Regression (OLS) - overview
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Regression (OLS) | Binomial test for a single proportion | Paired sample $t$ test | Binomial test for a single proportion | Two sample $z$ test |
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Independent variables | Independent variable | Independent 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 | None | One categorical with 2 independent 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 categorical with 2 independent groups | One quantitative of interval or ratio level | |
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: $\pi = \pi_0$
Here $\pi$ is the population proportion of 'successes', and $\pi_0$ is the population proportion of successes according to the null hypothesis. | H0: $\mu_1 = \mu_2$
Here $\mu_1$ is the population mean for group 1, and $\mu_2$ is the population mean for group 2. | |
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 two sided: $\pi \neq \pi_0$ H1 right sided: $\pi > \pi_0$ H1 left sided: $\pi < \pi_0$ | H1 two sided: $\mu_1 \neq \mu_2$ H1 right sided: $\mu_1 > \mu_2$ H1 left sided: $\mu_1 < \mu_2$ | |
Assumptions | Assumptions | Assumptions | Assumptions | Assumptions | |
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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 $\mu_0$. | $X$ = number of successes in the sample | $z = \dfrac{(\bar{y}_1 - \bar{y}_2) - 0}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}} = \dfrac{\bar{y}_1 - \bar{y}_2}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}}$
Here $\bar{y}_1$ 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, and $n_2$ is the sample size of group 2. The 0 represents the difference in population means according to the null hypothesis. 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 $\mu_2 > \mu_1$. | |
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 $X$ 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 | Binomial($n$, $P$) distribution.
Here $n = N$ (total sample size), and $P = \pi_0$ (population proportion according to the null hypothesis). | Standard normal distribution | |
Significant? | Significant? | Significant? | Significant? | Significant? | |
$F$ test:
| Two sided:
| Two sided:
| Two sided:
| Two sided:
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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 $\mu_1 - \mu_2$ | |
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. | - | $(\bar{y}_1 - \bar{y}_2) \pm z^* \times \sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}$
where the critical value $z^*$ 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. | |
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 $\mu_0.$ | - | - | |
Visual representation | n.a. | Visual representation | n.a. | Visual representation | |
Regression equations with: | - | - | |||
ANOVA table | n.a. | n.a. | n.a. | n.a. | |
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n.a. | n.a. | Equivalent to | n.a. | n.a. | |
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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 the proportion of smokers amongst office workers different from $\pi_0 = 0.2$? | 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. | |
SPSS | SPSS | SPSS | SPSS | n.a. | |
Analyze > Regression > Linear...
| Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
| Analyze > Compare Means > Paired-Samples T Test...
| Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
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Jamovi | Jamovi | Jamovi | Jamovi | n.a. | |
Regression > Linear Regression
| Frequencies > 2 Outcomes - Binomial test
| T-Tests > Paired Samples T-Test
| Frequencies > 2 Outcomes - Binomial test
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Practice questions | Practice questions | Practice questions | Practice questions | Practice questions | |