MEDD8815 Chap.6 Inference for Proportions and Two-Way Tables
Inference for Proportions and Two-Way Tables
Categorical responses
When the response is yes or no, the statistic is a sample proportion. Session 6 applies the reasoning of Chapter 4 to proportions and then extends it, through the chi-square statistic, to whole tables of counts.
The large-sample procedures rest on the Normal approximation, so each comes with a count condition that must be checked before the arithmetic.
One proportion and two proportions
A confidence interval for one proportion uses the sample proportion in its standard error, while a significance test uses the hypothesised value, because the test is carried out as if the null hypothesis were true.
For two groups the same logic produces an unpooled standard error for the interval and a pooled proportion for the test of equal proportions. The chapter works both on AskSia-authored data and shows how to plan a survey size for a chosen margin of error, taking 0.5 as the safe guess.
Two-way tables and chi-square
A two-way table cross-classifies cases by two categorical variables.
The chi-square test compares each observed count with the count expected if the variables were unrelated, found from the row and column totals, and adds the scaled squared differences. The chapter works a two-by-three table through expected counts, the statistic, its degrees of freedom and the Table F bracket, then follows up by locating the cells that contribute most.
It explains the two sampling designs the same test serves and the count conditions that make it valid.
Links and software
For a two-by-two table the chi-square statistic equals the square of the two-proportion z statistic, so the two-sided tests agree, while only the z test can handle a one-sided alternative.
The SPSS Crosstabs procedure, with Chi-square and Expected counts requested, the Weight Cases step for summarised tables, and the goodness-of-fit dialog are described as the session handout presents them.
What this chapter covers
- 01
Sampling distribution of a sample proportion
- 02
Confidence interval for one proportion
- 03
The z test for one proportion and its count condition
- 04
Sample size for a chosen margin of error
- 05
Comparing two proportions
- 06
Pooled and unpooled standard errors
- 07
Expected counts and the chi-square statistic
- 08
Degrees of freedom, validity and goodness of fit
Worked example · free
One proportion: interval and test from the same survey
- 2Compute the sample proportion and the standard error for the interval.
- 2Check the counts and compute the test statistic with the null value.
- 2Find the two-sided P-value and connect it to the interval.
Key terms
- Sample Proportion
- The fraction of a sample with a characteristic of interest, the count of successes divided by the sample size, written p-hat.
- Pooled Proportion
- The combined proportion of successes in two samples, used in the standard error of the test of equal proportions.
- Two-Way Table
- A table of counts that cross-classifies cases by two categorical variables, one forming the rows and the other the columns.
- Expected Count
- The count a cell would hold if the two variables were unrelated, equal to its row total times its column total divided by the table total.
- Chi-Square Statistic
- The sum over all cells of the squared difference between observed and expected counts divided by the expected count.
- Goodness of Fit
- A chi-square test comparing the observed counts of one categorical variable with the counts expected under hypothesised proportions, with degrees of freedom one less than the number of categories.
Inference for Proportions and Two-Way Tables FAQ
When can I use the large-sample procedures for a proportion?
For an interval, the sample should contain at least 10 successes and 10 failures. For a test, the expected numbers under the null value should both be at least 10. When the counts are smaller, the Normal approximation is unreliable and an exact method is needed.
How many degrees of freedom does a chi-square test have?
When a table has r rows and c columns, the degrees of freedom are r minus 1 times c minus 1, so a two-by-three table has 2. For a goodness-of-fit test on k categories they are k minus 1.
What should I report after a significant chi-square test?
Say how the variables are related, not only that they are. Compare the conditional distributions across rows or columns, compare observed with expected counts, and point to the cells with the largest contributions to the statistic; those cells show where the departure from independence lies.
Assessment move
For every proportion problem this week, write two headings before any arithmetic: interval standard error and test standard error, and put the right proportion under each. For every table, compute one expected count by hand and check it against the SPSS Expected row. Those two habits cover the errors this session's homework is built to catch.
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