Regression (OLS): overview

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Regression (OLS)
Independent variables
One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variables
Dependent variable
One quantitative of interval or ratio level
Null 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 $
Alternative 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: $\beta_k < 0$
Assumptions
  • 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
Test 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 person $j$, $N$ 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 $\beta_1$
Sample standard deviation of the residuals $s$
$\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 $F$:
  • $F$ 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
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:
$C\%$ confidence interval for $\beta_k$ and for $\mu_y$; $C\%$ prediction interval for $y_{new}$
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).
Effect 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 $x_k$
ANOVA table
ANOVA table regression analysis
Example context
Can mental health be predicted from fysical health, economic class, and gender?
Pratice questions