhile.
• Examples of significant figures
• 22.34 (4), 25 (2), 0.065 (2), 0.1065 (4), 14.212 (5), 14,000 (2)
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115. Derived variables
• A variable expressed as a relation of two or more
independently measured variables (e.g., ratios,
percentages, or rates)
• These type of variables are very common in the
field of biology; often times their construction is
the only way to gain an understanding of some
observed phenomena. However, they present
certain disadvantages when it comes to analysis.
• These are related to their inaccuracy (compounded
when independent variables are combined) and
their tendency to not be distributed normally.
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116. Experimental study
• Experiment: a process that studies the effect on a variable of
varying the value(s) of another variable or variables, while keeping all
other things equal.
• A typical experiment contains both a treatment group and a control
group.
• The treatment group consists of those individuals or things that
receive the treatment(s) being studied. The control group consists of
those individuals or things that do not receive the treatment(s) being
studied.
Proper experiments are either single-blind or doubleblind.
• A study is a single-blind experiment if only the researcher conducting
the study knows the identities of the members of the treatment and
control groups. If neither the researcher nor study participants know
who is in the treatment group and who is in the control group, the
study is a double-blind experiment.
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117. Survey study
• A process that uses questionnaires or similar
means to gather values for the responses
from a set of participants.
• Surveys are either informal, open to anyone
who wishes to participate; targeted, directed
toward a specific group of individuals; or
include people chosen at random.
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118. Research Design May be sub divided into,
1) Sampling design: Deals with, the method of
‘selecting relevant items’ for the study.
2) Observational design: Relates to the condition
under which the observations are to be made.
3) Statistical Design: Deals with the “number of
items” selected for the study and how the
selected data will be collected and analysed.
4) Operation design: The technique by which the
sampling, observational and statistical designs can
be carried out. or How the above three are
carried out.
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119. Sampling:
• Sample Frame: the list of all items in the
population from which samples will be
selected.
• You should always be careful to make sure
your frame completely represents a
population; otherwise any sample selected
will be biased, and the results generated by
analyses of that sample will be inaccurate.
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121. Why do sampling?
•Sampling is done because you usually cannot
gather data from the entire population.
•Even in relatively small populations, the data
may be needed urgently, and including
everyone in the population in your data
collection may take too long.
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124. Sample Selection Methods
• Proper sampling can be done with or without
replacement.
Sampling With Replacement:
• A sampling method in which each selected item is
returned to the frame from which it was selected
so that it has the same probability of being selected
again.
• EXAMPLE Selecting entries from a fishbowl and
returning each entry to the fishbowl after it is
drawn.
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125. • Sampling Without Replacement: A sampling method in
which each selected item is not returned to the frame
from which it was selected. Using this technique, an item
can be selected no more than one time.
• EXAMPLES Selecting numbers in state lottery games,
selecting cards from a deck of cards during games of
chance such as Blackjack.
• Sampling without replacement means that an item can
be selected no more than one time. You should choose
sampling without replacement over sampling with
replacement, because statisticians generally consider the
former to produce more desirable samples.
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132. Selecting participants by finding one or two participants and then asking them to
refer you to other.
Example: Meeting a homeless person, interviewing that person , and then asking
him/her to introduce you to other homeless people you might interview.
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140. Simple Random sampling….
• If the sample is not representative of the
population, the random variation is called
sampling error.
•One of the most obvious limitations of simple
random sampling method is its need of a
complete list of all the members of the
population.
•Please keep in mind that the list of the population
must be complete and up-to-date.
•This list is usually not available for large
populations. In cases as such, it is wiser to use
other sampling techniques.
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144. 144
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In a stratified sample, the items in the frame are first subdivided into separate
subpopulations, or strata, and a simple random sample is conducted within
each of the strata.
146. 146
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In a cluster sample, the items in the frame are divided into
several clusters so that each cluster is representative of the entire
population.
A random sampling of clusters is then taken, and all the items in
each selected cluster or a sample from each cluster are then
studied.
156. Sample size determination
In addition to the purpose of the study and
population size, three criteria usually will need
to be specified to determine the appropriate
sample size:
1. the level of precision,
2. the level of confidence or risk, and
3. the degree of variability in the attributes being
measured (Miaoulis and Michener, 1976).
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157. The Level of Precision/sampling error:
• The level of precision, sometimes called sampling
error, is the range in which the true value of the
population is estimated to be.
• This range is often expressed in percentage
points, (e.g., ±5 percent), in the same way that
results for political campaign polls are reported
by the media.
• Thus, if a researcher finds that 60% of farmers in
the sample have adopted a recommended
practice with a precision rate of ±5%, then he or
she can conclude that between 55% and 65% of
farmers in the population have adopted the
practice.
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158. The Level of Precision/sampling error…
• A sample with the smallest sampling error will always
be considered a good representative of the population.
• Bigger samples have lesser sampling errors. When the
sample survey becomes the census survey, the
sampling error becomes zero.
• On the other hand, smaller samples may be easier to
manage and have less non-sampling error.
• Handling of bigger samples is more expensive than
smaller ones.
• The non-sampling error increases with the increase in
sample size
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160. The Confidence Level:
• The confidence or risk level is ascertained through the well established
probability model called the normal distribution and an associated theorem
called the Central Limit theorem.
• The key idea encompassed in the Central Limit Theorem is that when a
population is repeatedly sampled, the average value of the attribute obtained
by those samples is equal to the true population value.
• In a normal distribution, approximately 95% of the sample values are within
two standard deviations of the true population value (e.g., mean).
• Central limit theorem states that if samples of size n taken from any arbitrary
population (with any arbitrary distribution) and calculate x , then sampling
distribution of x will approach the normal distribution as the sample size n
increases with mean
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161. • The confidence level tells how confident one can be that the
error toleration does not exceed what was planned for in the
precision specification.
• Usually 95% and 99% of probability are taken as the two
known degrees of confidence for specifying the interval
within which one may ascertain the existence of population
parameter (e.g. mean).
• 95% confidence level means if an investigator takes 100
independent samples from the same population, then 95
out of the 100 samples will provide an estimate within the
precision set by him. There is always a chance that the
sample you obtain does not represent the true population
value.
• This risk is reduced for 99% confidence levels and increased
for 90% (or lower) confidence levels.
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165. Degree Of Variability
• The third criterion, the degree of variability in the attributes
being measured refers to the distribution of attributes in the
population.
• The more heterogeneous a population, the larger the sample
size required to obtain a given level of precision. The less
variable (more homogeneous) a population, the smaller the
sample size.
• Note that a proportion of 50% indicates a greater level of
variability than either 20% or 80%.
• This is because 20% and 80% indicate that a large majority do
not or do, respectively, have the attribute of interest.
• Because a proportion of 0.5 indicates the maximum variability in
a population, it is often used in determining a more conservative
sample size, that is, the sample size may be larger than if the
true variability of the population attribute were used.
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166. STRATEGIES FOR DETERMINING SAMPLE SIZE
• There are several approaches to determining the sample size. There are
different formulae for determination of appropriate sample size when
different techniques of sampling are used.
• In this lecture we discuss formulae for determining representative
sample size when simple random sampling technique is used.
• Simple random sampling is the most common and the simplest method
of sampling. Each unit of the population has the equal chance of being
drawn in the sample.
• Therefore, it is a method of selecting n units out of a population of size N
by giving equal probability to all units.
These include
1. using a census for small populations,
2. imitating a sample size of similar studies,
3. using published tables, and
4. applying formulas to calculate a sample size.
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167. (a) Formula for proportions:
i) Cochran’s formula for calculating sample size when
the population is infinite:
• Cochran (1977) developed a formula to calculate a
representative sample for proportions as
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168. • Where n= required sample size, p = proportion of the
population having the characteristic, q = 1-p and d = the
degree of precision. The proportion of the population (p) may
be known from prior research or other sources; if it is
unknown use p = 0.5 which assumes maximum heterogeneity
(i.e. a 50/50 split). The degree of precision (d) is the margin of
error that is acceptable. Setting d = 0.02, for example, would
give a margin of error of plus or minus 2%.
• More accurately the formula is
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where z=1.96 at 95% confidence interval or
Since (1.96)2=4
174. iii) Yamane’s or Slovin's formula for calculating sample size
• Yamane (1967) suggested another simplified formula for
calculation of sample size from a population which is an
alternative to Cochran’s formula. According to him, for a 95%
confidence level and p = 0.5 , size of the sample should be
Zekeria Yusuf (PhD) 174
where, N is the population size and e is the level of precision
Though other formulae are also available in different literatures, the above two formulae
(Cochran and Yamane’s) are used extensively in comparison to the others. And also both
formulas are equally important.
175. Sample size determination for stratified random sampling
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N represents the population size. In previous example, N = 13191; n= 384.
177. The above formula can be represented in a simplified form as
follow:
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178. summary
• There are different formulae given by different researchers for the
determination of appropriate sample sizes.
• The researcher should choose the formula according to their needs and
convenience.
• In choosing the right one, the researcher has to take into consideration
about the maximum budget, time limit, nature of the study along with
desired level of precision, confidence level and variability within the
population of interest.
• Using an adequate sample along with high quality data collection will
result in more reliable and valid results.
• Finally, the sample size determination techniques provide the number
of responses that need to be obtained. Many researchers commonly
add 10% to the sample size to compensate for persons that the
researcher is unable to contact. The sample size is also often increased
by 30% to compensate for no-response.
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179. Summary…
• If descriptive statistics are to be used, e.g., mean,
frequencies, then nearly any sample size will suffice.
• An adjustment in the sample size may be needed to
accommodate a comparative analysis of subgroups.
• Sudman (1976) suggested that a minimum of 100
elements were required for each major group or
subgroup in the sample and for each minor subgroup, a
sample of 20 to 50 elements was necessary.
• According to Kish ( Kish, 1965) 30 to 200 elements are
sufficient when the attribute is present 20 to 80
percent of the time if the distribution approaches
normality.
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180. Type I and Type II errors
• Type I error, also known as a “false positive”:
the error of rejecting a null hypothesis when it is
actually true.
In other words, this is the error of accepting an
alternative hypothesis (the real hypothesis of
interest) when the results can be attributed to
chance.
it occurs when we are observing a difference when
in truth there is none (or more specifically - no
statistically significant difference).
So the probability of making a type I error in a test
with rejection region R is P(R | H0 is true).
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184. Type II error
• Type II error, also known as a "false negative":
the error of not rejecting a null hypothesis when
the alternative hypothesis is the true state of nature.
• In other words, this is the error of failing to accept
an alternative hypothesis when you don't have
adequate power.
• it occurs when we are failing to observe a
difference when in truth there is one.
• So the probability of making a type II error in a test
with rejection region R is 1 ( | is true) a − P R H .
• The power of the test can be ( | is true) a P R H .
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186. Type III Error
• Wrong model, but right answer.
Type III Error had two foci:
• H0 considerations [specific]
• Reasoning errors (outside H0) that concerned model
building—>statistical evaluation [generalized].
Common Influences on Type III Error would be:
• Incorrect operationalization of variables
• Poor theory (e.g., ad hoc explanations of findings)
•Mis-identifying causal architecture (Schwartz & Carpenter,
1999).
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187. Type IV Error
• Right model but wrong answer,
Common Influences on Type IV Error would be:
• Collinearity among predictors
• Aggregation bias
• Wrong test for data structure
• In most situations, two-tailed testing is the norm, although it
requires a larger sample size. One-tailed testing is acceptable
only if one can be sure that change or difference can only be in
one direction and not in either direction.
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189. Major Data Analysis Techniques
• Mean separation
• ANOVA
• Correlation Analysis;
• Regression Analysis;
• Factor Analysis;
• Cluster Analysis;
• Correspondence Analysis (Brand Mapping);
• Discriminant /Logistic Regression Analysis;
• Biplot analysis
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190. Measures of central tendency
• The extent to which the observations cluster
around a central location is described by the
central tendency and the spread towards the
extremes is described by the degree of dispersion.
• The measures of central tendency are mean,
median and mode.
• Mean (or the arithmetic average) is the sum of all
the scores divided by the number of scores. Mean
may be influenced profoundly by the extreme
variables.
• The extreme values are called outliers. The
formula for the mean is
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202. Measures of spread
• There are several measures that provide an indication of the spread
of observations about the center of the distribution
• The sample range
• = the difference between the highest and lowest observations in a
• data set provides information on the boundaries of the sample data
(but is a relatively crude measure of dispersion, and is a biased
estimate of the population range)
• Interquartile range (IQR)= 75th percentile – 25th percentile
• This measure indicates the boundaries of the majority of the
sample data and is less sensitive to outliers
• The IQR is the default box edge when constructing a box plot
• Other percentiles (e.g., 90th-10th, 95th-5th) can also be used
Zekeria Yusuf (PhD) 202
206. DF…
• So what exactly do we mean by Degrees of
freedom?
• The true definition actually stems from multi-
dimensional geometry and sampling theory and
is related to the restriction of random vectors to
lie in linear subspaces…………….
• For our purposes, the definition used by Gotelli
and Ellison (2004) will suffice: the number of
independent pieces of information (i.e., n) in a
data set that can be used to estimate statistical
parameters.
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213. Parametric tests
• The parametric tests assume that the data are on
a quantitative (numerical) scale, with a normal
distribution of the underlying population.
• The samples have the same variance
(homogeneity of variances).
• The samples are randomly drawn from the
population, and the observations within a group
are independent of each other.
• The commonly used parametric tests are the
Student’s t-test, analysis of variance (ANOVA) and
repeated measures ANOVA.
Zekeria Yusuf (PhD) 213
215. Comparison Between Treatment Means
• The two most commonly used test procedues for
pair comparisons in agricultural research are the
least significant difference (LSD) test which is
• suited for a planned pair comparison, and
Duncan's multiple range test (DMRT) which is
applicable to an unplanned pair comparison.
Other test procedures, such as the honestly
significant difference (HSD) test and the Student-
Newman-Keuls' multiple range test, can be found
in Steel and Torrie, 1980,* and Snedecor and
Cochran, 1980.
Zekeria Yusuf (PhD) 215
216. Means comparison
Three categories:
1. Pair-wise comparisons (Post-Hoc Comparison)
2. Comparison specified prior to performing the experiment
(Planned comparison)
3. Comparison specified after observing the outcome of the
experiment (Un-planned comparison)
Statistical inference procedures of pair-wise
comparisons:
– Fisher’s least significant difference (LSD) method
– Duncan’s Multiple Range Test (DMRT)
– Student Newman Keul Test (SNK)
– Tukey’s HSD (“Honestly Significantly Different”) Procedure
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217. Pair Comparison
Suppose there are t means
An F-test has revealed that there are significant
differences amongst the t means
Performing an analysis to determine precisely where
the differences exist.
t
x
x
x ,
,
2
,
1
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218. Pair Comparison
Two means are considered different if the
difference between the corresponding sample
means is larger than a critical number. Then, the
larger sample mean is believed to be associated
with a larger population mean.
Conditions common to all the methods:
– The ANOVA model is the one way analysis of variance
– The conditions required to perform the ANOVA are
satisfied.
– The experiment is fixed-effect.
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219. Comparing Pair-comparison methods
With the exception of the F-LSD test, there is no good
theoretical argument that favors one pair-comparison method
over the others. Professional statisticians often disagree on
which method is appropriate.
In terms of Power and the probability of making a Type I
error, the tests discussed can be ordered as follows:
Tukey HSD Test
Student-Newman-Keuls Test
Duncan Multiple Range Test
Fisher LSD Test
MORE Power HIGHER P[Type I Error]
Pairwise comparisons are traditionally considered as “post hoc” an
not “a priori”, if one needs to categorize all comparisons into
one of the two groups 219
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220. Fisher Least Significant Different (LSD) Method
This method builds on the equal variances t-test of
the difference between two means.
The test statistic is improved by using MSE rather
than sp
2.
It is concluded that mi and mj differ (at a%
significance level if |mi - mj| > LSD, where
)
1
1
(
,
2
j
i
dfe
n
n
MSE
t
LSD
a
220
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221. Critical t for a test about equality = ta(2), 221
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222. Duncan’s Multiple Range Test
The Duncan Multiple Range test uses different Significant Difference
values for means next to each other along the real number line, and
those with 1, 2, … , a means in between the two means being
compared.
The Significant Difference or the range value:
where ra,p, is the Duncan’s Significant Range Value with parameters
p
(= range-value) and (= MSE degree-of-freedom), and experiment-
wise alpha level a (= ajoint).
n
MSE
r
R p
p
a ,
,
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223. Duncan’s Multiple Range Test
MSE is the mean square error from the ANOVA table and n
is the number of observations used to calculate the means
being compared.
The range-value is:
– 2 if the two means being compared are adjacent
– 3 if one mean separates the two means being compared
– 4 if two means separate the two means being compared
– …
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225. Tukey HSD Procedure
The test procedure:
– Assumes equal number of observation per populations.
– Find a critical number w as follows:
g
n
MSE
dfe
dft
q )
,
(
a
w
dft = treatment degrees of freedom
=degrees of freedom = dfe
ng = number of observations per population
a = significance level
qa(dft,) = a critical value obtained from the studentized range table
225
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229. Tests to analyse the categorical data
• Chi-square test, Fischer’s exact test and McNemar’s test are
used to analyse the categorical or nominal variables.
• The Chi-square test compares the frequencies and tests
whether the observed data differ significantly from that of the
expected data if there were no differences between groups (i.e.,
the null hypothesis).
• It is calculated by the sum of the squared difference between
observed (O) and the expected (E) data (or the deviation, d)
divided by the expected data by the following formula:
Zekeria Yusuf (PhD) 229
A Yates correction factor is used when the sample size is small. Fischer’s exact test is
used to determine if there are non-random associations between two categorical
variables.
230. • The χ2 (chi-square) distribution is used for a
variable that is distributed as the square of
values from a standard normal distribution.
• Variances tend to follow a χ2 distribution and
we use the distribution to test for differences
between observed and expected outcomes
from a model (Categorical Data Analysis).
Zekeria Yusuf (PhD) 230
231. Analyzing Frequencies: The Chi-Square Test
• The t-test is used to compare the sample means of two sets of data.
The chi-square test is used to determine how the observed results
compare to an expected or theoretical result.
• For example, you decide to flip a coin 50 times. You expect a
proportion of 50% heads and 50% tails. Based on a 50:50 probability,
you predict 25 heads and 25 tails. These are the expected values. You
would rarely get exactly 25 and 25, but how far off can these
numbers be without the results being significantly different from
what you expected? After you conduct your experiment, you get 21
heads and 29 tails (the observed values). Is the difference between
observed and expected results purely due to chance? Or could it be
due to something else, such as something might be wrong with the
coin? The chi-square test can help you answer this question.
• The statistical null hypothesis is that the observed counts will be
equal to that expected, and the alternative hypothesis is that the
observed numbers are different from the expected.
Zekeria Yusuf (PhD) 231
232. • Note that this test must be used on raw
categorical data. Values need to be simple counts,
not percentages or proportions. The size of the
sample is an important aspect of the chi-square
test—it is more difficult to detect a statistically
significant difference between experimental and
observed results in a small sample than in a large
sample. Two common applications of this test in
biology are in analyzing the outcomes of a
genetic cross and the distribution of organisms in
response to an environmental factor of interest.
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237. Example 2
• Students just learned in their biology class that pill bugs use
gill-like structures to breathe oxygen. The students
• hypothesized that the pill bugs’ gills require them to live in
wet environments for their survival. To test the
• hypothesis, they wanted to determine whether pill bugs
show a preference for living in wet or dry
• environments.
• The students placed 15 pill bugs on the dry side of a two-
sided choice chamber, and 15 pill bugs on the wet
• side of the chamber. Fifteen minutes later, 26 pill bugs were
on the wet side and 4 on the dry side. The data
• are shown in Table 12.
Zekeria Yusuf (PhD) 237
240. What is ANOVA
• Statistical technique specially designed to test
whether the means of more than 2
quantitative populations are equal.
• Developed by Sir Ronald A. Fisher in 1920’s.
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242. Assumptions of ANOVA
1) Normality: The values in each group are
normally distributed.
2) Homogeneity of variances: The variance within
each group should be equal for all groups.
3) Independence of error: The error(variation of
each value around its own group mean) should
be independent for each value.
242
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243. ANOVA Assumptions
(1) Y-values and their error terms are normally
distributed for each level of the predictor variables.
ANOVA is generally robust to violations of this
assumption if sample sizes and variances are similar
across levels.
(2) Y-values and their error terms have the same
variance at each level of the predictor variables (i.e.
homogeneity of variance). Unequal variances can
be a big problem, but can be addressed using robust
ANOVA techniques.
(3) Y-values and their error terms are independent.
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244. Homogeneity of variance (HOV) test
The statistical validity of many commonly used
tests such as the t-test & ANOVA depends on the
extent to which the data conform to the
assumption of homogeneity of variance (HOV).
When a research design involves groups that
have very different variances, the p value
accompanying the test statistics, such as t & F,
may be too lenient or too harsh.
Furthermore, substantive research often requires
investigation of cross- or within-group fluctuation
in dispersion.
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245. 1. Methods based on variance, or the variance ratio
e.g., the one-sample x2 test, two-sample folded
form F test, Hartley's Fmax test with David's
multiple comparison procedure, Cochran's G test,
t-test for two correlated samples and Fr test for
two correlated samples.
Unfortunately, this approach is also most
sensitive to symmetry and kurtosis.
Those tests are easy but not robust.
Many of those tests cannot deal with unbalanced
designs.
245
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246. I. One-Sample HOV Test
• A convenient chi-square test can determine whether the
difference betweena sample variance and a known or posited
population variance is large enough to reject the null
hypothesis, Ho:
The x2 test has (n-1) degrees of freedom. The critical value for a
chosensignificance level can be found in the x2 table available in most statistics
textbooks.The test is not accurate when the population deviates from
normality and thesample size is small.
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247. • II. Two-Sample HOV Test
• This test, known as the folded form F test, is automatically
conducted whenPROC TTEST is invoked. The folded form F
test uses the ratio of the largervariance to the smaller
variance to test the null hypothesis,
247
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248. • The following SAS statements, with GROUP as the independent variable & SCORE
as the dependent variable, produce, among other things, the folded form F':
PROC TTTEST;
CLASS GROUP;
VAR SCORE;
RUN;
• The test has (n1-1) and (ns-1) degrees of freedom for the numerator and
the denominator respectively. Because the larger variance is always taken
to be thenumerator, F' is always larger than 1. In other words, only one
direction of the F distribution is considered. SAS/STAT adjusts for the
directional tail and prints out the correct p value. Should anyone try to
conduct the test by hand and refer to theconventional F table, he or she
needs to remember that the listed critical F at thesignificance level of 0.05
actually means a significant level of approximately 0.10in the case of the
folded form F test (Ferguson, 1981, pp. 189-192). The test is very sensitive
to deviations from the normal distribution.
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249. III. HOV Tests Involving Two Or More Samples
The test is not available from SAS and requires equal or roughly equal sample sizes under
the assumption of normality. The table of critical F. values for various combinations of k
(number of groups) and n (if all the groups have the same size) can be found in Kanji
(1993) or Rosenthal and Rosnow (1992). When the groups have slightly different sample
sizes, the harmonic mean may serve as the adjusted sample size n' (Rosenthal & Rosnow,
1991).
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250. • Even though the F. test can reject the overall null, it cannot
pinpoint between which two groups heterogeneity of variance
occurs.
• For that purpose, the researcher needs a multiple test
analogous to Duncan's test following the rejection of the
overall null in one-way ANOVA.
• David's multiple test (1954) extends thefolded form F test a
step further to pairwise comparisons among k groups,
alwaysplacing the larger variance over the smaller one, as is
done in the folded form F'formula.
• For critical values for the Duncan-type multiple HOV test, see
Tietjen &Beckman's maximum F-ratio table (1972).
• This test requires equal or roughlyequal group sizes and is very
sensitive to departures from the normal distribution.
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251. • As an improvement on Hartley's F,,, test, which involves only the
maximum and the minimum variances, Cochran's G test, also known as
Cochran's C test, uses the dispersion information in all the k groups. It is
appropriate for equal or roughly equal size groups and is typically used in
the situation where one group seems to be drastically more spread out
than all the other groups sharing more or less the same variance. In that
sense, it is a test to identify an outlier in terms of variance.
251
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252. • Cochran's 0, or C, is basically a variance ratio, except that the
denominator is the product of k (number of groups) and the pooled
within-group variance, often referred to as MS within or MSerror,
available from the one-way ANOVA print out. SAS does not have an
option for the test, but it can be done indirectly through ANOVA
plus a little bit of calculation.
• The following SAS statement generates, among other things,
MSerror:
PROC GLM;
CLASS GROUP;
MODEL SCORE = GROUP; RUN;
• This test requires a special table of critical values for various
combinations of k and n (Rosenthal & Rosnow, 1991; Winer 1971).
The harmonic mean may be adopted as the adjusted n' if the
groups have roughly equal sizes.
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253. 2. HOV tests based on the natural log transformation
The Bartlett-Kendall test and Bartlett x2 test relies on the natural log
transformation of the variance because the log variance approximates the
normal distribution quite well.
The Bartlett-Kendall test uses the log transformation of the variance, because
the sampling distribution of the log variance is normally distributed. The
numerator in the formula is the log of a variance ratio.
Likelihood ratio tests based on log variance are more robust than variance ratio
tests.
However, many statisticians still feel that they are quite vulnerable to
deviations from normality.
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254. Bartlett-Kendall test
• This test applies to equal size samples.
• In case of samples with roughly equal sizes, the
arithmetic average of the sample sizes is used in
the formula.
• A special table is needed for critical values
(Bartlett & Kendall, 1946; Pearson & Hartley,
1970).
• The Bartlett-Kendall test and Hartley's Fmax test,
one using log transformation and the other using
the variance ratio, produce practically identical
results.
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255. Another test that involves log transformation of the variance is
the Bartlett X2 test. The transformation allows the X2
distribution to serve as the basis for rejection of the null. The
log transformation also improves (though not
much)robustness in case of departures from the normal
distribution, but in doing so, reduces power slightly.
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256. • The numerator is essentially based on the negative log of the ratio
between thegroup variance and the geometric mean of the k group
variances.
• The denominatoris a correction factor to improve approximation to the x2
distribution. The chi-square test has (k-1) degrees of freedom.
• This likelihood test is sensitive todepartures from the normal distribution.
Preferably, the samples have comparablesizes.
• The Bartlett test does not have a subsequence multiple comparison
procedure.
• The following SAS statements conduct the Bartlett test:
PROC GLM;
CLASS GROUP;
MODEL SCORE = GROUP;
MEANS GROUP / HOVTEST = BARTLETT; RUN;
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257. Levene's test
• When the groups have different sizes, Levene's test is recommended.
• The test has two options. For Option One, group means are calculated first. For each person,
the absolute deviation of the person's score from the mean of the group to which the
person belongs is calculated, Ixy -x1I.
• This absolute deviation represents how far the person is displaced or spread out from the
group mean. Such variables are known as spread or dispersion variables.
• Since the variance of each group is related to the sum of the absolute deviations within the
group, testing the differences among the group means of the absolute deviations through
the regular one-way ANOVA is tantamount to testing homogeneity of variance.
• Option One is recommended for highly skewed data.
• The SAS statements for Option One are included:
PROC GLM;
CLASS GROUP;
MODEL SCORE = GROUP;
MEANS GROUP / HOVTEST = LEVENE TYPE = ABS; RUN;
• Option Two shares the same logic with Option One, but the spread variable is the square of
the absolute deviation. (xy - x1)2.
• SAS runs Option Two by default. One can also specify TYPE = SQUARE in the program above
to call up Option Two.
• One weakness of Levene's test is that it may allow a higher Type I error rate than it should.
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258. • An improvement on Levene's test is the Brown-Forsythe
test, which follows the same logic underlying one-way
ANOVA except that the spread variable becomes the square
of the deviation from the group median, rather than the
group mean.
• When all the distributions are normal, the Brown-Forsythe
test and the Levene's test Option Two are identical.
• The SAS Institute recommends the Brown-Forsythe test as
the most powerful "to detect variance differences while
protecting the Type I error probability" (1997).
• It is not yet clear what multiple comparison options are
appropriate following Levene's test or the Brown-Forsythe
test.
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259. • Another ANOVA-based test is the O'Brien test,
which relies on yet another spread variable r
through a formula that allows the statistician
to choose a weight(w) between 0 and 1 to
adjust the transformation:
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260. • The most commonly adopted w is 0.5 to offset the
anticipated moderate departurefrom kurtosis=0. The
actual kurtosis is almost never known, and the choice of
wother than 0.5 rarely makes a critical difference in
practice. When no w isspecified, SAS, by default, coverts
the dependent variable into r using w=0.5 andthen
subjects r to the regular one-way ANOVA. The following
SAS statementsaccomplish the O'Brien test:
• PROC GLM;
• CLASS GROUP;
• MODEL SCORE = GROUP;
• MEANS GROUP /HOVTEST = OBRIEN W = 0.5; RUN;
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266. SAS does not exactly perform the Sidney-Tukey test. When only two groups are
involved, SAS runs the Wilcoxon rank sums test (Sidney & Castellan, 1988), with more
than two groups, it switches to Friedman two-way analysis of variance by ranks (Sidney
& Caste Ilan, 1988).
Their results are comparable to those of the modified Sidney-Tukey test.
Because ranks, rather than absolute deviations, form the basis of the analysis, the test
has less power. The reported x2, can be conveniently converted into F using the formula
below:
The F test has (k-1), (N-k) degrees of freedom. Type I error rate tends to be slightly higher
when F approximation is adopted than when x2 is used. However, the difference is negligible
(Conover, Johnson & Johnson, 1981). 266
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267. IV. HOV Test for Factorial Designs
For the two-way ANOVA fixed-effect factorial design, O'Brien
proposed a robust procedure to test HOV (1979, 1981). The beauty
of it is that it can attribute differences in variance to the main
effects of independent variables A and B and the interaction effect
AXB.
It works with both balanced and unbalanced designs and allows
subsequent multiple comparisons for more detailed analysis. It is
the O'Brien test for one-way ANOVA generalized to the two-way
situation.
For the purpose of this paper, it is called the generalized O'Brien
test, even though it is exactly the same test as the one explained
above. The generalized O'Brien test is simply a two-way analysis of
variance of the transformed variable, r, and Welch ANOVA can be
conducted for pairwise comparisons with the significance level
adjusted down through the Bonforroni method.
The transformation to the spread variable r follows the formula:
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269. • SAS statements for two-way ANOVA with the
transformed variable SCORE as the dependent
variable are listed below:
• PROC GLM;
• CLASS A B;
• MODEL SCORE = A B A*B;
• RUN;
• O'Brien recommended Welch ANOVA for
subsequent multiple comparisons(1981).
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270. V. HOV Tests for Two Correlated Samples
Correlated samples are typically involved in pre-post designs or studies that
match the two subjects in each pair.
HOV tests for such situations need to take into consideration the correlation
between the two sets of scores. A positive correlation plus a statistically
significant increase in variability indicates greater dispersion of prior
differences.
A positive correlation plus a statistically significant decrease in variability
means reduction in prior differences.
However, when a negative correlation occurs, the researcher may have to
reconsider the research question and search for reasons other than the
treatment to account for the reversal of the direction of individual
differences.
Should a zero correlation occur, the matching process has failed its purpose.
The groups might as well be treated as independent samples.
• The t-test for the difference between the variance of two correlated
samples is not available from SAS. Fortunately, it is simple enough for
hand calculation. The West has (n-2) degrees of freedom.
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271. 3. HOV test based on logic of ANOVA
• This approach applies the logic of ANOVA to transformed variables.
• Tests, such as Levene's test, Brown-Forsythe test, and O'Brien test with
Welch ANOVA serving as a prudent procedure for multiple
comparisons, have a strong appeal to non-statistician researchers and
compare favorably to all the other approaches in terms of power and
robustness.
• Among the three tests, the Brown-Forsythe test and the O'Brien test
may have overall advantage over Levene's test.
• The O'Brien test is particularly appealing because it applies to both
one-way and two-way ANOVA and comes with a handy Welch-type
procedure for multiple comparisons, all of which can be accomplished
within SAS/STAT.
• Methodologically, it is also more sophisticated because it allows
kurtosis to come into play through the weight w.
• The ANOVA approach is often recommended, since HOV is typically
discussed in conjunction with ANOVA, ANOVA on differences among
means and ANOVA on differences among variances share the same
logic.
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272. HOV test….
• The last major approach to HOV testing is the
nonparametric alternative represented by the
modified Sidney-Tukey test.
• In the past, many attempts were made to
conduct HOV testing by way of ranks to simplify
computation.
• All of them use the chi-square approximation.
With the easy access to computers today, those
methods do not seem to have much to
recommend themselves for, and they are not
available from most of the software packages.
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273. ANOVA
One way ANOVA Two way ANOVA Three way ANOVA
Eg. The effect of light
on the rate of
photosynthesis
Eg. The effect of light and
temperature on the rate of
photosynthesis
e.g. the effect of light,
temperature and CO2
on the rate of
photosynthesis
ANOVA with repeated measures - comparing >=3 group means where the
participants are same in each group. E.g. Group of subjects is measured more
than twice, generally over time, such as patients weighed at baseline and every
month after a weight loss program.
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274. Steps in ANOVA
1. State null & alternative hypotheses
2. State Alpha
3. Calculate degrees of Freedom
4. State decision rule
5. Calculate test statistic
• - Calculate variance between samples
• - Calculate variance within the samples
• - Calculate F statistic
• - If F is significant, perform post hoc test
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277. Calculating variance between samples
1. Calculate the mean of each sample.
2. Calculate the Grand average
3. Take the difference between means of various
samples & grand average.
4. Square these deviations & obtain total which
will give sum of squares between samples (SSC)
5. Divide the total obtained in step 4 by the
degrees of freedom to calculate the mean sum
of square between samples (MSC).
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278. Calculating Variance within Samples
1. Calculate mean value of each sample
2. Take the deviations of the various items in a
sample from the mean values of the respective
samples.
3. Square these deviations & obtain total which
gives the sum of square within the samples (SSE)
4. Divide the total obtained in 3rd step by the
degrees of freedom to calculate the mean sum of
squares within samples (MSE).
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279. CORRELATION ANALYSIS
• Correlation analysis, expressed by correlation
coefficients, measures the degree of linear relationship
between two variables.
• Feature of Correlation coefficient:
• Between + and – 1;
• The sign of the correlation coefficient (+, -) defines the
direction of the relationship, +ve or –ve;
• A positive correlation coefficient means that as the
value of one variable increases, the value of the other
variable also increases; as one decreases the other
decreases; and
• A negative correlation coefficient indicates that as one
variable increases, the other decreases, and vice versa.
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280. • The absolute value of the correlation coefficient
measures the strength of the relationship.
• A correlation coefficient of r=0.50 indicates a
stronger degree of linear relationship than one of
r=0.40.
• Correlation coefficient of zero (r=0.0) indicates
the absence of a linear relationship.
• Correlation coefficients of r=+1.0 and r=-1.0
indicate a perfect linear relationship.
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281. • Measuring Correlations and Analyzing Linear Regression Correlations can
suggest relationships between sets of data. The correlation coefficient
(A) provides a measure of how related two variables are, and it is
expressed as a value between +1 and −1. The closer the value is to 0, the
weaker the correlation.
• For example, if you plot the width of an oak (Quercus sp.) leaf (Y) on an xy
scatter plot as a function of the leaf’s length (X), the correlation coefficient
( A) indicates how much width depends on length. An A-value equal to +1
would indicate a perfect positive correlation between width and length. In
other words, the longer an oak leaf, the wider it is.
• An A-value of −1 would indicate a perfect negative correlation—the longer
an oak leaf, the narrower it is. If there is no correlation between two
variables, the A-value equals 0, which would mean that there is no
relationship between oak leaf length and width. The null hypothesis (H0)
for a correlation is that there is no correlation and A = 0.
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288. Regression analysis
Regression analysis measures the:
• strength of a relationship between a variable (e.g.
overall customer satisfaction) one or more explaining
variables (e.g. satisfaction with product quality and
price).
• Correlation provides a single numeric summary of a
relation (called the correlation coefficient), while
regression analysis results in a "prediction" equation.
• The regression equation describes the relation
between the variables. If the relationship is strong
(expressed by the Rsquare value), it can be used to
predict values of one variable given the other variables
have known values.
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289. Factor Analysis
• Factor analysis aims to
describe a large number of
variables or questions by
only using a reduced set of
underlying variables, called
factors.
• It explains a pattern of
similarity between observed
variables.
• Questions which belong to
one factor are highly
correlated with each other.
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290. Use of Factor Analysis
• Factor analysis is often used in customer satisfaction
studies to identify underlying service dimensions, and in
profiling studies to determine core attitudes.
• For example, as part of a national survey on political
opinions, respondents may answer three separate
questions regarding environmental policy, reflecting
issues at the local, regional and national level.
• Factor analysis can be used to establish whether the
three measures do, in fact, measure the same thing.
• It is can also prove to be useful when a lengthy
questionnaire needs to be shortened, but still retain key
questions.
• Factor analysis will indicate which questions can be
omitted without losing too much information.
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291. DISCRIMINANT / LOGISTIC REGRESSION ANALYSIS
• Discriminant and logistic regression analysis are
statistical techniques that point out the
differences between two or more groups based
on several characteristics (most often rating
scales when Discriminant analysis, while logistic
regression can handle any type of variable).
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292. PCA/ MULTIDIMENSIONAL SCALING
• Multidimensional scaling (MDS) can be considered to be an
alternative to factor analysis.
• In general, the goal of the analysis is to detect meaningful
underlying dimensions that allow the researcher to explain
observed similarities or dissimilarities between the investigated
objects.
• In factor analysis, the similarities between objects (e.g.
variables) are expressed in the correlation matrix.
• With MDS one may analyse any kind of similarity or
dissimilarity matrix, in addition to correlation matrices.
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293. Probability distributions
• All random variables will have an associated probability
distribution with a range of values of the variable on the x-axis
and the relative probabilities of each value on the y-axis
• Most of the statistical procedures that you will use in the
study of biology make some assumptions about the
probability distribution of the variable you have measured (or
about the distribution of the statistical errors).
• We also use probability distributions to generate models and
make predictions, so they are very important to what we do.
• Many (too many) probability distributions have been defined
mathematically and there are several that work well in
describing biological phenomena. We will focus on a few of
the major ones.
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294. • Recall that a variable can be either discrete or
continuous in its distribution, which creates some
important differences in the probability distributions:
1. For discrete variables, the probability distribution will
include measurable probabilities for each possible
outcome
2. For continuous variables, there are an infinite number
of possible outcomes. Thus, the probability distribution
is what we call a probability density function (pdf),
and it is used to estimate the probability associated
with a range of values since the probability of any
single value = 0.
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298. Continuous probability distributions
• As we mentioned earlier, continuous variables are not limited to take on
integer values, but instead can take on an infinite number of values.
Therefore, we can’t estimate the probability of any single outcome and
instead estimate the probability than an outcome will fall within a specific
interval.
• The probability distribution is now termed a probability density function
(pdf), and we use it to estimate the probability of a variable falling within a
certain range of values. Through integration, we can estimate the area under
the curve (the curve is the pdf) for any range of values. Generally the pdf is
normalized so that the area under the curve representing the total probability
is approximately 1.
• We can also generate cumulative density functions (cdf) to examine the
probability of a variable being less than or greater than some value (Yi < Y).
These represent tail probabilities, which is here our familiar P-values come
from.
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300. Zekeria Yusuf (PhD) 300
The conversion of any normal random variable to a standard normal
random variable is what enables us to test hypotheses about the
mean, which will be the first hypothesis tests that we perform.
302. • Continuous variables are not always distributed
symmetrically. Many biological variables show right- or
positive skewness, with long tails that include larger
observations that occur with less frequency. The
• lognormal distribution, in which the log
transformation of the variable is normally distributed,
describes many biological data of this sort (i.e.,
measurement data that cannot be negative such as
lengths and weights). Another asymmetric distribution
observed, although less frequently in biology, for
continuous random variables is the Exponential
distribution
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304. t-distribution (or student’s t-distribution)
• The t-distribution (or student’s t-distribution)
is also used to test hypotheses concerning
differences between sample statistics and
population parameters. However, it accounts
for the fact that we are estimating the
standard deviation of the population
parameter using our sample data.
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308. F-distribution
• The F-distribution is a probability distribution
for a variable that is distributed as the ratio of
two χ2 distributions and is used for testing
hypotheses about the ratio of variances (this is
a very important distribution for testing
hypotheses using linear models, e.g.,
regression and ANOVA).
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310. Framing and Testing Hypotheses
• Hypotheses can be simply defined as possible
explanations for our observations. They often
stem directly from our observations, the existing
scientific literature, theoretical model predictions,
intuition and reasoning, or all of the above
• Good hypotheses:
1) Must be testable
2) Should generate unique predictions
• It is important to note that the statistical
treatment of data is generally the same for each
type of study. The difference is in the confidence
• we place in the inferences we make.
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311. Statistical vs. Scientific Hypotheses
• A statistical hypothesis tests for pattern in the
data. The statistical null hypothesis would be
one of “no pattern”, meaning no difference
between parameter estimates or no
relationship between a variable and some
measured factor. The statistical alternative
hypothesis would be that “some pattern
exists”
**But, how do these patterns relate to the
scientific hypothesis? Through significance/
nonsignificance differences.
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312. Statistical Significance and the P-value
• the statistical null hypothesis: that the difference
represents random variation( the difference is due to
chance)
• HO = some specific mechanism does not operate to
produce the observed differences
• We can then define one or more statistical alternative
hypotheses
• HA = observed difference is too large to be due to
random variation alone(= the difference is not due to
chance instead some other factors are responsible)
• The set of HA’s can be broadly defined as “not HO” (the
reverse of Ho)
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