Two-proportion Z-test

The Two-proportion Z-test is a statistical method used to determine whether the difference between the proportions of two groups, coming from a binomial distribution is statistically significant. ^[1] This approach relies on the assumption that the sample proportions follow a normal distribution under the Central Limit Theorem, allowing the construction of a z-test for hypothesis testing and confidence interval estimation. It is used in various fields to compare success rates, response rates, or other proportions across different groups.

Hypothesis test

The z-test for comparing two proportions is a Statistical hypothesis test for evaluating whether the proportion of a certain characteristic differs significantly between two independent samples. This test leverages the property that the sample proportions (which is the average of observations coming from a Bernoulli distribution) are asymptotically normal under the Central Limit Theorem, enabling the construction of a z-test.

The test involves two competing hypotheses:

• Null hypothesis (H₀): The proportions in the two populations are equal, i.e., ${\displaystyle p_{1}=p_{2}}$.
• Alternative hypothesis (H₁): The proportions in the two populations are not equal, i.e., ${\displaystyle p_{1}\neq p_{2}}$ (two-tailed) or ${\displaystyle p_{1}>p_{2}}$ / ${\displaystyle p_{1}<p_{2}}$ (one-tailed).

The z-statistic for comparing two proportions is computed using: ^[2]

${\displaystyle z={\frac {{\hat {p}}_{1}-{\hat {p}}_{2}}{\sqrt {{\hat {p}}(1-{\hat {p}})\left({\frac {1}{n_{1}}}+{\frac {1}{n_{2}}}\right)}}}}$

Where:

• ${\displaystyle {\hat {p}}_{1}}$ = sample proportion in the first sample
• ${\displaystyle {\hat {p}}_{2}}$ = sample proportion in the second sample
• ${\displaystyle n_{1}}$ = size of the first sample
• ${\displaystyle n_{2}}$ = size of the second sample
• ${\displaystyle {\hat {p}}}$ = pooled proportion, calculated as ${\displaystyle {\hat {p}}={\frac {x_{1}+x_{2}}{n_{1}+n_{2}}}}$, where ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$ are the counts of successes in the two samples.

The pooled proportion is used to estimate the shared probability of success under the null hypothesis, and the standard error accounts for variability across the two samples.

The z-test determines statistical significance by comparing the calculated z-statistic to a critical value. E.g., for a significance level of ${\displaystyle \alpha =0.05}$ we reject the null hypothesis if ${\displaystyle |z|>1.96}$ (for a two-tailed test). Or, alternatively, by computing the p-value and rejecting the null hypothesis if ${\displaystyle p<\alpha }$.

Confidence interval

The confidence interval for the difference between two proportions, based on the definitions above, is:

${\displaystyle ({\hat {p}}_{1}-{\hat {p}}_{2})\pm z_{\alpha /2}{\sqrt {{\frac {{\hat {p}}_{1}(1-{\hat {p}}_{1})}{n_{1}}}+{\frac {{\hat {p}}_{2}(1-{\hat {p}}_{2})}{n_{2}}}}}}$

Where:

• ${\displaystyle z_{\alpha /2}}$ is the critical value of the standard normal distribution (e.g., 1.96 for a 95% confidence level).

This interval provides a range of plausible values for the true difference between population proportions.

Using the z-test confidence intervals for hypothesis testing would give the same results as the chi-squared test for a two-by-two contingency table. ^{[3] : 216–7  [4] : 875} Fisher’s exact test is more suitable for when the sample sizes are small.

Notice how the variance estimation is different between the hypothesis testing and the confidence intervals. The first uses a pooled variance (based on the null hypothesis), while the second has to estimate the variance using each sample separately (so as to allow for the confidence interval to accommodate a range of differences in proportions). This difference may lead to slightly different results if using the confidence interval as an alternative to the hypothesis testing method.

Minimum detectable effect (MDE)

The minimum detectable effect (MDE) is the smallest difference between two proportions (${\displaystyle p_{1}}$ and ${\displaystyle p_{2}}$) that a statistical test can detect for a chosen Type I error level (${\displaystyle \alpha }$), statistical power (${\displaystyle 1-\beta }$), and sample sizes (${\displaystyle n_{1}}$ and ${\displaystyle n_{2}}$). It is commonly used in study design to determine whether the sample sizes allows for a test with sufficient sensitivity to detect meaningful differences.

The MDE for when using the (two-sided) z-test formula for comparing two proportions, incorporating critical values for ${\displaystyle \alpha }$ and ${\displaystyle 1-\beta }$, and the standard errors of the proportions: ^{[5] [6]}

${\displaystyle {\text{MDE}}=|p_{1}-p_{2}|=z_{1-\alpha /2}{\sqrt {p_{0}(1-p_{0})\left({\frac {1}{n_{1}}}+{\frac {1}{n_{2}}}\right)}}+z_{1-\beta }{\sqrt {{\frac {p_{1}(1-p_{1})}{n_{1}}}+{\frac {p_{2}(1-p_{2})}{n_{2}}}}}}$

Where:

• ${\displaystyle z_{1-\alpha /2}}$: Critical value for the significance level.
• ${\displaystyle z_{1-\beta }}$: Quantile for the desired power.
• ${\displaystyle p_{0}=p_{1}=p_{2}}$: When assuming the null is correct.

The MDE depends on the sample sizes, baseline proportions (${\displaystyle p_{1},p_{2}}$), and test parameters. When the baseline proportions are not known, they need to be assumed or roughly estimated from a small study. Larger samples or smaller power requirements leads to a smaller MDE, making the test more sensitive to smaller differences. Researchers may use the MDE to assess the feasibility of detecting meaningful differences before conducting a study.

[Proof]

The Minimal Detectable Effect (MDE) is the smallest difference, denoted as ${\displaystyle \Delta =|p_{1}-p_{2}|}$, that satisfies two essential criteria in hypothesis testing:

1. The null hypothesis (${\displaystyle H_{0}:p_{1}=p_{2}}$) is rejected at the specified significance level (${\displaystyle \alpha }$).
2. Statistical power (${\displaystyle 1-\beta }$) is achieved under the alternative hypothesis (${\displaystyle H_{a}:p_{1}\neq p_{2}}$).

Given that the distribution is normal under the null and the alternative hypothesis, for the two criteria to happen, it is required that the distance of ${\displaystyle |p_{1}-p_{2}|}$ will be such that the critical value for rejecting the null (${\displaystyle X_{\text{critical}}}$) is exactly in the location in which the probability of exceeding this value, under the null, is (${\displaystyle \alpha }$), and also that the probability of exceeding this value, under the alternative, is ${\displaystyle 1-\beta }$.

The first criterion establishes the critical value required to reject the null hypothesis. The second criterion specifies how far the alternative distribution must be from ${\displaystyle X_{\text{critical}}}$ to ensure that the probability of exceeding it under the alternative hypothesis is at least ${\displaystyle 1-\beta }$. ^{[7] [8]}

Condition 1: Rejecting ${\displaystyle H_{0}}$

Under the null hypothesis, the test statistic is based on the pooled standard error (${\displaystyle {\text{SE}}_{\text{null}}}$): ${\displaystyle Z_{\text{test}}={\frac {|p_{1}-p_{2}|}{{\text{SE}}_{\text{null}}}},\quad {\text{where }}{\text{SE}}_{\text{null}}={\sqrt {p_{0}(1-p_{0})\left({\frac {1}{n_{1}}}+{\frac {1}{n_{2}}}\right)}}.}$

${\displaystyle p_{0}}$ might be estimated (as described above).

To reject ${\displaystyle H_{0}}$, the observed difference must exceed the critical threshold (${\displaystyle Z_{\text{critical}}=z_{\alpha /2}}$) after properly inflating it to the SE: ${\displaystyle |p_{1}-p_{2}|\geq X_{critical}=z_{\alpha /2}\cdot {\text{SE}}_{\text{null}}}$

If the MDE is defined solely as ${\displaystyle MDE=z_{\alpha /2}\cdot {\text{SE}}_{\text{null}}}$, the statistical power would be only 50% because the alternative distribution is symmetric about the threshold. To achieve a higher power level, an additional component is required in the MDE calculation.

Condition 2: Achieving power ${\displaystyle 1-\beta }$

Under the alternative hypothesis, the standard error is (${\displaystyle {\text{SE}}_{\text{alt}}={\sqrt {{\frac {p_{1}(1-p_{1})}{n_{1}}}+{\frac {p_{2}(1-p_{2})}{n_{2}}}}}}$). It means that if the alternative distribution was centered around some value (e.g., ${\displaystyle X_{\text{critical}}}$), then the minimal ${\displaystyle |p_{1}-p_{2}|}$ must be at least larger than ${\displaystyle z_{\alpha /2}\cdot {\text{SE}}_{\text{null}}}$ to ensure that the probability of detecting the difference under the alternative hypothesis is at least ${\displaystyle 1-\beta }$.

Combining conditions

To meet both conditions, the total detectable difference incorporates components from both the null and alternative distributions. The MDE is defined as: ${\displaystyle {\text{MDE}}=z_{1-\alpha /2}\cdot {\text{SE}}_{\text{null}}+z_{1-\beta }\cdot {\text{SE}}_{\text{alt}}.}$

By summing the critical thresholds from the null and adding to it the relevant quantile from the alternative distributions, the MDE ensures the test satisfies the dual requirements of rejecting ${\displaystyle H_{0}}$ at significance level ${\displaystyle \alpha }$ and achieving statistical power of at least ${\displaystyle 1-\beta }$.

Assumptions and conditions

To ensure valid results, the following assumptions must be met:

1. Independent random samples: The samples must be drawn independently from the populations of interest.
2. Large sample sizes: Typically, ${\displaystyle n_{1}}$ and ${\displaystyle n_{2}}$ should exceed 30. ^{[ citation needed ]}
3. Success/failure condition: ^{[ citation needed ]}
   1. ${\displaystyle n_{1}{\hat {p}}_{1}>10}$ and ${\displaystyle n_{1}(1-{\hat {p}}_{1})>10}$
   2. ${\displaystyle n_{2}{\hat {p}}_{2}>10}$ and ${\displaystyle n_{2}(1-{\hat {p}}_{2})>10}$

The z-test is most reliable when sample sizes are large, and all assumptions are satisfied.

Software implementation

R

Use prop.test() with continuity correction disabled:

prop.test(x=c(120,150),n=c(1000,1000),correct=FALSE)

Output includes z-test equivalent results: chi-squared statistic, p-value, and confidence interval:

 2-sample test for equality of proportions without continuity correction  data:  c(120, 150) out of c(1000, 1000) X-squared = 3.8536, df = 1, p-value = 0.04964 alternative hypothesis: two.sided 95 percent confidence interval:  -5.992397e-02 -7.602882e-05 sample estimates: prop 1 prop 2    0.12   0.15 

Python

Use proportions_ztest from statsmodels:

fromstatsmodels.stats.proportionimportproportions_ztestz,p=proportions_ztest([120,150],[1000,1000],0)# For CI: from statsmodels.stats.proportion import proportions_diff_confint_indep

See also

• Two-sample hypothesis testing
• A/B testing

References

1. Hypothesis Test: Difference Between Proportions
2. How can we determine whether two processes produce the same proportion of defectives?
3. Confidence Intervals for the Difference Between Two Proportions
4. Newcombe, R. G. 1998. 'Interval Estimation for the Difference Between Independent Proportions: Comparison of Eleven Methods.' Statistics in Medicine, 17, pp. 873-890.
5. COOLSerdash (https://stats.stackexchange.com/users/21054/coolserdash), Two proportion sample size calculation, URL (version: 2023-04-14): https://stats.stackexchange.com/q/612894
6. Chow S-C, Shao J, Wang H, Lokhnygina Y (2018): Sample size calculations in clinical research. 3rd ed. CRC Press.
7. Calculating Sample Sizes for A/B Tests
8. Power, minimal detectable effect, and bucket size estimation in A/B tests (has some nice figures to illustrate the tradeoffs)

External links

• Inference for Proportions: Comparing Two Independent Samples (power calculator)