Regression (OLS) - overview

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Regression (OLS)
Kruskal-Wallis test
Marginal Homogeneity test / Stuart-Maxwell test
Independent variablesIndependent variableIndependent variable
One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variablesOne categorical with $I$ independent groups ($I \geqslant 2$)2 paired groups
Dependent variableDependent variableDependent variable
One quantitative of interval or ratio levelOne of ordinal levelOne categorical with $J$ independent groups ($J \geqslant 2$)
Null 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 $
If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in all $I$ populations:
  • The medians in the $I$ populations are equal
Else:
Formulation 1:
  • The scores in any of the $I$ populations are not systematically higher or lower than the scores in any of the other populations
Formulation 2:
  • P(an observation from population $g$ exceeds an observation from population $h$) = P(an observation from population $h$ exceeds an observation from population $g$), for each pair of groups.
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.
For each category $j$ of the dependent variable:

$\pi_j$ in the first paired group = $\pi_j$ in the second paired group

Here $\pi_j$ is the population proportion for category $j$
Alternative 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: $\beta_k < 0$
If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in all $I$ populations:
  • Not all of the medians in the $I$ populations are equal
Else:
Formulation 1:
  • The scores in some populations are systematically higher or lower than the scores in other populations
Formulation 2:
  • For at least one pair of groups:
    P(an observation from population $g$ exceeds an observation from population $h$) $\neq$ P(an observation from population $h$ exceeds an observation from population $g$)
For some categories of the dependent variable, $\pi_j$ in the first paired group $\neq$ $\pi_j$ in the second paired group
AssumptionsAssumptionsAssumptions
  • 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
Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2, $\ldots$, group $I$ sample is an independent SRS from population $I$. That is, within and between groups, observations 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
Test 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 $\beta_1$

$H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i} - 3(N + 1)$

Here $N$ is the total sample size, $R_i$ is the sum of ranks in group $i$, and $n_i$ is the sample size of group $i$. Remember that multiplication precedes addition, so first compute $\frac{12}{N (N + 1)} \times \sum \frac{R^2_i}{n_i}$ and then subtract $3(N + 1)$.

Note: if ties are present in the data, the formula for $H$ is more complicated.
Computing the test statistic is a bit complicated and involves matrix algebra. You probably won't need to calculate it by hand (unless you are following a technical course)
Sample standard deviation of the residuals $s$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 $H$ if H0 were trueSampling distribution of the test statistic 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

For large samples, approximately the chi-squared distribution with $I - 1$ degrees of freedom.

For small samples, the exact distribution of $H$ should be used.

Approximately a chi-squared distribution with $J - 1$ degrees of freedom
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:
For large samples, the table with critical $X^2$ values can be used. If we denote $X^2 = H$:
  • 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 $\alpha$
If we denote the test statistic as $X^2$:
  • 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 $\alpha$
$C\%$ confidence interval for $\beta_k$ and for $\mu_y$; $C\%$ prediction interval for $y_{new}$n.a.n.a.
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).
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Effect sizen.a.n.a.
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 $x_k$
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ANOVA tablen.a.n.a.
ANOVA table regression analysis
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Example contextExample contextExample context
Can mental health be predicted from fysical health, economic class, and gender?Do people from different religions tend to score differently on social economic status? Subjects are asked to taste three different types of mayonnaise, and to indicate which of the three types of mayonnaise they like best. They then have to drink a glass of beer, and taste and rate the three types of mayonnaise again. Does drinking a beer change which type of mayonnaise people like best?
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 > K Independent Samples...
  • Put your dependent variable in the box below Test Variable List and your independent (grouping) variable in the box below Grouping Variable
  • Click on the Define Range... button. If you can't click on it, first click on the grouping variable so its background turns yellow
  • Fill in the smallest value you have used to indicate your groups in the box next to Minimum, and the largest value you have used to indicate your groups in the box next to Maximum
  • Continue and click OK
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 Marginal Homogeneity test
JamoviJamovin.a.
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'
ANOVA > One Way ANOVA - Kruskal-Wallis
  • Put your dependent variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable
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