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CHAPTER 6 COMPARING SAMPLES (WHEN
NORMAL DISTRIBUTIONS CANNOT BE USED)
• This chapter introduces:
– the ways for comparing the means
from different samples where
distributions are not matching a
normal distribution.
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PARAMETRIC AND NONPARAMETRIC TESTS
• Parametric tests assumes that the samples are drawn from
predictable (and often Normal) population .
• In non-parametric test, we do not assume any particular distribution
and is commonly used for categorical data.
• Advantages of non-parametric tests
– They do not require us to make the assumption that a population
is distributed in the shape of a normal curve or another specific
shape
– Generally easier to compute and understand
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PARAMETRIC AND NONPARAMETRIC TESTS
• Disadvantages of non-parametric tests
– They ignore a certain amount of information
Example: by replacing values with rankings
– They are often not as efficient or “sharp” as parametric tests
The estimate of an interval at the 95% confidence level using
a non-parametric test may be twice as large as the estimate
using a parametric test
Trade off: lose sharpness in estimating intervals but gain the
ability to use less information and to calculate faster.
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INDEPENDENT SUBJECTS &
REPEATED / MATCHED SUBJECTS
• Before carrying out statistical tests, we need to know whether the 2
samples to be compared contain independent subjects or repeated /
matched subjects.
– The statistical formula for each case is different
•
Independent subjects
– One group of cases or subjects forms one sample and a
completely different group forms the second sample
– Subjects are randomly allocated to each group
– Eg planning applications from different local authorities
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INDEPENDENT SUBJECTS &
REPEATED / MATCHED SUBJECTS
• Repeated measures
– Each subject is measured twice
– Example in both control and experimental condition, as before and after
treatment
• Matched pairs
– On the basis of pre-selection, subjects are sorted into matched pairs on
the variable to be measured.
– Example pairs of equal ability
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STATISTICAL TESTS FOR
DIFFERENT SITUATIONS & DATA
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TESTING FOR SIGNIFICANCE
Step 1 NULL HYPOTHESIS, H0
• Assumption: 2 samples belong to the same population no real
difference between them. Observed differences (if any) are random
and is to be expected
Step 2
• Carry out statistical test to get a value for the statistic
Step 3
• Look up statistical tables for no. of subjects and find the value of the
statistic (corresponding to the significance level decided to use,
normally p = 5% or p = 1% level)
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TESTING FOR SIGNIFICANCE
Step 4
• If the calculated statistical value is bigger than that in the table
– reject H0 (hypothesis of no difference) and
– conclude that the two samples are different
• If calculation value is smaller than that in the tables,
– keep H0 and
– conclude that the 2 samples do belong to the same population
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THE SIGN TEST (Nominal data;
Repeated Measures / Matched Pair)
• Test the claim that there will be a significant difference between
subjects’ scores on an arithmetic test after drinking 4 pints of beer
• Arithmetic test is administered to each person with and without beer.
The scores are shown in the table below.
• There are:
– 9 “+”
– 1 “–”; and
– 2 zeros
• H0 : Drinking beer does not impair arithmetic
performance
• H1 : Drinking beer impairs arithmetic performance
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THE SIGN TEST
• Count the number of
occurrences of the least
frequent sign. Call this X
– (only one ‘-’ X = 1)
• Count the total number of
signs. Label this as N
– (1 ‘-’ plus 9 ‘+’ N =
10)
• From table (left) find the
critical value (table) for
your N
– (for N = 10, critical X
value in the table = 1)
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THE SIGN TEST (Nominal data;
Repeated Measures / Matched Pair)
• In this example, as there is only
1 “-” sign and it falls on the
critical value, the null hypothesis
is therefore rejected.
• Conclusion: drinking beer does
impair arithmetic performance
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WILCOXON TEST (Ordinal data;
Repeated Measures / Matched Pair)
• It is a less crude test than the sign test but it is applied to similar
data.
• Same data as previous example of Sign Test:
Sort from smallest to largest
Ignore “+”
or “-” sign
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(A-B) value Frequency Ranking
1
3
(1+2+3)/3 = 2
2
1
4/1 = 4
3
2
(5+6)/2 = 5.5
4
0
0
5
4
(7+8+9+10)/4 =12
8.5
WILCOXON TEST (Ordinal data;
Repeated Measures / Matched Pair)
B is better than A
Denotes A better than B
• Sum the value in ranking row for subjects doing better under A (= 53)
(5.5 + 8.5 + 8.5 + 2 + 8.5 + 5.5 + 8.5 + 2 + 4 = 53)
• Sum the value in ranking row for subjects doing better under B (= 2)
• The test statistic is the smaller value of these 2 (smaller of 53 & 2 = 2)
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WILCOXON TEST (Ordinal data;
Repeated Measures / Matched Pair)
• T = 2, N = 10,
– critical value T = 8 at 5%
level ; &
– critical value T = 3 at 1%
level
• Since T = 2 is smaller than
either of the critical values,
the difference is significant at
both 5% & 1% level
• Conclusion: Reject Null
hypothesis
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MANN WHITNEY U TEST (Ordinal
data; Independent Subjects)
• Compare truancy days off school from families in 2 housing estates
A & B.
• Note that these are data from 2 independent samples
How the Rank scores are derived
Value Frequency Ranking
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1
1
14
1
2
18
1
3
22
1
4
23
3
(5+6+7)/3 = 6
25
1
8
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2
(9+10)/2 = 9.5
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MANN WHITNEY U TEST (Ordinal
data; Independent Subjects)
• Rank the scores across both estates as if they are a single group.
• Add up the rank score for both estates
• Compute U1 and U2
Where:
R = Total of ranks in smallest group (R = 66)
N1 = Number of cases in smallest group (N1 = 10)
N2 = Number of cases in largest group (N2 = 10)
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MANN WHITNEY U
TEST
• Take the smaller of U1
and U2
– If this value is ≤ the
critical values in the
statistical tables,
then the difference is
significant
• U2 = 11 < U1 = 89 and
critical value = 23
• Since U2 = 11 < critical
value = 23, result is
significant. Truancy
rates on the two estates
are significantly different
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UNRELATED T-TEST (Interval /
Ratio data; Independent Subjects)
• The way of assessing differences between means in chapter 5 is
called the z test
– Accurate only for large samples (≥ 30)
• If sample size < 30, sample std dev is not reliable as estimate of
population std dev
• use t-test (or students t-test)
– t-distribution is quite similar to normal distribution but shape is
flatter & wider
– There is a different t-distribution for every possible sample size
– Shape approaching normal distribution as sample size increases
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UNRELATED T-TEST (Interval /
Ratio data; Independent Subjects)
Note that t-test is a parametric test
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UNRELATED T-TEST (Interval /
Ratio data; Independent Subjects)
Example
• Does an average box of cereal contain 368 grams of cereal? A random
sample of 36 boxes had a mean of 372.5 grams and a standard deviation
of 12 grams. Test at the 0.05 level.
H0: µ = 368
t=
sample _ mean − µ
n
H1: µ ≠ 368
Degree of freedom = 36-1 = 35
σ
t=
372.5 − 368
= 2.25
12
36
Critical t = 2.03 (from t-distribution table)
Therefore reject H0. There is evidence that the population average weight
is not 368 gram.
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RELATED T-TEST (Interval / Ratio data;
Repeated Measures / Matched Pairs )
• Concluding concept is similar to unrelated t-test
• In most statistical packages t values will be given, with the
associated probability level quoted.
• If the probability value is LESS than 0.05 or 0.01 then the result is
significant at the 5% or 1% level respectively
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SPSS Exercise 5
• Hand-in Requirements: open a word file to store your analysis
results for hand in. Please type your name at the top of first page.
• Data file - The file name is ‘uk house price 1980 – 2005’. It is saved
on VISION.
Part 1: Simple analysis of interval and ratio data
• To make a preliminary examination of your data, use Explore to
examine the variable ‘achange1’.
Analyse
Descriptive Statistics
Explore
• Move the variable achange1 into the dependent box and click OK.
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SPSS Exercise 5
a
b
c
a. Valid - This refers to the non-missing cases. In this column, the N is given,
which is the number of non-missing cases; and the Percent is given, which is
the percent of non-missing cases.
b. Missing - This refers to the missing cases. In this column, the N is given,
which is the number of missing cases; and the Percent is given, which is the
percent of the missing cases.
c. Total - This refers to the total number cases, both non-missing and missing. In
this column, the N is given, which is the total number of cases in the data set;
and the Percent is given, which is the total percent of cases in the data set.
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SPSS Exercise 5
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
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SPSS Exercise 5
a. Statistic - These are the descriptive statistics.
b. Std. Error - These are the standard errors for the descriptive
statistics. The standard error gives some idea about the variability
possible in the statistic.
c. Mean - This is the arithmetic mean across the observations. It is the
most widely used measure of central tendency. It is commonly
called the average. The mean is sensitive to extremely large or
small values.
d. 95% Confidence Interval for Mean Lower Bound - This is the
lower (95%) confidence limit for the mean. If we repeatedly drew
samples of 200 students' writing test scores and calculated the
mean for each sample, we would expect that 95% of them would fall
between the lower and the upper 95% confidence limits. This gives
you some idea about the variability of the estimate of the true
population mean.
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SPSS Exercise 5
e. 95% Confidence Interval for Mean Upper Bound - This is the
upper (95%) confidence limit for the mean.
f. 5% Trimmed Mean - This is the mean that would be obtained if the
lower and upper 2.5% of values of the variable were deleted. If the
value of the 5% trimmed mean is very different from the mean, this
indicates that there are some outliers. However, you cannot assume
that all outliers have been removed from the trimmed mean.
g. Median - This is the median. The median splits the distribution such
that half of all values are above this value, and half are below.
h. Variance - The variance is a measure of variability. It is the sum of
the squared distances of data value from the mean divided by the
variance divisor. We don't generally use variance as an index of
spread because it is in squared units. Instead, we use standard
deviation.
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SPSS Exercise 5
i. St. Deviation - Standard deviation is the square root of the variance.
It measures the spread of a set of observations. The larger the
standard deviation is, the more spread out the observations are.
j. Minimum - This is the minimum, or smallest, value of the variable.
k. Maximum - This is the maximum, or largest, value of the variable.
l. Range - The range is a measure of the spread of a variable. It is
equal to the difference between the largest and the smallest
observations. It is easy to compute and easy to understand.
However, it is very insensitive to variability.
m. Interquartile Range - The interquartile range is the difference
between the upper and the lower quartiles. It measures the spread
of a data set. It is robust to extreme observations.
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SPSS Exercise 5
n. Skewness - Skewness measures the degree and direction of
asymmetry. A symmetric distribution such as a normal distribution
has a skewness of 0, and a distribution that is skewed to the left,
e.g. when the mean is less than the median, has a negative
skewness.
o. Kurtosis - Kurtosis is a measure of the heaviness of the tails of a
distribution. A normal distribution has kurtosis 0. Extremely nonnormal distributions may have high positive or negative kurtosis
values, while nearly normal distributions will have kurtosis values
close to 0. Kurtosis is positive if the tails are "heavier" than for a
normal distribution and negative if the tails are "lighter" than for a
normal distribution.
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SPSS Exercise 5
a. Frequency - This is the frequency of the leaves.
b. Stem - This is the stem. It is the number in the 10s place of the value of the variable (stem
width). Eg in the 2nd last line, the stem is 2 and leaves are 5, 7 & 9. The value of the variable is
25, 27 & 29.
c. Leaf - This is the leaf. It is the number in the 1s place of the value of the variable. The
number of leaves tells you how many of these numbers is in the variable. For example, on the
2nd line, there is two 5, two 6 and two 8 (hence, the frequency is six).
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SPSS Exercise 5
a
b
c
d
e
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SPSS Exercise 5
a. This is the maximum score unless there are values more than 1.5
times the inter quartile range above Q3, in which, it is the line
extends to a maximum of 1.5 times the inter quartile range
b. This is the third quartile (Q3), also known as the 75th percentile.
c. This is the median (Q2), also known as the 50th percentile. If the
median line within the box is not equidistant from the edges of the
box, then the data is skewed
d. This is the first quartile (Q1), also known as the 25th percentile.
e. This is the minimum score unless there are values less than 1.5
times the inter quartile range below Q1, in which case, line extends
to a maximum of 1.5 times the inter quartile range
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SPSS Exercise 5
• Histogram
Graph
Legacy Dialogs,
Histogram,
Use the achange1 variable
(Please tick the Display
normal curve option when
you run the histogram).
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SPSS Exercise 5
• Hand in Requirement 1
Copy the histogram of achange1 you just produced to the word file,
and answer these questions:
a) How many percentage points does each bar represent in this
diagram? 2.5%
b) What are the minimum and the maximum percentage changes
represented by the tallest bar? Max = 12.5%; Min = 10%
c) What are the possible minimum and maximum quarterly house price
changes? Min = -12.5% ; Max = 32.5%
c) Is this distribution a normal distribution? Why? Not normal
distribution as it does not resemble a bell curve.
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SPSS Exercise 5
d) What are the main difference between a bar chart you produced in
the previous exercise and a histogram?
The bar graph is often used to show a visual comparison of discrete
elements, while the histogram is used to show the frequency of nondiscrete, continuous items
The items in the histogram are usually numbers that are grouped or
categorized in such a way that they are considered to be ranges. In
regards to bar graphs, the items are taken as separate entities
A bar graph is drawn in such a way that a bar representing the
frequency of an item is not touching the bar of the next item. There
is a visible space between the bars. The bars in the histogram are
always the touching the next one. There are no spaces in between
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SPSS Exercise 5
• Analyse quantitative data using Report
Analyse
Report
Case summarise
Move index1, price1 and achange1 in the Variables box; and
move year into the Grouping Variable(s) box.
Hand in Requirement 2
– Repeat the report procedure. You could include more variables
in Variables box, but do not use year this time as grouping
variable, use period instead. Before you run the process, make
sure you have not selected the Display Cases box.
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SPSS Exercise 5
Part 2: Comparing Means: Parametric and Non-parametric Tests
Independent samples t-test (Parametric)
• Suppose we are interested in the comparison of housing price
changes during the 1980s and the 1990s, we can use the comparing
means analysis.
• Our null hypothesis will be:
At the 5% of level of significance, there is no significant
difference between the mean quarterly house price changes
during the 1980s and the 1990s
• To use this parametric test, we need to confirm that the data is a
normal distribution by using histogram and normal curve function.
For this exercise, we presume the data confirms the normal
distribution
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SPSS Exercise 5
• Analyse data using Compare Means
Analyse
Compare Means
Independent Sample t-test
The test variable is achange1 and the Grouping Variable would
be period. You also need to define groups: the value 1 (1980s)
for group 1 and the value 2 (1990s) for group 2.
• Interpretation of Test Results: The second table contains two set
of test results:
(1) Levene’s Test for Equality of Variances and
(2) t-test for Equality of Means.
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SPSS Exercise 5
• You should check, firstly, the significance level of the equality of
variances of the two groups using Levene’s test for equality of
variances.
• If Levene’s Sig. is greater than 0.05, equal variances of the groups are
assumed.
• if Levene’s Sig. is smaller than 0.05, equal variances of the groups are
not assumed.
Equality of variance assumed
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SPSS Exercise 5
•
•
•
t-test for Equality of Means includes two rows of results: the first row results
for the assumed equal variances, and the second row for the unequal
variances.
Should use the row for assumed equal variance since Sig F is > 0.05 (see
previous slide)
After you have decided which row of the t-test result is appropriate for your
data, then you should look at the Sig. (2-tailed) level.
– If this value is greater than 0.05 (5%), the mean quarterly house price
changes of the two periods are not significantly different;
– if this value is smaller than 0.05 (5%), the mean quarterly house price
changes of the two groups are significantly different.
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Value is < 5%; population is significantly different
SPSS Exercise 5
Hand in Requirement 3
Examine the t test tables (from SPSS output window, you don’t need
to copy the table over to word) and answer these questions in your
word document:
• Looking at the first table (Group Statistics), what can we say about
house price changes over the two periods?
– The average quarterly price increase for all houses has
decreased to 1.6725% from 12.2475%
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• Are variances of house price changes over the two periods equal?
Why?
– The variances of house price changes over the 2 periods are
equal as the Levene’s test for Equality of Variances cannot reject
the null hypothesis.
• What is our Null Hypothesis for this test?
– The mean quarterly price change for the period 1980-1989 is
equal to the mean quarterly price change for the period 19901999 at 5% significant level
• What are the test results (values) for significant level, mean
differences, standard error of differences, and 95% confident interval
of difference?
– Significant level = 5%; mean differences = 10.575; standard error of
differences = 1.58875; 95% confident interval of difference = 7.41205 to
13.73795
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SPSS Exercise 5
•
Does the result of this parametric t-test accept or reject our null hypothesis?
Why?
– The t-test rejects the null hypothesis as its significant value is <5%
Hand in Requirement 4
•
We have assumed that our data is normally distributed for the above
analysis. If our data is not normally distributed, we should use the nonparametric Mann-Whitney U test.
•
Analyse data using Compare Means
Analyse
Non parametric test
Legacy Dialogs
2 Independent Sample
The test variable is achange1 and the Grouping Variable would be
period. You also need to define groups: the value 1 (1980s) for group 1
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and the value 2 (1990s) for group 2.
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SPSS Exercise 5
The significance level (p) is the last row of the
second table.
If this value is greater than 0.05 (5%), the
mean quarterly house price changes of the
two periods are not significantly different;
if this value is smaller than 0.05 (5%), the
mean age of the two groups are significantly
different.
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Hand in Requirement 5
• If both test variables have a normal distribution, we can use the
Paired samples t-test (Parametric).
• Run this test using nchange2 and ochange4.
Analyse
Compare Means
• Discuss the result in your Word Document:
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SPSS Exercise 5
•
Looking at the first table (Paired Samples Statistics), what can we say
about new and old house price changes over between 1980 and 2005?
–
•
The average quarterly price change for older houses are greater than
new houses
What is our Null Hypothesis for this paired samples t test?
–
–
The average quarterly price change for new houses are equal to the
average quarterly price change for old houses ; or
The difference between the average quarterly price change between
old and new houses are zero.
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SPSS Exercise 5
•
What are the test results (values) for significant level, mean
differences, standard error of differences, and 95% confident
interval of difference?
–
–
–
Significant level = 5%;
- mean differences = -1.22376
standard error of differences = 0.40366
95% confident interval = -2.02462 to -0.42291
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SPSS Exercise 5
• Does the result of this parametric paired sample t-test accept or reject
our null hypothesis? Why?
– The null hypothesis should be rejected.
– The value of 0.003 is the two-tailed p-value computed using the t
distribution. It is the probability of observing a greater absolute
value of t under the null hypothesis. The p-value is less than the
pre-specified alpha level (5% here). We can conclude that mean
difference is statistically significantly different from zero.
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THE END
Any questions?
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