Chapter Eighteen
Data Transformations
and Nonparametric
Tests of Significance
PowerPoint Presentation created by
Dr. Susan R. Burns
Morningside College
Smith/Davis (c) 2005 Prentice Hall
Assumptions of Inferential Statistical
Tests and the Fmax Test
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The assumptions researchers make when they
conduct parametric tests (i.e., used to attempt to
estimate population parameters). Are as follows:
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The populations from which you have drawn your samples
are normally distributed (i.e., the populations have the
shape of a normal curve).
The populations from which you have drawn your samples
have equivalent variances. Most researchers refer to this
assumption as homogeneity of variance.
Smith/Davis (c) 2005 Prentice Hall
Assumptions of Inferential Statistical
Tests and the Fmax Test
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If your samples are equal in size, you can use them to test the
assumption that the population variances are equivalent (using
the Fmax).
For example, assume that you conducted a multi-group
experiment and obtained the following variances (n = 11).
Group 1 θ2 = 22.14, Group 2 θ2 = 42.46, Group 3 θ2 = 34.82,
Group 4 θ2 = 52.68. Select the smallest and largest variances
and plug them into the Fmax formula.
To evaluate this statistic, you use the appropriate F table,
entering at the n – 1 row and the k (number of samples in the
study) column.
Smith/Davis (c) 2005 Prentice Hall
Data Transformations
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Using data transformations is one technique
researchers use to deal with violations of the
assumptions of inferential tests.
It is a legitimate procedure that involves the
application of an accepted mathematical procedure
to all scores in a data set. Common data
transformations include:
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Square root – research calculates the square root of each
score.
Logarithmic – researcher calculates the logarithm of each
score.
Reciprocal – researcher calculates the reciprocal (i.e., 1/X)
of each score.
Smith/Davis (c) 2005 Prentice Hall
Data Transformations
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What Data Transformations Do and Not Do:
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Data transformations change certain
characteristics of distributions.
For example, they can have the effect of
normalizing distributions or equalizing variances
between (among) distributions.
These transformations are desirable if you are in
violation of the assumptions for inferential
statistics.
They do not alter the relative position of the data
in the sample.
Smith/Davis (c) 2005 Prentice Hall
Data Transformations
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Using Data Transformations and a Caveat
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Assuming that your data transformation corrected the problem that
prompted you to use it, you can proceed with your planned
analysis.
However, if you perform more than one type of analysis on your
data, remember to use the transformed data for all of your
analyses.
Also, you will need to be careful when interpreting the results of
your statistical analysis and draw conclusions.
Your interpretations and conclusions need to be stated in terms of
the transformed data, not in terms of the original data you
gathered.
You really can only compare your data to other research projects
that have used the same transformations.
Smith/Davis (c) 2005 Prentice Hall
Nonparametric Tests of Significance
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Nonparametric Tests of Significance are
significance tests that do not attempt to
estimate population parameters such as
means and variances.
Smith/Davis (c) 2005 Prentice Hall
Chi Square Tests
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Chi-Square Tests are procedures that compare the fit
or mismatch between two or more distributions.
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Chi Square Test for Two Levels of a Single Nominal Variable
– When we have one nominal variable, there is an expected
frequency distribution and an observed frequency
distribution.
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The expected frequency distribution is the anticipated
distribution of frequencies into categories because of previous
research results or a theoretical prediction.
The observed frequency distribution is the actual distribution of
frequencies that you obtain when you conduct your research.
The Chi Square Test for Two Levels of a Single Nominal
Variable is the simplest form of this test.
Smith/Davis (c) 2005 Prentice Hall
Calculation of the Chi Square Test
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Calculation of the chi-square test statistic is based on the
following formula:
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Step 1. Subtract the appropriate expected value from each
corresponding observed value. Square the difference and then
divide this squared value by the expected value.
Step 2. Sum all the products you calculated in Step 1.
To determine significance, you will need to calculate the
degrees of freedom and then look up the critical value in the
appropriate table. To calculate the degrees of freedom, use the
following formula:
df = Number of Categories – 1
Smith/Davis (c) 2005 Prentice Hall
Chi Square Example
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An example given in your
textbook examines democratic
and republican candidate
preference in the current
election.
A newspaper says that the two
are absolutely even in terms of
voter preferences.
You conduct a voter preference
survey of 100 people in your
town and find that 64 people
prefer the democratic
candidate, whereas 36 prefer
the republican candidate.
Smith/Davis (c) 2005 Prentice Hall
Chi Square Test
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Writing up your result in a research report would look
something like this:
2(Number of categories, N) = #.##, p < .05
A significant result with a chi square test is interpreted
differently than a significance test with an inferential
statistic.
When you have a significant chi square, that means that
the two distributions you are tests are not similar.
In a two-group design, you would be able to say anything
about the two groups differing from each other.
It is the expected and observed distributions that are
differing from each other.
Smith/Davis (c) 2005 Prentice Hall
Chi-Square Test for More Than Two
Levels of a Single Nominal Variable
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Again, you are comparing an observed and an expected
distribution when you are comparing more than two levels of a
single nominal variable.
Thus, the appropriate formula still is as follows:
2 = Σ(O – E)2/E
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knowing that you may have different expected values (E) for each
level.
When your nominal data has three or more levels/categories, it
is not readily apparent where the discrepancy between the
observed and expected frequencies exists.
Significant chi squares with more than two groups require
follow-up tests.
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Most researchers use a series of smaller, more focused chisquare tests.
Smith/Davis (c) 2005 Prentice Hall
Chi-Square Test for Two Nominal
Variables
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This situation is analogous to a multiple IV research design.
When you have two nominal variables, you will display y our
data in a contingency table that shows the distribution of
frequencies and totals for two nominal variables.
To find the expected frequency for any of the cells of your
contingency table, use the following formula:
Expected Frequency = (row total X column total)/grand total
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You still use the chi square formula to test your contingency
table:
2 = Σ(O – E)2/E
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The calculation for degrees of freedom for a contingency table
is as follows:
df = (number of rows – 1)(number of columns – 1)
Smith/Davis (c) 2005 Prentice Hall
Chi-Square Test for Two Nominal
Variables
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The interpretation of a chi square for a contingency table is a bit
different that our interpretation for one nominal variable.
The chi square for a contingency table tests to determine if the
distributions in the table have similar or different patterns.
If the chi square is not significant, then the distributions are
similar (i.e., the distributions are independent).
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If the chi square is significant, then the distributions are
dissimilar, and researchers say that are dependent.
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Another way of saying this is the pattern for one category is
essentially the same as the pattern for the other level(s).
Meaning, the pattern for one category is not the same pattern for
the other level(s). The nature of the pattern on one nominal
variable depends on the specific category or level of the other
nominal variable.
These types of tests are also referred to as chi-square tests for
independence.
Smith/Davis (c) 2005 Prentice Hall
Rank Order Tests
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Rank-Order Tests are non-parametric tests that are appropriate
to use when you have ordinal data.
The underlying rational for the ordinal-data tests involves
ranking all the scores, disregarding specific group membership,
then you compare the ranks for the various groups.
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If the groups were drawn from the same population, then the ranks
should not differ noticeably between (among) the groups.
If the IV was effective, then the ranks will not be evenly distributed
between (among) the groups; smaller ranks will be associated with
one group and larger ranks will be associated with another
group(s).
Smith/Davis (c) 2005 Prentice Hall
Mann-Whitney U Test
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Mann-Whitney U Test is used with two independent
groups that are relatively small (n = 20 or less).
Step 1. Rank all scores in your data set (disregarding
the group membership); the 1 lowest score is
assigned the rank of 1.
Step 2. Sum the ranks for each group
Step 3. Compute a U value for each group according
to the following formulas:
U1 = (n1) (n2) + n1(n1 + 1)/2 – ΣR1
U2 = (n1) (n2) + n2(n2 + 1)/2 – ΣR2
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Where: ΣR1 = Sum of Ranks Group 1; ΣR2 = Sum of Ranks
Group 2; n = number of participants in the appropriate group.
Smith/Davis (c) 2005 Prentice Hall
Mann-Whitney U Test Example
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You have designed a new
method for teaching spelling to
second graders.
You conduct an experiment to
determine if your new method is
superior to the method being
used.
You randomly assign 12 second
graders to two equal groups.
You teach group 1 in the
traditional manner, whereas
group 2 learns to spell with your
new method,
After two months, both groups
complete the same 30-word
spelling test.
Smith/Davis (c) 2005 Prentice Hall
Mann-Whitney U Test
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Step 4. Determine which U Value (U1 or U2) to use to
test for significance. For a two-tailed test, you will
use the smaller U. For a one-tailed test, you will use
the U for the group you predict will have the larger
sum of ranks.
Step 5. Obtain the critical U value from the
appropriate Table.
Step 6. Compare your calculated U value to the
critical value.
Smith/Davis (c) 2005 Prentice Hall
Rank Sums Test
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Rank Sums Test is used when you have two
independent groups and the n in one or both groups
is greater than 20.
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Step 1. Rank order all the scores in your data
set (disregarding group membership). The
lowest scores is assigned the rank of 1.
Step 2. Select one group and sum the ranks
(ΣR).
Step 3. Use the following formula to calculate
the expected sum of ranks:
ΣRexp = n(n + 1)/2
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Smith/Davis (c) 2005 Prentice Hall
Rank Sums Test
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Step 4. Use the ΣR and ΣRexp for the selected group to
calculate the rank sums z statistic according to the following
formula:
Zrank sums = ΣR - ΣRexp/√(n1) (n2) (n2 + 1)/N
Step 5. Use the appropriate table to determine the critical z
value for the .05 level. If you have a directional hypothesis (i.e.,
one-tailed test), you will find the z that occurs 45% from the
mean. For non-directional (two-tail tests), you will find the z that
occurs 47.5% from the mean. Remember you split the alpha
level equally for a two-tailed test.
Step 6. Disregard the sign of your z statistic and compare it to
the critical value.
Step 7. Calculate an effect size. η2 = (z rank scores)2/N-1
Smith/Davis (c) 2005 Prentice Hall
Wilcoxon T Test
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Wilcoxon T Test – is used when you have two related groups
and ordinal data.
A researcher believes that viewing a video showing the benefits
of recycling will result in more positive recycling attitudes and
behaviors. Ten volunteers complete a recycling questionnaire
(higher scores = more positive attitudes) before and after
viewing the video.
Smith/Davis (c) 2005 Prentice Hall
Wilcoxon T Test
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Step 1. Find the difference between the scores for each pair.
It doesn’t matter which score is subtracted, just be
consistent throughout all the pairs.
Step 2. Assign ranks to all nonzero differences. (Disregard
the sign of the difference. The smallest difference = rank 1.
Tied differences receive the average of the ranks they are
tied for.).
Step 3. Determine the ranks that are based on positive
differences and the ranks that are based on negative
differences.
Step 4. Sum the positive and negative ranks. These sums
are the T values you will use to determine significance.
Smith/Davis (c) 2005 Prentice Hall
Wilcoxon T Test
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Step 5. If you have a non-directional hypothesis (two-tailed
test), you will use the smaller of the two sums of ranks. If
you have a directional hypothesis (one-tailed) you will have
to determine which sum of ranks your hypothesis predicts
will be smaller.
Step 6. N for the Wilcoxon T Test equals the number of
nonzero differences.
Step 7. Use the appropriate table to check the critical value
for your test. To be significant, the calculated value must be
equal to or less than the table value.
Smith/Davis (c) 2005 Prentice Hall
Kruskal-Wallis H Test
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Kruskal-Wallis H Test is appropriate when you have more than
two groups and ordinal data.
For example, a sport psychologist compared methods of
teaching putting to beginning golfers. The methods were visual
imagery and repetitive practice, and the combination of visual
imagery ad repetitive practice.
The researcher randomly assigned 21 equally inexperience
golfers to three equal groups.
Following equivalent amounts of training, each golfer attempted
to sink 25 putts from the same location on the putting green.
The researcher rank-ordered (smaller rank = lower score) all
the final performances.
Smith/Davis (c) 2005 Prentice Hall
Kruskal-Wallis H Test
Smith/Davis (c) 2005 Prentice Hall
Kruskal-Wallis H Test
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Step 1. Rank all scores (1 = lowest score).
Step 2. Sum the ranks for each group/condition.
Step 3. Square the sum of ranks for each
group/condition.
Step 4. Use the following formula, calculate the
sum of squares between groups:
Smith/Davis (c) 2005 Prentice Hall
Kruskal-Wallis H Test
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Step 5. Use the following formula to calculate
Smith/Davis (c) 2005 Prentice Hall
Kruskal-Wallis H Test
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Step 6. Use the chi square table to find the critical
value for H. The degrees of freedom for the H test =
k – 1, where k is the number of groups/conditions.
Step 7. Calculate an effect size:
Step 8. If appropriate conduct post hoc tests. The
procedure for conducting such tests is similar to
conducting follow-up tests when you have nominal
data. In short, you dives simpler follow-up analyses.
Smith/Davis (c) 2005 Prentice Hall