Logistic regression  overview
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Logistic regression  Spearman's rho  Goodness of fit 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  None  
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 
 
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: $\rho_s < 0$ 
 
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{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.  $X^2 = \sum{\frac{(\mbox{observed cell count}  \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
where the expected cell count for one cell = $N \times \pi_j$, the observed cell count is the observed sample count in that same cell, and the sum is over all $J$ cells  
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 $X^2$ 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:

 
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?  Is the proportion of people with a low, moderate, and high social economic status in the population different from $\pi_{low}$ = .2, $\pi_{moderate}$ = .6, and $\pi_{high}$ = .2?  
SPSS  SPSS  SPSS  
Analyze > Regression > Binary Logistic...
 Analyze > Correlate > Bivariate...
 Analyze > Nonparametric Tests > Legacy Dialogs > Chisquare...
 
Jamovi  Jamovi  Jamovi  
Regression > 2 Outcomes  Binomial
 Regression > Correlation Matrix
 Frequencies > N Outcomes  $\chi^2$ Goodness of fit
 
Practice questions  Practice questions  Practice questions  