z test for the difference between two proportions - overview
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$z$ test for the difference between two proportions | Spearman's rho | Logistic regression |
You cannot compare more than 3 methods |
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Independent/grouping variable | Variable 1 | Independent variables | |
One categorical with 2 independent groups | One of ordinal level | One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variables | |
Dependent variable | Variable 2 | Dependent variable | |
One categorical with 2 independent groups | One of ordinal level | One categorical with 2 independent groups | |
Null hypothesis | Null hypothesis | Null hypothesis | |
H0: $\pi_1 = \pi_2$
Here $\pi_1$ is the population proportion of 'successes' for group 1, and $\pi_2$ is the population proportion of 'successes' for group 2. | H0: $\rho_s = 0$
Here $\rho_s$ is the Spearman correlation in the population. The Spearman correlation is a measure for the strength and direction of the monotonic relationship between two variables of at least ordinal measurement level. In words, the null hypothesis would be: H0: there is no monotonic relationship between the two variables in the population. | Model chi-squared test for the complete regression model:
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Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |
H1 two sided: $\pi_1 \neq \pi_2$ H1 right sided: $\pi_1 > \pi_2$ H1 left sided: $\pi_1 < \pi_2$ | H1 two sided: $\rho_s \neq 0$ H1 right sided: $\rho_s > 0$ H1 left sided: $\rho_s < 0$ | Model chi-squared test for the complete regression model:
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Assumptions | Assumptions | Assumptions | |
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Test statistic | Test statistic | Test statistic | |
$z = \dfrac{p_1 - p_2}{\sqrt{p(1 - p)\Bigg(\dfrac{1}{n_1} + \dfrac{1}{n_2}\Bigg)}}$
Here $p_1$ is the sample proportion of successes in group 1: $\dfrac{X_1}{n_1}$, $p_2$ is the sample proportion of successes in group 2: $\dfrac{X_2}{n_2}$, $p$ is the total proportion of successes in the sample: $\dfrac{X_1 + X_2}{n_1 + n_2}$, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. Note: we could just as well compute $p_2 - p_1$ in the numerator, but then the left sided alternative becomes $\pi_2 < \pi_1$, and the right sided alternative becomes $\pi_2 > \pi_1.$ | $t = \dfrac{r_s \times \sqrt{N - 2}}{\sqrt{1 - r_s^2}} $ Here $r_s$ is the sample Spearman correlation and $N$ is the sample size. The sample Spearman correlation $r_s$ is equal to the Pearson correlation applied to the rank scores. | Model chi-squared test for the complete regression model:
The wald statistic can be defined in two ways:
Likelihood ratio chi-squared test for individual $\beta_k$:
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Sampling distribution of $z$ if H0 were true | Sampling distribution of $t$ if H0 were true | Sampling distribution of $X^2$ and of the Wald statistic if H0 were true | |
Approximately the standard normal distribution | Approximately the $t$ distribution with $N - 2$ degrees of freedom | Sampling distribution of $X^2$, as computed in the model chi-squared test for the complete model:
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Significant? | Significant? | Significant? | |
Two sided:
| Two sided:
| For the model chi-squared test for the complete regression model and likelihood ratio chi-squared test for individual $\beta_k$:
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Approximate $C\%$ confidence interval for $\pi_1 - \pi_2$ | n.a. | Wald-type approximate $C\%$ confidence interval for $\beta_k$ | |
Regular (large sample):
| - | $b_k \pm z^* \times SE_{b_k}$ where the critical value $z^*$ is the value under the normal curve with the area $C / 100$ between $-z^*$ and $z^*$ (e.g. $z^*$ = 1.96 for a 95% confidence interval). | |
n.a. | n.a. | Goodness of fit measure $R^2_L$ | |
- | - | $R^2_L = \dfrac{D_{null} - D_K}{D_{null}}$ There are several other goodness of fit measures in logistic regression. In logistic regression, there is no single agreed upon measure of goodness of fit. | |
Equivalent to | n.a. | n.a. | |
When testing two sided: chi-squared test for the relationship between two categorical variables, where both categorical variables have 2 levels. | - | - | |
Example context | Example context | Example context | |
Is the proportion of smokers different between men and women? Use the normal approximation for the sampling distribution of the test statistic. | Is there a monotonic relationship between physical health and mental health? | Can body mass index, stress level, and gender predict whether people get diagnosed with diabetes? | |
SPSS | SPSS | SPSS | |
SPSS does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chi-squared test instead. The $p$ value resulting from this chi-squared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to:
Analyze > Descriptive Statistics > Crosstabs...
| Analyze > Correlate > Bivariate...
| Analyze > Regression > Binary Logistic...
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Jamovi | Jamovi | Jamovi | |
Jamovi does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chi-squared test instead. The $p$ value resulting from this chi-squared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to:
Frequencies > Independent Samples - $\chi^2$ test of association
| Regression > Correlation Matrix
| Regression > 2 Outcomes - Binomial
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Practice questions | Practice questions | Practice questions | |