Simple Mathematics in Psychological Research

8/6/2019 Simple Mathematics in Psychological Research

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Simple Mathematics in

Psychological Research

Dr George Varvatsoulias


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Starting generally...

Type I error factors:

1. Lenient levels of significance

2. Poor experimental design3. Problems with confounding variables

Type II error factors:

1. Stringent levels of significance

2. Poor experimental design

3. Problems with confounding variables


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One-variable G (Chi-square) or goodness-

of-fit testHypothesis: Same brands of training shoes are not equally

chosen from people

Example: 110 participants were asked to choose a pair of

training shoes of their likeness

Number of people choosing different brands of

training shoes

Group A Group B Group C Group D

20pts 60pts 10pts 20pts


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of-fit test If the null hypothesis is true (the opposite to

the one suggested in the previous slide) thenwe should expect 110/4 to be in each

category, so 110/4=27.5 In other words, equal number of people are

evenly distributed over the training shoes

brands Observed frequencies: 20;60;10;20

Expected frequencies: 27.5;27.5;27.5;27.5


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of-fit testObserved Expected = Difference

20 27.5 = -7.5

60 27.5 = 32.5

10 27.5 = -17.5

20 27.5 = -7.5

Total: 110 Total: 110


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of-fit test

We square all the numbers

7.5=56.25

32.5=1056.25

-17.5=306.25-7.5=56.25


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of-fit testWe divide by a measure of variance: in this case

it is 27.5

56.25/27.5=2.051056.25/27.5=38.41

306.25/27.5=11.14

56.25/27.5=2.05The figures are then added to give a total of

53.65, the Gof which is 53.65 (53.7)


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of-fit test

...finally, the degrees of freedom (df) is one less

than the number of categories, which in thiscase is DF=3 (4-1)

In conclusion, G=53.65, if the null hypothesis

were true, i.e. if all training shoes brands are

preferred equally


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Mann-Whitney test (the value ofU)

Score Rank 1 Score Rank 2

7.00 6 4.00 2.5

15.00 11 6.00 4

14.00 10 11.00 93.00 1 7.00 6

17.00 12 9.00 8

n=5 =40 3 is the lowest

score, so we rank it

as 1

4.00

7.00

2.5

6

Mean rank=8 n=7 =38

Mean rank=5.4

Eg. For the second

rank which is 2.5 (4;

4: ranks

2&3=2+3/2=2.5)


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Mann-Whitney test (the value ofU)

Observed value ofU: The lowest total number ofranks

Critical value ofU: Then lowest total number of

the observed valueIn other words: Observed value U=38

Critical value U=6 (calculatedbythe number ofparticipants in each group)

Since the critical value (6) is less than the observedvalue (38) we can reject the null hypothesis


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Critical values at the 5% level for one-

tailed test


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Critical values at the 5% level for two-

tailed test


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Wilcoxon-test (1)

Participant 1st set ofscores 2nd set of

scores

Difference Rank

1 12 8 4 2

2 7 9 -2 1

3 18 13 5 3

4 33 4 29 7

5 11 19 -8 4

6 25 45 -20 6

7 3 17 -14 5


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Wilcoxon-test (2)

T= is the sum of the ranks of the less frequent

sign, i.e. T= 4+5+29=38

N

= 7, critical value is either 3 or 2 dependingon one- or two-tailed test

since the observed value (38) is greater than

the critical value (3 or 2), the null hypothesis

can be retained...


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Some critical values ofTat the 5%

level for theWilcoxon-testN One-tailed-test Two-tailed test

5 T 0

6 2 0

7 3 2

8 5 3

9 8 5

10 11 8

11 13 10


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Spearmans rank correlation test (rho)

(1)Participan

t

Memory

score

GCSE

score

Rank A Rank B Diff erence

between

rank and

rank B

d

1 18 8 10 2.5 7.5 56.25

2 14 16 5.5 9 -3.5 12.25

3 17 10 9 5 4.0 16.0

4 13 9 4 4 0 0

5 10 15 3 8 -5.0 25.0

6 8 14 1 7 -6.0 36.0

7 15 12 7 6 1 1.0


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Spearmans rank correlation test (rho)

(2)

d (sum of differences squared) = 261.0(hypothetically taken out ofN=10)

Observed value rho = 1- (6 d /N(N-1) = 1-

6X261.0/10X(100-1) = 1-1566/990 = 1-1.58 = -0.58

Critical value for 0ne- or two-tailed test with a N =10 is 0.564 or 0.648

Since the observed value (0.58) is greater thanthe critical value (0.564), the null hypothesis canbe rejected


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Examples of critical values ofrho at 5%

levelN One-tailed test Two-tailed test

4 1.000 -

5 0.900 1.000

6 0.829 0.886

7 0.714 0.786

8 0.643 0.738

9 0.600 0.700

10 0.564 0.648


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Under which preconditions, experimental

designs qualify for the above tests (1)

Spearmans rank correlation test:

1. The hypothesis predicts a correlation

between two variables2. The two sets of data are pairs of scores from

one person or thing = related

3. The data are ordin

al(numbers in order) orinterval(a set containing all points real

numbers between two given endpoints)


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The Man-Whitney U test:

1. The hypothesis predicts a difference between

two sets of data2. The two sets of data are from separate

groups of participants = independentdesigns

3. The data are ordin

alor in

terval


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Chi-square (x):

1. The hypothesis predicts a difference between two

conditions or an association between variables

2. The sets of data must be independent(no individual

should have a score in more than one cell)

3. The data are in frequencies (nominal)

Explanation: The test is unreliable ifexpectedfrequencies fall below 5 in any cell, i.e. we need at

least 20 participants for a 2X2 contingency table


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TheWilcoxon matched pairs signed ranks test

(T):

1. The hypothesis predicts a differenc

e betweentwo sets of data

2. The two sets of data are pairs of scores from

one person (or a matched pair) = related

3. The data are ordinalor interval

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Simple Mathematics in Psychological Research

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