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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
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.
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.
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.
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).
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.