Chi Square- Contingency and Analysis of residuals Flashcards
What are three things to know about the Chi Square test for contingency?
- It asks whether observed frequencies reflect the independence of two qualitative variables
- It compares the actual observed frequencies of some phenomenon (in our sample) with the frequencies we would expect if there were no relationship at all between the two variables in the larger population.
- Two variables are independent if knowledge of the value of one variable provides no information about the value of another.
What are the steps for the chi square test of contingency?
- State the research question
- State the statistical hypothesis (H1 and H0). E.g. there is an association or there is no association.
- Set the decision rule (Reject H0 if…)
- Calculate the degrees of freedom= (c-1)*(r-1)
- Calculate the test statistic using the chi square formula.
- Decide if the result is significant.
- Interpret the results in terms of expectations.
What is the formula for calculating the expected values in a multidimensional table?
(row total/overall total)*column total.
What do we know about isolating the source of association?
- A significant chi tells us that there is an association between the variables, however it does not tell us where the association comes from
- We need to determine in which cell(s) the significance comes from.
- To locate the source of significance we compute the residual.
What is a residual in a contingency table?
It is the deviation of the observed frequency from the expected frequency.
What is the relationship between the size of the deviation and the size of the sample? And what does this mean for our analysis?
Cells with the largest expected values have the largest residuals, therefore the residual needs to standardized in order to control for sample/cell size.
What is the standardised residual?
The standardised residual is a residual that controls for the size of the expected cell by dividing by the root of the expected frequency.
e=(O-E)/sqrt(E)
What does the standardised residual indicate?
- The standardised residual indicates each cell’s relative contribution to the significance of the chi square value.
- It also shows you how far the observed value is from the expected value in terms of standard deviations.
What does the size of a standard residual indicate?
- Larger standardised residuals indicate a stronger contribution to the chi-squre statistic.
What does the sign (+ or -) of a standardised residual indicate?
- Positive values indicate that the observed count is greater than expected.
- Negative values indicate that the observed count is less than expected.
Why can’t we determine significance from the standardised residual?
- the standard normal distribution: mean=0, stdv=1
- the standardised residuals have a stdv less than 1, so it underestimates the size of the variance.
- therefore we look at the adjusted residual to determine the significance of the contribution of the observed values.
What is the adjusted residual?
d= e/sqrt(1-nrow/ntotal)*(1-ncol/ntotal)
How are adjusted residuals interpreted?
- Adjusted residuals are interpreted in the same way as z-scores.
- The critical value for z when a=0.05 is 1.96 or -1.96
- Therefore, any adjusted residual larger than 1.96 or smaller than -1.96 is statistically significant.
How do you word your analysis of residuals?
- The number of heart attacks were significantly less than expected in those who took aspirin.
- The number of heart attacks were significantly more than expected in those who took placebo.
- A significantly higher than expected number of people with no heart attack took aspirin.
- A significantly lower than expected number of people with no heart attack took placebo.
Note on the interpreting adjusted residuals.
- Interpret the strength and direction of association using the standardized residuals.
- Significance should be interpreted in terms of what was expected.
