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# Hypothesis Testing (1)

advertisement ```Learning objectives
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What is hypothesis?
Types of hypothesis
Normal distribution curve
Hypothesis testing
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Level of significance
Types of errors
p value
One &amp; two tail tests
Degree of freedom
Data analysis
What is a Hypothesis?
•
An educated guess
•
A tentative point of view
•
A proposition not yet tested
•
A preliminary explanation
•
A preliminary Postulate
•
A hypothesis is a claim (assumption) about a
population parameter
Various authors
• “A hypothesis is a conjectural statement of the relation between two or
more variables”. (Kerlinger, 1956)
• “Hypothesis are single tentative guesses, good hunches – assumed for
use in devising theory or planning experiments intended to be given a
direct experimental test when possible”. (Eric Rogers, 1966)
• “Hypothesis is a formal statement that presents the expected relationship
between an independent and dependent variable.”(Creswell, 1994)
• A hypothesis is a logical supposition, a reasonable guess, an educated
conjecture. It provides a tentative explanation for a phenomenon under
investigation.&quot; (Leedy and Ormrod, 2001).
A Hypothesis :
•
•
•
•
•
•
•
•
must make a prediction
must identify at least two variables
should have an elucidating power
should strive to furnish an acceptable explanation or accounting of
a fact
must be falsifiable meaning hypotheses must be capable of being
refuted based on the results of the study
must be formulated in simple, understandable terms
should correspond with existing knowledge
In general, a hypothesis needs to be unambiguous, specific,
quantifiable, testable and generalizable.
Characteristics of a Testable Hypothesis
1. A Hypothesis must be conceptually clear
- concepts should be clearly defined
- the definitions should be commonly accepted
- the definitions should be easily communicable
2. The hypothesis should have empirical reference
- Variables in the hypothesis should be empirical realities
- If they are not it would not be possible to make the observation
and ultimately the test
3. The Hypothesis must be specific
- Place, situation and operation
Characteristics of a Testable Hypothesis
4. A hypothesis should be related to available techniques of
research
- Either the techniques are already available or
- The researcher should be in a position to develop suitable
techniques
5. The hypothesis should be related to a body of theory
- Hypothesis has to be supported by theoretical argumentation
- It should depend on the existing body of knowledge
In this way
the study could benefit from the existing knowledge and
later on through testing the hypothesis could contribute to the reservoir of
knowledge
Categorizing Hypotheses
Can be categorized in different ways
1. Based on their formulation
• Null Hypotheses and Alternate Hypotheses
2. Based on direction
• Directional and Non-directional Hypothesis
3. Based on their derivation
• Inductive and Deductive Hypotheses
The Null Hypothesis, H0
• States the claim or assertion to be tested
• Is always about a population parameter, not about a sample
statistic
• Begin with the assumption that the null hypothesis is true
– Similar to the notion of innocent until
proven guilty
• Refers to the status quo
• Always contains “=” , “≤” or “” sign
• May or may not be rejected
• It states that independent variable has no effect and there
will be no difference b/w the two groups.
The Alternative Hypothesis, H1
•
•
•
•
•
Is the opposite of the null hypothesis
Challenges the status quo
Never contains the “=” , “≤” or “” sign
May or may not be proven
Is generally the hypothesis that the researcher is trying to
prove
• It states that independent variable has an effect and
there will be a difference b/w the two groups.
Categorizing Hypotheses (Cont…)
2. Directional Hypothesis and Non-directional Hypothesis
• Simply based on the wording of the hypothesis we can tell
the difference between directional and non-directional
– If the hypothesis simply predicts that there will be a difference between
the two groups, then it is a non-directional hypothesis. It is nondirectional because it predicts that there will be a difference but does not
specify how the groups will differ.
– If, however, the hypothesis uses so-called comparison terms, such as
“greater,”“less,”“better,” or “worse,” then it is a directional hypothesis. It
is directional because it predicts that there will be a difference between
the two groups and it specifies how the two groups will differ
3. Inductive and Deductive Hypotheses(Theory
Building and Theory Testing)
• classified in terms of how they were derived:
- Inductive hypothesis - a generalization based on
observation
Observation
Pattern
Hypothesis
Theory
- Deductive hypothesis - derived from theory
Theory
Hypothesis
Observation
Confirmation
Normal Distribution Curve
•A normal distribution curve is symmetrical, bellshaped curve defined by the mean and standard
deviation of a data set.
•The normal curve is a probability distribution with a
total area under the curve of 1.
•The mean of the data in a standard normal distribution
is 0 and the standard deviation is 1.
•A standard normal distribution is the set of all z-scores
One standard deviation away from the mean (  ) in either
direction on the horizontal axis accounts for around 68 percent
of the data. Two standard deviations away from the mean
accounts for roughly 95 percent of the data with three standard
deviations representing about 99.7 percent of the data.
6 Steps in Hypothesis Testing
1.
State the null hypothesis, H0 and the alternative hypothesis, H1
2.
Choose the level of significance, , and the sample size, n
3.
Determine the appropriate test statistic (two-tail, one-tail, and Z or t
distribution) and sampling distribution
4.
Determine the critical values(mainly three criteria, (i) significance
level,(ii) degree of freedom,(iii) One or two tailed test,that divide the
rejection and non rejection regions
5.
Collect data and compute the value of the test statistic
6.
Make the statistical decision and state the managerial conclusion. If
the test statistic falls into the non rejection region, do not reject the null
hypothesis H0. If the test statistic falls into the rejection region, reject
the null hypothesis. ExpresCshatph9e-1m5anagerial conclusion in the context
of the problem
Steps in Hypothesis Testing
Problem Definition
Clearly state the null and
alternate hypotheses.
Choose the relevant test and
the appropriate probability
distribution
Determine the
significance level
Choose the critical value
Determine the
degrees of
freedom
Compute relevant
test statistic
Compare test statistic and
critical value
Decide if one-or
two-tailed test
Does the test statistic fall in
the critical region?
Yes
Reject null
No
Do not reject null
Level of Significance, 
• Defines the unlikely values of the sample statistic if the null
hypothesis is true
• Indicates the percentage of sample means that is outside the cut-off
limits (critical value)
• It is the max. value of probablity of rejecting null hypothesis when it
is true.
– Defines rejection region of the sampling distribution
• Is designated by  , (level of significance)
– Typical values are 0.01, 0.05, or 0.10
• Is selected by the researcher at the beginning
• Provides the critical value(s) of the test
Level of Significance and the Rejection Region
Level of significance =

//2
H0: μ = 3
H1: μ ≠ 3 Two-tail test
Represents
critical value
//2
Rejection
region is
shaded
0

H0: μ ≤ 3
H1: μ &gt; 3Upper-tail test
0
H 0: μ ≥ 3

H1: μ &lt; 3Lower-tail test
0
Errors in Making
Decisions
• Type I Error
– Reject a true null hypothesis
– Considered a serious type of error
• The probability of Type I Error is 
• Called level of significance of
the test
• Set by the researcher in
advance
• Type II Error
– Fail to reject a false null
hypothesis
• The probability of Type II Error is β
Testing of hypotheses
Type I and Type II Errors
No study is perfect,
there is always the chance for error
Decision
Accept H0 /
reject HA
Reject H0
/accept HA
H0 true / HA false
H0 false / HA true
Type II error ()
OK
p=1-
Type I error ()
p=
p=
OK
p=1-
 - level of
1- - power of
significance the test
Testing of hypotheses
Type I and Type II Errors
α =0.05
there is only 5 chance in 100 that the result
termed &quot;significant&quot; could occur by chance
alone
The probability of making a Type I (α) can be decreased by altering
the level of significance.
it will be more difficult to find a significant result
the power of the test will be decreased
the risk of a Type II error will be increased
Type I &amp; II Error Relationship
 Type I and Type II errors cannot happen at the same
time

Type I error can only occur if H0 is true

Type II error can only occur if H0 is false
If Type I error probability (  )
Type II error probability ( β )
, then
Factors affecting type II error
All else equal:
– β
when the difference between hypothesized
parameter and its true value
– β
when 
– β
when σ
– β
when n
Testing of hypotheses
Type I and Type II Errors
The probability of making a Type II () can be decreased by
increasing the level of significance.
it will increase the chance of a Type I error
To which type of error you are willing to risk ?
Degree of Freedom
• The number or bits of &quot;free&quot; or unconstrained data used in
calculating a sample statistic or test statistic
• It refers to the scores in a distribution that are free to change
without changing the mean of distribution.
• A sample mean (X) has `n' degree of freedom
• A sample variance (s2) has (n-1) degrees of freedom
• This no. is used to determine power ,because the more
subjects the greater the power
One-Tail Test
• In many cases, the alternative hypothesis focuses on a particular
direction
• Determines whether a particular population parameter is larger or
smaller than some predefined value
• Uses one critical value of test statistic
H0: μ ≥ 3
H1 : μ &lt; 3
H0: μ ≤ 3
H1: μ &gt; 3
This is a lower-tail test since the alternative
hypothesis is focused on the lower tail below the
mean of 3
This is an upper-tail test since the alternative
hypothesis is focused on the upper tail above the
mean of 3
Two tailed test
• Two-tailed Test
the
• Determines
likelihood that a
population parameter
is within certain upper
and lower bounds
• May use one or two
critical values
Confidence interval and significance test
Null hypothesis
is accepted
A value for null hypothesis
within the 95% CI
p-value &gt; 0.05
Null hypothesis
is rejected
A value for null hypothesis
outside of 95% CI
p-value &lt; 0.05
p-Value Approach to Testing
• p-value: Probability of
obtaining a test statistic
more extreme ( ≤ or  )
than the observed sample
value given H0 is true
• Also called observed level
of significance
• Smallest value of  for
which H0 can be rejected
p-Value Approach to Testing
(continued)
• Convert Sample Statistic (e.g., X ) to Test
Statistic (e.g., Z statistic )
• Obtain the p-value from a table or computer
• Compare the p-value with 
– If p-value &lt;  , reject H0
– If p-value   , do not reject H0
The First Question
After examining your data, ask: does what you're testing
seem to be a question of relatedness or a question of
difference?
•If relatedness (between your control and your experimental
samples or between you dependent and independent variable),
We will be using tests for correlation (positive or negative)
or regression.
•If difference (your control differs from your experimental),
we will be testing for independence between distributions,
means or variances. Different tests will be employed if
your data show parametric or non-parametric properties.
Parametric or Non-parametric
Parametric tests: to estimate at least one population parameter
from sample statistics and are restricted to data that:
1) show a normal distribution
2) are independent of one another
3) are on the same continuous scale of measurement
4) require certain assumptions about the parameters of
the
population such as knowing μ and 
 Non-parametric tests : are used on data that:
1) show an other-than normal distribution
2) are dependent or conditional on one another
3) in general, do not have a continuous scale of
measurement
4) does not require assumptions about the parameters of
population such as knowing μ and  are not needed
the
Parametric and nonparametric tests of
significance
N o n p a r a m e t r i c tests
One group
Two
unrelated
groups
T w o related
groups
K-unrelated
groups
K-related
groups
Nominal
data
Chi square
goodness
o f fit
Chi square
Ordinal data
Wilcoxon
signed rank test
Wilcoxon rank
s u m t e s t , MannWhitney t e s t
McNemar’s t e s t W i l c o x o n
signed rank test
Chi square
Kruskal -Wallis
test
one way
analysis of
variance
Friedman
matched
samples
Pa ra metri c tests
Ordinal, interval,
ratio data
O n e g r o u p t-test
Student’s t-test
Paired Student’s
t-test
A N O V A
A N O V A with
repeated
measurements
Types of Parametric tests
1. Large sample tests
 Z-test
2. Small sample tests
 t-test
* Independent/ unpaired t-test
* Paired t-test
ANOVA (Analysis of variance)
* One way ANOVA
* Two way ANOVA
41
Z test:
• It is used to test the null hypothesis for a single sample
when the population variance is known.
• A z-test is used for testing the mean of a population versus
a standard, or comparing the means of two populations, with
large (n ≥ 30) samples whether you know the population
standard deviation or not
• It is used to judge the significance of several statistical
measures ,particularly mean.
• It compares a sample mean with the sampling distribution,
i.e the sample is part of the sampling distribution
• It is also used for testing the proportion of some characteristic
versus a standard proportion, or comparing the proportions of
two populations.
Ex. Comparing the average engineering salaries of men
versus women.
Ex. Comparing the fraction defectives from two production
lines.
44
Formula in Computing the Test Statistic Using Z Test (Two Sample
Mean Test)
• when the given means
are sample means.
𝒙₁ − 𝒙₂
𝒛=
𝒔₁&sup2; 𝒔₂&sup2;
𝒏₁ + 𝒏₂
• when the given means
are population means.
𝒛=
𝝁₁ − 𝝁₂
𝝈₁&sup2; 𝝈₂&sup2;
+
𝒏₁ 𝒏₂
𝒙₁ = mean of the 1st sample
𝒙₂ = mean of the 2nd sample
𝝁₁ = mean of the 1st population
𝝁₂ = mean of the 2nd population
𝐬₁ = standard deviation of the 1st
sample
𝐬₂ = standard deviation of the 2nd
sample
𝝈₁ = standard deviation of the 1st
population
𝝈₂ = standard deviation of the 2nd
population
𝒏₁ = size of the 1st sample or
population
𝒏₂ = size of the 2nd sample or
population
One tailed Z test:
• A directional test in which a prediction is made that the
population represented by sample is either below or
above the general population
Ha: μ 0 &lt; μ 1 or Ha: μ 0 &gt; μ 1
Two tailed Z test:
• A non directional test in which a prediction is made that
the population represented by sample will differ from the
general population,but thre direction of the difference is
not predicted
Ha: μ 0 ≠ μ 1
Example
•
Ho: Children who learn whole language approach do not statistically
significantly differ from the average child in word recognition (&micro; = 75%, σ =
5%).
In symbols: Ho: &micro; = 75%.
H1: Children who learn whole language approach statistically significantly differ
from the average child with respect to word recognition (&micro; = 75%, σ = 5%).
In symbols: H1: &micro; ≠ 75%.
•α = 0.05, thus the critical values (C.V.) are &plusmn; 1.96. Sample
mean= 78% Population mean = 75% σ = 5% n = 50
Z=
x 
.
= 78 - .75 = 0.03
= 4.24(This is the test statistic which is a z –
.05 /√50 .05 /√50
score (unit: standard deviation
n
We reject the null hypothesis and conclude that children who learn the whole
language statistically significantly differ from the average child in word
recognition, z = 4.24, p &lt; .05.

t test: Derived by W S Gosset in 1908
• It is based on t-distribution
• It is the indicator of the no. of standard deviation units
the sample mean is from the mean of the sampling
distribution
• Used to judge the significance of a smaple mean or for
judging the difference b/w the means of 2 samples in
case of small sample(usually &lt; 30) when population
variance is not known,
• Properties of t distribution:
i. It has mean 0
ii. It has variance greater than one
iii. It is bell shaped symmetrical distribution about mean
• Assumption for t test:
i. Sample must be random, observations independent
ii. Standard deviation is not known
iii. Normal distribution of population
Uses of t test:
i.
The mean of the sample
ii. The difference between means or to compare two samples
iii. Correlation coefficient
Types of t test:
a. Paired t test
b. Unpaired t test
Paired t test:
• Consists of a sample of matched pairs of similar units, or one
group of units that has been tested twice (a &quot;repeated
measures&quot; t-test).
• Ex. where subjects are tested prior to a treatment, say for
high blood pressure, and the same subjects are tested again
after treatment with a blood-pressure lowering medication
Unpaired t test:
• When two separate sets
of independent and identically
distributed samples are obtained, one
from each of the two populations
being compared.
• Ex: 1. compare the height of girls and
boys.
2. compare 2 stress reduction
interventions
when one group practiced
mindfulness meditation while the other
learned progressive muscle
relaxation.
One tailed t test:
• A directional test in which a prediction is made that the
population represented by sample is either below or
above the general population
Ha: μ 0 &lt; μ 1 or Ha: μ 0 &gt; μ 1
Two tailed t test:
• A non directional test in which a prediction is made that
the population represented by sample will differ from the
general population, but the direction of the difference is
not predicted
Ha: μ 0 ≠ μ 1
ANOVA
• Prof R. A fisher was the first to use the term variance and
developed a theory concerning ANOVA
• ANOVA (Analysis of Variance) compares the means of two or
more parametric samples.
• It tests the difference among different groups of data for
homogenity
• Basic principle of ANOVA is to test for differences among the
means of the populations by examining the amount of variation
within each of these samples, relative to the amount of variation
b/w samples.
• The statistic for ANOVA is called the F statistic, which we get
from the F Test
F = Estimate of population variance based on b/w sample variance
Estimate of population variance based on within sample variance
• If we take one factor and investigate the differences amongst it
various categories we use one way ANOVA
• In case we investigate 2 factors at the same time ,then we use two
way ANOVA
• The ANOVA test has 2 degrees of freedom:
– N-I (Total number sampled – Number of Groups)
– I-1 (Number of Groups – 1)
• Assumption for ANOVA test:
i. Normal distribution of population
ii. 3 or more groups, iii. Variables are independent
iv. Data is interval or ratio, v. Homogenity of variance
Difference between one &amp; two way ANOVA
• An example of when a one-way ANOVA could be used is if
we want to determine if there is a difference in the mean
height of stalks of three different types of seeds. Since there
is more than one mean, we can use a one-way ANOVA
since there is only one factor that could be making the
heights different.
• Now, if we take these three different types of seeds, and
then add the possibility that three different types of fertilizer
is used, then we would want to use a two-way ANOVA.
• The mean height of the stalks could be different for a
combination of several reasons
• The types of seed could cause the change,
the types of fertilizer could cause the change, and/or there is
an interaction between the type of seed and the type of
fertilizer.
• There are two factors here (type of seed and type of
fertilizer), so, if the assumptions hold, then we can use a
two-way ANOVA.
Summary of parametric tests applied for different
type of data
Sl no
Type of Group
Parametric test
1.
Comparison of two paired groups
Paired ‘t’ test
2.
Comparison of two unpaired groups
Unpaired ‘t’ test
3.
Comparison of three or more matched groups
Two way ANOVA
4.
Comparison of three or more matched groups
One way ANOVA
5.
Correlation between two variables
Pearson correlation
60
Commonly used non parametric tests
• Commonly used Non Parametric Tests are:
− Chi Square test
− The Sign Test
− Wilcoxon Signed-Ranks Test
− Mann–Whitney U or Wilcoxon rank sum test
− The Kruskal Wallis or H test
− Friedman ANOVA
− The Spearman rank correlation test
− Cochran's Q test
Chi Square test
• First used by Karl Pearson
• Simplest &amp; most widely used non-parametric test in statistical
work.
• Calculated using the formulaχ2
= ∑ ( O – E )2
E
O = observed frequencies
E = expected frequencies
• Greater the discrepancy b/w observed &amp; expected frequencies,
greater shall be the value of χ2.
• Calculated value of χ2 is compared with table value of χ2 for
given degrees of freedom.
Chi Square test
• Application of chi-square test:
– Test of association (smoking &amp; cancer, treatment &amp; outcome of
disease, vaccination &amp; immunity)
– Test of proportions (compare frequencies of diabetics &amp; nondiabetics in groups weighing 40-50kg, 50-60kg, 60-70kg &amp; &gt;70kg.)
– The chi-square for goodness of fit (determine if actual numbers are
similar to the expected/theoretical numbers)
Chi Square test
• Attack rates among vaccinated &amp; unvaccinated children
against measles :
Group
Result
Total
Attacked
Not-attacked
Vaccinated
(observed)
(a)10
(b) 90
(a+b)100
Unvaccinated
(observed)
(c) 26
(d) 74
(c+d) 100
Total
(a+c) 36
(b+d) 164
200
• Prove protective value of vaccination by χ2 test at 5% level of
significance
Chi Square test
Group
Total
Result
Attacked
Not-attacked
Vaccinated
(Expected)
18
82
100
Unvaccinated
(Expected)
18
82
100
Total
36
164
200
Chi Square test
 χ2 value = ∑ (O-E)2/E
 (10-18)2 + (90-82)2 + (26-18)2 +(74-82)2
18
82
18
82
64 + 64
+ 64
+ 64
18
82
18
82 =8.67
 calculated value (8.67) &gt; 3.84 (expected value corresponding to P=0.05)
 Direct formula = (ad-bc)2 * N
(a+b)(c+d)(a+c)(b+d)
 Null hypothesis is rejected. Vaccination is protective.
• Yates’ correction: applies when we have two categories (one degree of
•
•
•
freedom)
Used when sample size is ≥ 40, and expected frequency of &lt;5 in one cell
Subtracting 0.5 from the difference between each observed value and its expected
value in a 2 &times; 2 contingency table
χ2 = ∑ [O- E-0.5]2
E
The Chi-Square Test for Goodness-of-Fit (cont.)
• The null hypothesis specifies
the proportion of the
population that should be in
each category.
• The proportions from the null
hypothesis are used to
compute expected
frequencies that describe
how the sample would
appear if it were in perfect
agreement with the null
hypothesis.
The Chi-Square Test for Independence
• The second chi-square test, the chi-square test for independence,
can be used and interpreted in two different ways:
1. Testing hypotheses about the relationship between two
variables in a population, or
2. Testing hypotheses about differences between proportions
for two or more populations.
Sign Test
•
•
•
•
•
•
Used for paired data, can be ordinal or continuous
Simple and easy to interpret
Makes no assumptions about distribution of the data
Not very powerful
To evaluate H0 we only need to know the signs of the differences
If half the differences are positive and half are negative, then the
median = 0 (H0 is true).
• If the signs are more unbalanced, then that is evidence against H0.
Sign Test
– Children in an orthodontia
study were asked to rate how
they felt about their teeth on a 5
point scale.
– Survey administered before and
after treatment.
How do you feel about your
teeth?
1. Wish I could change
them
2. Don’t like, but can put up
with them
3. No particular feelings one
way or the other
4. I am satisfied with them
5. Consider myself fortunate
in this area
child
Rating
before
Rating
after
1
1
5
2
1
4
3
3
1
4
2
3
5
4
4
6
1
4
7
3
5
8
1
5
9
1
4
10
4
4
11
1
1
12
1
4
13
1
4
14
2
4
15
1
4
16
2
5
17
1
4
18
1
5
19
4
4
20
3
5
• Use the sign test to evaluate
whether these data provide
evidence that orthodontic
treatment improves children’s
image of their teeth.
child
Rating
before
Rating
after
change
1
1
5
4
2
1
4
3
3
3
1
-2
4
2
3
1
5
4
4
0
6
1
4
3
difference between the two
7
3
5
2
ratings
8
1
5
4
9
1
4
3
10
4
4
0
11
1
1
0
12
1
4
3
13
1
4
3
14
2
4
2
15
1
4
3
16
2
5
3
17
1
4
3
18
1
5
4
19
4
4
0
20
3
5
2
• First, for each child, compute the
child
Rating
before
Rating
after
change
sign
1
1
5
4
+
2
1
4
3
+
3
3
1
-2
-
4
2
3
1
+
5
4
4
0
0
6
1
4
3
+
7
3
5
2
+
8
1
5
4
+
9
1
4
3
+
10
4
4
0
0
11
1
1
0
0
12
1
4
3
+
13
1
4
3
+
14
2
4
2
+
15
1
4
3
+
16
2
5
3
+
17
1
4
3
+
18
1
5
4
+
19
4
4
0
0
20
3
5
2
+
• The sign test looks at the signs
of the differences
– 15 children felt better
about their teeth (+
difference in ratings)
– 1 child felt worse (- diff.)
– 4 children felt the same
(difference = 0)
• If H0 were true we’d expect an
equal number of positive and
negative differences.
(P value from table 0.004)
Wilcoxon signed-rank test
• Nonparametric equivalent of the paired t-test.
• Similar to sign test, but take into consideration the magnitude of difference
among the pairs of values. (Sign test only considers the direction of difference
but not the magnitude of differences.) For eg
• The 14 difference scores in BP among hypertensive patients after giving drug
A were:
-20, -8, -14, -12, -26, +6, -18, -10, -12, -10, -8, +4, +2, -18
• The statistic T is found by calculating the sum of the positive ranks, and the
sum of the negative ranks.
• The smaller of the two values is considered.
74
Wilcoxon signed-rank test
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Score
+2
+4
+6
-8
-8
-10
-10
-12
-14
-16
-18
-18
-20
-26
Rank
1
2
3
4.5
4.5
6.5
6.5
8
9
10
11.5
11.5
13
14
Sum of positive ranks = 6
Sum of negative ranks = 99
T= 6
For N = 14, and α = .05, the critical
value of T = 21.
If T is equal to or less than T
critical, then null hypothesis is
rejected i.e., drug A decreasesthe
BP among hypertensive patients.
Mann-Whitney U test
• Mann-Whitney U – similar to Wilcoxon signed-ranks test except
that the samples are independent and not paired.
• Null hypothesis: the population means are the same for the two
groups.
• Rank the combined data values for the two groups. Then find the
average rank in each group.
• Then the U value is calculated using formula
• U= N1*N2+ Nx(Nx+1) _ Rx (where Rx is larger rank
total)
2
• To be statistically significant, obtained U has to be equal to or
LESS than this critical value.
Example
• 10 dieters following Atkin’s diet vs. 10 dieters following Jenny Craig diet
Hypothetical RESULTS:
• Atkin’s group loses an average of 34.5 lbs.
• J. Craig group loses an average of 18.5 lbs.
• Conclusion: Atkin’s is better?
• When individual data is seen
• Atkin’s, change in weight (lbs):
+4, +3, 0, -3, -4, -5, -11, -14, -15, -300
•J. Craig, change in weight (lbs)
-8, -10, -12, -16, -18, -20, -21, -24, -26, -30
• RANK the values, 1 being the least weight loss and 20 being the most
weight loss.
• Atkin’s
– +4, +3, 0, -3, -4, -5, -11, -14, -15, -300
– 1, 2, 3, 4, 5, 6, 9, 11, 12, 20
• J. Craig
− -8, -10, -12, -16, -18, -20, -21, -24, -26, -30
− 7, 8, 10, 13, 14, 15, 16, 17, 18, 19
• Sum of Atkin’s ranks: 1+ 2 + 3 + 4 + 5 + 6 + 9 + 11+ 12 + 20=73
• Sum of Jenny Craig’s ranks: 7 + 8 +10+ 13+ 14+ 15+16+ 17+
18+19=137
• Jenny Craig clearly ranked higher.
• Calculated U value (18) &lt; table value (27), Null hypothesis is rejected.
Kruskal-Wallis One-way ANOVA
• It’s more powerful than Chi-square test.
• It is computed exactly like the Mann-Whitney test, except that
there are more groups (&gt;2 groups).
• Applied on independent samples with the same shape (but not
necessarily normal).
Friedman ANOVA
• Friedman ANOVA: When either a matched-subjects or repeatedmeasure design is used and the hypothesis of a difference
among three or more (k) treatments is to be tested, the Friedman
ANOVA by ranks test can be used.
Spearman rank-order correlation
• Use to assess the relationship between two ordinal variables or
two skewed continuous variables.
• Nonparametric equivalent of the Pearson correlation.
• It is a relative measure which varies from -1 (perfect negative
relationship) to +1 (perfect positive relationship).
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