1. Overview of Lecture Parametric vs Non-Parametric Statistical Tests.
Single Sample Chi-Square
Multi-Sample Chi-Square
Analysing Chi-Square Residuals

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.

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.

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.

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).

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

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

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

9. Single Sample Chi-Square Statistic - Method

10. Single Sample Chi-Square Statistic - Formula
The test statistic, , is calculated by:
Where
is the observed frequency
is the expected frequency

11. c = ( - 4 ) 14 + 9 16 1 Observed-Expected Frequencies 2
Research
Academic
Clinical
Occupational
Educational
Total
Job
Observed
Frequency
Expected
Observed - 10 5 30 15 70 14 -4 -9 16 1 c 2 = ( - 4 ) 14 + 9 16 1
Expected Frequencies

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

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

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

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

16. Multi-Sample Chi-Square Statistic - Method

17. Multi-Sample Chi-Square Statistic - Formula
The test statistic, , is calculated by:
Where
is the observed frequency
is the expected frequency

18. c = ( - 5 ) 15 + . 7 10 20 17 Observed-Expected Frequencies 2
Research
Academic
Clinical
Occupational
Educational
Total
Job
Females
Expected
Frequency
Observed -
Males 10 5 30 15 70
7.5 20
17.5 -5
-2.5
2.5
-10 40 35
140 c 2 = ( - 5 ) 15 + . 7 10 20 17
Expected Frequencies

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

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

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

22. Analysing Residuals - Formula
A residual is calculated by:
Where
is the observed frequency
is the expected frequency

23. Analysing Residuals - Interpretation
When a residual  |±1.96|
There is a significance difference between the observed and expected frequencies

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.