# 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 not assumed
Regression (OLS)
Two sample $t$ test - equal variances assumed
Independent variablesIndependent variableIndependent variableIndependent variableIndependent variablesIndependent variable
One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variablesNoneNoneOne categorical with 2 independent groupsOne 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 groups
Dependent variableDependent variableDependent variableDependent variableDependent variableDependent variable
One quantitative of interval or ratio levelOne categorical with $J$ independent groups ($J \geqslant 2$)One quantitative of interval or ratio levelOne quantitative of interval or ratio levelOne quantitative of interval or ratio levelOne quantitative of interval or ratio level
Null hypothesisNull 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 $\mu_y = \beta_0 + \beta_1 \times x_1 + \beta_2 \times x_2 + \ldots + \beta_K \times x_K$
• The population proportions in each of the $J$ conditions are $\pi_1$, $\pi_2$, $\ldots$, $\pi_J$
or equivalently
• The probability of drawing an observation from condition 1 is $\pi_1$, the probability of drawing an observation from condition 2 is $\pi_2$, $\ldots$, the probability of drawing an observation from condition $J$ is $\pi_J$
$\mu = \mu_0$
$\mu$ 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
$F$ 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 $\mu_y = \beta_0 + \beta_1 \times x_1 + \beta_2 \times x_2 + \ldots + \beta_K \times x_K$
$\mu_1 = \mu_2$
$\mu_1$ is the unknown mean in population 1, $\mu_2$ is the unknown mean in population 2
Alternative hypothesisAlternative 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: $\beta_k < 0$
• The population proportions are not all as specified under the null hypothesis
or equivalently
• The probabilities of drawing an observation from each of the conditions are not all as specified under the null hypothesis
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: $\mu_1 < \mu_2$
$F$ 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: $\mu_1 \neq \mu_2$
Right sided: $\mu_1 > \mu_2$
Left sided: $\mu_1 < \mu_2$
AssumptionsAssumptionsAssumptionsAssumptionsAssumptionsAssumptions
• 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
Often ignored additional assumption:
• Variables are measured without error
Also pay attention to:
• Multicollinearity
• Outliers
• Sample size is large enough for $X^2$ to be approximately chi-squared distributed. Rule of thumb: all $J$ expected cell counts are 5 or more
• Sample is a simple random sample from the population. That is, observations are independent of one another
• Scores are normally distributed in the population
• Population standard deviation $\sigma$ is known
• Sample is a simple random sample from the population. That is, observations are independent of one another
• Within each population, the scores on the dependent variable are normally distributed
• 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
• 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
Often ignored additional assumption:
• Variables are measured without error
Also pay attention to:
• Multicollinearity
• Outliers
• Within each population, the scores on the dependent variable are normally distributed
• The standard deviation of the scores on the dependent variable is the same in both populations: $\sigma_1 = \sigma_2$
• 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
Test statisticTest 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 $\beta_1$
$X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
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 $\mu_0$
$t = \dfrac{(\bar{y}_1 - \bar{y}_2) - 0}{\sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}} = \dfrac{\bar{y}_1 - \bar{y}_2}{\sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}}$
$\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $s^2_1$ is the sample variance in group 1, $s^2_2$ is the sample variance in group 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 $\sqrt{\frac{s^2_1}{n_1} + \frac{s^2_2}{n_2}}$ 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 $\mu_2 > \mu_1$
$F$ 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 $\beta_1$
$t = \dfrac{(\bar{y}_1 - \bar{y}_2) - 0}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}} = \dfrac{\bar{y}_1 - \bar{y}_2}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}}$
$\bar{y}_1$ 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 $\mu_2 > \mu_1$
Sample standard deviation of the residuals $s$n.a.n.a.n.a.Sample standard deviation of the residuals $s$Pooled standard deviation
\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}---\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 trueSampling distribution of X^2 if H0 were trueSampling distribution of z if H0 were trueSampling distribution of t if H0 were trueSampling distribution of F and of t if H0 were trueSampling distribution of t 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 a chi-squared distribution with J - 1 degrees of freedomStandard normalApproximately a t distribution with k degrees of freedom, with k equal to k = \dfrac{\Bigg(\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}\Bigg)^2}{\dfrac{1}{n_1 - 1} \Bigg(\dfrac{s^2_1}{n_1}\Bigg)^2 + \dfrac{1}{n_2 - 1} \Bigg(\dfrac{s^2_2}{n_2}\Bigg)^2} or k = the smaller of n_1 - 1 and n_2 - 1 First definition of k is used by computer programs, second definition is often used for hand calculations 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 t distribution with n_1 + n_2 - 2 degrees of freedom Significant?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: • Check if X^2 observed in sample is equal to or larger than critical value X^{2*} or • Find p value corresponding to observed X^2 and check if it is equal to or smaller than \alpha Two sided: Right sided: Left sided: Two sided: Right sided: Left sided: 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: C\% confidence interval for \beta_k and for \mu_y; C\% prediction interval for y_{new}n.a.C\% confidence interval for \muApproximate C\% confidence interval for \mu_1 - \mu_2$$C\% confidence interval for $\beta_k$ and for $\mu_y$; $C\%$ prediction interval for y_{new}$$C\% confidence interval for \mu_1 - \mu_2 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). -\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 \sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}} where the critical value t^* is the value under the t_{k} 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. 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). (\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. Effect sizen.a.Effect sizen.a.Effect sizeEffect size 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)$$Per independent variable: • Correlation squared r^2_k: the proportion of the total variance in the dependent variable y that is explained by the independent variable x_k, not corrected for the other independent variables in the model • Semi-partial correlation squared sr^2_k: the proportion of the total variance in the dependent variable y that is uniquely explained by the independent variable x_k, beyond the part that is already explained by the other independent variables in the model • Partial correlation squared pr^2_k: the proportion of the variance in the dependent variable y not explained by the other independent variables, that is uniquely explained by the independent variable x_k -Cohen's d: Standardized difference between the sample mean and \mu_0:$$d = \frac{\bar{y} - \mu_0}{\sigma}$$Indicates how many standard deviations \sigma the sample mean \bar{y} is removed from \mu_0 -Complete model: • Proportion variance explained R^2: Proportion variance of the dependent variable y explained by the sample regression equation (the independent variables):$$ \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)$$Per independent variable: • Correlation squared r^2_k: the proportion of the total variance in the dependent variable y that is explained by the independent variable x_k, not corrected for the other independent variables in the model • Semi-partial correlation squared sr^2_k: the proportion of the total variance in the dependent variable y that is uniquely explained by the independent variable x_k, beyond the part that is already explained by the other independent variables in the model • Partial correlation squared pr^2_k: the proportion of the variance in the dependent variable y not explained by the other independent variables, that is uniquely explained by the independent variable x_k Cohen's d: Standardized difference between the mean in group 1 and in group 2:$$d = \frac{\bar{y}_1 - \bar{y}_2}{s_p}$Indicates how many standard deviations$s_p$the two sample means are removed from each other n.a.n.a.Visual representationVisual representationn.a.Visual representation --- ANOVA tablen.a.n.a.n.a.ANOVA tablen.a. --- - n.a.n.a.n.a.n.a.n.a.Equivalent to -----One way ANOVA with an independent variable with 2 levels ($I$= 2): • two sided two sample$t$test equivalent to ANOVA$F$test when$I$= 2 • two sample$t$test equivalent to$t$test for contrast when$I$= 2 • two sample$t$test equivalent to$t$test multiple comparisons when$I$= 2 OLS regression with one categorical independent variable with 2 levels: • two sided two sample$t$test equivalent to$F$test regression model • two sample$t$test equivalent to$t$test for regression coefficient$\beta_1$Example contextExample contextExample contextExample contextExample contextExample context Can mental health be predicted from fysical health, economic class, and gender?Is the proportion of people with a low, moderate, and high social economic status in the population different from$\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$= 3.Is the average mental health score different between men and women?Can mental health be predicted from fysical health, economic class, and gender?Is the average mental health score different between men and women? Assume that in the population, the standard deviation of mental health scores is equal amongst men and women. SPSSSPSSn.a.SPSSSPSSSPSS Analyze > Regression > Linear... • Put your dependent variable in the box below Dependent and your independent (predictor) variables in the box below Independent(s) Analyze > Nonparametric Tests > Legacy Dialogs > Chi-square... • Put your categorical variable in the box below Test Variable List • Fill in the population proportions / probabilities according to$H_0$in the box below Expected Values. If$H_0$states that they are all equal, just pick 'All categories equal' (default) -Analyze > Compare Means > Independent-Samples T Test... • Put your dependent (quantitative) variable in the box below Test Variable(s) and your independent (grouping) variable in the box below Grouping Variable • Click on the Define Groups... button. If you can't click on it, first click on the grouping variable so its background turns yellow • Fill in the value you have used to indicate your first group in the box next to Group 1, and the value you have used to indicate your second group in the box next to Group 2 • Continue and click OK Analyze > Regression > Linear... • Put your dependent variable in the box below Dependent and your independent (predictor) variables in the box below Independent(s) Analyze > Compare Means > Independent-Samples T Test... • Put your dependent (quantitative) variable in the box below Test Variable(s) and your independent (grouping) variable in the box below Grouping Variable • Click on the Define Groups... button. If you can't click on it, first click on the grouping variable so its background turns yellow • Fill in the value you have used to indicate your first group in the box next to Group 1, and the value you have used to indicate your second group in the box next to Group 2 • Continue and click OK JamoviJamovin.a.JamoviJamoviJamovi Regression > Linear Regression • Put your dependent variable in the box below Dependent Variable and your independent variables of interval/ratio level in the box below Covariates • If you also have code (dummy) variables as independent variables, you can put these in the box below Covariates as well • Instead of transforming your categorical independent variable(s) into code variables, you can also put the untransformed categorical independent variables in the box below Factors. Jamovi will then make the code variables for you 'behind the scenes' Frequencies > N Outcomes -$\chi^2$Goodness of fit • Put your categorical variable in the box below Variable • Click on Expected Proportions and fill in the population proportions / probabilities according to$H_0$in the boxes below Ratio. If$H_0\$ states that they are all equal, you can leave the ratios equal to the default values (1)
-T-Tests > Independent Samples T-Test
• Put your dependent (quantitative) variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable
• Under Tests, select Welch's
• Under Hypothesis, select your alternative hypothesis
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'
T-Tests > Independent Samples T-Test
• Put your dependent (quantitative) variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable
• Under Tests, select Student's (selected by default)
• Under Hypothesis, select your alternative hypothesis
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