c82mcp diploma statistics school of psychology university of nottingham 1 overview of lecture...
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![Page 1: C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Overview of Lecture Parametric vs Non-Parametric Statistical Tests. Single Sample](https://reader035.vdocuments.us/reader035/viewer/2022062619/5515d08e550346dd6f8b45ec/html5/thumbnails/1.jpg)
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
1
Overview of Lecture
• Parametric vs Non-Parametric Statistical Tests.• Single Sample Chi-Square• Multi-Sample Chi-Square• Analysing Chi-Square Residuals
![Page 2: C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Overview of Lecture Parametric vs Non-Parametric Statistical Tests. Single Sample](https://reader035.vdocuments.us/reader035/viewer/2022062619/5515d08e550346dd6f8b45ec/html5/thumbnails/2.jpg)
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
2
Parametric Vs Non-Parametric Statistical Tests.
• Many statistical tests make assumptions about the population from which the scores are taken.
• The most common assumption is that the data is normally distributed.
• Some statistical tests don't make assumptions about the population from which the scores are taken.
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C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
3
Parametric Tests
• Parametric tests test hypotheses about specific parameters such as the mean or the variance.
• They make the assumption that these parameters are central to our research hypotheses.
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C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
4
Parametric Test Assumptions.
• Parametric tests usually (thought not always) make the following assumptions:• The scores must be independent. In other words the
selection of any particular score must not bias the chance of any other case for inclusion.
• The observations must be drawn from normally distributed populations.
• The populations (if comparing two or more groups) must have the same variance.
• The variables must have been measured in at least an interval scale so that is is possible to interpret the results.
![Page 5: C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Overview of Lecture Parametric vs Non-Parametric Statistical Tests. Single Sample](https://reader035.vdocuments.us/reader035/viewer/2022062619/5515d08e550346dd6f8b45ec/html5/thumbnails/5.jpg)
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
5
Non-Parametric Tests
• Non-parametric tests on the other hand are based on a statistical model that has only very few assumptions.
• None of these assumptions include making assumptions about the form of the population distribution from which the sample was taken.
• Whenever we look at categorical or ordinal data we usually use non-parametric tests.
• Furthermore, if we can show that the data is not normally distributed we should also use non-parametric tests (but there are exceptions).
![Page 6: C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Overview of Lecture Parametric vs Non-Parametric Statistical Tests. Single Sample](https://reader035.vdocuments.us/reader035/viewer/2022062619/5515d08e550346dd6f8b45ec/html5/thumbnails/6.jpg)
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
6
Nominal/Categorical Scale Data
• Numbers are used to divide different behaviours into different classes without implying that the different classes are numerically related to each other.
• Whenever we look at nominal or categorical data we usually use non-parametric tests
• These non-parametric tests focus on the frequencies or counts of membership of categories or nominal groups
![Page 7: C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Overview of Lecture Parametric vs Non-Parametric Statistical Tests. Single Sample](https://reader035.vdocuments.us/reader035/viewer/2022062619/5515d08e550346dd6f8b45ec/html5/thumbnails/7.jpg)
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
7
Single Sample Chi-Square Statistic - Rationale
• When the Null Hypothesis is true• The observed differences in frequencies will be due to
chance• When the Null Hypothesis is false
• The differences in frequencies will reflect actual differences in the population
![Page 8: C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Overview of Lecture Parametric vs Non-Parametric Statistical Tests. Single Sample](https://reader035.vdocuments.us/reader035/viewer/2022062619/5515d08e550346dd6f8b45ec/html5/thumbnails/8.jpg)
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
8
Single Sample Chi-Square Statistic - Method
• Arrange the data in a table• Each category has a separate entry• The number of members of each category are counted
• Calculate the frequencies expected by chance• Find the difference between the observed & expected
frequencies
![Page 9: C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Overview of Lecture Parametric vs Non-Parametric Statistical Tests. Single Sample](https://reader035.vdocuments.us/reader035/viewer/2022062619/5515d08e550346dd6f8b45ec/html5/thumbnails/9.jpg)
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
9
Single Sample Chi-Square Statistic - Method
Research
Academic
Clinical
Occupational
Educational
Total
Job Observed Frequency
Expected Frequency
Observed - Expected
10
5
30
15
10
70
14
14
14
14
14
-4
-9
16
1
-4
ExpectedTotal Number of CasesNumber of Categories
![Page 10: C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Overview of Lecture Parametric vs Non-Parametric Statistical Tests. Single Sample](https://reader035.vdocuments.us/reader035/viewer/2022062619/5515d08e550346dd6f8b45ec/html5/thumbnails/10.jpg)
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
10
Single Sample Chi-Square Statistic - Formula
• The test statistic, , is calculated by:
• Where is the observed frequency is the expected frequency
2 ( fo fe)
fe
2
fo
fe
2
![Page 11: C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Overview of Lecture Parametric vs Non-Parametric Statistical Tests. Single Sample](https://reader035.vdocuments.us/reader035/viewer/2022062619/5515d08e550346dd6f8b45ec/html5/thumbnails/11.jpg)
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
11
Research
Academic
Clinical
Occupational
Educational
Total
Job Observed Frequency
Expected Frequency
Observed - Expected
10
5
30
15
10
70
14
14
14
14
14
70
-4
-9
16
1
-4
2( 4)214
( 9)2
14 (16)2
14(1)2
14 ( 4)2
14
Expected Frequencies
Observed-Expected Frequencies
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C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
12
Single Sample Chi-Square Statistic - Significance• In order to test the null hypothesis that the distribution of
frequencies is equal (i.e. occurred by chance) we look up a critical value of chi-square in tables
• To do this we need to know the degrees of freedom associated with the chi-square• degrees of freedom = number of categories-1
• We reject the null hypothesis when
• For this data we can reject the null hypothesis
observed2 critical
2
![Page 13: C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Overview of Lecture Parametric vs Non-Parametric Statistical Tests. Single Sample](https://reader035.vdocuments.us/reader035/viewer/2022062619/5515d08e550346dd6f8b45ec/html5/thumbnails/13.jpg)
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
13
Single Sample Chi-Square Statistic - Interpretation• Rejecting the null hypothesis
• This means that the frequencies associated with each of the categories did not represent only chance fluctuations in the data
• Failing to reject the null hypothesis• This means that the differences in the frequencies
associated with each of the categories was due to chance fluctuations in the data
![Page 14: C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Overview of Lecture Parametric vs Non-Parametric Statistical Tests. Single Sample](https://reader035.vdocuments.us/reader035/viewer/2022062619/5515d08e550346dd6f8b45ec/html5/thumbnails/14.jpg)
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
14
Multi-Sample Chi-Square Statistic - Rationale
• Used when we look at the relationship between two independent variables and their effects on frequencies
• Under the null hypothesis• The differences in the observed frequencies are due to
chance• When the null hypothesis is false
• The difference in the observed frequencies are due to the effects of the two variables
![Page 15: C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Overview of Lecture Parametric vs Non-Parametric Statistical Tests. Single Sample](https://reader035.vdocuments.us/reader035/viewer/2022062619/5515d08e550346dd6f8b45ec/html5/thumbnails/15.jpg)
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
15
Multi-Sample Chi-Square Statistic - Method
• Arrange the data in a table• Each category has a separate entry• The number of members of each category are counted
• Calculate the frequencies expected by chance• Find the difference between the observed & expected
frequencies
![Page 16: C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Overview of Lecture Parametric vs Non-Parametric Statistical Tests. Single Sample](https://reader035.vdocuments.us/reader035/viewer/2022062619/5515d08e550346dd6f8b45ec/html5/thumbnails/16.jpg)
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
16
Multi-Sample Chi-Square Statistic - Method
Research
Academic
Clinical
Occupational
Educational
Total
Job Females Expected Frequency
Observed - Expected
Males Expected Frequency
Observed - Expected
Total
10
5
30
15
10
70
15
7.5
20
17.5
10
-5
-2.5
10
-2.5
0
20
10
10
20
10
70
15
7.5
20
17.5
10
5
2.5
-10
2.5
0
30
15
40
35
20
140
ExpectedRow Total x Column TotalGrand Total
![Page 17: C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Overview of Lecture Parametric vs Non-Parametric Statistical Tests. Single Sample](https://reader035.vdocuments.us/reader035/viewer/2022062619/5515d08e550346dd6f8b45ec/html5/thumbnails/17.jpg)
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
17
Multi-Sample Chi-Square Statistic - Formula
• The test statistic, , is calculated by:
• Where is the observed frequency is the expected frequency
2
2 ( fo fe)
fe
2
fo
fe
![Page 18: C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Overview of Lecture Parametric vs Non-Parametric Statistical Tests. Single Sample](https://reader035.vdocuments.us/reader035/viewer/2022062619/5515d08e550346dd6f8b45ec/html5/thumbnails/18.jpg)
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
18
Research
Academic
Clinical
Occupational
Educational
Total
Job Females Expected Frequency
Observed - Expected
Males Expected Frequency
Observed - Expected
Total
10
5
30
15
10
70
15
7.5
20
17.5
10
-5
-2.5
10
-2.5
0
20
10
10
20
10
70
15
7.5
20
17.5
10
5
2.5
-10
2.5
0
30
15
40
35
20
140
2( 5)215
( 2.5)2
7.5 (10)2
20......( 10)2
20 (2.5)2
17.5 (0)2
10
Observed-Expected Frequencies
Expected Frequencies
![Page 19: C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Overview of Lecture Parametric vs Non-Parametric Statistical Tests. Single Sample](https://reader035.vdocuments.us/reader035/viewer/2022062619/5515d08e550346dd6f8b45ec/html5/thumbnails/19.jpg)
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
19
Multi-Sample Chi-Square Statistic - Significance
• In order to test the null hypothesis that the distribution of frequencies is equal (i.e. occurred by chance) we look up a critical value of chi-square in tables
• To do this we need to know the degrees of freedom associated with the chi-square• degrees of freedom = (rows-1)(columns-1)
• We reject the null hypothesis when
• For this data we can reject the null hypothesis
observed2 critical
2
![Page 20: C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Overview of Lecture Parametric vs Non-Parametric Statistical Tests. Single Sample](https://reader035.vdocuments.us/reader035/viewer/2022062619/5515d08e550346dd6f8b45ec/html5/thumbnails/20.jpg)
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
20
Multi-Sample Chi-Square Statistic - Interpretation• Rejecting the null hypothesis
• This means that the frequencies associated with cell in the design did not represent only chance fluctuations in the data.
• Failing to reject the null hypothesis• This means that the differences in the frequencies
associated with each cell in the design was due to chance fluctuations in the data
![Page 21: C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Overview of Lecture Parametric vs Non-Parametric Statistical Tests. Single Sample](https://reader035.vdocuments.us/reader035/viewer/2022062619/5515d08e550346dd6f8b45ec/html5/thumbnails/21.jpg)
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
21
Chi-Square Statistic - Analysing Residuals
• Since the Chi-Square Statistic is calculated using all the information from the experiment:• it tell us that at least one of the cell frequencies is
different from chance• it cannot tell which cell frequency is different from chance
• To find out which cells differ from what we would expect by chance we analyse the residuals• residuals - what is left over after we have removed the
effect of chance
![Page 22: C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Overview of Lecture Parametric vs Non-Parametric Statistical Tests. Single Sample](https://reader035.vdocuments.us/reader035/viewer/2022062619/5515d08e550346dd6f8b45ec/html5/thumbnails/22.jpg)
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
22
Analysing Residuals - Formula
• A residual is calculated by:
• Where is the observed frequency is the expected frequency
Residual( fo fe)
fe
fo
fe
![Page 23: C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Overview of Lecture Parametric vs Non-Parametric Statistical Tests. Single Sample](https://reader035.vdocuments.us/reader035/viewer/2022062619/5515d08e550346dd6f8b45ec/html5/thumbnails/23.jpg)
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
23
Analysing Residuals - Interpretation
• When a residual |±1.96|• There is a significance difference between the observed
and expected frequencies
![Page 24: C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Overview of Lecture Parametric vs Non-Parametric Statistical Tests. Single Sample](https://reader035.vdocuments.us/reader035/viewer/2022062619/5515d08e550346dd6f8b45ec/html5/thumbnails/24.jpg)
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
24
Pearson's Chi-Square - Assumptions
• The categories must be mutually exclusive. In other words no single subject can contribute a score to more than one category.
• The observations must be independent. A particular score cannot influence any other score.
• Both the observed and the expected frequencies must be greater than or equal to 5.