Regression (OLS)  overview
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Regression (OLS)  $z$ test for the difference between two proportions  Paired sample $t$ test  Sign test  Two sample $z$ test  Spearman's rho 


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  One categorical with 2 independent groups  2 paired groups  2 paired groups  One categorical with 2 independent groups  One of ordinal level  
Dependent variable  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  One of ordinal level  
Null hypothesis  Null hypothesis  Null hypothesis  Null hypothesis  Null hypothesis  Null hypothesis  
$F$ test for the complete regression model:
 $\pi_1 = \pi_2$
$\pi_1$ is the unknown proportion of "successes" in population 1; $\pi_2$ is the unknown proportion of "successes" in population 2  $\mu = \mu_0$
$\mu$ 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 
 $\mu_1 = \mu_2$
$\mu_1$ is the unknown mean in population 1, $\mu_2$ is the unknown mean in population 2  $\rho_s = 0$
$\rho_s$ is the unknown Spearman correlation in the population. In words: there is no monotonic relationship between the two variables in the population  
Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  
$F$ test for the complete regression model:
 Two sided: $\pi_1 \neq \pi_2$ Right sided: $\pi_1 > \pi_2$ Left sided: $\pi_1 < \pi_2$  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: $\rho_s \neq 0$ Right sided: $\rho_s > 0$ Left sided: $\rho_s < 0$  
Assumptions  Assumptions  Assumptions  Assumptions  Assumptions  Assumptions  


 Sample of pairs is a simple random sample from the population of pairs. That is, pairs 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
Note: this assumption is only important for the significance test, not for the correlation coefficient itself. The correlation coefficient itself just measures the strength of the monotonic relationship between two variables.  
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 $\beta_1$  $z = \dfrac{p_1  p_2}{\sqrt{p(1  p)\Bigg(\dfrac{1}{n_1} + \dfrac{1}{n_2}\Bigg)}}$
$p_1$ is the sample proportion of successes in group 1: $\dfrac{X_1}{n_1}$, $p_2$ is the sample proportion of successes in group 2: $\dfrac{X_2}{n_2}$, $p$ is the total proportion of successes in the sample: $\dfrac{X_1 + X_2}{n_1 + n_2}$, $n_1$ is the sample size of group 1, $n_2$ is the sample size of group 2 Note: we could just as well compute $p_2  p_1$ in the numerator, but then the left sided alternative becomes $\pi_2 < \pi_1$, and the right sided alternative becomes $\pi_2 > \pi_1$  $t = \dfrac{\bar{y}  \mu_0}{s / \sqrt{N}}$
$\bar{y}$ 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 $\mu_0$  $W = $ number of difference scores that is larger than 0  $z = \dfrac{(\bar{y}_1  \bar{y}_2)  0}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}} = \dfrac{\bar{y}_1  \bar{y}_2}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}}$
$\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $\sigma^2_1$ is the population variance in population 1, $\sigma^2_2$ is the population variance in population 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 $\sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}}$ is the standard deviation of the sampling distribution of $\bar{y}_1  \bar{y}_2$. The $z$ value indicates how many of these standard deviations $\bar{y}_1  \bar{y}_2$ 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 $\mu_2 > \mu_1$  $t = \dfrac{r_s \times \sqrt{N  2}}{\sqrt{1  r_s^2}} $ where $r_s$ is the sample Spearman correlation and $N$ is the sample size. The sample Spearman correlation $r_s$ is equal to the Pearson correlation applied to the rank scores.  
Sample standard deviation of the residuals $s$  n.a.  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 $z$ if H0 were true  Sampling distribution of $t$ if H0 were true  Sampling distribution of $W$ if H0 were true  Sampling distribution of $z$ if H0 were true  Sampling distribution of $t$ if H0 were true  
Sampling distribution of $F$:
 Approximately standard normal  $t$ distribution with $N  1$ degrees of freedom  The exact distribution of $W$ under the null hypothesis is the Binomial($n$, $p$) distribution, with $n =$ number of positive differences $+$ number of negative differences, and $p = 0.5$.
If $n$ is large, $W$ is approximately normally distributed under the null hypothesis, with mean $np = n \times 0.5$ and standard deviation $\sqrt{np(1p)} = \sqrt{n \times 0.5(1  0.5)}$. Hence, if $n$ is large, the standardized test statistic $$z = \frac{W  n \times 0.5}{\sqrt{n \times 0.5(1  0.5)}}$$ follows approximately a standard normal distribution if the null hypothesis were true.  Standard normal  Approximately a $t$ distribution with $N  2$ degrees of freedom  
Significant?  Significant?  Significant?  Significant?  Significant?  Significant?  
$F$ test:
 Two sided:
 Two sided:
 If $n$ is small, the table for the binomial distribution should be used: Two sided:
If $n$ is large, the table for standard normal probabilities can be used: Two sided:
 Two sided:
 Two sided:
 
$C\%$ confidence interval for $\beta_k$ and for $\mu_y$; $C\%$ prediction interval for $y_{new}$  Approximate $C\%$ confidence interval for $\pi_1  \pi_2$  $C\%$ confidence interval for $\mu$  n.a.  $C\%$ confidence interval for $\mu_1  \mu_2$  n.a.  
Confidence interval for $\beta_k$:
 Regular (large sample):
 $\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$
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.    $(\bar{y}_1  \bar{y}_2) \pm z^* \times \sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}$
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 $\mu_1  \mu_2$ can also be used as significance test.    
Effect size  n.a.  Effect size  n.a.  n.a.  n.a.  
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 $\mu_0$        
n.a.  n.a.  Visual representation  n.a.  Visual representation  n.a.  
        
ANOVA table  n.a.  n.a.  n.a.  n.a.  n.a.  
          
n.a.  Equivalent to  Equivalent to  Equivalent to  n.a.  n.a.  
  When testing two sided: chisquared test for the relationship between two categorical variables, where both categorical variables have 2 levels  One sample $t$ test on the difference scores
Repeated measures ANOVA with one dichotomous within subjects factor 
Two sided sign test is equivalent to
     
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 smokers different between men and women? Use the normal approximation for the sampling distribution of the test statistic.  Is the average difference between the mental health scores before and after an intervention different from $\mu_0$ = 0?  Do people tend to score higher on mental health after a mindfulness course?  Is the average mental health score different between men and women? Assume that in the population, the standard devation of the mental health scores is $\sigma_1$ = 2 amongst men and $\sigma_2$ = 2.5 amongst women.  Is there a monotonic relationship between physical health and mental health?  
SPSS  SPSS  SPSS  SPSS  n.a.  SPSS  
Analyze > Regression > Linear...
 SPSS does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chisquared test instead. The $p$ value resulting from this chisquared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to:
Analyze > Descriptive Statistics > Crosstabs...
 Analyze > Compare Means > PairedSamples T Test...
 Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
   Analyze > Correlate > Bivariate...
 
Jamovi  Jamovi  Jamovi  Jamovi  n.a.  Jamovi  
Regression > Linear Regression
 Jamovi does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chisquared test instead. The $p$ value resulting from this chisquared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to:
Frequencies > Independent Samples  $\chi^2$ test of association
 TTests > Paired Samples TTest
 Jamovi does not have a specific option for the sign test. However, you can do the Friedman test instead. The $p$ value resulting from this Friedman test is equivalent to the two sided $p$ value that would have resulted from the sign test. Go to:
ANOVA > Repeated Measures ANOVA  Friedman
   Regression > Correlation Matrix
 
Practice questions  Practice questions  Practice questions  Practice questions  Practice questions  Practice questions  