Logistic regression  overview
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Logistic regression  Spearman's rho  Marginal Homogeneity test / StuartMaxwell test 


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 of ordinal level  2 paired groups  
Dependent variable  Dependent variable  Dependent variable  
One categorical with 2 independent groups  One of ordinal level  One categorical with $J$ independent groups ($J \geqslant 2$)  
Null hypothesis  Null hypothesis  Null hypothesis  
Model chisquared test for the complete regression model:
 $\rho_s = 0$
$\rho_s$ is the unknown Spearman correlation in the population. In words: there is no monotonic relationship between the two variables in the population  For each category $j$ of the dependent variable:
$\pi_j$ in the first paired group = $\pi_j$ in the second paired group Here $\pi_j$ is the population proportion for category $j$  
Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  
Model chisquared test for the complete regression model:
 Two sided: $\rho_s \neq 0$ Right sided: $\rho_s > 0$ Left sided: $\rho_s < 0$  For some categories of the dependent variable, $\pi_j$ in the first paired group $\neq$ $\pi_j$ in the second paired group  
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.  Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another  
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{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.  Computing the test statistic is a bit complicated and involves matrix algebra. You probably won't need to calculate it by hand (unless you are following a technical course)  
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 the test statistic 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 $N  2$ degrees of freedom  Approximately a chisquared distribution with $J  1$ 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:
 If we denote the test statistic as $X^2$:
 
Waldtype approximate $C\%$ confidence interval for $\beta_k$  n.a.  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)      
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.      
Example context  Example context  Example context  
Can body mass index, stress level, and gender predict whether people get diagnosed with diabetes?  Is there a monotonic relationship between physical health and mental health?  Subjects are asked to taste three different types of mayonnaise, and to indicate which of the three types of mayonnaise they like best. They then have to drink a glass of beer, and taste and rate the three types of mayonnaise again. Does drinking a beer change which type of mayonnaise people like best?  
SPSS  SPSS  SPSS  
Analyze > Regression > Binary Logistic...
 Analyze > Correlate > Bivariate...
 Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
 
Jamovi  Jamovi  n.a.  
Regression > 2 Outcomes  Binomial
 Regression > Correlation Matrix
   
Practice questions  Practice questions  Practice questions  