Regression (OLS)  overview
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Regression (OLS)  Goodness of fit test  One sample $z$ test for the mean  Two sample $t$ test  equal variances assumed  One sample $z$ test for the mean  Friedman test 


Independent variables  Independent variable  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  None  One categorical with 2 independent groups  None  One within subject factor ($\geq 2$ related groups)  
Dependent variable  Dependent variable  Dependent variable  Dependent variable  Dependent variable  Dependent variable  
One quantitative of interval or ratio level  One categorical with $J$ independent groups ($J \geqslant 2$)  One quantitative of interval or ratio level  One quantitative of interval or ratio level  One quantitative of interval or ratio level  One of ordinal level  
Null hypothesis  Null hypothesis  Null hypothesis  Null hypothesis  Null hypothesis  Null hypothesis  
$F$ test for the complete regression model:

 $\mu = \mu_0$
$\mu$ is the unknown population mean; $\mu_0$ is the population mean according to the null hypothesis  $\mu_1 = \mu_2$
$\mu_1$ is the unknown mean in population 1, $\mu_2$ is the unknown mean in population 2  $\mu = \mu_0$
$\mu$ is the unknown population mean; $\mu_0$ is the population mean according to the null hypothesis  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.  
Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  
$F$ test for the complete regression model:

 Two sided: $\mu \neq \mu_0$ Right sided: $\mu > \mu_0$ Left sided: $\mu < \mu_0$  Two sided: $\mu_1 \neq \mu_2$ Right sided: $\mu_1 > \mu_2$ Left sided: $\mu_1 < \mu_2$  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  
Assumptions  Assumptions  Assumptions  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  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 $\beta_1$  $X^2 = \sum{\frac{(\mbox{observed cell count}  \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
where the expected cell count for one cell = $N \times \pi_j$, the observed cell count is the observed sample count in that same cell, and the sum is over all $J$ cells  $z = \dfrac{\bar{y}  \mu_0}{\sigma / \sqrt{N}}$
$\bar{y}$ 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 $\mu_0$  $t = \dfrac{(\bar{y}_1  \bar{y}_2)  0}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}} = \dfrac{\bar{y}_1  \bar{y}_2}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}}$
$\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $s_p$ is the pooled standard deviation, $n_1$ 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 $s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$ 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 $\mu_2 > \mu_1$  $z = \dfrac{\bar{y}  \mu_0}{\sigma / \sqrt{N}}$
$\bar{y}$ 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 $\mu_0$  $Q = \dfrac{12}{N \times k(k + 1)} \sum R^2_i  3 \times N(k + 1)$
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.  
Sample standard deviation of the residuals $s$  n.a.  n.a.  Pooled standard deviation  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} $      $s_p = \sqrt{\dfrac{(n_1  1) \times s^2_1 + (n_2  1) \times s^2_2}{n_1 + n_2  2}}$      
Sampling distribution of $F$ and of $t$ if H0 were true  Sampling distribution of $X^2$ if H0 were true  Sampling distribution of $z$ if H0 were true  Sampling distribution of $t$ if H0 were true  Sampling distribution of $z$ if H0 were true  Sampling distribution of $Q$ if H0 were true  
Sampling distribution of $F$:
 Approximately a chisquared distribution with $J  1$ degrees of freedom  Standard normal  $t$ distribution with $n_1 + n_2  2$ degrees of freedom  Standard normal  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?  Significant?  Significant?  Significant?  Significant?  
$F$ test:

 Two sided:
 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 $X^2 = Q$:
 
$C\%$ confidence interval for $\beta_k$ and for $\mu_y$; $C\%$ prediction interval for $y_{new}$  n.a.  $C\%$ confidence interval for $\mu$  $C\%$ confidence interval for $\mu_1  \mu_2$  $C\%$ confidence interval for $\mu$  n.a.  
Confidence interval for $\beta_k$:
   $\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
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.  $(\bar{y}_1  \bar{y}_2) \pm t^* \times s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$
where the critical value $t^*$ is the value under the $t_{n_1 + n_2  2}$ 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.  $\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
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  Effect size  Effect size  n.a.  
Complete model:
   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 $\mu_0$  Cohen's $d$: Standardized difference between the mean in group $1$ and in group $2$: $$d = \frac{\bar{y}_1  \bar{y}_2}{s_p}$$ Indicates how many standard deviations $s_p$ the two sample means are removed from each other  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  Visual representation  Visual representation  n.a.  
      
ANOVA table  n.a.  n.a.  n.a.  n.a.  n.a.  
          
n.a.  n.a.  n.a.  Equivalent to  n.a.  n.a.  
      One way ANOVA with an independent variable with 2 levels ($I$ = 2):
OLS regression with one categorical independent variable with 2 levels:
     
Example context  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 of people with a low, moderate, and high social economic status in the population different from $\pi_{low}$ = .2, $\pi_{moderate}$ = .6, and $\pi_{high}$ = .2?  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.  Is the average mental health score different between men and women? Assume that in the population, the standard deviation of mental health scores is equal amongst men and women.  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.  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  n.a.  SPSS  n.a.  SPSS  
Analyze > Regression > Linear...
 Analyze > Nonparametric Tests > Legacy Dialogs > Chisquare...
   Analyze > Compare Means > IndependentSamples T Test...
   Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
 
Jamovi  Jamovi  n.a.  Jamovi  n.a.  Jamovi  
Regression > Linear Regression
 Frequencies > N Outcomes  $\chi^2$ Goodness of fit
   TTests > Independent Samples TTest
   ANOVA > Repeated Measures ANOVA  Friedman
 
Practice questions  Practice questions  Practice questions  Practice questions  Practice questions  Practice questions  