ANCOVA: overview
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ANCOVA  $z$ test for the difference between two proportions 


Independent variables  Independent variable  
One or more categorical with independent groups, and one or more quantitative control variables of interval or ratio level (covariates)  One categorical with 2 independent groups  
Dependent variable  Dependent variable  
One quantitative of interval or ratio level  One categorical with 2 independent groups  
THIS TABLE IS YET TO BE COMPLETED  Null hypothesis  
  $\pi_1 = \pi_2$
$\pi_1$ is the unknown proportion of "successes" in population 1; $\pi_2$ is the unknown proportion of "successes" in population 2  
n.a.  Alternative hypothesis  
  Two sided: $\pi_1 \neq \pi_2$ Right sided: $\pi_1 > \pi_2$ Left sided: $\pi_1 < \pi_2$  
n.a.  Assumptions  
 
 
n.a.  Test statistic  
  $z = \dfrac{p_1  p_2}{\sqrt{p(1  p)\Bigg(\dfrac{1}{n_1} + \dfrac{1}{n_2}\Bigg)}}$
$p_1$ 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, $n_2$ 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 $\pi_2 > \pi_1$  
n.a.  Sampling distribution of $z$ if H0 were true  
  Approximately standard normal  
n.a.  Significant?  
  Two sided:
 
n.a.  Approximate $C\%$ confidence interval for $\pi_1  \pi_2$  
  Regular (large sample):
 
n.a.  Equivalent to  
  When testing two sided: chisquared test for the relationship between two categorical variables, where both categorical variables have 2 levels  
n.a.  Example context  
  Is the proportion smokers different between men and women? Use the normal approximation for the sampling distribution of the test statistic.  
n.a.  SPSS  
  SPSS does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chisquared test instead. The $p$ value resulting from this chisquared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to:
Analyze > Descriptive Statistics > Crosstabs...
 
n.a.  Jamovi  
  Jamovi does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chisquared test instead. The $p$ value resulting from this chisquared 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
 
Practice questions  Practice questions  