Logistic regression  overview
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Logistic regression  Two sample $t$ test  equal variances not assumed  Spearman's rho 


Independent variables  Independent variable  Independent 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  One of ordinal level  
Dependent variable  Dependent variable  Dependent variable  
One categorical with 2 independent groups  One quantitative of interval or ratio level  One of ordinal level  
Null hypothesis  Null hypothesis  Null hypothesis  
Model chisquared test for the complete regression model:
 $\mu_1 = \mu_2$
$\mu_1$ is the unknown mean in population 1, $\mu_2$ is the unknown mean in population 2  $\rho_s = 0$
$\rho_s$ is the unknown Spearman correlation in the population. In words: there is no monotonic relationship between the two variables in the population  
Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  
Model chisquared test for the complete regression model:
 Two sided: $\mu_1 \neq \mu_2$ Right sided: $\mu_1 > \mu_2$ Left sided: $\mu_1 < \mu_2$  Two sided: $\rho_s \neq 0$ Right sided: $\rho_s > 0$ Left sided: $\rho_s < 0$  
Assumptions  Assumptions  Assumptions  

 Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
Note: this assumption is only important for the significance test, not for the correlation coefficient itself. The correlation coefficient itself just measures the strength of the monotonic relationship between two variables.  
Test statistic  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 $\beta_k$:
 $t = \dfrac{(\bar{y}_1  \bar{y}_2)  0}{\sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}} = \dfrac{\bar{y}_1  \bar{y}_2}{\sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}}$
$\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $s^2_1$ is the sample variance in group 1, $s^2_2$ is the sample variance in group 2, $n_1$ is the sample size of group 1, $n_2$ is the sample size of group 2. The 0 represents the difference in population means according to H0. The denominator $\sqrt{\frac{s^2_1}{n_1} + \frac{s^2_2}{n_2}}$ is the standard error of the sampling distribution of $\bar{y}_1  \bar{y}_2$. The $t$ value indicates how many standard errors $\bar{y}_1  \bar{y}_2$ 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 $\mu_2 > \mu_1$  $t = \dfrac{r_s \times \sqrt{N  2}}{\sqrt{1  r_s^2}} $ where $r_s$ 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.  
Sampling distribution of $X^2$ and of the Wald statistic if H0 were true  Sampling distribution of $t$ if H0 were true  Sampling distribution of $t$ if H0 were true  
Sampling distribution of $X^2$, as computed in the model chisquared test for the complete model:
 Approximately a $t$ distribution with $k$ degrees of freedom, with $k$ equal to $k = \dfrac{\Bigg(\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}\Bigg)^2}{\dfrac{1}{n_1  1} \Bigg(\dfrac{s^2_1}{n_1}\Bigg)^2 + \dfrac{1}{n_2  1} \Bigg(\dfrac{s^2_2}{n_2}\Bigg)^2}$ or $k$ = the smaller of $n_1$  1 and $n_2$  1 First definition of $k$ is used by computer programs, second definition is often used for hand calculations  Approximately a $t$ distribution with $N  2$ degrees of freedom  
Significant?  Significant?  Significant?  
For the model chisquared test for the complete regression model and likelihood ratio chisquared test for individual $\beta_k$:
 Two sided:
 Two sided:
 
Waldtype approximate $C\%$ confidence interval for $\beta_k$  Approximate $C\%$ confidence interval for $\mu_1  \mu_2$  n.a.  
$b_k \pm z^* \times SE_{b_k}$ where $z^*$ 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)  $(\bar{y}_1  \bar{y}_2) \pm t^* \times \sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}$
where the critical value $t^*$ is the value under the $t_{k}$ distribution with the area $C / 100$ between $t^*$ and $t^*$ (e.g. $t^*$ = 2.086 for a 95% confidence interval when df = 20) The confidence interval for $\mu_1  \mu_2$ can also be used as significance test.    
Goodness of fit measure $R^2_L$  n.a.  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  n.a.  
    
Example context  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?  Is there a monotonic relationship between physical health and mental health?  
SPSS  SPSS  SPSS  
Analyze > Regression > Binary Logistic...
 Analyze > Compare Means > IndependentSamples T Test...
 Analyze > Correlate > Bivariate...
 
Jamovi  Jamovi  Jamovi  
Regression > 2 Outcomes  Binomial
 TTests > Independent Samples TTest
 Regression > Correlation Matrix
 
Practice questions  Practice questions  Practice questions  