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**Overview of Lecture Parametric vs Non-Parametric Statistical Tests.**

Single Sample Chi-Square Multi-Sample Chi-Square Analysing Chi-Square Residuals

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**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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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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**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.

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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).

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**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

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**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

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**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

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**Single Sample Chi-Square Statistic - Method**

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**Single Sample Chi-Square Statistic - Formula**

The test statistic, , is calculated by: Where is the observed frequency is the expected frequency

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**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

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**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

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**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

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**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

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**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

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**Multi-Sample Chi-Square Statistic - Method**

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**Multi-Sample Chi-Square Statistic - Formula**

The test statistic, , is calculated by: Where is the observed frequency is the expected frequency

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**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

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**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

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**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

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**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

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**Analysing Residuals - Formula**

A residual is calculated by: Where is the observed frequency is the expected frequency

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**Analysing Residuals - Interpretation**

When a residual |±1.96| There is a significance difference between the observed and expected frequencies

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**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.

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