Regression (OLS) - overview
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Regression (OLS) | Binomial test for a single proportion | Paired sample $t$ test | Friedman test | One sample $t$ test for the mean | Cochran's Q test |
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Independent variables | Independent variable | Independent variable | Independent/grouping variable | Independent variable | Independent/grouping 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) | 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 2 independent groups | One quantitative of interval or ratio level | One of ordinal level | One quantitative of interval or ratio level | One categorical with 2 independent groups | |
Null hypothesis | Null hypothesis | Null hypothesis | Null hypothesis | Null hypothesis | Null hypothesis | |
$F$ test for the complete regression model:
| H0: $\pi = \pi_0$
Here $\pi$ is the population proportion of 'successes', and $\pi_0$ is the population proportion of successes according to the null hypothesis. | H0: $\mu = \mu_0$
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. | H0: the population scores in any of the related groups are not systematically higher or lower than the population scores in any of the other related groups
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. | H0: $\mu = \mu_0$
Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis. | H0: $\pi_1 = \pi_2 = \ldots = \pi_I$
Here $\pi_1$ is the population proportion of 'successes' for group 1, $\pi_2$ is the population proportion of 'successes' for group 2, and $\pi_I$ is the population proportion of 'successes' for group $I.$ | |
Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |
$F$ test for the complete regression model:
| H1 two sided: $\pi \neq \pi_0$ H1 right sided: $\pi > \pi_0$ H1 left sided: $\pi < \pi_0$ | H1 two sided: $\mu \neq \mu_0$ H1 right sided: $\mu > \mu_0$ H1 left sided: $\mu < \mu_0$ | H1: the population scores in some of the related groups are systematically higher or lower than the population scores in other related groups | H1 two sided: $\mu \neq \mu_0$ H1 right sided: $\mu > \mu_0$ H1 left sided: $\mu < \mu_0$ | H1: not all population proportions are equal | |
Assumptions | Assumptions | Assumptions | Assumptions | Assumptions | Assumptions | |
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Test statistic | Test statistic | Test statistic | Test statistic | Test statistic | Test statistic | |
$F$ test for the complete regression model:
Note 2: if there is only one independent variable in the model ($K = 1$), 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 | $t = \dfrac{\bar{y} - \mu_0}{s / \sqrt{N}}$
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 $\mu_0$. | $Q = \dfrac{12}{N \times k(k + 1)} \sum R^2_i - 3 \times N(k + 1)$
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. | $t = \dfrac{\bar{y} - \mu_0}{s / \sqrt{N}}$
Here $\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to the null hypothesis, $s$ is the sample standard deviation, and $N$ is the sample size. 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 $\mu_0$. | If a failure is scored as 0 and a success is scored as 1:
$Q = k(k - 1) \dfrac{\sum_{groups} \Big (\mbox{group total} - \frac{\mbox{grand total}}{k} \Big)^2}{\sum_{blocks} \mbox{block total} \times (k - \mbox{block total})}$ Here $k$ is the number of related groups (usually the number of repeated measurements), a group total is the sum of the scores in a group, a block total is the sum of the scores in a block (usually a subject), and the grand total is the sum of all the scores. Before computing $Q$, first exclude blocks with equal scores in all $k$ groups. | |
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 $X$ 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 $Q$ 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 chi-squared 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 | If the number of blocks (usually the number of subjects) is large, approximately the chi-squared distribution with $k - 1$ degrees of freedom | |
Significant? | 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:
| If the number of blocks is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
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$C\%$ 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 $\mu$ | n.a. | |
Confidence interval for $\beta_k$:
| - | $\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$
where the critical value $t^*$ is the value under the $t_{N-1}$ 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 $t_{N-1}$ 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 | n.a. | |
Complete model:
| - | Cohen's $d$: Standardized difference between the sample mean of the difference scores and $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{s}$$ Cohen's $d$ 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 $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{s}$$ Cohen's $d$ indicates how many standard deviations $s$ the sample mean $\bar{y}$ is removed from $\mu_0.$ | - | |
Visual representation | n.a. | Visual representation | n.a. | Visual representation | n.a. | |
Regression equations with: | - | - | - | |||
ANOVA table | n.a. | n.a. | n.a. | n.a. | n.a. | |
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n.a. | n.a. | Equivalent to | n.a. | n.a. | Equivalent to | |
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| - | - | Friedman test, with a categorical dependent variable consisting of two independent groups. | |
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 smokers amongst office workers different from $\pi_0 = 0.2$? | 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 (pre-intervention), measurement point 2 (1 week post-intervention), and measurement point 3 (6 weeks post-intervention)? | Is the average mental health score of office workers different from $\mu_0 = 50$? | Subjects perform three different tasks, which they can either perform correctly or incorrectly. Is there a difference in task performance between the three different tasks? | |
SPSS | SPSS | SPSS | SPSS | SPSS | SPSS | |
Analyze > Regression > Linear...
| Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
| Analyze > Compare Means > Paired-Samples T Test...
| Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
| Analyze > Compare Means > One-Sample T Test...
| Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
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Jamovi | Jamovi | Jamovi | Jamovi | Jamovi | Jamovi | |
Regression > Linear Regression
| Frequencies > 2 Outcomes - Binomial test
| T-Tests > Paired Samples T-Test
| ANOVA > Repeated Measures ANOVA - Friedman
| T-Tests > One Sample T-Test
| Jamovi does not have a specific option for the Cochran's Q test. However, you can do the Friedman test instead. The $p$ value resulting from this Friedman test is equivalent to the $p$ value that would have resulted from the Cochran's Q test. Go to:
ANOVA > Repeated Measures ANOVA - Friedman
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Practice questions | Practice questions | Practice questions | Practice questions | Practice questions | Practice questions | |