Logistic regression  overview
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Logistic regression  Two sample $z$ test 


Independent variables  Independent/grouping variable  
One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variables  One categorical with 2 independent groups  
Dependent variable  Dependent variable  
One categorical with 2 independent groups  One quantitative of interval or ratio level  
Null hypothesis  Null hypothesis  
Model chisquared test for the complete regression model:
 H_{0}: $\mu_1 = \mu_2$
Here $\mu_1$ is the population mean for group 1, and $\mu_2$ is the population mean for group 2.  
Alternative hypothesis  Alternative hypothesis  
Model chisquared test for the complete regression model:
 H_{1} two sided: $\mu_1 \neq \mu_2$ H_{1} right sided: $\mu_1 > \mu_2$ H_{1} left sided: $\mu_1 < \mu_2$  
Assumptions  Assumptions  

 
Test statistic  Test statistic  
Model chisquared test for the complete regression model:
The wald statistic can be defined in two ways:
Likelihood ratio chisquared test for individual $\beta_k$:
 $z = \dfrac{(\bar{y}_1  \bar{y}_2)  0}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}} = \dfrac{\bar{y}_1  \bar{y}_2}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}}$
Here $\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $\sigma^2_1$ is the population variance in population 1, $\sigma^2_2$ is the population variance in population 2, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. The 0 represents the difference in population means according to the null hypothesis. The denominator $\sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}}$ is the standard deviation of the sampling distribution of $\bar{y}_1  \bar{y}_2$. The $z$ value indicates how many of these standard deviations $\bar{y}_1  \bar{y}_2$ is removed from 0. Note: we could just as well compute $\bar{y}_2  \bar{y}_1$ in the numerator, but then the left sided alternative becomes $\mu_2 < \mu_1$, and the right sided alternative becomes $\mu_2 > \mu_1$.  
Sampling distribution of $X^2$ and of the Wald statistic if H_{0} were true  Sampling distribution of $z$ if H_{0} were true  
Sampling distribution of $X^2$, as computed in the model chisquared test for the complete model:
 Standard normal distribution  
Significant?  Significant?  
For the model chisquared test for the complete regression model and likelihood ratio chisquared test for individual $\beta_k$:
 Two sided:
 
Waldtype approximate $C\%$ confidence interval for $\beta_k$  $C\%$ confidence interval for $\mu_1  \mu_2$  
$b_k \pm z^* \times SE_{b_k}$ 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).  $(\bar{y}_1  \bar{y}_2) \pm z^* \times \sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}$
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). The confidence interval for $\mu_1  \mu_2$ can also be used as significance test.  
Goodness of fit measure $R^2_L$  n.a.  
$R^2_L = \dfrac{D_{null}  D_K}{D_{null}}$ 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.    
n.a.  Visual representation  
  
Example context  Example context  
Can body mass index, stress level, and gender predict whether people get diagnosed with diabetes?  Is the average mental health score different between men and women? Assume that in the population, the standard devation of the mental health scores is $\sigma_1 = 2$ amongst men and $\sigma_2 = 2.5$ amongst women.  
SPSS  n.a.  
Analyze > Regression > Binary Logistic...
   
Jamovi  n.a.  
Regression > 2 Outcomes  Binomial
   
Practice questions  Practice questions  