- Z-Test - Assumptions
- SPSS Z-Tests Dialogs
- SPSS Z-Test Output
- SPSS Z-Tests - Strengths & Weaknesses
- APA Reporting Z-Tests

A z-test for independent proportions tests if 2 subpopulations

score similarly on a dichotomous variable.
Example: are the proportions (or percentages) of correct answers equal between male and female students?

Although z-tests are widely used in the social sciences, they were only introduced in SPSS version 27. So let's see how to run them and interpret their output. We'll use exam-questions.sav -partly shown below- throughout this tutorial.

Now, before running the actual z-tests, we first need to make sure we meet their assumptions.

## Z-Test - Assumptions

Z-tests for independent proportions require 2 assumptions:

- independent observations and
- sufficient sample sizes.

Regarding this second assumption, Agresti and Franklin (2014)^{2} propose that both outcomes should occur at least 10 times in both samples. That is,

$$p_a n_a \ge 10, (1 - p_a) n_a \ge 10, p_b n_b \ge 10, (1 - p_b) n_b \ge 10$$

where

- \(n_a\) and \(n_b\) denote the sample sizes of groups a and b and
- \(p_a\) and \(p_b\) denote the proportions of “successes” in both groups.

Note that some other textbooks^{3,4} suggest that smaller sample sizes may be sufficient. If you're not sure about meeting the sample sizes assumption, run a minimal CROSSTABS command as in
crosstabs v1 to v5 by sex.
As shown below, note that all 5 exam questions easily meet the sample sizes assumption.

For insufficient sample sizes, Agresti and Caffo (2000)^{1} proposed a simple adjustment for computing confidence intervals: simply add one observation for each outcome to each group (4 observations in total) and proceed as usual with these adjusted sample sizes.

## SPSS Z-Tests Dialogs

First off, let's navigate to

and fill out the dialogs as shown below.Clicking “Paste” results in the SPSS syntax below. Let's run it.

***Z-tests for independent proportions (requires SPSS 27+).**

PROPORTIONS

/INDEPENDENTSAMPLES v1 BY sex SELECT=LEVEL(0 ,1 ) CITYPES=AGRESTI_CAFFO WALD TESTTYPES=WALDH0

/SUCCESS VALUE=LEVEL(1 )

/CRITERIA CILEVEL=95

/MISSING SCOPE=ANALYSIS USERMISSING=EXCLUDE.

## SPSS Z-Test Output

The first table shows the observed proportions for male and female students. Note that female students seem to perform somewhat better: a proportion of .768 (or 76.8%) answered correctly as compared to .720 for male students.

The second output table shows that the difference between our sample proportions is -.048.

The “normal” 95% confidence interval for this difference (denoted as Wald) is [-.141, .044]. Note that this CI encloses zero: male and female populations performing equally well is within the range of likely values.

I don't recommend reporting the Agresti-Caffo corrected CI unless your data don't meet the sample sizes assumption.

The third table shows the z-test results. First note that p(2-tailed) = .309. As a rule of thumb, we
reject the null hypothesis if p < 0.05
which is not the case here. Conclusion: we do *not* reject the null hypothesis that the population difference is zero. That is: the sample difference of -.048 is *not* statistically significant.

Finally, note that SPSS reports the **wrong standard error** for this test. The correct standard error is 0.0475 as computed in this Googlesheet (read-only) shown below.

## SPSS Z-Tests - Strengths and Weaknesses

What's good about z-tests in SPSS is that

- you can analyze many dependent variables in one go;
- both the independent and dependent variables may be either string variables or numeric variables;We also tested SPSS z-tests on a mixture of string and numeric dependent variables. Although doing so is very awkward, the results were correct.
- many corrections -such as Agresti-Caffo- are available.

However, what I really don't like about SPSS z-tests is that

- no warning is issued if the sample sizes assumption isn't met;
- no effect size measures are available. Cohen’s H seems completely absent from SPSS and phi coefficients are available from CROSSTABS or CORRELATIONS;
- SPSS reports the wrong standard error for the actual z-test;
- z-tests and confidence intervals are reported in separate tables. I'd rather see these as different columns in a single table with one row per dependent variable.

## APA Reporting Z-Tests

The APA guidelines don't explicitly mention how to report z-tests. However, it makes sense to report something like
the difference between males and females

was not significant, z = -1.02, p(2-tailed) = .309.
You should obviously report the actual proportions and sample sizes as well. If you analyzed multiple dependent variables, you may want to create a table showing

- both proportions being compared;
- the difference between the proportions and its confidence interval;
- z and p(2-tailed) for the null hypothesis of equal population proportions;
- some effect size measure.

## References

- Agresti, A & Caffo, B. (2000). Simple and Effective Confidence Intervals for Proportions and Differences of Proportions.
*The American Statistician, 54(4),*280-288. - Agresti, A. & Franklin, C. (2014).
*Statistics. The Art & Science of Learning from Data.*Essex: Pearson Education Limited. - Twisk, J.W.R. (2016).
*Inleiding in de Toegepaste Biostatistiek*[Introduction to Applied Biostatistics]. Houten: Bohn Stafleu van Loghum. - Van den Brink, W.P. & Koele, P. (2002).
*Statistiek, deel 3*[Statistics, part 3]. Amsterdam: Boom.