Logistic regression  overview
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Logistic regression  Friedman 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 within subject factor ($\geq 2$ related groups)  
Dependent variable  Dependent variable  
One categorical with 2 independent groups  One of ordinal level  
Null hypothesis  Null hypothesis  
Model chisquared test for the complete regression model:
 H_{0}: the population scores in any of the related groups are not systematically higher or lower than the population scores in any of the other related groups
Usually the related groups are the different measurement points. Several different formulations of the null hypothesis can be found in the literature, and we do not agree with all of them. Make sure you (also) learn the one that is given in your text book or by your teacher.  
Alternative hypothesis  Alternative hypothesis  
Model chisquared test for the complete regression model:
 H_{1}: the population scores in some of the related groups are systematically higher or lower than the population scores in other related groups  
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 $\beta_k$:
 $Q = \dfrac{12}{N \times k(k + 1)} \sum R^2_i  3 \times N(k + 1)$
Here $N$ is the number of 'blocks' (usually the subjects  so if you have 4 repeated measurements for 60 subjects, $N$ equals 60), $k$ is the number of related groups (usually the number of repeated measurements), and $R_i$ is the sum of ranks in group $i$. Remember that multiplication precedes addition, so first compute $\frac{12}{N \times k(k + 1)} \times \sum R^2_i$ and then subtract $3 \times N(k + 1)$. Note: if ties are present in the data, the formula for $Q$ is more complicated.  
Sampling distribution of $X^2$ and of the Wald statistic if H_{0} were true  Sampling distribution of $Q$ if H_{0} were true  
Sampling distribution of $X^2$, as computed in the model chisquared test for the complete model:
 If the number of blocks $N$ is large, approximately the chisquared distribution with $k  1$ degrees of freedom.
For small samples, the exact distribution of $Q$ should be used.  
Significant?  Significant?  
For the model chisquared test for the complete regression model and likelihood ratio chisquared test for individual $\beta_k$:
 If the number of blocks $N$ is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
 
Waldtype approximate $C\%$ confidence interval for $\beta_k$  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.  
$R^2_L = \dfrac{D_{null}  D_K}{D_{null}}$ 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  
Can body mass index, stress level, and gender predict whether people get diagnosed with diabetes?  Is there a difference in depression level between measurement point 1 (preintervention), measurement point 2 (1 week postintervention), and measurement point 3 (6 weeks postintervention)?  
SPSS  SPSS  
Analyze > Regression > Binary Logistic...
 Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
 
Jamovi  Jamovi  
Regression > 2 Outcomes  Binomial
 ANOVA > Repeated Measures ANOVA  Friedman
 
Practice questions  Practice questions  