chi-square test (test 1)

the difference between two population proportions: chi-square test

• remember...the most common research design encountered in biomedical research involves the comparison of outcomes in two groups

• when the outcome is dichotomous, we are usually interested in comparing the proportions from two groups (or populations) and asking whether we have sufficient evidence to conclude that they are different.

• example: we may ask whether the proportion experiencing a cure in a group taking a new medication is different than the proportion experiencing a cure in another group taking an older medication

when are chi-square tests appropriate?

when the outcome is discrete (dichotomous, ordinal, or categorical)

• data has been counted and divided into categories

pearson chi square test

groups are independent (or unrelated to one another)

• we can use the chi-square test to help us calculate p values for hypothesis testing.

• we can also construct confidence intervals for the difference between two population proportion

when examining the relationship between 2 variables and both variables are nominal

chi square test of independence (or association)

assess whether 2 nominal variables of any number of categories (2 or more) are independent

chi square test of homogeneity

appropriate statistical test to assess whether 2 proportions are =

chi square homogeneity and independence are

interchangeable (identical mathematically)

• When we do a chi-square test, there are certain ____ we make to ensure the validity of the results of our hypothesis testing procedure:

• The observations within each group and between each group are ____, meaning that knowing the value of any one observation tells us nothing about the value of another observation.

• The sample sizes are “____ ____” in each group

• assumptions

• independent

• large enough

relative risk RR

Incidence Proportion (Exposed) / Incidence Proportion (Unexposed)

odds ratio OR

Odds (Exposed) / Odds (Unexposed)

absolute risk reduction ARR

Incidence Proportion (Exposed) - Incidence Proportion (Unexposed)

relative risk reduction RRR

[Incidence Proportion (Exposed) - Incidence Proportion (Unexposed)] / [Incidence Proportion (Unexposed)]

2 independent samples

• chi square

• fisher’s exact

related or paired samples

mcnemar’s test

3 or more independent samples

chi square for k independent samples

3 or more related samples

cochran q

fishers exact test

• One of our assumptions when comparing two proportions using the Pearson chi-square test was that the sample sizes are “large enough” in each group.

  • “Large enough” usually means that the expected frequency of each cell (i.e., under the null hypothesis of independence) is at least 5.

  • The tests that we have talked about so far don’t work well when this assumption is not met.

• Sir Ronald Fisher developed a test that can be used when the sample size requirements are not met.

• This test has come to be called Fisher’s exact test.

  • It is called exact because we don’t have to rely on approximations, but rather we can calculate the exact probability of obtaining the observed results or results that are more extreme.

  • It can be used to test the same hypotheses we discussed earlier (i.e., two nominal variables are independent, two nominal variables are not associated, proportions are the same in two different groups, the populations are homogeneous, odds are the same, risk is the same, etc.).

mcnemar’s test and cochran’s q test

• Recall that repeated-measures ANOVA, like the paired t test, can be used to test the equality of means when all members of a random sample are measured under a number of different conditions.

• The difference between repeated-measures ANOVA and the paired t test is that with the paired t test we are comparing two means and with repeated-measures ANOVA we are comparing 3 or more means.

• Similarly, the difference between Cochran’s Q test and McNemar’s test is that with McNemar’s test we are comparing two proportions and with Cochran’s Q test we are comparing 3 or more proportions.

• Thus, the paired t test and McNemar’s test are similar in that they both compare two dependent values: the paired t test compares dependent means and McNemar’s test compares dependent proportions.

• Repeated-measures ANOVA extends the paired t test when one is interested in more than two dependent means and Cochran’s Q test extends McNemar’s test when one is interested in more than 2 dependent proportions