Chapter Eighteen

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Chapter Eighteen

• Data Transformations and Nonparametric Tests of
  Significance

PowerPoint Presentation created by Dr. Susan R.
BurnsMorningside College
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Assumptions of Inferential Statistical Tests and
the Fmax Test

• The assumptions researchers make when they
  conduct parametric tests (i.e., used to attempt
  to estimate population parameters). Are as
  follows
• 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.

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Assumptions of Inferential Statistical Tests and
the Fmax Test

• 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 ?2 22.14, Group 2
  ?2 42.46, Group 3 ?2 34.82, Group 4 ?2
  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.

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Data Transformations

• 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
• 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.

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Data Transformations

• What Data Transformations Do and Not Do
• 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.

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Data Transformations

• Using Data Transformations and a Caveat
• 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.

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Nonparametric Tests of Significance

• Nonparametric Tests of Significance are
  significance tests that do not attempt to
  estimate population parameters such as means and
  variances.

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Chi Square Tests

• Chi-Square Tests are procedures that compare the
  fit or mismatch between two or more
  distributions.
• 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.
• 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.

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Calculation of the Chi Square Test

• Calculation of the chi-square test statistic is
  based on the following formula
• 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

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Chi Square Example

• 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.

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Chi Square Test

• Writing up your result in a research report would
  look something like this
• ?2(Number of categories, N) ., p lt .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.

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Chi-Square Test for More Than Two Levels of a
Single Nominal Variable

• 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 S(O E)2/E
• 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.
• Most researchers use a series of smaller, more
  focused chi-square tests.

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Chi-Square Test for Two Nominal Variables

• 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
• You still use the chi square formula to test your
  contingency table
• ?2 S(O E)2/E
• The calculation for degrees of freedom for a
  contingency table is as follows
• df (number of rows 1)(number of columns
  1)

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Chi-Square Test for Two Nominal Variables

• 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).
• Another way of saying this is the pattern for one
  category is essentially the same as the pattern
  for the other level(s).
• If the chi square is significant, then the
  distributions are dissimilar, and researchers say
  that are dependent.
• 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.

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Rank Order Tests

• 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.
• 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).

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Mann-Whitney U Test

• 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 SR1
• U2 (n1) (n2) n2(n2 1)/2 SR2
• Where SR1 Sum of Ranks Group 1 SR2 Sum of
  Ranks Group 2 n number of participants in the
  appropriate group.

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Mann-Whitney U Test Example

• 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.

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Mann-Whitney U Test

• 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.

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Rank Sums Test

• Rank Sums Test is used when you have two
  independent groups and the n in one or both
  groups is greater than 20.
• 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 (SR).
• Step 3. Use the following formula to calculate
  the expected sum of ranks
• SRexp n(n 1)/2

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Rank Sums Test

• Step 4. Use the SR and SRexp for the selected
  group to calculate the rank sums z statistic
  according to the following formula
• Zrank sums SR - SRexp/v(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. ?2 (z rank
  scores)2/N-1

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Wilcoxon T Test

• 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.

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Wilcoxon T Test

• Step 1. Find the difference between the scores
  for each pair. It doesnt 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.

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Wilcoxon T Test

• 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.

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Kruskal-Wallis H Test

• 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.

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Kruskal-Wallis H Test
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Kruskal-Wallis H Test

• 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

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Kruskal-Wallis H Test

• Step 5. Use the following formula to calculate

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Kruskal-Wallis H Test

• 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.