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  3. University Of Cape Town (UCT)
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  5. Research Psychology PSY3007S (PSY3007S)

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Summary Chi-square

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These notes include a brief outline of chi-square tests (goodness-of-fit and test of contigency), as well as a step-by-step guide and Excel example. The step-by-step guide includes stating hypotheses, to analysis of residuals and interpretation.

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  • Tutorial 19.
  • November 12, 2020
  • 5
  • 2020/2021
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Wednesday, 11 November 2020

Statistics

Chi-square

Outline:

1. Goodness-of-fit (GOF),

2. Test of Contingency,

3. Effect size (how different the observed and expected values are?) and analysis of residuals
(where do the significant associations lie?),

4. Step-by-step guide.



What is chi-square:

• A test that is able to analyse frequencies,

• It deals with categorical variables (gender, sex) and sometimes ordinal.



Types of chi-square tests:

1. Goodness-of-fit

- Used when you would like to ask if what you observe ‘fits’ with what you expected to see —
concerned with unidimensional data,

- E.g. are as many people right-handed as we expect?
2. Test of contingency

- Used when you would like to ask if one variable is associated with another (or ore).
Concerned with multi-dimensional data,

- E.g. is handedness associated with creativity?


Assumptions for a goodness-of-fit test:

1. Categories should be exhaustive and mutually exclusive,

2. Observations must be independent — data must come from different participants.



Assumptions for test of contingency:

1. Categories should be exhaustive and mutually exclusive,



1

, Wednesday, 11 November 2020

2. Observations must be independent — data must come from different participants,

3. You must have at least 20 data points and a minimum of 5 observations per cell for
expected values — you need a sufficiently big sample size.



Important to note:

- P-values are not enough to tell you about significant — you need to look at it in conduction
with the effect size (Cramer’s V),

- P-values cannot tell you where the association is — you need to do this through analysis of
residuals.



What is a residual?

- It is the deviation of the observed from the expected frequency,
- Residual = O - E.
However, the size of the deviation is related to the size of the sample. Therefore, cells with
larger E values have larger residuals. To account for this, we standardise the residuals.
Standardised residuals indicate each cell’s relative contribution to the significant of the chi-
square value.




Following the above, we need to calculate the adjusted residual — this is because it
underestimates the size of variance.




- e = standard residual
- nrow = row total
- ntotal = grand total
- ncol = column total

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