Regression (OLS)  overview
This page offers structured overviews of one or more selected methods. Add additional methods for comparisons by clicking on the dropdown button in the righthand column. To practice with a specific method click the button at the bottom row of the table
Regression (OLS)  Binomial test for a single proportion  Paired sample $t$ test  Friedman test  Paired sample $t$ test 


Independent variables  Independent variable  Independent variable  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  None  2 paired groups  One within subject factor ($\geq 2$ related groups)  2 paired groups  
Dependent variable  Dependent variable  Dependent variable  Dependent variable  Dependent variable  
One quantitative of interval or ratio level  One categorical with 2 independent groups  One quantitative of interval or ratio level  One of ordinal level  One quantitative of interval or ratio level  
Null hypothesis  Null hypothesis  Null hypothesis  Null hypothesis  Null hypothesis  
$F$ test for the complete regression model:
 $\pi = \pi_0$
$\pi$ is the population proportion of "successes"; $\pi_0$ is the population proportion of successes according to the null hypothesis  $\mu = \mu_0$
$\mu$ is the unknown population mean of the difference scores; $\mu_0$ is the population mean of the difference scores according to the null hypothesis, which is usually 0  The scores in any of the related groups are not systematically higher or lower than the scores in any of the other related groups
Note: 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.  $\mu = \mu_0$
$\mu$ is the unknown population mean of the difference scores; $\mu_0$ is the population mean of the difference scores according to the null hypothesis, which is usually 0  
Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  
$F$ test for the complete regression model:
 Two sided: $\pi \neq \pi_0$ Right sided: $\pi > \pi_0$ Left sided: $\pi < \pi_0$  Two sided: $\mu \neq \mu_0$ Right sided: $\mu > \mu_0$ Left sided: $\mu < \mu_0$  The scores in some of the related groups are systematically higher or lower than the scores in other related groups  Two sided: $\mu \neq \mu_0$ Right sided: $\mu > \mu_0$ Left sided: $\mu < \mu_0$  
Assumptions  Assumptions  Assumptions  Assumptions  Assumptions  
 Sample is a simple random sample from the population. That is, observations are independent of one another 
 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  Test statistic  Test statistic  
$F$ test for the complete regression model:
Note 2: if only one independent variable ($K = 1$), the $F$ test for the complete regression model is equivalent to the two sided $t$ test for $\beta_1$  $X$ = number of successes in the sample  $t = \dfrac{\bar{y}  \mu_0}{s / \sqrt{N}}$
$\bar{y}$ is the sample mean of the difference scores, $\mu_0$ is the population mean of the difference scores according to H0, $s$ is the sample standard deviation of the difference scores, $N$ 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 $\mu_0$  $Q = \dfrac{12}{N \times k(k + 1)} \sum R^2_i  3 \times N(k + 1)$
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.  $t = \dfrac{\bar{y}  \mu_0}{s / \sqrt{N}}$
$\bar{y}$ is the sample mean of the difference scores, $\mu_0$ is the population mean of the difference scores according to H0, $s$ is the sample standard deviation of the difference scores, $N$ 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 $\mu_0$  
Sample standard deviation of the residuals $s$  n.a.  n.a.  n.a.  n.a.  
$\begin{aligned} s &= \sqrt{\dfrac{\sum (y_j  \hat{y}_j)^2}{N  K  1}}\\ &= \sqrt{\dfrac{\mbox{sum of squares error}}{\mbox{degrees of freedom error}}}\\ &= \sqrt{\mbox{mean square error}} \end{aligned} $          
Sampling distribution of $F$ and of $t$ if H0 were true  Sampling distribution of $X$ if H0 were true  Sampling distribution of $t$ if H0 were true  Sampling distribution of $Q$ if H0 were true  Sampling distribution of $t$ if H0 were true  
Sampling distribution of $F$:
 Binomial($n$, $p$) distribution
Here $n = N$ (total sample size), and $p = \pi_0$ (population proportion according to the null hypothesis)  $t$ distribution with $N  1$ degrees of freedom  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.  $t$ distribution with $N  1$ degrees of freedom  
Significant?  Significant?  Significant?  Significant?  Significant?  
$F$ test:
 Two sided:
 Two sided:
 If the number of blocks $N$ is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
 Two sided:
 
$C\%$ confidence interval for $\beta_k$ and for $\mu_y$; $C\%$ prediction interval for $y_{new}$  n.a.  $C\%$ confidence interval for $\mu$  n.a.  $C\%$ confidence interval for $\mu$  
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.    $\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$
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.  
Effect size  n.a.  Effect size  n.a.  Effect size  
Complete model:
   Cohen's $d$: Standardized difference between the sample mean of the difference scores and $\mu_0$: $$d = \frac{\bar{y}  \mu_0}{s}$$ Indicates how many standard deviations $s$ the sample mean of the difference scores $\bar{y}$ is removed from $\mu_0$    Cohen's $d$: Standardized difference between the sample mean of the difference scores and $\mu_0$: $$d = \frac{\bar{y}  \mu_0}{s}$$ Indicates how many standard deviations $s$ the sample mean of the difference scores $\bar{y}$ is removed from $\mu_0$  
n.a.  n.a.  Visual representation  n.a.  Visual representation  
      
ANOVA table  n.a.  n.a.  n.a.  n.a.  
        
n.a.  n.a.  Equivalent to  n.a.  Equivalent to  
    One sample $t$ test on the difference scores
Repeated measures ANOVA with one dichotomous within subjects factor    One sample $t$ test on the difference scores
Repeated measures ANOVA with one dichotomous within subjects factor  
Example context  Example context  Example context  Example context  Example context  
Can mental health be predicted from fysical health, economic class, and gender?  Is the proportion smokers amongst office workers different from $\pi_0 = .2$?  Is the average difference between the mental health scores before and after an intervention different from $\mu_0$ = 0?  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)?  Is the average difference between the mental health scores before and after an intervention different from $\mu_0$ = 0?  
SPSS  SPSS  SPSS  SPSS  SPSS  
Analyze > Regression > Linear...
 Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
 Analyze > Compare Means > PairedSamples T Test...
 Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
 Analyze > Compare Means > PairedSamples T Test...
 
Jamovi  Jamovi  Jamovi  Jamovi  Jamovi  
Regression > Linear Regression
 Frequencies > 2 Outcomes  Binomial test
 TTests > Paired Samples TTest
 ANOVA > Repeated Measures ANOVA  Friedman
 TTests > Paired Samples TTest
 
Practice questions  Practice questions  Practice questions  Practice questions  Practice questions  