Merge pull request #104 from StatProofBook/master

JoramSoch · web-flow · commit d1c3da6ae0ba · 2022-12-23T16:45:32.000+01:00
update to master
diff --git a/D/iass.md b/D/iass.md
@@ -42,4 +42,4 @@ $$ \label{eq:iass}
 \end{split}
 $$
 
-Here, $\bar{y}_{i j \bullet}$ is the mean for the $(i,j)$-th cell (out of $a \times b$ cells), computed from $n_{ij}$ values $y_{ijk}$, $\bar{y}_{i \bullet \bullet}$ and $\bar{y}_{\bullet j \bullet}$ are the level means for the two factors and and $\bar{y}_{\bullet \bullet \bullet}$ is the mean across all values $y_{ijk}$.
+Here, $\bar{y} _{i j \bullet}$ is the mean for the $(i,j)$-th cell (out of $a \times b$ cells), computed from $n_{ij}$ values $y_{ijk}$, $\bar{y} _{i \bullet \bullet}$ and $\bar{y} _{\bullet j \bullet}$ are the level means for the two factors and and $\bar{y} _{\bullet \bullet \bullet}$ is the mean across all values $y_{ijk}$.
diff --git a/D/trss.md b/D/trss.md
@@ -39,4 +39,4 @@ $$ \label{eq:trss}
 \mathrm{SS}_\mathrm{treat} = \sum_{i=1}^{k} \sum_{j=1}^{n_i} (\bar{y}_i - \bar{y})^2 \; .
 $$
 
-Here, $\bar{y} _i$ is the mean for the $i$-th level of the factor (out of $k$ levels), computed from $n_i$ values $y_{ij}$, and $\bar{y}$ is the mean across all values $y_{ij}$.
+Here, $\bar{y}_i$ is the mean for the $i$-th level of the factor (out of $k$ levels), computed from $n_i$ values $y_{ij}$, and $\bar{y}$ is the mean across all values $y_{ij}$.
diff --git a/I/DbA.md b/I/DbA.md
@@ -8,7 +8,7 @@ title: "Definition by Author"
 
 - [Sample covariance](/D/cov-samp)
 
-### JoramSoch (178 definitions)
+### JoramSoch (180 definitions)
 
 - [Akaike information criterion](/D/aic)
 - [Alternative hypothesis](/D/h1)
@@ -65,6 +65,7 @@ title: "Definition by Author"
 - [Expected value of a random vector](/D/mean-rvec)
 - [Explained sum of squares](/D/ess)
 - [Exponential distribution](/D/exp)
+- [F-contrast for contrast-based inference in multiple linear regression](/D/fcon)
 - [F-distribution](/D/f)
 - [Family evidence](/D/fe)
 - [Flat, hard and soft prior distribution](/D/prior-flat)
@@ -173,6 +174,7 @@ title: "Definition by Author"
 - [Statistical hypothesis](/D/hyp)
 - [Statistical hypothesis test](/D/test)
 - [Statistical independence](/D/ind)
+- [t-contrast for contrast-based inference in multiple linear regression](/D/tcon)
 - [t-distribution](/D/t)
 - [Test statistic](/D/tstat)
 - [Total sum of squares](/D/tss)
diff --git a/I/DbN.md b/I/DbN.md
@@ -191,3 +191,5 @@ title: "Definition by Number"
 | D182 | anova2 | [Two-way analysis of variance](/D/anova2) | JoramSoch | 2022-11-06 |
 | D183 | trss | [Treatment sum of squares](/D/trss) | JoramSoch | 2022-12-14 |
 | D184 | iass | [Interaction sum of squares](/D/iass) | JoramSoch | 2022-12-14 |
+| D185 | tcon | [t-contrast for contrast-based inference in multiple linear regression](/D/tcon) | JoramSoch | 2022-12-16 |
+| D186 | fcon | [F-contrast for contrast-based inference in multiple linear regression](/D/fcon) | JoramSoch | 2022-12-16 |
diff --git a/I/DbT.md b/I/DbT.md
@@ -80,6 +80,7 @@ title: "Definition by Topic"
 
 ### F
 
+- [F-contrast for contrast-based inference in multiple linear regression](/D/fcon)
 - [F-distribution](/D/f)
 - [Family evidence](/D/fe)
 - [Flat, hard and soft prior distribution](/D/prior-flat)
@@ -229,6 +230,7 @@ title: "Definition by Topic"
 
 ### T
 
+- [t-contrast for contrast-based inference in multiple linear regression](/D/tcon)
 - [t-distribution](/D/t)
 - [Test statistic](/D/tstat)
 - [Total sum of squares](/D/tss)
diff --git a/I/PbA.md b/I/PbA.md
@@ -8,7 +8,7 @@ title: "Proof by Author"
 
 - [Covariance matrix of the multinomial distribution](/P/mult-cov)
 
-### JoramSoch (369 proofs)
+### JoramSoch (375 proofs)
 
 - [Accuracy and complexity for the univariate Gaussian](/P/ug-anc)
 - [Accuracy and complexity for the univariate Gaussian with known variance](/P/ugkv-anc)
@@ -83,9 +83,11 @@ title: "Proof by Author"
 - [Differential entropy of the normal distribution](/P/norm-dent)
 - [Differential entropy of the normal-gamma distribution](/P/ng-dent)
 - [Distribution of parameter estimates for simple linear regression](/P/slr-olsdist)
+- [Distribution of residual sum of squares in multiple linear regression with weighted least squares](/P/mlr-rssdist)
 - [Distribution of the inverse general linear model](/P/iglm-dist)
 - [Distribution of the transformed general linear model](/P/tglm-dist)
 - [Distributional transformation using cumulative distribution function](/P/cdf-dt)
+- [Distributions of estimated parameters, fitted signal and residuals in multiple linear regression upon weighted least squares](/P/mlr-wlsdist)
 - [Effects of mean-centering on parameter estimates for simple linear regression](/P/slr-meancent)
 - [Entropy of the Bernoulli distribution](/P/bern-ent)
 - [Entropy of the binomial distribution](/P/bin-ent)
@@ -112,12 +114,15 @@ title: "Proof by Author"
 - [F-test for interaction in two-way analysis of variance](/P/anova2-fia)
 - [F-test for main effect in one-way analysis of variance](/P/anova1-f)
 - [F-test for main effect in two-way analysis of variance](/P/anova2-fme)
+- [F-test for multiple linear regression using contrast-based inference](/P/mlr-f)
 - [First central moment is zero](/P/momcent-1st)
 - [First raw moment is mean](/P/momraw-1st)
 - [Full width at half maximum for the normal distribution](/P/norm-fwhm)
 - [Gamma distribution is a special case of Wishart distribution](/P/gam-wish)
 - [Gaussian integral](/P/norm-gi)
 - [Gibbs' inequality](/P/gibbs-ineq)
+- [Independence of estimated parameters and residuals in multiple linear regression](/P/mlr-ind)
+- [Independence of products of multivariate normal random vector](/P/mvn-indprod)
 - [Inflection points of the probability density function of the normal distribution](/P/norm-infl)
 - [Invariance of the covariance matrix under addition of constant vector](/P/covmat-inv)
 - [Invariance of the differential entropy under addition of a constant](/P/dent-inv)
@@ -353,6 +358,7 @@ title: "Proof by Author"
 - [Symmetry of the covariance](/P/cov-symm)
 - [Symmetry of the covariance matrix](/P/covmat-symm)
 - [t-distribution is a special case of multivariate t-distribution](/P/t-mvt)
+- [t-test for multiple linear regression using contrast-based inference](/P/mlr-t)
 - [The p-value follows a uniform distribution under the null hypothesis](/P/pval-h0)
 - [The regression line goes through the center of mass point](/P/slr-comp)
 - [The residuals and the covariate are uncorrelated in simple linear regression](/P/slr-rescorr)
diff --git a/I/PbN.md b/I/PbN.md
@@ -394,3 +394,9 @@ title: "Proof by Number"
 | P386 | mult-mll | [Maximum log-likelihood for multinomial observations](/P/mult-mll) | JoramSoch | 2022-12-02 |
 | P387 | mult-lbf | [Log Bayes factor for multinomial observations](/P/mult-lbf) | JoramSoch | 2022-12-02 |
 | P388 | mult-pp | [Posterior probability of the alternative model for multinomial observations](/P/mult-pp) | JoramSoch | 2022-12-02 |
+| P389 | mlr-wlsdist | [Distributions of estimated parameters, fitted signal and residuals in multiple linear regression upon weighted least squares](/P/mlr-wlsdist) | JoramSoch | 2022-12-13 |
+| P390 | mlr-rssdist | [Distribution of residual sum of squares in multiple linear regression with weighted least squares](/P/mlr-rssdist) | JoramSoch | 2022-12-13 |
+| P391 | mlr-t | [t-test for multiple linear regression using contrast-based inference](/P/mlr-t) | JoramSoch | 2022-12-13 |
+| P392 | mlr-f | [F-test for multiple linear regression using contrast-based inference](/P/mlr-f) | JoramSoch | 2022-12-13 |
+| P393 | mlr-ind | [Independence of estimated parameters and residuals in multiple linear regression](/P/mlr-ind) | JoramSoch | 2022-12-13 |
+| P394 | mvn-indprod | [Independence of products of multivariate normal random vector](/P/mvn-indprod) | JoramSoch | 2022-12-13 |
diff --git a/I/PbT.md b/I/PbT.md
@@ -91,9 +91,11 @@ title: "Proof by Topic"
 - [Differential entropy of the normal distribution](/P/norm-dent)
 - [Differential entropy of the normal-gamma distribution](/P/ng-dent)
 - [Distribution of parameter estimates for simple linear regression](/P/slr-olsdist)
+- [Distribution of residual sum of squares in multiple linear regression with weighted least squares](/P/mlr-rssdist)
 - [Distribution of the inverse general linear model](/P/iglm-dist)
 - [Distribution of the transformed general linear model](/P/tglm-dist)
 - [Distributional transformation using cumulative distribution function](/P/cdf-dt)
+- [Distributions of estimated parameters, fitted signal and residuals in multiple linear regression upon weighted least squares](/P/mlr-wlsdist)
 
 ### E
 
@@ -127,6 +129,7 @@ title: "Proof by Topic"
 - [F-test for interaction in two-way analysis of variance](/P/anova2-fia)
 - [F-test for main effect in one-way analysis of variance](/P/anova1-f)
 - [F-test for main effect in two-way analysis of variance](/P/anova2-fme)
+- [F-test for multiple linear regression using contrast-based inference](/P/mlr-f)
 - [First central moment is zero](/P/momcent-1st)
 - [First raw moment is mean](/P/momraw-1st)
 - [Full width at half maximum for the normal distribution](/P/norm-fwhm)
@@ -139,6 +142,8 @@ title: "Proof by Topic"
 
 ### I
 
+- [Independence of estimated parameters and residuals in multiple linear regression](/P/mlr-ind)
+- [Independence of products of multivariate normal random vector](/P/mvn-indprod)
 - [Inflection points of the probability density function of the normal distribution](/P/norm-infl)
 - [Invariance of the covariance matrix under addition of constant vector](/P/covmat-inv)
 - [Invariance of the differential entropy under addition of a constant](/P/dent-inv)
@@ -419,6 +424,7 @@ title: "Proof by Topic"
 ### T
 
 - [t-distribution is a special case of multivariate t-distribution](/P/t-mvt)
+- [t-test for multiple linear regression using contrast-based inference](/P/mlr-t)
 - [The p-value follows a uniform distribution under the null hypothesis](/P/pval-h0)
 - [The regression line goes through the center of mass point](/P/slr-comp)
 - [The residuals and the covariate are uncorrelated in simple linear regression](/P/slr-rescorr)
diff --git a/P/anova2-pss.md b/P/anova2-pss.md
@@ -54,7 +54,7 @@ $$ \label{eq:anova2-tss}
 \mathrm{SS}_\mathrm{tot} = \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} (y_{ijk} - \bar{y}_{\bullet \bullet \bullet})^2
 $$
 
-where $\bar{y}_{\bullet \bullet \bullet}$ is the mean across all values $y_{ijk}$. This can be rewritten as
+where $\bar{y} _{\bullet \bullet \bullet}$ is the mean across all values $y_{ijk}$. This can be rewritten as
 
 $$ \label{eq:anova2-pss-s1}
 \begin{split}
diff --git a/P/mlr-f.md b/P/mlr-f.md
@@ -171,7 +171,7 @@ F &\overset{\eqref{eq:mlr-f-s1}}{=} \left( \hat{\gamma} -  C^\mathrm{T} \hat{\be
 &\overset{\eqref{eq:mlr-f-h0}}{=} \hat{\gamma}^\mathrm{T} \left( \frac{n-p}{\hat{\varepsilon}^\mathrm{T} \hat{\varepsilon}} Q \right) \hat{\gamma} / q \\
 &\overset{\eqref{eq:fcon}}{=} \hat{\beta}^\mathrm{T} C \left( \frac{n-p}{\hat{\varepsilon}^\mathrm{T} \hat{\varepsilon}} Q \right) C^\mathrm{T} \hat{\beta} / q \\
 &\overset{\eqref{eq:g-est-tau-dist-cond}}{=} \hat{\beta}^\mathrm{T} C \left( \frac{n-p}{\hat{\varepsilon}^\mathrm{T} \hat{\varepsilon}} \left( C^\mathrm{T} (X^\mathrm{T} V^{-1} X)^{-1} C \right)^{-1} \right) C^\mathrm{T} \hat{\beta} / q \\
-&\overset{\eqref{eq:g-est-tau-dist-cond}}{=} \hat{\beta}^\mathrm{T} C \left( \frac{n-p}{(y-X\hat{\beta})^\mathrm{T} V^{-1} (y-X\hat{\beta})} \left( C^\mathrm{T} (X^\mathrm{T} V^{-1} X)^{-1} C \right)^{-1} \right) C^\mathrm{T} \hat{\beta} / q \\
+&\overset{\eqref{eq:mlr-rss}}{=} \hat{\beta}^\mathrm{T} C \left( \frac{n-p}{(y-X\hat{\beta})^\mathrm{T} V^{-1} (y-X\hat{\beta})} \left( C^\mathrm{T} (X^\mathrm{T} V^{-1} X)^{-1} C \right)^{-1} \right) C^\mathrm{T} \hat{\beta} / q \\
 &\overset{\eqref{eq:mlr-est}}{=} \hat{\beta}^\mathrm{T} C \left( \frac{1}{\hat{\sigma}^2} \left( C^\mathrm{T} (X^\mathrm{T} V^{-1} X)^{-1} C \right)^{-1} \right) C^\mathrm{T} \hat{\beta} / q \\
 &= \hat{\beta}^\mathrm{T} C \left( \hat{\sigma}^2 C^\mathrm{T} (X^\mathrm{T} V^{-1} X)^{-1} C \right)^{-1} C^\mathrm{T} \hat{\beta} / q \; .
 \end{split}
diff --git a/P/mlr-rssdist.md b/P/mlr-rssdist.md
@@ -46,7 +46,7 @@ $$ \label{eq:mlr}
 y = X\beta + \varepsilon, \; \varepsilon \sim \mathcal{N}(0, \sigma^2 V)
 $$
 
-and consider estimation using [weighted least squares](/P/mlr-wls). Then, the [residual sum of squares](/D/rss) $\hat{\varepsilon}^\mathrm{T} \hat{\varepsilon}$, divided by the true [error variance](/D/mlr) $\sigma^2$, follows a [chi-squared distribution] with $n-p$ [degrees of freedom](/D/dof)
+and consider estimation using [weighted least squares](/P/mlr-wls). Then, the [residual sum of squares](/D/rss) $\hat{\varepsilon}^\mathrm{T} \hat{\varepsilon}$, divided by the true [error variance](/D/mlr) $\sigma^2$, follows a [chi-squared distribution](/D/chi2) with $n-p$ [degrees of freedom](/D/dof)
 
 $$ \label{eq:mlr-rss-dist}
 \frac{\hat{\varepsilon}^\mathrm{T} \hat{\varepsilon}}{\sigma^2} \sim \chi^2(n-p)
@@ -79,7 +79,7 @@ $$ \label{eq:y-tilde-dist}
 \tilde{y} \sim \mathcal{N}(\tilde{X} \beta, \sigma^2 I_n) \; .
 $$
 
-With that, we have obtained a [linear regression model](/D/mlr) with independent observations. [Cochran's theorem for multivariate normal variables](/P/mvn-cochran) states that, for an $n \times 1$ [normal random vector] whose [covariance matrix](/D/covmat) is a scalar multiple of the identity matrix, a specific squared form will follow a [non-central chi-squared distribution](/D/ncchi2) where the degrees of freedom and the non-centrality paramter depend on the matrix in the quadratic form:
+With that, we have obtained a [linear regression model](/D/mlr) with independent observations. [Cochran's theorem for multivariate normal variables](/P/mvn-cochran) states that, for an $n \times 1$ [normal random vector](/D/mvn) whose [covariance matrix](/D/covmat) is a scalar multiple of the identity matrix, a specific squared form will follow a [non-central chi-squared distribution](/D/ncchi2) where the degrees of freedom and the non-centrality paramter depend on the matrix in the quadratic form:
 
 $$ \label{eq:mvn-cochran}
 x \sim \mathcal{N}(\mu, \sigma^2 I_n) \quad \Rightarrow \quad y = x^\mathrm{T} A x /\sigma^2 \sim \chi^2\left( \mathrm{tr}(A), \mu^\mathrm{T} A \mu \right) \; .
diff --git a/P/mvn-indprod.md b/P/mvn-indprod.md
@@ -48,8 +48,8 @@ $$
 
 Then, due to the [linear transformation theorem](/P/mvn-ltt), we have
 
-$$ \label{eq:CZ}
-CZ = \left[ \begin{array}{c} AZ \\ BZ \end{array} \right] \sim \mathcal{N}\left( \left[ \begin{array}{c} A\mu \\ B\mu \end{array} \right], C \Sigma C^\mathrm{T} \right)
+$$ \label{eq:CX}
+CX = \left[ \begin{array}{c} AX \\ BX \end{array} \right] \sim \mathcal{N}\left( \left[ \begin{array}{c} A\mu \\ B\mu \end{array} \right], C \Sigma C^\mathrm{T} \right)
 $$
 
 with the combined [covariance matrix](/D/covmat)
@@ -58,7 +58,7 @@ $$ \label{eq:CSC}
 C \Sigma C^\mathrm{T} = \left[ \begin{array}{cc} A \Sigma A^\mathrm{T} & A \Sigma B^\mathrm{T} \\ B \Sigma A^\mathrm{T} & B \Sigma B^\mathrm{T} \end{array} \right] \; .
 $$
 
-We know that [the necessary and sufficient condition for two components of a multivariate normal random vector to be independent is that their entries in the covariance matrix are zero](/P/mvn-ind). Thus, $AZ$ and $BZ$ are [independent](/D/ind), if and only if
+We know that [the necessary and sufficient condition for two components of a multivariate normal random vector to be independent is that their entries in the covariance matrix are zero](/P/mvn-ind). Thus, $AX$ and $BX$ are [independent](/D/ind), if and only if
 
 $$ \label{eq:mvn-indprod-qed}
 A \Sigma B^\mathrm{T} = (B \Sigma A^\mathrm{T})^\mathrm{T} = 0_{kl}
