Regression (OLS)  overview
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Regression (OLS)  Binomial test for a single proportion  Paired sample $t$ test  Marginal Homogeneity test / StuartMaxwell test  One sample $z$ test for the mean 


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  2 paired groups  None  
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 categorical with $J$ independent groups ($J \geqslant 2$)  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  For each category $j$ of the dependent variable:
$\pi_j$ in the first paired group = $\pi_j$ in the second paired group Here $\pi_j$ is the population proportion for category $j$  $\mu = \mu_0$
$\mu$ is the unknown population mean; $\mu_0$ is the population mean according to the null hypothesis  
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: $\mu < \mu_0$  For some categories of the dependent variable, $\pi_j$ in the first paired group $\neq$ $\pi_j$ in the second paired group  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 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  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 $\mu_0$  Computing the test statistic is a bit complicated and involves matrix algebra. You probably won't need to calculate it by hand (unless you are following a technical course)  $z = \dfrac{\bar{y}  \mu_0}{\sigma / \sqrt{N}}$
$\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to H0, $\sigma$ is the population standard deviation, $N$ is the sample size. The denominator $\sigma / \sqrt{N}$ is the standard deviation of the sampling distribution of $\bar{y}$. The $z$ value indicates how many of these standard deviations $\bar{y}$ 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 the test statistic if H0 were true  Sampling distribution of $z$ 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  Approximately a chisquared distribution with $J  1$ degrees of freedom  Standard normal  
Significant?  Significant?  Significant?  Significant?  Significant?  
$F$ test:
 Two sided:
 Two sided:
 If we denote the test statistic as $X^2$:
 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.    $\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
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) 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 and $\mu_0$: $$d = \frac{\bar{y}  \mu_0}{\sigma}$$ Indicates how many standard deviations $\sigma$ the sample mean $\bar{y}$ 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.  n.a.  
    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?  Subjects are asked to taste three different types of mayonnaise, and to indicate which of the three types of mayonnaise they like best. They then have to drink a glass of beer, and taste and rate the three types of mayonnaise again. Does drinking a beer change which type of mayonnaise people like best?  Is the average mental health score of office workers different from $\mu_0$ = 50? Assume that the standard deviation of the mental health scores in the population is $\sigma$ = 3.  
SPSS  SPSS  SPSS  SPSS  n.a.  
Analyze > Regression > Linear...
 Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
 Analyze > Compare Means > PairedSamples T Test...
 Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
   
Jamovi  Jamovi  Jamovi  n.a.  n.a.  
Regression > Linear Regression
 Frequencies > 2 Outcomes  Binomial test
 TTests > Paired Samples TTest
     
Practice questions  Practice questions  Practice questions  Practice questions  Practice questions  