Regression (OLS) - overview

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Regression (OLS)
Chi-squared test for the relationship between two categorical variables
Regression (OLS)
One sample $z$ test for the mean
One way ANOVA
Independent variablesIndependent /column variableIndependent variablesIndependent variableIndependent/grouping 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 ($I \geqslant 2$)One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variablesNoneOne categorical with $I$ independent groups ($I \geqslant 2$)
Dependent variableDependent /row 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 level
Null hypothesisNull hypothesisNull hypothesisNull hypothesisNull hypothesis
$F$ test for the complete regression model:
  • H0: $\beta_1 = \beta_2 = \ldots = \beta_K = 0$
    or equivalenty
  • H0: the variance explained by all the independent variables together (the complete model) is 0 in the population, i.e. $\rho^2 = 0$
$t$ test for individual regression coefficient $\beta_k$:
  • H0: $\beta_k = 0$
in the regression equation $ \mu_y = \beta_0 + \beta_1 \times x_1 + \beta_2 \times x_2 + \ldots + \beta_K \times x_K$. Here $ x_i$ represents independent variable $ i$, $\beta_i$ is the regression weight for independent variable $ x_i$, and $\mu_y$ represents the population mean of the dependent variable $ y$ given the scores on the independent variables.
H0: there is no association between the row and column variable

More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
  • H0: the distribution of the dependent variable is the same in each of the $I$ populations
If there is one random sample of size $N$ from the total population:
  • H0: the row and column variables are independent
$F$ test for the complete regression model:
  • H0: $\beta_1 = \beta_2 = \ldots = \beta_K = 0$
    or equivalenty
  • H0: the variance explained by all the independent variables together (the complete model) is 0 in the population, i.e. $\rho^2 = 0$
$t$ test for individual regression coefficient $\beta_k$:
  • H0: $\beta_k = 0$
in the regression equation $ \mu_y = \beta_0 + \beta_1 \times x_1 + \beta_2 \times x_2 + \ldots + \beta_K \times x_K$. Here $ x_i$ represents independent variable $ i$, $\beta_i$ is the regression weight for independent variable $ x_i$, and $\mu_y$ represents the population mean of the dependent variable $ y$ given the scores on the independent variables.
H0: $\mu = \mu_0$

Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis.
ANOVA $F$ test:
  • H0: $\mu_1 = \mu_2 = \ldots = \mu_I$
    $\mu_1$ is the population mean for group 1; $\mu_2$ is the population mean for group 2; $\mu_I$ is the population mean for group $I$
$t$ Test for contrast:
  • H0: $\Psi = 0$
    $\Psi$ is the population contrast, defined as $\Psi = \sum a_i\mu_i$. Here $\mu_i$ is the population mean for group $i$ and $a_i$ is the coefficient for $\mu_i$. The coefficients $a_i$ sum to 0.
$t$ Test multiple comparisons:
  • H0: $\mu_g = \mu_h$
    $\mu_g$ is the population mean for group $g$; $\mu_h$ is the population mean for group $h$
Alternative hypothesisAlternative hypothesisAlternative hypothesisAlternative hypothesisAlternative hypothesis
$F$ test for the complete regression model:
  • H1: not all population regression coefficients are 0
    or equivalenty
  • H1: the variance explained by all the independent variables together (the complete model) is larger than 0 in the population, i.e. $\rho^2 > 0$
$t$ test for individual regression coefficient $\beta_k$:
  • H1 two sided: $\beta_k \neq 0$
  • H1 right sided: $\beta_k > 0$
  • H1 left sided: $\beta_k < 0$
H1: there is an association between the row and column variable

More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
  • H1: the distribution of the dependent variable is not the same in all of the $I$ populations
If there is one random sample of size $N$ from the total population:
  • H1: the row and column variables are dependent
$F$ test for the complete regression model:
  • H1: not all population regression coefficients are 0
    or equivalenty
  • H1: the variance explained by all the independent variables together (the complete model) is larger than 0 in the population, i.e. $\rho^2 > 0$
$t$ test for individual regression coefficient $\beta_k$:
  • H1 two sided: $\beta_k \neq 0$
  • H1 right sided: $\beta_k > 0$
  • H1 left sided: $\beta_k < 0$
H1 two sided: $\mu \neq \mu_0$
H1 right sided: $\mu > \mu_0$
H1 left sided: $\mu < \mu_0$
ANOVA $F$ test:
  • H1: not all population means are equal
$t$ Test for contrast:
  • H1 two sided: $\Psi \neq 0$
  • H1 right sided: $\Psi > 0$
  • H1 left sided: $\Psi < 0$
$t$ Test multiple comparisons:
  • H1 - usually two sided: $\mu_g \neq \mu_h$
AssumptionsAssumptionsAssumptionsAssumptionsAssumptions
  • 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 under the null hypothesis. Rule of thumb:
    • 2 $\times$ 2 table: all four expected cell counts are 5 or more
    • Larger than 2 $\times$ 2 tables: average of the expected cell counts is 5 or more, smallest expected cell count is 1 or more
  • There are $I$ independent simple random samples from each of $I$ populations defined by the independent variable, or there is one simple random sample from the total population
  • 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
  • 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
  • The standard deviation of the scores on the dependent variable is the same in each of the populations: $\sigma_1 = \sigma_2 = \ldots = \sigma_I$
  • 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
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 $SE_{b_k}$ is more complicated.
Note 1: mean square model is also known as mean square regression, and mean square error is also known as mean square residual.
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 $\beta_1.$
$X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
Here for each cell, the expected cell count = $\dfrac{\mbox{row total} \times \mbox{column total}}{\mbox{total sample size}}$, the observed cell count is the observed sample count in that same cell, and the sum is over all $I \times J$ cells.
$F$ 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 $SE_{b_k}$ is more complicated.
Note 1: mean square model is also known as mean square regression, and mean square error is also known as mean square residual.
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 $\beta_1.$
$z = \dfrac{\bar{y} - \mu_0}{\sigma / \sqrt{N}}$
Here $\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to the null hypothesis, $\sigma$ is the population standard deviation, and $N$ 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 $\mu_0$.
ANOVA $F$ test:
  • $\begin{aligned}[t] F &= \dfrac{\sum\nolimits_{subjects} (\mbox{subject's group mean} - \mbox{overall mean})^2 / (I - 1)}{\sum\nolimits_{subjects} (\mbox{subject's score} - \mbox{its group mean})^2 / (N - I)}\\ &= \dfrac{\mbox{sum of squares between} / \mbox{degrees of freedom between}}{\mbox{sum of squares error} / \mbox{degrees of freedom error}}\\ &= \dfrac{\mbox{mean square between}}{\mbox{mean square error}} \end{aligned} $
    where $N$ is the total sample size, and $I$ is the number of groups.
    Note: mean square between is also known as mean square model, and mean square error is also known as mean square residual or mean square within.
$t$ Test for contrast:
  • $t = \dfrac{c}{s_p\sqrt{\sum \dfrac{a^2_i}{n_i}}}$
    Here $c$ is the sample estimate of the population contrast $\Psi$: $c = \sum a_i\bar{y}_i$, with $\bar{y}_i$ the sample mean in group $i$. $s_p$ is the pooled standard deviation based on all the $I$ groups in the ANOVA, $a_i$ is the contrast coefficient for group $i$, and $n_i$ is the sample size of group $i$.
    Note that if the contrast compares only two group means with each other, this $t$ statistic is very similar to the two sample $t$ statistic (assuming equal population standard deviations). In that case the only difference is that we now base the pooled standard deviation on all the $I$ groups, which affects the $t$ value if $I \geqslant 3$. It also affects the corresponding degrees of freedom.
$t$ Test multiple comparisons:
  • $t = \dfrac{\bar{y}_g - \bar{y}_h}{s_p\sqrt{\dfrac{1}{n_g} + \dfrac{1}{n_h}}}$
    $\bar{y}_g$ is the sample mean in group $g$, $\bar{y}_h$ is the sample mean in group $h$, $s_p$ is the pooled standard deviation based on all the $I$ groups in the ANOVA, $n_g$ is the sample size of group $g$, and $n_h$ is the sample size of group $h$.
    Note that this $t$ statistic is very similar to the two sample $t$ statistic (assuming equal population standard deviations). The only difference is that we now base the pooled standard deviation on all the $I$ groups, which affects the $t$ value if $I \geqslant 3$. It also affects the corresponding degrees of freedom.
Sample standard deviation of the residuals $s$n.a.Sample standard deviation of the residuals $s$n.a.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} $-$ \begin{aligned} s_p &= \sqrt{\dfrac{(n_1 - 1) \times s^2_1 + (n_2 - 1) \times s^2_2 + \ldots + (n_I - 1) \times s^2_I}{N - I}}\\ &= \sqrt{\dfrac{\sum\nolimits_{subjects} (\mbox{subject's score} - \mbox{its group mean})^2}{N - I}}\\ &= \sqrt{\dfrac{\mbox{sum of squares error}}{\mbox{degrees of freedom error}}}\\ &= \sqrt{\mbox{mean square error}} \end{aligned} $

Here $s^2_i$ is the variance in group $i.$
Sampling distribution of $F$ and of $t$ if H0 were trueSampling distribution of $X^2$ if H0 were trueSampling distribution of $F$ and of $t$ if H0 were trueSampling distribution of $z$ if H0 were trueSampling 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
Approximately the chi-squared distribution with $(I - 1) \times (J - 1)$ degrees of freedomSampling 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
Standard normal distributionSampling distribution of $F$:
  • $F$ distribution with $I - 1$ (df between, numerator) and $N - I$ (df error, denominator) degrees of freedom
Sampling distribution of $t$:
  • $t$ distribution with $N - I$ degrees of freedom
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 $\alpha$
$F$ 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: $F$ 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 $\alpha$ (e.g. .01 < $p$ < .025 when $F$ = 3.91, df between = 4, and df error = 20)

$t$ Test for contrast two sided: $t$ Test for contrast right sided: $t$ Test for contrast left sided:
$t$ Test multiple comparisons two sided:
  • Check if $t$ observed in sample is at least as extreme as critical value $t^{**}$. Adapt $t^{**}$ according to a multiple comparison procedure (e.g., Bonferroni) or
  • Find two sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$. Adapt the $p$ value or $\alpha$ according to a multiple comparison procedure
$t$ Test multiple comparisons right sided
  • Check if $t$ observed in sample is equal to or larger than critical value $t^{**}$. Adapt $t^{**}$ according to a multiple comparison procedure (e.g., Bonferroni) or
  • Find right sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$. Adapt the $p$ value or $\alpha$ according to a multiple comparison procedure
$t$ Test multiple comparisons left sided
  • Check if $t$ observed in sample is equal to or smaller than critical value $t^{**}$. Adapt $t^{**}$ according to a multiple comparison procedure (e.g., Bonferroni) or
  • Find left sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$. Adapt the $p$ value or $\alpha$ according to a multiple comparison procedure
$C\%$ confidence interval for $\beta_k$ and for $\mu_y$, $C\%$ prediction interval for $y_{new}$n.a.$C\%$ confidence interval for $\beta_k$ and for $\mu_y$, $C\%$ prediction interval for $y_{new}$$C\%$ confidence interval for $\mu$$C\%$ confidence interval for $\Psi$, for $\mu_g - \mu_h$, and for $\mu_i$
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).
-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 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 $\mu$ can also be used as significance test.
Confidence interval for $\Psi$ (contrast):
  • $c \pm t^* \times s_p\sqrt{\sum \dfrac{a^2_i}{n_i}}$
    where the critical value $t^*$ is the value under the $t_{N - I}$ distribution with the area $C / 100$ between $-t^*$ and $t^*$ (e.g. $t^*$ = 2.086 for a 95% confidence interval when df = 20). Note that $n_i$ is the sample size of group $i$, and $N$ is the total sample size, based on all the $I$ groups.
Confidence interval for $\mu_g - \mu_h$ (multiple comparisons):
  • $(\bar{y}_g - \bar{y}_h) \pm t^{**} \times s_p\sqrt{\dfrac{1}{n_g} + \dfrac{1}{n_h}}$
    where $t^{**}$ depends upon $C$, degrees of freedom ($N - I$), and the multiple comparison procedure. If you do not want to apply a multiple comparison procedure, $t^{**} = t^* = $ the value under the $t_{N - I}$ distribution with the area $C / 100$ between $-t^*$ and $t^*$. Note that $n_g$ is the sample size of group $g$, $n_h$ is the sample size of group $h$, and $N$ is the total sample size, based on all the $I$ groups.
Confidence interval for single population mean $\mu_i$:
  • $\bar{y}_i \pm t^* \times \dfrac{s_p}{\sqrt{n_i}}$
    where $\bar{y}_i$ is the sample mean in group $i$, $n_i$ is the sample size of group $i$, and the critical value $t^*$ is the value under the $t_{N - I}$ distribution with the area $C / 100$ between $-t^*$ and $t^*$ (e.g. $t^*$ = 2.086 for a 95% confidence interval when df = 20). Note that $n_i$ is the sample size of group $i$, and $N$ is the total sample size, based on all the $I$ groups.
Effect sizen.a.Effect sizeEffect 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 $x_k$
-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 $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{\sigma}$$ Cohen's $d$ indicates how many standard deviations $\sigma$ the sample mean $\bar{y}$ is removed from $\mu_0.$
  • Proportion variance explained $\eta^2$ and $R^2$:
    Proportion variance of the dependent variable $y$ explained by the independent variable: $$ \begin{align} \eta^2 = R^2 &= \dfrac{\mbox{sum of squares between}}{\mbox{sum of squares total}} \end{align} $$ Only in one way ANOVA $\eta^2 = R^2.$ $\eta^2$ (and $R^2$) is the proportion variance explained in the sample. It is a positively biased estimate of the proportion variance explained in the population.

  • Proportion variance explained $\omega^2$:
    Corrects for the positive bias in $\eta^2$ and is equal to: $$\omega^2 = \frac{\mbox{sum of squares between} - \mbox{df between} \times \mbox{mean square error}}{\mbox{sum of squares total} + \mbox{mean square error}}$$ $\omega^2$ is a better estimate of the explained variance in the population than $\eta^2.$

  • Cohen's $d$:
    Standardized difference between the mean in group $g$ and in group $h$: $$d_{g,h} = \frac{\bar{y}_g - \bar{y}_h}{s_p}$$ Cohen's $d$ indicates how many standard deviations $s_p$ two sample means are removed from each other.
Visual representationn.a.Visual representationVisual representationn.a.
Regression equations with: -Regression equations with:
One sample z test
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ANOVA tablen.a.ANOVA tablen.a.ANOVA table
ANOVA table regression analysis
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ANOVA table regression analysis
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ANOVA table

Click the link for a step by step explanation of how to compute the sum of squares.
n.a.n.a.n.a.n.a.Equivalent to
----OLS regression with one categorical independent variable transformed into $I - 1$ code variables:
  • $F$ test ANOVA is equivalent to $F$ test regression model
  • $t$ test for contrast $i$ is equivalent to $t$ test for regression coefficient $\beta_i$ (specific contrast tested depends on how the code variables are defined)
Example contextExample contextExample contextExample contextExample context
Can mental health be predicted from fysical health, economic class, and gender?Is there an association between economic class and gender? Is the distribution of economic class different between men and women?Can mental health be predicted from fysical health, economic class, and gender?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 people from a low, moderate, and high economic class?
SPSSSPSSSPSSn.a.SPSS
Analyze > Regression > Linear...
  • Put your dependent variable in the box below Dependent and your independent (predictor) variables in the box below Independent(s)
Analyze > Descriptive Statistics > Crosstabs...
  • Put one of your two categorical variables in the box below Row(s), and the other categorical 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 > 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 > One-Way ANOVA...
  • Put your dependent (quantitative) variable in the box below Dependent List and your independent (grouping) variable in the box below Factor
or
Analyze > General Linear Model > Univariate...
  • Put your dependent (quantitative) variable in the box below Dependent Variable and your independent (grouping) variable in the box below Fixed Factor(s)
JamoviJamoviJamovin.a.Jamovi
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 > Independent Samples - $\chi^2$ test of association
  • Put one of your two categorical variables in the box below Rows, and the other categorical variable in the box below Columns
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 > ANOVA
  • Put your dependent (quantitative) variable in the box below Dependent Variable and your independent (grouping) variable in the box below Fixed Factors
Practice questionsPractice questionsPractice questionsPractice questionsPractice questions