Logistic regression  overview
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Logistic regression  Spearman's rho  Sign 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 of ordinal level  
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.  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.  $W = $ number of difference scores that is larger than 0  
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 $W$ 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  The exact distribution of $W$ under the null hypothesis is the Binomial($n$, $p$) distribution, with $n =$ number of positive differences $+$ number of negative differences, and $p = 0.5$.
If $n$ is large, $W$ is approximately normally distributed under the null hypothesis, with mean $np = n \times 0.5$ and standard deviation $\sqrt{np(1p)} = \sqrt{n \times 0.5(1  0.5)}$. Hence, if $n$ is large, the standardized test statistic $$z = \frac{W  n \times 0.5}{\sqrt{n \times 0.5(1  0.5)}}$$ follows approximately a standard normal distribution if the null hypothesis were true.  
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 $n$ is small, the table for the binomial distribution should be used: Two sided:
If $n$ is large, the table for standard normal probabilities can be used: 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.      
n.a.  n.a.  Equivalent to  
   
Two sided sign test is equivalent to
 
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?  Do people tend to score higher on mental health after a mindfulness course?  
SPSS  SPSS  SPSS  
Analyze > Regression > Binary Logistic...
 Analyze > Correlate > Bivariate...
 Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
 
Jamovi  Jamovi  Jamovi  
Regression > 2 Outcomes  Binomial
 Regression > Correlation Matrix
 Jamovi does not have a specific option for the sign test. However, you can do the Friedman test instead. The $p$ value resulting from this Friedman test is equivalent to the two sided $p$ value that would have resulted from the sign test. Go to:
ANOVA > Repeated Measures ANOVA  Friedman
 
Practice questions  Practice questions  Practice questions  