simple mathematics in psychological research
TRANSCRIPT
-
8/6/2019 Simple Mathematics in Psychological Research
1/22
Simple Mathematics in
Psychological Research
Dr George Varvatsoulias
-
8/6/2019 Simple Mathematics in Psychological Research
2/22
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
-
8/6/2019 Simple Mathematics in Psychological Research
3/22
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
-
8/6/2019 Simple Mathematics in Psychological Research
4/22
One-variable G (Chi-square) or goodness-
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
-
8/6/2019 Simple Mathematics in Psychological Research
5/22
One-variable G (Chi-square) or goodness-
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
-
8/6/2019 Simple Mathematics in Psychological Research
6/22
One-variable G (Chi-square) or goodness-
of-fit test
We square all the numbers
7.5=56.25
32.5=1056.25
-17.5=306.25-7.5=56.25
-
8/6/2019 Simple Mathematics in Psychological Research
7/22
One-variable G (Chi-square) or goodness-
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)
-
8/6/2019 Simple Mathematics in Psychological Research
8/22
One-variable G (Chi-square) or goodness-
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
-
8/6/2019 Simple Mathematics in Psychological Research
9/22
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)
-
8/6/2019 Simple Mathematics in Psychological Research
10/22
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
-
8/6/2019 Simple Mathematics in Psychological Research
11/22
Critical values at the 5% level for one-
tailed test
-
8/6/2019 Simple Mathematics in Psychological Research
12/22
Critical values at the 5% level for two-
tailed test
-
8/6/2019 Simple Mathematics in Psychological Research
13/22
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
-
8/6/2019 Simple Mathematics in Psychological Research
14/22
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...
-
8/6/2019 Simple Mathematics in Psychological Research
15/22
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
-
8/6/2019 Simple Mathematics in Psychological Research
16/22
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
-
8/6/2019 Simple Mathematics in Psychological Research
17/22
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
-
8/6/2019 Simple Mathematics in Psychological Research
18/22
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
-
8/6/2019 Simple Mathematics in Psychological Research
19/22
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)
-
8/6/2019 Simple Mathematics in Psychological Research
20/22
Under which preconditions, experimental
designs qualify for the above tests (2)
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
-
8/6/2019 Simple Mathematics in Psychological Research
21/22
Under which preconditions, experimental
designs qualify for the above tests (3)
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
-
8/6/2019 Simple Mathematics in Psychological Research
22/22
Under which preconditions, experimental
designs qualify for the above tests (4)
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