NonParametric Tests

1
Chapter_13

• Non-Parametric Tests
• Field_2005

2
What are non-parametric tests?

• They do not make any parametric assumptions about
  the data such as normality, homogeneity of
  variances, etc.
• They are therefore also called 'assumption-free'
  tests
• They work on the principle of ranking data. The
  lowest score receives the rank 1, the next
  highest score the rank 2, etc., without implying
  that the intervals between the ranks are equal.
  Low scores will be represented as low ranks, high
  scores as high ranks. The analysis is then
  carried out on the ranks and not on the original
  scores.
• 4 tests will be considered here
• - Wilcoxon rank-sum test -Kruskal-Wallis test
• (Mann-Whitney test)
• - Wilcoxon signed-rank test - Friedman's test

3
Outlook Terminology
4
Wilcoxon rank-sum test and Mann-Whitney Test

• With these two tests you can compare 2
  independent conditions.
• They are equivalent to an independent t-test
• Example The effect of Ecstasy vs. Alcohol shall
  be measured, using the Beck Depression Inventory
  (BDI).

5
The data effect of Ecstasy vs. Alcohol
Depression scores were obtained one day after
taking the drug (Sunday)? and three days later
(Wednesday)? to find out if there is a
development of depression over time
6
The theory

• The scores are translated into ranks.
• The lowest score gets the lowest rank, the next
  higher score the next higher rank up to the
  highest rank.
• If there is no difference in the depression level
  between Ecstasy and Alcohol, a similar number of
  low and high ranks should be found in each group.
  If we add up the ranks, the summed total of ranks
  in each group should be about the same.
• If there is a difference between the two groups,
  e.g., Ecstasy produces higher levels of
  depression, one would find higher ranks in the
  Ecstasy group and lower ranks in the Alcohol
  group.

7
Ranking of the data (Sunday and Wednesday)?
Same scores share 'tied ranks'. The actual value
of a tied rank is the average of the ranks that
constitute it. E.g., the actual rank for the tied
ranks 3 and 4 for the 2 occurrences of the score
6 in the Wednesday data is 3,5. The ranks are
summed up for each group and day (AW59 EW151
AS90.5 ES119.5). The lowest of the sums serves
as test statistics.For Wednesday this is WS59.
For Sunday, it is WS90.5.
8
The test statistics, mean and SE(Wilcoxon
rank-sum test)

• Lower sum of ranks for Wednesday WS 59
• (WS Wilcoxon sum)?
• Lower sum of ranks for Sunday WS 90.5
• Mean of the test statistics (mean of Wilcoxon
  sum, WS)
• __
• WS n1(n1n21) 10(10101) 105
• 2 2
• SE of the test statistics (SE of ?WS)
• SEWS ?n1 n2(n1n21)/12
• ???10x10)(10101)/12 13.23

Why 12?
9
Test statistic as z-score, significance

• _ __
• Z X-?X WS - WS
• s SEWS
• __
• zSunday WS - WS 90.5 105 -1.10 ns
• SEWS 13.23
• __
• zWednesday WS - WS 59 105 -3.48
• SEWS 13.23
• If the z-scores are 1.96 (irrespective of or
  -), then the test is significant.
• ? The group difference for Sunday is n.s.,
  whereas
• ? The group difference for Wednesday is
  significant

10
Mann-Whitney (U) test

• The Mann-Whitney test is similar to the Wilcoxon
  rank-sum test but uses the U test statistic.
• U N1N2 N1(N11) - R1
• 2
• USunday (10x10) 10(11) - 119.5 35.50
• 2
• UWednesday (10x10) 10(11) -151.0 4.00
• 2
• SPSS produces both statistics. Since they are
  related they always say the same. Choose yourself!

R1 sum of ranks of Group 1, here
Ecstasy 119.5
R1 sum of ranks of Group 1, here Ecstasy 119.5
R1 sum of ranks of Group 1, here Ecstasy 151
11
Data input Ecstacy_Alcand provisional analysis

• For a between subjects test, we need a coding
  variable (as in a between subjects t-test), e.g.
• 'drug' 1ecstasy 2alcohol
• We then have a column for the dependent variable
  BDI on Sunday (sunbdi) and one for BDI on
  Wednesday (wedbdi).

12
Before running the Analysis Run a test of
normalityAnalyze ? Descriptive Statistics ?
Explore

• Sunbdi and wedbdi go to the dependent list
• 'drug administered' goes to the factor list
• In the plots, tick 'test of normality' for the
  test of normality

13
Test of Normality
ns ns
ns ns
Non- normal
Normal
Normal
Non- ormal

• Both the K-S test and the Shapiro-Wilk test tell
  us that the distribution for Ecstasy-sunbdi and
  Alcohol-wedbdi are not normal.

14
Decision for a non-parametric test

• As we have seen, some of the distributions are
  non-normal. What can you do?
• Transform the data (z-, logarithmic, etc.)?
• Choose a non-parametric test

15
Homogeneity of variances
The homogeneity test you can request in the
'Options' of a simple One-way ANOVA. It also
comes automatically if you run a t-test for
independent samples.

• Levene's test is n.s. ? the variances of the
  Sunday and Wednesday data are equal

16
Further DescriptivesAnalyze ? Descriptive
Statistics ? Frequencies or ?
Descriptives

• Request basic descriptive statistics such as the
  mean, median, SD, variance.
• Note that for a non-parametric test, the median
  is a better indicator of the central tendency
  than the mean.

17
Running the analysis(using your own
Ecstasy_Alc.sav)?

• Analyze ? Nonparametric Test ? 2 independent
  samples

Tranfer Sunbdi and Wedbdi to the 'Test Variable
List' and 'drug' to the 'grouping variable'
window.
Tranfer Sunbdi and Wedbdi to the 'Test Variable
List' and 'drug' to the 'grouping variable'
window.
Exact...
18
Specifying the dialog boxes
Define the two levels of the grouping variable
Request 'Descriptives' in the 'Options' box
If you have installed 'Exact Tests', you can
request such an 'Exact Test'
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Exact Test
Exact...

• You may or may not have 'Exact...' in your Main
  Dialog Box (I haven't). The 'Ecact Test' is an
  extra module of SPSS which needs to be installed.
• It enables an Exact test of the significance of
  the Kruskal-Wallis test, which is a good thing to
  have for small samples. However, it is a very
  time-demanding procedure (can take really
  long...).
• Instead of an 'Exact Test', a less intense test
  can be requested based on the 'Monte Carlo'
  Method.
• In the Monte-Carlo-Method, a distribution similar
  to the sample is found and then many samples (up
  to 10.000) are created for which the mean
  significance value and Confidence Intervals are
  computed.

20
Other options for the Mann-Whitney test
Do not confuse the K-S test for normality with
the K-S Z-test!!!

• Kolmogorov-Smirnov Z The K-S Z-test test whether
  two samples have been drawn from the same
  population. In sofar, it does the same as the M-W
  test. The K-S Z-test has even better power for
  small samples (n
• Moses Extreme Reactions This test compares the
  variability of scores in the two groups, hence
  like a non-parametric Levene test.
• Wald-Wolfowitz runs This is a variant of the
  M-W-test which looks for 'runs' of scores in a
  row from the same group
• AAAAAAAAAAAAEEEEAEEEEEEEE
• If the groups are different, runs or ranks for
  each group should cluster at different ends of
  the distribution.

21
Output from Mann-Whitney Test
Mean Ranks Sum of ranks/n
Sum of Ranks are all ranks summed up
22
Test statistics of the M-W test
Wilcoxon W the lower WS for sunbdi and wedbdi.
For the Sunday-BDI, there are no differences
between the two groups Ecstasy vs. Alcohol. For
the Wednesday-BDI, there are significant
diffe- rences, though the average rank is higher
in the Ecstasy users (15.1)than in the alcohol
users (5.9)?
For the Sunday-BDI, there are no differences
between the two groups Ecstasy vs. Alcohol. For
the Wednesday-BDI, there are significant
diffe- rences, though the average rank is higher
in the Ecstasy users (15.1) than in the alcohol
users (5.9)?
23
Comparison with t-test for independent samples
Levene's tests for homogeneity of variances OK
Levene's tests for homogeneity of variances OK
T-test statistics

• ? Wilcoxon rank sum test and t-test yield the
  same results

24
Calculating the effect sizes

• The effect size r can easily be calculated from
  the z-scores.
• r z
• ?n
• rSunday -1.11 -.25
• ???
• rWednesday -3.48 -.78
• ???

Medium effect
Huge effect
25
Reporting the results(Field_2005_532)?

• Ecstasy users (Mdn17.5) didn't seem to differ in
  depression levels from alcohol users (Mdn16) the
  day after the drugs were taken, U35.5, ns,
  r-25. However, by Wednesday, ecstasy users
  (Mdn33.5) were significantly more depressed than
  alcohol users (Mdn7.5), U4, p

Mdn Median
26
Non-parametric tests and statistical power

• With a non-parametric test we avoid the
  assumptions of a parametric test, esp. normality.
  However, by ranking the scores rather than
  computing the scores directly, we lose
  information about the magnitude of the difference
  between the scores (remember, two ranks do not
  tell you anymore how far, numerically, the two
  original scores were apart). Therefore, we may
  lose statistical power, i.e., we may not detect
  an effect which is genuinely there.
• However, non-parametric tests are only less
  powerful if parametric assumptions are met. Thus,
  if you run a parametric and a non-parametric test
  over normally-distributed data, then the
  non-parametric test may be weaker.

27
Non-parametric tests and statistical power

• The problem
• For normally-distributed data Type 1-error rate
  is 5.
• For non-normally-distributed data we would not
  know where 5 of the non-normal distribution are.
  It depends on the shape of the distribution.

28
Terminology
29
Comparing two related conditions The Wilcoxon
Signed-rank test

• The Wilcoxon signed-rank test is used when you
  want to compare 2 conditions but within the same
  subject.
• It is the non-parametric equivalent to the
  dependent t-test.
• Expl Measuring the differences between the
  depression scores on Sunday and Wednesday, from
  the previous example.
• Note before, we had only tested the difference
  between the two groups of Ecstasy and Alcohol
  users (between subjects design).

30
The theory

• First, the differences between the scores in the
  two conditions are obtained, then they are
  ranked.
• Additionally, the sign (positive/negative) of the
  difference is assigned to the rank.

31
Rankingthe data
The test statistics is the smaller of the summed
ranks T0 for Ecstasy and T8 for Alcohol
32
Calculating significance

• General formulas
• ?
• T n(n1) Test statistics
• 4
• SET ?n(n1)(2n1)/24 SE of Test statistics

33
Calculating significance

• ?T n(n1)/4 Test statistic
• SET ?n(n1)(2n1)/24 SE of Test statistics
• ?TEcstasy 8(81)/4 18
• SET Ecstasy ??(81)(161)/24 7.14
• ?TAlcohol 10(101)/4 27.5
• SET Alcohol ???(101)(201)/24 9.81

Note that there are only n8 in the Ecstacy group
now.
34
z-scores
TEcstasy 0 ?TEcstasy 18 SET
Ecstasy 7.14 TAlcohol 8 ?TAlcohol
27.5 SET Ecstasy 9.81

• Z X-?X T - ?T
• s SET
• zEcstasy 0-18 -2.52
• 7.14
• zAlcohol 8-27.5 -1.99
• 9.81
• ? Both values are is a significant difference in depression scores
  between Sunday and Wednesday for both drugs,
  Ecstasy and Alcohol

35
Before running the analysis (using your own
Ecstasy_Alc.sav)?

• Before running the analysis, you have to split
  the file for the Ecstasy and the Alcohol group
• Data ? Split File ? Organize output by groups

36
Running Wilcoxon signed-rank test(using your own
Ecstasy_Alc.sav)?

• Analyze ? Non-parametric tets ? 2-related samples

Exact...
If you have 'Exact' choose 'Asymptotic only'
37
Alternatives for Wilcoxon signed-rank test

• In the main dialog window, there are 3
  alternative tests which you may choose instead of
  Wilcoxon
• 1. Sign It only considers the direction of the
  differences (pos or neg), irrespective of
  magnitude of change. Therefore, it looses power.
• 2. McNemar Good for nominal (not ordinal) data,
  i.e., two related dichotomous variables.
• 3. Marginal Homogeneity Extension of the McNemar
  for ordinal data. Equivalent to Wilcoxon.
• (My version of SPSS does not have this option)?

38
Aside Request Descriptive StatisticsAnalyze ?
Descriptive Statistics ? FrequenciesBefore,
split the files according to 'kind of drug'!
Later, when you report the results, you will
need esp. the Median (Mdn) which is a better
suited value of the central tendency as the mean
for non-parametric data.
Later, when you report the results, you will
need esp. the Median (Mdn) which is a better
suited value of the central tendency as the mean
for non-parametric data. These are the
outputs for the split data (Ecstasy and Alcohol
separatly)?
39
You can also request the medians from the
descriptive statistics of the signed-rank test by
clicking on quartiles in the options
Ecstasy
Alcohol
40
Output for Ecstacy

• SPSS first gives the results for the Ecstasy group

There were no neg differences so that Wed Sunday There were 8 pos diff, so that Wed
Sun There were 2 ties (same values) which are
excluded
? All included differences (8 out of 10, since
the 2 ties were excluded) were positive, i.e.,
depression scores were always higher on Wednesday
than they were on Sunday, for Ecstasy.
The difference (z-score)? between Sun-Wed is
significant! The z-score is based on the neg
ranks since they are the smaller
41
Output for Alcohol

• SPSS then gives the results for the Alcohol group

There were 9 neg differences so that Wed Sunday There was 1 pos diff, so that Wed
Sun There were 0 ties (same values)
? 9 out of 10 differences were negative, and 1
was positive, i.e., depression scores were lower
on Wednesday than they were on Sunday, for
Alcohol.
The difference (z-score)? between Sun-Wed is
significant! The z-score is based on the pos
ranks since they are the smaller ones
42
Effects of Ecstasy vs. Alcohol

• For Ecstasy, depression increases from Sunday to
  Wednesday.
• For Alcohol, depression decreases from Sunday to
  Wednesday.
• This reverse effect is an interaction!

43
Calculating the effect sizes

• Effect sizes for the Wilcoxon signed-rank test
  can be calculated from the z-scores
• r z
• ?n
• rEcstasy -2.53 -.57
• ???
• rAlcohol 1.99 -.44
• ???

Note although 2 ties in the Ecstasy group
were excluded, here, all 10 subjects are included
in the calculation of the effect size (????
Medium to large effects
Medium to large effects
44
Reporting the results(Field_2005_541)?

• For Ecstasy users, depression levels were
  significantly higher on Wednesday (Mdn33.5) than
  on Sunday (Mdn17.50), T0,p
• For Alcohol users, the opposite was true
  depression levels were significantly lower on
  Wednesday (Mdn7.5) than on Sunday (Mdn16), T8,
  p

45
Terminology
46
Differences between several independent groups
The Kruskal-Wallis Test

• The Kruskal-Wallis Test is the non-parametric
  equivalent to a Simple One-way independent ANOVA.
• Example
• Background It has been claimed that the chemical
  'genistein' which naturally occurs in soya
  products decreases the number of sperms in males.
• Research question Do groups of male subject who
  eat various amounts of soya meals per week have
  different amounts of sperm after a year's period ?

47
The variables

• Independent variable number of soya meals
• (1) no soya meals (control condition) 0 per
  year
• (2) 1 soya meal per week - 52 per year
• (3) 4 soya meals per week - 208 per year
• (4) 7 soya meals per week - 364 per year
• Each group consisted of 20 different male
  individuals.
• Dependent variable number of sperms

48
The Theory of the Kruskal-Wallis Test

• As the other non-parametric tests, the K-W Test
  is also based on ranked data.
• First, the scores are ranked,irrespective of
  group memebership.
• Then, for each group, their ranks are added. The
  sum of ranks for each group is Ri.

49
Ranked data for the soya experiment
50
The Test Statistic H

• k
• H 12 Si1 R2i - 3 (N1)
• N(N1) ni
• H 12 9272 8832 8832 5472
  - 3(81)
• 80(81) 20 20 20 20
• 12 (42,966.45 38,984.45 38,984.45
  14,960.45) -243 6480
• 0.0019 (135,895.8) 243
• 251.66 243 8.659

H has a ?2 distribution Its df k-1 where k
of groups, hence df 4-13
H Hcritical (3) 7.81, pdistribution)?
51
Data input

• As for a One-Way ANOVA, we code the different
  groups with a dummy coding variable 'Soya' in the
  1st column
• (1) no soya
• (2) 1 soya meal
• (3) 4 soya meals
• (4) 7 soya meals
• The dependent variable 'sperm' goes in the 2nd
  column

52
The data in SPSS(Soya.sav)?
Dummy coding 1,2,3,4
Dep Var 'sperm'
ranks
53
Exploratory analyses

• Analyze ? Descriptive Statistics ? Explore
• tick 'Test of Normality' in 'Statistics'

Most groups show non-normal data distributions
Analyze ? General Linear Model ? Univariate, tick
'Homogeneity test' in 'options'
Levene's test is significant ? heterogeneous varia
nces
The data violate two parametric assumptions
Normal distribution of data Homogeneous
variances. Therefore, a non-parametric test is
advised.
54
Running the Kruskal-Wallis test

• Analyze ? Nonparametric Tests ?
• K-Independent Samples...

Define the range of the independent Var 1-4
levels of soya meals
Exact...
Jonckheere-Terpstra
If you have, choose 'Monte Carlo'
If you have, select 'Jonckheere-Terpstra'. This
is for a linear trend in the data
55
Output Kruskal-Wallis Test
Mean Ranks for all levels of '' of soya meals'
Main Test Statistics H (here, called
Chi-Square)? H is significant If you have
requested 'Monte Carlo', the result will also be
displayed here.

• ? Number of soya meals has a significant effect
  on sperm count, overall.
• However, we do not know where the difference is
  exactly located.

56
Boxplots for the 4 groupsGraphs ? Boxplots

• Visual inspection
• The Medians for groups 1-3
• seem rather similar however,
• the Median for group 4
• seems somewhat lower

How can we know which particular difference(s)
brought about the overall difference?
57
1. Posthoc Tests for Kruskal-Wallis

• 1. Posthoc tests in nonparametric tests can be
  done with the Mann-Whitney test (for pairs of
  unrelated samples).
• If we want to do Posthoc tests, we risk inflating
  Type I error.
• In order to correct for family-wise error
  inflation, we may use the Bonferroni correction.
  However, then we loose power.
• 2. Posthoc tests in nonparametric tests can be
  done by hand

58
1. Posthoc Tests for Kruskal-Wallis

• ? Compromise do only a few promising
  comparisons, e.g. Each level against the control
  condition (as in 'simple' contrasts)?
• Test 1 no soya vs. 1 soya meal
• Test 2 no soya vs. 4 soya meals
• Test 3 no soya vs. 7 soya meals
• With 3 tests, we have to divide our ?-level by 3,
• .05/3 .0167
• So we are doing our Posthoc tests on this more
  rigorous level.

59
1. Single Mann-Whitney tests forthe three
comparisons
Analyze ? Nonparametric Tests ? 2-Independent
tests, define the groups in the grouping
variables window group 1 vs 2 1 vs 3 1 vs 4.
(Here, the contrast 1 vs 4 is requested.)? Each
Mann-Whitney test is carried out indepdently
60
1. Output of the Single Mann-Whitney tests for
the three comparisons
Group 1 vs 2 (no vs 1 soya meal) n.s
Group 1 vs 3 (no vs 4 soya meals) n.s
Group 1 vs 4 (no vs 7 soya meals)
? Eating only 1 to 4 soya meals a week does not
affect number of sperms as compared to not eating
soya meals at all. However, eating 7 soya meals a
week significantly diminishes number of sperms.
61
2. Posthoc Tests in nonparametric tests(for
nerds)?

• You can also calculate the differences for all
  pairs of contrasts by hand.
• You take the difference between the mean ranks of
  the different groups and compare them to a value
  based on the value of z (corrected for the number
  of comparisons you make) and a constant based on
  the total sample size and the sample size in the
  2 groups being compared.
• ??Ru - ?Rv??z??k(k-1) ? N(N1) /12 ((1/nu)
  1/nv))?

K number of groups (4)? N total sample size
(80)? nu number of subj in 1st group (20)? nv
number of subj in 2nd group (20)?

• Difference between the
• mean rank of the 2 groups,
• ignoring the /- sign only the
• ?absolute value? is considered

62
Determining the critical difference for z

• ??Ru - ?Rv??z??k(k-1) ? N(N1) /12 ((1/nu)
  1/nv))?
• In order to know the value for z??k(k-1) , we
  need to determine the ??level. Normally, it is
  .05. This level needs to be divided by 12 which
  is k(k-1) where k is the number of groups, that
  is, 4x312. The ??level therefore is .05/12
  .00417. Now, z??k(k-1) means 'the value of z for
  which only .00417 other values of z are bigger'.
• Looking up in Appendix A.1 (normal
  z-distribution) the smaller portion for .00417
  (actually, .00145), we find the value of z2.64.
  This is the crit value.

63
Determining the critical difference for z

• ??Ru - ?Rv??z??k(k-1) ? N(N1) /12 ((1/nu)
  1/nv))?
• crit. Diff 2.64 ? (80(801)/12) (1/20 1/20)?
• crit. Diff 2.64 ? 540(0.1)
• crit. Diff 2.64 ? 54
• crit. Diff 19.4
• Since sample sizes for all groups are identical,
  this value holds for all comparisons.
• We now can test the actual differences in mean
  ranks for all comparisons against this critical
  difference. If a value is bigger, then the
  comparison is significant.

64
Testing individual differences in mean rank
against the critical difference (19.4)?

• According to this calculation, none of the
  differences is significant! However, in the
  previous Mann-Whitney test the 'No meals 7
  meals' had been significant. How come?

65
Significant or ns comparisons?

• In the old calculation we had to divide our
  overall ? level into 3 portions.
• In the old M-W test we had only conducted 3
  comparisons which yields a corrected ? of
  .05/3.0167. The ? of the 'no vs 7 meals'
  comparison had been .009 which is smaller than
  .0167.
• In the new comparison, however, we have an ? of
  .05/6 (for all 6 comparisons) .0083. Now .009
  .0083, hence the comparison is n.s.
• ? Better carry out only a few reasonable
  comparisons

66
Testing for trends the Jonckheere-Terpstra test

• This test looks at the differences between the
  medians of the groups, just as the
  Kruskall-Wallis test does.
• Additionally, it includes information about
  whether the medians are ordered.
• In our example, we predict an order for the
  number of sperms in the 4 groups, indeed
• no meal 1 meal 4 meals 7 meals
• In the coding variable, we have already encoded
  the order which we expect (1234)?

67
Output of the J-T test
If you have J-T in your version of SPSS,
it would look like this
Z-score (912-1200)/116.33-2.476
J-T test should always be 1-tailed (since we have
a directed hypo!) We compare -2.47 against 1.65
which is the z-value for an ?-level of 5 for a
1- tailed test. Since 2.471.65 the result is
significant. The negative sign means that medians
are in descending order (a positive sign would
have meant ascending order).
68
Calculating effect sizes

• Calculate only effect sizes for single focused
  comparisons
• r z
• ?2n
• rNoSoya 1 meal -0.243/?40 -.04
• rNoSoya 4 meal -0.325/?40 -.05
• rNoSoya 7 meal -2.597/?40 -.41
• rJonckheere -2.47/??0 -.28

Neglegible effects
Negligible effects
Medium effects
Medium effects
69
Reporting the results of the Kruskal-Wallis Test
(Field_2005_556)?

• Sperm counts were significantly affected by
  eating soya meals (H(3) 8.66, p Mann-Whitney Tests were used to follow up this
  finding. A Bonferroni corrrection was applied and
  so all effects are reported at a .0167 level of
  significance. It appeared that sperm counts were
  no different when one soya meal (U191, r-.04)
  or four soya meals (U188, r -.05) were eaten
  per week compared to none. However, when seven
  soya meals were eaten per week, sperm counts were
  significantly lower than when no soya was eaten
  (U104, r-.41).

70
Terminology
71
Differences between several related groups
Friedman's ANOVA

• Friedman's ANOVA is the non-parametric analogue
  to a repeated measure ANOVA (see chapter 11)
  where the same subjects have been subjected to
  various conditions.
• Example here Testing the effect of a new diet
  called 'Andikins diet' on n10 women. Their
  weight (in kg) was tested 3 times
• Start
• Month 1
• Month 2
• Would they loose weight in the course of the diet?

http//goc-frankfurt.de/images/dualit/personenwaag
e_450.jpg
72
Theory of Friedman's ANOVA

• Subject's weight on each of the 3 dates is listed
  in a separate column. Then ranks for the 3 dates
  are determined and listed in separate columns.
• Then, the ranks are summed up for each Condition
  (Ri)?

Always the 3 scores are compared The
smallest one gets 1, the next 2, and the
biggest one 3.
Diet data with ranks
73
The Test statistic Fr

• From the sum of ranks for each group, the test
  statistic Fr is derived
• k
• Fr 12/Nk (k1) Si1 R2i - 3N(k1)?
• (12/(10x3)(31)) (192 202 212))
  (3x10)(31)?
• 12/120 (361400441) 120
• 0.1 (1202) 120
• 120.2 - 120 0.2

74
Data Input and provisional analysis (using)
diet.sav

• First, test for normality
• Analyze ? Descriptive Statistics ? Explore, tick
  'Normality plots with tests' in the 'Plots' window

Data sheet
In the Shapiro-Wilk test (which is more accurate
than the K-S Test, two groups (Start, 1 month)
show non-normal distributions. This violation of
a parametric constraint justifies the choice of a
non-para-metric test.
75
Running Friedman's ANOVA

• Analyze ? Non-parametric Tests ? K Related
  Samples...

If you have 'Exact', tick 'Exact and limit
calculation time to 5 minutes.
Other options
Other options
Exact...
Request everything there is - it is not much...
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Other options

• Kendall's W Similar to Friedman's ANOVA, but
  looks specifically at agreement between raters.
  For example to what extent (from 0-1) women rate
  Justin Timberlake, David Beckham, or Tony Blair
  on their attractiveness. This is like a
  correlation coefficient.
• Cochran's Q This is an extension of NcNemar's
  test. It is like a Friedman's test for
  dichotomous data. For example, if women should
  judge whether they would like to kiss Justin
  Timberlake, David Beckham, or Tony Blair and they
  could only answer Yes or No.

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Output from Friedman's ANOVA
The F-Statistics is called Chi-Square, here. It
has df2 (k-1, where k is the of groups). The
statistics is n.s.
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Posthoc tests for Friedman's ANOVAWilcoxon
signed-rank tests but correcting for the numbers
of tests we do, here ? .05/3.0167.
Analyze ? Nonparametric Tests ? 2-Related Tests,
tick 'Wilcoxon', specify the 3 pairs of groups
Mean ranks and sum of ranks for all 3 comparisons
So, actually, we do not have to calculate any
further...
All comparisons are ns, as expected from the
overall ns effect.
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Posthoc tests for Friedman's ANOVA- calculation
by hand

• We take the difference between the mean ranks of
  the different groups and compare them to a value
  based on the value of z (corrected for the of
  comparions) and a constant based on the total
  sample size (n10) and the of conditions (k3)?
• ??Ru - ?Rv??z??k(k-1) ? k(k1)/6N
• z??k(k-1) .05/3(3-1) .00833
• If the difference is significant, it should have
  a higher value than the value of z for which only
  .00833 other values of z are bigger. As before,
  we look in the Appendix A.1 under the column
  Smaller Portion. The number corresponding to
  .00833 is the critical value it is between 2.39
  and 2.4.

k(k-1) 3 (3-1) 6
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Calculating the critical differences

• Critical difference z??k(k-1) ? k(k1)/6N
• crit. Diff 2.4 ? (3(31)/6x10
• crit. Diff 2.4 ? 12/60
• crit. Diff 2.4 ? 0.2
• crit. Diff 1.07
• ? If the differences between mean ranks are ?
  the critical difference 1.07, then that
  difference is significant.

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Calculating the differences between mean ranks
for diet data

• ? None of the differences is ? the critical
  difference 1.07, hence none of the comparisons is
  significant.

82
Calculating the effect size

• Again, we will only calculate the effect sizes
  for single comparisons

• r z
• ?2n
• rStart 1 month -0.051/??? -.01
• rStart 2 months -0.255/??0 -.06
• r1 month 2 months -0.153/??0 -.03

Tiny effects
Tiny effects
Tiny effects
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Reporting the results of Friedman's ANOVA
(Field_2005_566)?

• The weight of participants did not significantly
  change over the 2 months of the diet (?2(2)
  0.20, p .05). Wilcoxon tests were used to
  follow up on this finding. A Bonferroni
  correction was applied and so all effects are
  reported at a .0167 level of significance. It
  appeared that weight didn't significantly change
  from the start of the diet to 1 month, T27,
  r-.01, from the start of the diet to 2 months,
  T25, r-.06, or from 1 month to 2 months,
  T26,r-0.3. We can conclude that the Andikinds
  diet (...) is a complete failure.

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Summary Terminology