Logistic regression  overview
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Logistic regression  Two sample $t$ test  equal variances not assumed  Cochran's Q 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 categorical with 2 independent groups  One within subject factor ($\geq 2$ related groups)  
Dependent variable  Dependent variable  Dependent variable  
One categorical with 2 independent groups  One quantitative of interval or ratio level  One categorical with 2 independent groups  
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  $\pi_1 = \pi_2 = \ldots = \pi_I$
$\pi_1$ is the population proportion of 'successes' in group 1; $\pi_2$ is the population proportion of 'successes' in group 2; $\pi_I$ is the population proportion of 'successes' in group $I$  
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: $\mu_1 < \mu_2$  Not all population proportions are equal  
Assumptions  Assumptions  Assumptions  

 Sample of 'blocks' (usually the subjects) is a simple random sample from the population. That is, blocks 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 $\beta_k$:
 $t = \dfrac{(\bar{y}_1  \bar{y}_2)  0}{\sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}} = \dfrac{\bar{y}_1  \bar{y}_2}{\sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}}$
$\bar{y}_1$ 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 $\mu_2 > \mu_1$  If a failure is scored as 0 and a success is scored as 1:
$Q = k(k  1) \dfrac{\sum_{groups} \Big (\mbox{group total}  \frac{\mbox{grand total}}{k} \Big)^2}{\sum_{blocks} \mbox{block total} \times (k  \mbox{block total})}$ Here $k$ is the number of related groups (usually the number of repeated measurements), a group total is the sum of the scores in a group, a block total is the sum of the scores in a block (usually a subject), and the grand total is the sum of all the scores. Before computing $Q$, first exclude blocks with equal scores in all $k$ groups  
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 $Q$ 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  If the number of blocks (usually the number of subjects) is large, approximately the chisquared distribution with $k  1$ 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 the number of blocks is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
 
Waldtype approximate $C\%$ confidence interval for $\beta_k$  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.  
    
n.a.  n.a.  Equivalent to  
    Friedman test, with a categorical dependent variable consisting of two independent groups  
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?  Subjects perform three different tasks, which they can either perform correctly or incorrectly. Is there a difference in task performance between the three different tasks?  
SPSS  SPSS  SPSS  
Analyze > Regression > Binary Logistic...
 Analyze > Compare Means > IndependentSamples T Test...
 Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
 
Jamovi  Jamovi  Jamovi  
Regression > 2 Outcomes  Binomial
 TTests > Independent Samples TTest
 Jamovi does not have a specific option for the Cochran's Q test. However, you can do the Friedman test instead. The $p$ value resulting from this Friedman test is equivalent to the $p$ value that would have resulted from the Cochran's Q test. Go to:
ANOVA > Repeated Measures ANOVA  Friedman
 
Practice questions  Practice questions  Practice questions  