# Regression (OLS) - overview

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Regression (OLS)
$z$ test for the difference between two proportions
Paired sample $t$ test
Sign test
Pearson correlation
Independent variablesIndependent variableIndependent variableIndependent variableVariable 1
One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variablesOne categorical with 2 independent groups2 paired groups2 paired groupsOne quantitative of interval or ratio level
Dependent variableDependent variableDependent variableDependent variableVariable 2
One quantitative of interval or ratio levelOne categorical with 2 independent groupsOne quantitative of interval or ratio levelOne of ordinal levelOne quantitative of interval or ratio level
Null hypothesisNull hypothesisNull hypothesisNull hypothesisNull hypothesis
$F$ test for the complete regression model:
• $\beta_1 = \beta_2 = \ldots = \beta_K = 0$
or equivalenty
• The variance explained by all the independent variables together (the complete model) is 0 in the population: $\rho^2 = 0$
$t$ test for individual regression coefficient $\beta_k$:
• $\beta_k = 0$
in the regression equation $\mu_y = \beta_0 + \beta_1 \times x_1 + \beta_2 \times x_2 + \ldots + \beta_K \times x_K$
$\pi_1 = \pi_2$
$\pi_1$ 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
• P(first score of a pair exceeds second score of a pair) = P(second score of a pair exceeds first score of a pair)
If the dependent variable is measured on a continuous scale, this can also be formulated as:
• The median of the difference scores is zero in the population
$\rho = \rho_0$
$\rho$ is the unknown Pearson correlation in the population, $\rho_0$ is the correlation in the population according to the null hypothesis (usually 0)
Alternative hypothesisAlternative hypothesisAlternative hypothesisAlternative hypothesisAlternative hypothesis
$F$ test for the complete regression model:
• not all population regression coefficients are 0
or equivalenty
• The variance explained by all the independent variables together (the complete model) is larger than 0 in the population: $\rho^2 > 0$
$t$ test for individual $\beta_k$:
• Two sided: $\beta_k \neq 0$
• Right sided: $\beta_k > 0$
• Left sided: $\beta_k < 0$
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: P(first score of a pair exceeds second score of a pair) $\neq$ P(second score of a pair exceeds first score of a pair)
• Right sided: P(first score of a pair exceeds second score of a pair) > P(second score of a pair exceeds first score of a pair)
• Left sided: P(first score of a pair exceeds second score of a pair) < P(second score of a pair exceeds first score of a pair)
If the dependent variable is measured on a continuous scale, this can also be formulated as:
• Two sided: the median of the difference scores is different from zero in the population
• Right sided: the median of the difference scores is larger than zero in the population
• Left sided: the median of the difference scores is smaller than zero in the population
Two sided: $\rho \neq \rho_0$
Right sided: $\rho > \rho_0$
Left sided: $\rho < \rho_0$
AssumptionsAssumptionsAssumptionsAssumptionsAssumptions of tests for correlation
• In the population, the residuals are normally distributed at each combination of values of the independent variables
• In the population, the standard deviation $\sigma$ of the residuals is the same for each combination of values of the independent variables (homoscedasticity)
• In the population, the relationship between the independent variables and the mean of the dependent variable $\mu_y$ is linear. If this linearity assumption holds, the mean of the residuals is 0 for each combination of values of the independent variables
• The residuals are independent of one another
• Variables are measured without error
Also pay attention to:
• Multicollinearity
• Outliers
• Sample size is large enough for $z$ to be approximately normally distributed. Rule of thumb:
• Significance test: number of successes and number of failures are each 5 or more in both sample groups
• Regular (large sample) 90%, 95%, or 99% confidence interval: number of successes and number of failures are each 10 or more in both sample groups
• Plus four 90%, 95%, or 99% confidence interval: sample sizes of both groups are 5 or more
• Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2. That is, within and between groups, observations are independent of one another
• Difference scores are normally distributed in the population
• Sample of difference scores is a simple random sample from the population of difference scores. That is, difference scores are independent of one another
Population of difference scores can be conceived of as the difference scores we would find if we would apply our study (e.g., applying an intervention and measuring pre-post scores) to all individuals in the population.
Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
• In the population, the two variables are jointly normally distributed (this covers the normality, homoscedasticity, and linearity assumptions)
• Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
Note: these assumptions are only important for the significance test and confidence interval, not for the correlation coefficient itself. The correlation coefficient just measures the strength of the linear relationship between two variables.
Test statisticTest statisticTest statisticTest statisticTest statistic
$F$ test for the complete regression model:
• \begin{aligned}[t] F &= \dfrac{\sum (\hat{y}_j - \bar{y})^2 / K}{\sum (y_j - \hat{y}_j)^2 / (N - K - 1)}\\ &= \dfrac{\mbox{sum of squares model} / \mbox{degrees of freedom model}}{\mbox{sum of squares error} / \mbox{degrees of freedom error}}\\ &= \dfrac{\mbox{mean square model}}{\mbox{mean square error}} \end{aligned}
where $\hat{y}_j$ is the predicted score on the dependent variable $y$ of subject $j$, $\bar{y}$ is the mean of $y$, $y_j$ is the score on $y$ of subject $j$, $N$ is the total sample size, and $K$ is the number of independent variables
$t$ test for individual $\beta_k$:
• $t = \dfrac{b_k}{SE_{b_k}}$
• If only one independent variable:
$SE_{b_1} = \dfrac{\sqrt{\sum (y_j - \hat{y}_j)^2 / (N - 2)}}{\sqrt{\sum (x_j - \bar{x})^2}} = \dfrac{s}{\sqrt{\sum (x_j - \bar{x})^2}}$, with $s$ the sample standard deviation of the residuals, $x_j$ the score of subject $j$ on the independent variable $x$, and $\bar{x}$ the mean of $x$. For models with more than one independent variable, computing $SE_{b_k}$ becomes complicated
Note 1: mean square model is also known as mean square regression; mean square error is also known as mean square residual
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 0Test statistic for testing H0: $\rho = 0$:
• $t = \dfrac{r \times \sqrt{N - 2}}{\sqrt{1 - r^2}}$
where $r$ is the sample correlation $r = \frac{1}{N - 1} \sum_{j}\Big(\frac{x_{j} - \bar{x}}{s_x} \Big) \Big(\frac{y_{j} - \bar{y}}{s_y} \Big)$ and $N$ is the sample size
Test statistic for testing values for $\rho$ other than $\rho = 0$:
• $z = \dfrac{r_{Fisher} - \rho_{0_{Fisher}}}{\sqrt{\dfrac{1}{N - 3}}}$
• $r_{Fisher} = \dfrac{1}{2} \times \log\Bigg(\dfrac{1 + r}{1 - r} \Bigg )$, where $r$ is the sample correlation
• $\rho_{0_{Fisher}} = \dfrac{1}{2} \times \log\Bigg( \dfrac{1 + \rho_0}{1 - \rho_0} \Bigg )$, where $\rho_0$ is the population correlation according to H0
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 trueSampling distribution of $z$ if H0 were trueSampling distribution of $t$ if H0 were trueSampling distribution of $W$ if H0 were trueSampling distribution of $t$ and of $z$ if H0 were true
Sampling distribution of $F$:
• $F$ distribution with $K$ (df model, numerator) and $N - K - 1$ (df error, denominator) degrees of freedom
Sampling distribution of $t$:
• $t$ distribution with $N - K - 1$ (df error) degrees of freedom
Approximately standard normal$t$ distribution with $N - 1$ degrees of freedomThe 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 $\sqrt{np(1-p)} = \sqrt{n \times 0.5(1 - 0.5)}$. 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.
Sampling distribution of $t$:
• $t$ distribution with $N - 2$ degrees of freedom
Sampling distribution of $z$:
• Approximately standard normal
Significant?Significant?Significant?Significant?Significant?
$F$ test:
• Check if $F$ observed in sample is equal to or larger than critical value $F^*$ or
• Find $p$ value corresponding to observed $F$ and check if it is equal to or smaller than $\alpha$
$t$ Test two sided:
$t$ Test right sided:
$t$ Test left sided:
Two sided:
Right sided:
Left sided:
Two sided:
Right sided:
Left sided:
If $n$ is small, the table for the binomial distribution should be used:
Two sided:
• Check if $W$ observed in sample is in the rejection region or
• Find two sided $p$ value corresponding to observed $W$ and check if it is equal to or smaller than $\alpha$
Right sided:
• Check if $W$ observed in sample is in the rejection region or
• Find right sided $p$ value corresponding to observed $W$ and check if it is equal to or smaller than $\alpha$
Left sided:
• Check if $W$ observed in sample is in the rejection region or
• Find left sided $p$ value corresponding to observed $W$ and check if it is equal to or smaller than $\alpha$

If $n$ is large, the table for standard normal probabilities can be used:
Two sided:
Right sided:
Left sided:
$t$ Test two sided:
$t$ Test right sided:
$t$ Test left sided:
$z$ Test two sided:
$z$ Test right sided:
$z$ Test left 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 \mun.a.Approximate C% confidence interval for \rho Confidence interval for \beta_k: • b_k \pm t^* \times SE_{b_k} • If only one independent variable: SE_{b_1} = \dfrac{\sqrt{\sum (y_j - \hat{y}_j)^2 / (N - 2)}}{\sqrt{\sum (x_j - \bar{x})^2}} = \dfrac{s}{\sqrt{\sum (x_j - \bar{x})^2}} Confidence interval for \mu_y, the population mean of y given the values on the independent variables: • \hat{y} \pm t^* \times SE_{\hat{y}} • If only one independent variable: SE_{\hat{y}} = s \sqrt{\dfrac{1}{N} + \dfrac{(x^* - \bar{x})^2}{\sum (x_j - \bar{x})^2}} Prediction interval for y_{new}, the score on y of a future respondent: • \hat{y} \pm t^* \times SE_{y_{new}} • If only one independent variable: SE_{y_{new}} = s \sqrt{1 + \dfrac{1}{N} + \dfrac{(x^* - \bar{x})^2}{\sum (x_j - \bar{x})^2}} In all formulas, the critical value t^* is the value under the t_{N - K - 1} distribution with the area C / 100 between -t^* and t^* (e.g. t^* = 2.086 for a 95% confidence interval when df = 20). Regular (large sample): • (p_1 - p_2) \pm z^* \times \sqrt{\dfrac{p_1(1 - p_1)}{n_1} + \dfrac{p_2(1 - p_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) With plus four method: • (p_{1.plus} - p_{2.plus}) \pm z^* \times \sqrt{\dfrac{p_{1.plus}(1 - p_{1.plus})}{n_1 + 2} + \dfrac{p_{2.plus}(1 - p_{2.plus})}{n_2 + 2}} where p_{1.plus} = \dfrac{X_1 + 1}{n_1 + 2}, p_{2.plus} = \dfrac{X_2 + 1}{n_2 + 2}, and 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) \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. -First compute approximate C% confidence interval for \rho_{Fisher}: • lower_{Fisher} = r_{Fisher} - z^* \times \sqrt{\dfrac{1}{N - 3}} • upper_{Fisher} = r_{Fisher} + z^* \times \sqrt{\dfrac{1}{N - 3}} where r_{Fisher} = \frac{1}{2} \times \log\Bigg(\dfrac{1 + r}{1 - r} \Bigg ) and 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). Then transform back to get approximate C% confidence interval for \rho: • lower bound = \dfrac{e^{2 \times lower_{Fisher}} - 1}{e^{2 \times lower_{Fisher}} + 1} • upper bound = \dfrac{e^{2 \times upper_{Fisher}} - 1}{e^{2 \times upper_{Fisher}} + 1} Effect sizen.a.Effect sizen.a.Properties of the Pearson correlation coefficient Complete model: • Proportion variance explained R^2: Proportion variance of the dependent variable y explained by the sample regression equation (the independent variables):$$ \begin{align} R^2 &= \dfrac{\sum (\hat{y}_j - \bar{y})^2}{\sum (y_j - \bar{y})^2}\\ &= \dfrac{\mbox{sum of squares model}}{\mbox{sum of squares total}}\\ &= 1 - \dfrac{\mbox{sum of squares error}}{\mbox{sum of squares total}}\\ &= r(y, \hat{y})^2 \end{align} $$R^2 is the proportion variance explained in the sample by the sample regression equation. It is a positively biased estimate of the proportion variance explained in the population by the population regression equation, \rho^2. If there is only one independent variable, R^2 = r^2: the correlation between the independent variable x and dependent variable y squared. • Wherry's R^2 / shrunken R^2: Corrects for the positive bias in R^2 and is equal to$$R^2_W = 1 - \frac{N - 1}{N - K - 1}(1 - R^2)$$R^2_W is a less biased estimate than R^2 of the proportion variance explained in the population by the population regression equation, \rho^2 • Stein's R^2: Estimates the proportion of variance in y that we expect the current sample regression equation to explain in a different sample drawn from the same population. It is equal to$$R^2_S = 1 - \frac{(N - 1)(N - 2)(N + 1)}{(N - K - 1)(N - K - 2)(N)}(1 - R^2)$$Per independent variable: • Correlation squared r^2_k: the proportion of the total variance in the dependent variable y that is explained by the independent variable x_k, not corrected for the other independent variables in the model • Semi-partial correlation squared sr^2_k: the proportion of the total variance in the dependent variable y that is uniquely explained by the independent variable x_k, beyond the part that is already explained by the other independent variables in the model • Partial correlation squared pr^2_k: the proportion of the variance in the dependent variable y not explained by the other independent variables, that is uniquely explained by the independent variable x_k -Cohen's d: Standardized difference between the sample mean of the difference scores and \mu_0:$$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$- • The Pearson correlation coefficient is a measure for the linear relationship between two quantitative variables. • The Pearson correlation coefficient squared reflects the proportion of variance explained in one variable by the other variable. • The Pearson correlation coefficient can take on values between -1 (perfect negative relationship) and 1 (perfect positive relationship). A value of 0 means no linear relationship. • The absolute size of the Pearson correlation coefficient is not affected by any linear transformation of the variables. However, the sign of the Pearson correlation will flip when the scores on one of the two variables are multiplied by a negative number (reversing the direction of measurement of that variable). For example: • the correlation between$x$and$y$is equivalent to the correlation between$3x + 5$and$2y - 6$. • the absolute value of the correlation between$x$and$y$is equivalent to the absolute value of the correlation between$-3x + 5$and$2y - 6$. However, the signs of the two correlation coefficients will be in opposite directions, due to the multiplication of$x$by$-3$. • The Pearson correlation coefficient does not say anything about causality. • The Pearson correlation coefficient is sensitive to outliers. n.a.n.a.Visual representationn.a.n.a. ---- ANOVA tablen.a.n.a.n.a.n.a. ---- n.a.Equivalent toEquivalent toEquivalent toEquivalent to -When testing two sided: chi-squared test for the relationship between two categorical variables, where both categorical variables have 2 levelsOne sample$t$test on the difference scores Repeated measures ANOVA with one dichotomous within subjects factor Two sided sign test is equivalent to OLS regression with one independent variable: •$b_1 = r \times \frac{s_y}{s_x}$• Results significance test ($t$and$p$value) testing$H_0$:$\beta_1 = 0$are equivalent to results significance test testing$H_0$:$\rho = 0$Example contextExample contextExample contextExample contextExample 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 there a linear relationship between physical health and mental health? SPSSSPSSSPSSSPSSSPSS Analyze > Regression > Linear... • Put your dependent variable in the box below Dependent and your independent (predictor) variables in the box below Independent(s) SPSS does not have a specific option for the$z$test for the difference between two proportions. However, you can do the chi-squared test instead. The$p$value resulting from this chi-squared test is equivalent to the two sided$p$value that would have resulted from the$z$test. Go to: Analyze > Descriptive Statistics > Crosstabs... • Put your independent (grouping) variable in the box below Row(s), and your dependent variable in the box below Column(s) • Click the Statistics... button, and click on the square in front of Chi-square • Continue and click OK Analyze > Compare Means > Paired-Samples T Test... • Put the two paired variables in the boxes below Variable 1 and Variable 2 Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples... • Put the two paired variables in the boxes below Variable 1 and Variable 2 • Under Test Type, select the Sign test Analyze > Correlate > Bivariate... • Put your two variables in the box below Variables JamoviJamoviJamoviJamoviJamovi Regression > Linear Regression • Put your dependent variable in the box below Dependent Variable and your independent variables of interval/ratio level in the box below Covariates • If you also have code (dummy) variables as independent variables, you can put these in the box below Covariates as well • Instead of transforming your categorical independent variable(s) into code variables, you can also put the untransformed categorical independent variables in the box below Factors. Jamovi will then make the code variables for you 'behind the scenes' Jamovi does not have a specific option for the$z$test for the difference between two proportions. However, you can do the chi-squared test instead. The$p$value resulting from this chi-squared 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 • Put your independent (grouping) variable in the box below Rows, and your dependent variable in the box below Columns T-Tests > Paired Samples T-Test • Put the two paired variables in the box below Paired Variables, one on the left side of the vertical line and one on the right side of the vertical line • Under Hypothesis, select your alternative hypothesis 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
• Put the two paired variables in the box below Measures
Regression > Correlation Matrix
• Put your two variables in the white box at the right
• Under Correlation Coefficients, select Pearson (selected by default)
• Under Hypothesis, select your alternative hypothesis
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