Term 2 L7: Chi square and Contingency Tables

Question 1

Two types of chi square test:

Answer 1

for goodness of fit;

of independence

Question 2

What level of data are chi square

Answer 2

The tests are often useful for

nominal or ordinal variables.

Question 3

Both tests designed to test Hypotehses about:

Answer 3

frequencies (counts).

Question 4

Goodness of Fit - what is it answering?

Answer 4

whether the deviation of observed results from expected values is large enough for us to reject the null hypothesis?

Is the difference between observed and expected values statistically significant?

Question 5

The Chi-Square (χ2) Statistic (goodness-of-fit test)

Answer 5

𝜒
2 = ∑(𝑂−𝐸)2 / 𝐸

O - observed frequencies;
E - expected frequencies;
Σ - sum (to be taken over all cells in the table

Question 6

Degrees of freedom for the goodness-of-fit test

Answer 6

df = ‘Number of cells’ – 1.

Question 7

The distribution of the χ

2 statistic depends on….

Answer 7

the associated degrees of

freedom (df, denoted by k in the graph above)

Question 8

Interpreting results of chi square goodness of fit

Answer 8

called a goodness-of-fit test,
because we are testing whether the distribution of a variable (here: accuracy of guesses) fits an expected distribution (under
the null hypothesis).

In this case, we saw that the data do not fit the expectation. Practitioners of Therapeutic Touch performed worse than their
theory would have led to suppose (and also worse than chance)

Question 9

How do you report the result of chi square statstic?

Answer 9

χ2 (1) = 4.128

Question 10

Chi-square test of independence

Analysis of Contingency Tables

Answer 10

Contingency tables are frequently used in the analysis of

categorical variables

Question 11

what does Chi-square test of independence test?

Answer 11

whether the null hypothesis
that the row variable and the column variable are independent
(i.e. that there is no association between the two variables)

Question 12

Give an example of a hypothesis for a Chi square test of independence

Answer 12

Does the table
constitute evidence for an effect of “type of treatment” on the likelihood of suicide attempt, (or could we have obtained the numbers in the table by chance, even if there was no effect in the
population)?

Question 13

Marginals

Answer 13

allow us to work out expected outcomes

and if we have one of the observed values we can caluate all the observed values

Question 14

Degrees of freedom for χ2 test of independence

Answer 14

𝑑𝑓 = (𝑟 − 1) × (𝑐 − 1)

Question 15

What to do with small expected frequencies? for test of independence

SPSS vs Howell what are their minimum values?

Answer 15

Small expected frequencies
The chi-square test can be inaccurate if one or more of the expected values are very small.
One convention is to set a minimum value to 5. This convention is overly cautious.

• For a 2×2 table, a sample size of n=10 will ensure reasonable accuracy (see Howell 2013, p. 152). This corresponds to an average expected value of 2.5!

• For larger tables, all cells should have expected values greater
than 1.

Question 16

Measures of effect size

is chi sqare a measure of effect size?
why?

Answer 16

The chi-square statistic is not a measure of effect size.

Chi-square is sensitive to sample size. That means that you cannot
compare chi-square values computed on different tables, if the sample
sizes of the two tables are different.
• For example, if two samples have the same effect size, but one
sample is twice as large as the other, then the value for chi-square
for the larger sample will be twice as high as that of the smaller
sample.

Also: If the sample is large, the chi-square statistic may be significant
even if the actual effect is very small. This is an issue of power:
• If the sample is large, the chi-square test is very powerful, which
means that it may detect small effects that exist in the population –
including effects that are too small to be meaningful.
• If the sample is small, the test has little power, which means that
even large effects may not lead to statistically significant results.

Question 17

There are several ways to measure the effect size in contingency
tables (4)

Answer 17

Absolute risk reduction:

Using MBT, by how many percentage
points do we reduce the risk of a suicide attempt compared to TaU?

• Numbers Needed To Treat:

How many patients do we need to treat
with MBT to avoid one suicide attempt (compared to TaU)?

Relative risk:

How much higher is the risk of a suicide attempt under
TaU compared to Mentalization-based Treatment MBT?

Odds Ratio:

How much higher are the odds of a suicide attempt
under TaU compared to MBT?

Question 18

Interpreting odds ration

Answer 18

As before, let’s define: 𝑂𝑅 =
𝑂𝑑𝑑𝑠𝑇𝑎𝑈/ 𝑂𝑑𝑑𝑠𝑀𝐵𝑇

Then the observed odds ratio can be interpreted as follows:
OR = 1 → No effect (no difference between groups; odds
are the same).
OR > 1 → “Positive” effect. Higher odds for TaU than MBT.
OR < 1 → “Negative” effect. Lower odds for TaU than MBT.

Examples
OR = 2 → The odds of having a suicide attempt are twice as high under TaU as under MBT.
OR = 1.5 → The odds of having a suicide attempt are 50% higher under TaU compared to MBT.

OR = 0.5 → The odds of having a suicide attempt are halved under TaU compared to MBT.

Question 19

CIs for Effect Sizes: Odds Ratios

What is effect size?
why calculate CIs? (SE)

Answer 19

Effect sizes computed from a sample are estimates of the ‘true’ effect size. Like all statistics, effect size estimates are subject to sampling error. It is therefore good practice to
report confidence intervals for effect size estimates

This means that we are 95% confident that the interval
between .94 and 12.97 contains the true odds ratio. This
confidence interval is rather wide, reflecting the uncertainty of
our estimate.

Question 20

CIs is 1 or 0 an important figure?

Answer 20

1

We are 95% confident that the interval between
0.94 and 12.97 contains the true odds ratio.

• Note that the confidence interval contains the value 1.
• In the case of OR, 1 signifies “no effect”.

• This agrees with the non-significant result of the chi-square test on these data (which led us to conclude that we had no evidence to reject the null hypothesis of ‘no effect’).