Paired sample t test  overview
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Paired sample $t$ test  Logistic regression 


Independent variable  Independent variables  
2 paired groups  One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variables  
Dependent variable  Dependent variable  
One quantitative of interval or ratio level  One categorical with 2 independent groups  
Null hypothesis  Null hypothesis  
H_{0}: $\mu = \mu_0$
Here $\mu$ is the population mean of the difference scores, and $\mu_0$ is the population mean of the difference scores according to the null hypothesis, which is usually 0. A difference score is the difference between the first score of a pair and the second score of a pair.  Model chisquared test for the complete regression model:
 
Alternative hypothesis  Alternative hypothesis  
H_{1} two sided: $\mu \neq \mu_0$ H_{1} right sided: $\mu > \mu_0$ H_{1} left sided: $\mu < \mu_0$  Model chisquared test for the complete regression model:
 
Assumptions  Assumptions  

 
Test statistic  Test statistic  
$t = \dfrac{\bar{y}  \mu_0}{s / \sqrt{N}}$
Here $\bar{y}$ is the sample mean of the difference scores, $\mu_0$ is the population mean of the difference scores according to the null hypothesis, $s$ is the sample standard deviation of the difference scores, and $N$ is the sample size (number of difference scores). The denominator $s / \sqrt{N}$ is the standard error of the sampling distribution of $\bar{y}$. The $t$ value indicates how many standard errors $\bar{y}$ is removed from $\mu_0$.  Model chisquared test for the complete regression model:
The wald statistic can be defined in two ways:
Likelihood ratio chisquared test for individual $\beta_k$:
 
Sampling distribution of $t$ if H_{0} were true  Sampling distribution of $X^2$ and of the Wald statistic if H_{0} were true  
$t$ distribution with $N  1$ degrees of freedom  Sampling distribution of $X^2$, as computed in the model chisquared test for the complete model:
 
Significant?  Significant?  
Two sided:
 For the model chisquared test for the complete regression model and likelihood ratio chisquared test for individual $\beta_k$:
 
$C\%$ confidence interval for $\mu$  Waldtype approximate $C\%$ confidence interval for $\beta_k$  
$\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$
where the critical value $t^*$ is the value under the $t_{N1}$ 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 $\mu$ can also be used as significance test.  $b_k \pm z^* \times SE_{b_k}$ where the critical value $z^*$ 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).  
Effect size  Goodness of fit measure $R^2_L$  
Cohen's $d$: Standardized difference between the sample mean of the difference scores and $\mu_0$: $$d = \frac{\bar{y}  \mu_0}{s}$$ Cohen's $d$ indicates how many standard deviations $s$ the sample mean of the difference scores $\bar{y}$ is removed from $\mu_0.$  $R^2_L = \dfrac{D_{null}  D_K}{D_{null}}$ 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.  
Visual representation  n.a.  
  
Equivalent to  n.a.  
   
Example context  Example context  
Is the average difference between the mental health scores before and after an intervention different from $\mu_0 = 0$?  Can body mass index, stress level, and gender predict whether people get diagnosed with diabetes?  
SPSS  SPSS  
Analyze > Compare Means > PairedSamples T Test...
 Analyze > Regression > Binary Logistic...
 
Jamovi  Jamovi  
TTests > Paired Samples TTest
 Regression > 2 Outcomes  Binomial
 
Practice questions  Practice questions  