Hypothesis testing
Chapter 6
Statistics for Marketing & Consumer Research
Copyright © 2008 - Mario Mazzocchi
1
Sampling and inference
1)
2)
3)
4)
5)
Exploit the probabilistic nature of sampling to show how
statements on the population parameters can be based
on sample estimates
Consider the degree of uncertainty (level of confidence)
in such an operation
Build a confidence interval, which is a range of values
around the sample mean which is expected to include the
true population mean at a given probability (confidence)
level
Compute the probability associated with a given
statement on the population parameters (or parameters
from different population)
Decide whether the statement is false depending on its
probability level (hypothesis testing)
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2
Direct and indirect problems
• Direct problem – If we knew the population mean and
the standard error then we would know the exact
probability of any sample mean
• Indirect problem –
• Confidence interval: Given the sample mean, we can
exploit the normal probability curve to find a range of
value which will contain the true population mean at a
given confidence level
• Hypothesis testing: Given the sample mean, we can
exploit the normal probability curve to check the
probability of an hypothesis on the true mean – then we
can decide whether to discard or retain that hypothesis
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3
Example
• Extract a simple random sample of 100 units out of
a population of 400 pub customers
• Measure the average expenditure on beer on a
sample
• The number of potential samples is huge
(2.24*1096) and we observe only one sample
• However, if extraction is random, the probability
distribution of all sample means is a normal curve
centered around the true population mean
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The normal distribution of the sample
means
• Once more the normal distribution...
• The central point is the
true population mean and
is the most likely sample
mean
• The larger the standard
error the flatter the curve
95% of
probability
2.5% of
probability
2.5% of
probability


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
• 95% of all possible
sample means fall within a
range of about two
standard errors from the
mean (1.96 to be precise)
5
Indirect problem
• We have only one sample mean (x-bar)
• 95% of sample means fall within the range
[ 1.96  ;  1.96  ]
• If 100% of the sample means were in that range we
could state with certainty that the true mean falls
in the range
[ x 1.96  ; x 1.96  ]
• However, we can affirm that only for 95% of the
sample means, thus we refer to a 95% confidence
level
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6
Confidence level
x  1.96 
95% of
probability
x  1.96 
2.5% of
probability
2.5% of
probability



If
the singleONLY
sample
we
extractMEANS
falls inFALL
this range,
the–
HOWEVER,
95%mean
OF THE
SAMPLE
IN THISthen
RANGE
true population
THUS
WE CAN STATE
meanTHAT
is also
THEincluded
TRUE POPULATION
in the interval
MEAN
between
IS
INCLUDED
THE
AND
x  1.96BETWEEN
x  1.96IN
x  1.96
x  1.96
 .
andINTERVAL
In fact, even
ifwe
get one
of the
.


WITH
A 95%itCONFIDENCE
extremes,
will still be atLEVEL
a distance of 1.96 from 
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Confidence levels and critical values
• The critical value 1.96 corresponds to a
probability of 95%, but it is possible to know
exact critical values for any confidence level
• For example, if we want a 99% confidence
level, the critical value based on the normal
curve is 2.58
• Most packages including Microsoft Excel
allow computation of critical values
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8
A further complication
• The standard error of the mean  is unknown
(sample estimate)
• This adds some further uncertainty on the top of
the sampling error
• Hence, we use an approximation of the normal
distribution which is more conservative, is flatter
and assigns higher probabilities to the tails
(extreme values) compared to the normal
distribution
• This distribution is the so-called Student-t
distribution and its critical values are different
from those of the normal distribution
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9
The Student-t distribution
0.4
Normal distribution
t(20)
Dotted lines show the
Student t distribution with
different degrees of
freedom (1, 5 and 20)
t(5)
0.35
0.3
t(1)
The bold line represents
the standard normal
distribution (with mean zero
and standard deviation
equal to one).
0.25
0.2
0.15
The degrees of freedom
are equal to the sample
size minus one.
0.1
0.05
0
-4
-3
-2
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-1
0
1
2
3
4
10
Critical values and sample size
Level of
confidence
99%
95%
90%
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Student t values
Normal
value
t according to sample
size
z
10
20
30
40
3.17
2.23
1.81
2.85
2.09
1.72
2.75
2.04
1.70
2.70
2.02
1.68
2.58
1.96
1.64
11
How to build a confidence interval from
a sample
1. Compute the sample mean x-bar and standard
deviation s
2. Estimate the standard error of the mean sx
3. Choose a confidence level a
4. Choose the appropriate coefficient for critical
values (see previous slide), using the Student t
approximation instead of the Normal (z) value if
the sample size is below fifty
5. Compute the lower and upper confidence limits
as:
 x  za / 2sx ; x  za / 2sx 
 x  ta / 2sx ; x  ta / 2sx 
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for the Normal distribution or
for the Student t approximation
12
SPSS confidence intervals
Click here
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13
Confidence interval
Descriptives
In a typical week how
much do you spend
on fresh or frozen
chicken (Euro)?
Mean
95% Confidence
Interval for Mean
5% Trimmed Mean
Median
Variance
Std. Deviation
Minimum
Maximum
Range
Interquartile Range
Skewness
Kurtosis
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Lower Bound
Upper Bound
Statistic
5.6677
5.2817
6.0537
5.2958
5.0000
17.089
4.13383
.00
30.00
30.00
4.50
2.084
8.005
Std. Error
.19640
Boundaries of
the c.i.
.116
.231
14
Hypothesis testing (HT)
• HT is a form of inference
• HT is a statistical tool to decide whether to
reject or not a statement about the target
population on the basis of statistics
computed on sample
• This is only possible if the sampling
distribution is known!
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Example (Trust data-set)
• The sample mean expenditure for chicken is £ 5.67
• Considering only the sub-sample of those belonging to
consumer organizations the mean is £ 5.04
• May we safely conclude that those who belong to consumer
organizations spend less?
• This depends on the precision of the estimates.
• the upper limit of the 99% confidence interval of the sub-sample of
respondents associated to consumer organizations is 6.84
• the upper limit of the confidence interval for the overall sample is
6.18
• There is a chance that those who belong to consumer organization
actually spend more than other people
• Statistical tests based on the sample help in deciding
whether the hypothesis should be rejected; for example,
that there is no difference between respondents who
belong to a consumer organization and those who don’t.
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Hypothesis testing
• The difference between the two means follows a known
statistical distribution
• If the hypothesis of mean equality is true, the difference
between the two means should be zero
• The actual difference is it is £ 0.63 (£ 5.67 minus £ 5.04), but
this difference may be generated by sampling error
• When no difference exists at the population level, what is the
probability that a sample with a difference of £ 0.63 is
extracted?
• If the probability is very high (say 90%) then it is wise not to reject the
hypothesis to equality
• Instead, a very low probability (e.g. 0.001%) means that it is
very unlikely that the difference is due sampling error, so one
should choose to reject the null hypothesis and conclude that
belonging to a consumer organization actually makes a
difference on chicken expenditure.
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The hypotheses
• The initial hypothesis is called the null
hypothesis, denoted by H0
• Contextually, the researcher sets an
alternative hypothesis (H1) which is
complementary to H0 and remains valid if H0 is
rejected.
• Two-tailed tests are those where the
alternative hypothesis can go in either
direction
• When the alternative hypothesis is formulated
in a unique direction (for example >6), the
test is one-tailed
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Significance and confidence
• The threshold probability level below which the null
hypothesis is rejected is the significance level
arbitrarily set by the researcher (usually at the 5% or
1% level).
• It is denoted by a and its complementary 1-a is the
confidence level of a test
• The smaller the significance level the smaller the
rejection region, (which is the set of values that leads
to rejection of the hypothesis) and the larger is the
acceptance region, (which is the set of values that lead
to non-rejection of the hypothesis)
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Errors confidence and power
The statistical power of a test is the probability of correctly
rejecting the null hypothesis when it is false and it is equal to
1-b. It can be estimated on sample data and depends on
• sample size
• significance level a
• effect size (a measure of “how wrong” is the null hypothesis)
When the null hypothesis is not rejected and power is above
80% then the conclusion is usually regarded as robust.
Reject H0
Non-reject H0
H0 is true
Type I Error
prob. =significance level (a)
Correct
prob. = confidence level (1-a)
H0 is false, H1 is true
Correct
prob. = power level (1-b)
Type II Error
prob. = b
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Hypothesis testing
1. Formulate the null hypothesis (H0) and the alternative
hypothesis (H1)
2. Determine the distribution of the sample test statistic
under H0
3. Choose a significance level a (i.e. a confidence level 1-a)
4. Compute the sample test statistic and its probability level
(p-value)
5. When p-value<a, then reject H0
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One mean test
• The null hypothesis is made on the population
mean
• Example:
• UK Department of Environment, Food and Rural Affairs
(DEFRA) figure on average weekly household chicken
consumption: 1.15 Kg
• UK household average in the Trust data-set: 1.75 Kg on
92 respondents
• The DEFRA figure is likely to be more representative; we
test the hypothesis that the population mean is actually
1.15
Null H0:  = 1.15 versus the alternative H1:  ≠ 1.15
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Probability distribution
• As for confidence intervals, the sample means
probabilities follow the normal distribution
• Under the null hypothesis H0, the sample mean is
to be distributed as a normal with mean 1.15 and
unknown variance
• The sample standard error of the mean is 0.28
• With a sample size of 92 we can use the normal
curve (with less than 50 observations we should
have referred to the Student t distribution)
• We set the significance level a=0.05 (which means
that the confidence level is 95%)
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Testing the hypothesis on a mean
(unknown variance)
• The test statistic is built as follows:
Sample mean
x  0
t=
sx
Population mean
under H0
Under H0 the standardized test statistic
follows the Standard Normal distribution
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Standard error of the
mean (estimated
through the sample)
24
Critical values
The test statistic is the sample mean, which need to be
standardized:
x   1.75  1.15
z=
=
= 2.14
sx
0.28
• This value needs to be compared with the critical values
which separate the acceptance and the rejection regions
• This is a two-tailed test, since the alternative hypothesis is
formulated in both direction and since  ≠ 1.15 holds for
either  < 1.15 or  > 1.15.
• Critical values are obtained like for confidence intervals:
-za/2 defines the left rejection region (negative values < 2.5% probability)
+za/2 defines the right rejection region (positive values < 2.5% probability)
• For a 5% significance level (or a 95% confidence level):
–z0.025=-1.96 and +z0.025=+1.96
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Acceptance and rejection areas
The test statistic
lies in the rejection
area
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Hypothesis testing in SPSS
Test value
here
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SPSS output
One-Sample Test
Test Value = 1.15
t
In a typical week how
much fresh or frozen
chicken do you buy
for your household
consumption (Kg.)?
2.183
df
91
Sig. (2-tailed)
Mean
Difference
.032
.60152
95% Confidence
Interval of the
Difference
Lower
Upper
.0541
1.1490
p value
The null hypothesis is rejected (as the p-value is smaller than 0.05)
• We reject the null hypothesis that the average
weekly consumption is Kg. 1.15
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One sided hypothesis
(one tailed test)
• We want to test whether average consumer evaluation of animal
welfare is larger than 4.9 (on a 7-point scale)
• It is convenient to formulate the hypothesis as follows: H 0 :   4.9
H1 :   4.9
• This is an one tailed test, as the alternative hypothesis is expressed
directionally: the rejection area lies on the right of the critical value
(all values on the left are consistent with the null hypothesis)
Sample mean: 5.01
5.01  4.90
z=
= 1.49
Standard deviation: 1.65
0.074
Standard error of the mean: 0.074
• instead of two critical values with a/2=0.025 as in the two-tailed test,
we only require a single critical value for a=0.05 (which corresponds to
the a=0.10 two-tailed critical values), which is 1.64
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One-tailed test
The test statistic
lies in the
acceptance area
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One mean one-tailed test in SPSS
• Exactly like two-tailed tests, but when interpreting the
output:
– One should only consider values larger than the positive critical
value for rejection (or smaller of the negative critical value if the
null direction is >)
– To get the correct critical value one should consider the a one
instead of a/2 (for example, z0.05 rather than z0.25)
– Thus, the critical value for a 5% significance level in a one-tailed
test corresponds to the critical value for a 10% two-tailed test. For
example z0.05=1.64
– Similarly, when looking at the two-tailed test p-value in SPSS, one
should consider a “double” threshold, i.e. reject the null at the 95%
level when p>0.10 (since SPSS always assumes two-tailed tests)
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SPSS output and one-sided test
One-Sample Test
Test Value = 4.9
Animal welfare
t
1.545
df
496
Sig. (2-tailed)
.123
Mean
Difference
.114
95% Confidence
Interval of the
Difference
Lower
Upper
-.03
.26
• The null hypothesis is not rejected at the 5% significance level
because Sig>0.10
• Note that differently from two two-tailed test here the
threshold is twice the chosen significance level.
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Test on two means
• Tests of equality on two means follow directly from the
single-mean test
• The difference of two normally distributed sample means is
still a normally distributed, provided that the samples are
unrelated or paired
• The mean of the difference distribution is zero under the
null hypothesis of mean equality
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Unrelated samples
• The sampled units belong to different populations
and are randomly extracted from each of the
population. The key condition is that the sampled
units are randomly assigned to the two groups
• This excludes the case where:
a) a single sample is drawn
b) the units are subdivided into two groups according to
some variable (gender, living in urban versus rural
areas, etc.)
c) the sampling process might have some selection bias
creating dependency between the two groups
• Most social studies consider the samples to be
unrelated if the units are randomly extracted and
the groups are mutually exclusive.
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Related and paired samples
• In related samples he same subjects may belong to both
groups.
• For example, if the same individual is interviewed in two
waves, the two samples are said to be related.
• Two sub-sets from the same sample are generally related
(not for stratified sampling)
• Paired samples are a special case where exactly the same
units appear in both samples
• In this case it is possible to compute the difference for
each of the sampled units and the result corresponds to a
single sample
• Matched samples are artificially paired samples, where
the two samples are matched according to some
characteristics.
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Example (Trust data-set)
• 71% of respondents are female
• the targeted population are people in charge of food
purchases; males and females in the Trust data-set are
associated by the fact that they are responsible for food
purchases
• This excludes those females and males who are not; any
gender comparison is conditional on this external factor
• We could not be conclusive in testing – for example –
whether males like chicken more than females
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Test for unrelated samples
• Example (Trust data-set)
• Italian versus UK respondents (extracted independently)
• Does the attitude towards chicken (question q9, “In my
household we like chicken”) differ between the two
countries?
H0: UK = ITA
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vs.
H1: UK ≠ ITA
37
Unrelated samples
• The two means are normally distributed
• Under the null hypothesis their difference is also normally
distributed
• Consider the difference variable D = UK - ITA
• The test becomes identical to the one mean test for D = 0
• However, a measure of the standard error for D is
necessary
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Standard error for
mean comparison testing
• In the (unlikely) event that the true standard
errors are known the joint standard error is:
   =
1
2
 12
n1

 22
n2
• Everything proceeds as for the one mean test, thus
the test statistic is:
x x
t=
1
  
1
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2
2
39
Test statistics (unknown standard
errors)
• With unknown but equal standard errors, the test statistics is
t=
x1  x2
1
1
sx

n1 n2
• Given that the standard error is estimated, additional
uncertainty requires the use of the t distribution with
n1+ n2-2 degrees of freedom
• With different standard errors the test statistic is
x1  x2
z=
sx2
sx2

n1
n2
• This statistic can only be applied to large samples and the
standard normal distribution is the reference
1
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40
How to decide whether the standard
errors are equal or different?
• There are appropriate hypothesis tests for the equality of
two variances (discussed later)
• SPSS shows the p-value for the Levene’s test (Brown and
Forsythe, 1974)
• SPSS provides the outcomes of both test, with and without
assuming equality of standard errors.
• In most cases these two tests provide consistent outcomes.
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Mean comparison with
unrelated samples
• The mean for the 100 UK respondents is 6.12 (standard
error 0.15)
• the mean for the 100 Italian respondents is 5.62 (standard
error 0.11)
• Is a mean difference of 0.5 significantly different from 0?
• Are the standard error equal?
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SPSS example
Target variable
here
Sub-groups defined here
through a grouping variable
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SPSS output
The
null hypothesis of equality of variance (Levene’s test) is not
CONCLUSION:
rejected
thea5%
s.l. (as p-value
is larger thandifference
0.05), thus the two
Thereatis
statistically
significant
standard errors could be regarded as equal
between the UK and Italy in terms of attitudes
towards chicken (as measured by q9)
Independent Samples Test
Levene's Test for
Equality of Variances
F
In my household
we like chicken
Equal variances
assumed
Equal variances
not assumed
Sig.
.243
.622
t-test for Equality of Means
t
df
Sig. (2-tailed)
Mean
Difference
Std. Error
Difference
95% Confidence
Interval of the
Difference
Lower
Upper
2.682
198
.008
.500
.186
.132
.868
2.682
183.426
.008
.500
.186
.132
.868
At any rate, the null hypothesis is rejected in both cases
(as the p-value is smaller than 0.05)
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Paired samples
• As a second case study, consider the situation where two
measures are taken on the same respondents.
• For example, all respondents were asked a second question
on their general attitude towards chicken, “A good diet
should always include chicken” (q10)
• Do the two questions measure the same item (general
attitude towards chicken)?
• Can we assume that the results are – on average – equal?
• In this case the samples are paired and it is possible to
compute a difference between the response to q9 and q10
for each of the sampled household.
• Everything goes back to one mean test discussed earlier
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Paired samples
• One may compute a third variable for each of the
respondent as the difference between q9 and q10, then
compute the mean and the standard error for this variable
• SPSS does this automatically
–
–
–
–
mean of 5.73 for q9 versus a mean of 5.50 for q10
average difference 0.23
estimated standard error 0.06.
The t test statistic is 3.84, largely above the two-tailed 99% critical
value,
• the null hypothesis of mean equality should be rejected.
• It is not safe to assume that the two questions are
measuring the same construct
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Paired samples
Select two variables from the
same data-set
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Output
Paired Samples Test
Paired Differences
Mean
Pair
1
In my household we like
chicken - A good diet
should include chicken
•
.231
Std. Deviation
1.348
Std. Error
Mean
.060
95% Confidence
Interval of the
Difference
Lower
Upper
.112
.350
t
3.824
df
Sig. (2-tailed)
497
.000
Note how mean comparison tests can be used in the post-editing phase to
check consistency (see lecture 4):
1) Try again with q9 and q12e, the two questions are targeted to measure
the same construct with slightly different wording
2) If the null hypothesis is not rejected, one can compute the difference
between q9 and q12 and look for outliers
3) Cases with outliers show an inconsistency
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Other type of tests
• Independent samples
• Standard t-test (as seen)
• Paired samples
• It becomes a one-sample t-test
• Related samples
• Non-parametric tests
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Related samples tests and
non-parametric statistics
• Do not require knowledge of the underlying distributions
(see Gibbons, 1993 for a good introduction to nonparametric testing).
• Non-parametric tests are also used in situations where the
variables to be tested are qualitative
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Non-parametric testing (NPT)
• Parametric methods assume knowledge of the distribution
(usually normal) and its parameters
• Problems: different distributions (income), small sample
sizes, etc.
• NPT – No prior assumption on the distribution (and its
parameters)
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Some non-parametric tests
One group
• Runs (Wald-Wolfowitz)
• Kolmogorov-Smirnov
• Chi-square test
• Binomial test
• Wilcoxon signed-rank
Two groups
• Runs (Wald-Wolfowitz) on two groups
• Mann-Whitney test (Wilcoxon-rank-sum test)
• Kolmogorov-Smirnov on two groups
• Wilcoxon paired sample test (on two paired samples)
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Runs test
• On one group
• Check whether the order in the sample is
random
• Check whether observations below and above a
cut-off point (e.g. median) follow a random
order
• On two groups
• Check whether observations from two groups
follow a random order (after being sorted
according to some variable)
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One-group tests
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Runs test with a single group
•
NULL HYPOTHESIS: the cases in a sample are ordered in a
random fashion
ALTERNATIVE HYPOTHESIS: the order is not random
CRITERION:
•
•
•
•
•
SEQUENCE:
•
•
•
•
A dichotomous variable (e.g. gender) OR
A cut-off point for a metric variable (e.g. income) which
generates a dicotomous variable
111111111122222222221111111111 unlikely to be random
112122112122121211212212111212 more likely to be random
Concentrations and sequences of ones and twos highlight nonrandomness
For large samples, it is possible to test for randomness in the
sequence using the normal distribution.
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Kolmogorov-Smirnov
• The null hypothesis is that a random sample is drawn from
a (user specified) given distribution
• SPSS allows to test whether a variable is distributed
according to a normal distribution, a Poisson curve, a
uniform or an exponential distribution
• Once the distribution and its parameters are known, it
becomes possible to estimate the population parameters
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Chi-square test
• Compare the frequency of the observed distribution with
the expected frequency from a theoretical distribution
(e.g. the normal curve)
• The more different are the frequencies, the less likely it is
that the empirical observations come from the theoretical
distribution
• The Chi-square statistics synthesizes the distance between
the observed and expected frequencies and is compared to
critical values from the Chi-square distribution
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Binomial test
• Applied to dichotomous variables, e.g. gender (males and
females)
• NULL HYPOTHESIS: the proportion of males and females in
the sample is imputable to sampling error only
• E.g. if we get 71% females and 29% males in the trust dataset, could this depend on sampling error only?
• Assuming a 50%-50% true distribution between males and
females, a 71-29 outcome leads to rejection of the null
hypothesis
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Wilcoxon signed rank test
• Non-parametric correspondent of the one-mean t-test
• NULL HYPOTHESIS: the median (or the difference between
two medians) is equal to some specified value, provided
that the distribution is symmetric
• CRITERION: based on ranking, which consists of assigning
increasing discrete values one, two, etc. to the cases once
they have been sorted in ascending order according to the
variable to be tested
• The observations which differ from the hypothesized
median are ranked according to their distance from the
median
• Ranks above and below the assumed median are summated
to build the test statistic
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Two groups tests
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Runs test with two groups
1.
2.
3.
4.
NULL HYPOTHESIS: the cases in a dataset are extracted from
two independent samples from the same population
ALTERNATIVE HYPOTHESIS: the samples are not independent
OR they do not belong to the same population
It corresponds to a mean comparison test, but it tests whether
they belong to the same population
CRITERION:
1)
2)
3)
5.
6.
7.
Merge the samples into a single data-set and
Order the variables according to a metric variable and
Proceed as for one group, using a dichotomous variable or a cutoff point
SEQUENCE: (as before)
111111111122222222221111111111 unlikely to be random
112122112122121211212212111212 more likely to be
random
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Mann-Whitney test
• Also known as the Wilcoxon-rank-sum test
• ASSUMPTION: the two samples are random and come from
population that follow the same distribution apart from a
translation k
• NULL HYPOTHESIS: k=0 (which means that the samples are
extracted from the same population)
• The null here is stronger than the null of mean equality
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Example (Trust) (1)
•
•
•
Trust in the European food safety authority for French
(F) and German (G) respondents (measured on a one to
seven scale)
Trust of Germans = Trust of French + k
NULL HYPOTHESIS H0: k= 0
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Trust (2)
1.
2.
3.
4.
5.
All cases are ranked by trust level independently from
the country and a rank number is assigned
Ranks for each group (F and G) are summated
The sums are compared (allowing for different group
sizes) through the U statistic
The U statistic is based on the frequency with whom the
first sample has a higher rank than the second sample
If the two samples come from the same distribution the
frequency should be random (similarly for the Run test)
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Other tests
Kolmogorov-Smirnov for two groups
• Two distributions are compared, instead of
simply comparing the theoretical
distribution with the empirical one
Wilcoxon paired sample test
• Wilcoxon signed rank test on the median
difference between the two samples
provided that the differences are
symmetrically distributed
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Non-parametric tests in SPSS
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Other tests
• Test on proportions
• Proceed like for means
• For large samples they correspond to test on
means (use the normal distribution)
• Test on variances
• F-test for the equality of variances
• Levene’s F-test
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Test on variances: F distribution
Under the null hypothesis of variance equality, if the two samples are
extracted from normal distributions or the sample sizes are large enough,
the ratio between two variances follows a distribution like this:
The F-distribution is characterized by two
values for degrees of freedom:
Number of obs of the first variance – 1
Number of obs of the second variance – 1
Notation: F(n1-1;n2-1)
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F-test
• Note that the F distribution only includes positive value; it is nonsymmetrical and for two-tailed tests the critical values are defined
in a slightly different manner when compared to the t and normal
distributions
• For a given significance level a,the critical value for the right
rejection region (the first variance is larger than the second) is
denoted by Fa/2 as usual, since we want to exclude F values larger
than Fa/2 because their probability is below a/2.
• Instead, the critical value for the left rejection area is denoted as
Fa/2 , since we set the probability that F is larger than the critical
value to a/2, which means that the probability that F is smaller
than the critical value is actually a/2, as desired.
• This difference is also relevant to one-tailed test where the
alternative hypothesis is 1 < 2, where the critical value will be Fa
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Example (Trust data)
•
Is the variance in chicken consumption for males
equal to the variance for females?
H0: M = F
vs
H1: M ≠ F
• Variance males: 0.87
• Variance females: 2.60
• F-statistic: 0.87/2.60=0.33
• Critical values:
F0.975(129,313)=0.74
F0.025(129,313)=1.33
• The null hypothesis is rejected
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Other tests on variances
• The standard F test is valid under the
assumption of normal distribution of the
populations
• Generalization for non-normal populations
(SPSS and SAS):
• Levene’s test for homogeneity of variances
(slightly different F statistic).
• Levene’s test can be biased by the presence of
outliers
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