Merge pull request #98 from StatProofBook/master

JoramSoch · web-flow · commit eaa1d40ced89 · 2022-11-11T17:16:14.000+01:00
update to master
diff --git a/I/DbA.md b/I/DbA.md
@@ -8,7 +8,7 @@ title: "Definition by Author"
 
 - [Sample covariance](/D/cov-samp)
 
-### JoramSoch (174 definitions)
+### JoramSoch (176 definitions)
 
 - [Akaike information criterion](/D/aic)
 - [Alternative hypothesis](/D/h1)
@@ -120,6 +120,7 @@ title: "Definition by Author"
 - [Null hypothesis](/D/h0)
 - [One-tailed and two-tailed hypothesis](/D/hyp-tail)
 - [One-tailed and two-tailed test](/D/test-tail)
+- [One-way analysis of variance](/D/anova1)
 - [p-value](/D/pval)
 - [Point and set hypothesis](/D/hyp-point)
 - [Poisson distribution](/D/poiss)
@@ -175,6 +176,7 @@ title: "Definition by Author"
 - [Test statistic](/D/tstat)
 - [Total sum of squares](/D/tss)
 - [Transformed general linear model](/D/tglm)
+- [Two-way analysis of variance](/D/anova2)
 - [Uniform and non-uniform prior distribution](/D/prior-uni)
 - [Uniform-prior log model evidence](/D/uplme)
 - [Univariate and multivariate random variable](/D/rvar-uni)
diff --git a/I/DbN.md b/I/DbN.md
@@ -187,3 +187,5 @@ title: "Definition by Number"
 | D178 | betabin-data | [Beta-binomial data](/D/betabin-data) | JoramSoch | 2022-10-20 |
 | D179 | me | [Model evidence](/D/me) | JoramSoch | 2022-10-20 |
 | D180 | fe | [Family evidence](/D/fe) | JoramSoch | 2022-10-20 |
+| D181 | anova1 | [One-way analysis of variance](/D/anova1) | JoramSoch | 2022-11-06 |
+| D182 | anova2 | [Two-way analysis of variance](/D/anova2) | JoramSoch | 2022-11-06 |
diff --git a/I/DbT.md b/I/DbT.md
@@ -160,6 +160,7 @@ title: "Definition by Topic"
 
 - [One-tailed and two-tailed hypothesis](/D/hyp-tail)
 - [One-tailed and two-tailed test](/D/test-tail)
+- [One-way analysis of variance](/D/anova1)
 
 ### P
 
@@ -231,6 +232,7 @@ title: "Definition by Topic"
 - [Test statistic](/D/tstat)
 - [Total sum of squares](/D/tss)
 - [Transformed general linear model](/D/tglm)
+- [Two-way analysis of variance](/D/anova2)
 
 ### U
 
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 (349 proofs)
+### JoramSoch (352 proofs)
 
 - [Accuracy and complexity for the univariate Gaussian](/P/ug-anc)
 - [Accuracy and complexity for the univariate Gaussian with known variance](/P/ugkv-anc)
@@ -105,6 +105,7 @@ title: "Proof by Author"
 - [Expression of the cumulative distribution function of the normal distribution without the error function](/P/norm-cdfwerf)
 - [Expression of the probability mass function of the beta-binomial distribution using only the gamma function](/P/betabin-pmfitogf)
 - [Extreme points of the probability density function of the normal distribution](/P/norm-extr)
+- [F-test for main effect in one-way analysis of variance](/P/anova1-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)
@@ -213,9 +214,11 @@ title: "Proof by Author"
 - [One-sample z-test for independent observations](/P/ugkv-ztest1)
 - [Ordinary least squares for multiple linear regression](/P/mlr-ols)
 - [Ordinary least squares for multiple linear regression](/P/mlr-ols2)
+- [Ordinary least squares for one-way analysis of variance](/P/anova1-ols)
 - [Ordinary least squares for simple linear regression](/P/slr-ols)
 - [Ordinary least squares for simple linear regression](/P/slr-ols2)
 - [Ordinary least squares for the general linear model](/P/glm-ols)
+- [Ordinary least squares for two-way analysis of variance](/P/anova2-ols)
 - [Paired t-test for dependent observations](/P/ug-ttestp)
 - [Paired z-test for dependent observations](/P/ugkv-ztestp)
 - [Parameter estimates for simple linear regression are uncorrelated after mean-centering](/P/slr-olscorr)
diff --git a/I/PbN.md b/I/PbN.md
@@ -374,3 +374,6 @@ title: "Proof by Number"
 | P366 | betabin-cdf | [Cumulative distribution function of the beta-binomial distribution](/P/betabin-cdf) | JoramSoch | 2022-10-22 |
 | P367 | me-der | [Derivation of the model evidence](/P/me-der) | JoramSoch | 2022-10-20 |
 | P368 | fe-der | [Derivation of the family evidence](/P/fe-der) | JoramSoch | 2022-10-20 |
+| P369 | anova1-ols | [Ordinary least squares for one-way analysis of variance](/P/anova1-ols) | JoramSoch | 2022-11-06 |
+| P370 | anova1-f | [F-test for main effect in one-way analysis of variance](/P/anova1-f) | JoramSoch | 2022-11-06 |
+| P371 | anova2-ols | [Ordinary least squares for two-way analysis of variance](/P/anova2-ols) | JoramSoch | 2022-11-06 |
diff --git a/I/PbT.md b/I/PbT.md
@@ -120,6 +120,7 @@ title: "Proof by Topic"
 
 ### F
 
+- [F-test for main effect in one-way analysis of variance](/P/anova1-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)
@@ -259,9 +260,11 @@ title: "Proof by Topic"
 - [One-sample z-test for independent observations](/P/ugkv-ztest1)
 - [Ordinary least squares for multiple linear regression](/P/mlr-ols)
 - [Ordinary least squares for multiple linear regression](/P/mlr-ols2)
+- [Ordinary least squares for one-way analysis of variance](/P/anova1-ols)
 - [Ordinary least squares for simple linear regression](/P/slr-ols)
 - [Ordinary least squares for simple linear regression](/P/slr-ols2)
 - [Ordinary least squares for the general linear model](/P/glm-ols)
+- [Ordinary least squares for two-way analysis of variance](/P/anova2-ols)
 
 ### P
 
diff --git a/I/PwS.md b/I/PwS.md
@@ -77,8 +77,10 @@ title: "Proofs without Source"
 - [Mode of the normal distribution](/P/norm-mode)
 - [Necessary and sufficient condition for independence of multivariate normal random variables](/P/mvn-ind)
 - [Normal-gamma distribution is a special case of normal-Wishart distribution](/P/ng-nw)
+- [Ordinary least squares for one-way analysis of variance](/P/anova1-ols)
 - [Ordinary least squares for simple linear regression](/P/slr-ols2)
 - [Ordinary least squares for the general linear model](/P/glm-ols)
+- [Ordinary least squares for two-way analysis of variance](/P/anova2-ols)
 - [Parameter estimates for simple linear regression are uncorrelated after mean-centering](/P/slr-olscorr)
 - [Posterior density is proportional to joint likelihood](/P/post-jl)
 - [Probability density function of the beta distribution](/P/beta-pdf)
diff --git a/P/bin-margcond.md b/P/bin-margcond.md
@@ -64,7 +64,7 @@ $$ \label{eq:Y-dist-s3}
 \mathrm{Pr}(Y = m) = \sum_{k=m}^{n} {k \choose m} \, q^m \, (1-q)^{k-m} \cdot {n \choose k} \, p^k \, (1-p)^{n-k} \; .
 $$
 
-Applying the binomial coefficient identity ${n \choose k} {k \choose m} = {n \choose m} {n-m \choose k-m}$ rearranging the terms, we have:
+Applying the binomial coefficient identity ${n \choose k} {k \choose m} = {n \choose m} {n-m \choose k-m}$ and rearranging the terms, we have:
 
 $$ \label{eq:Y-dist-s4}
 \mathrm{Pr}(Y = m) = \sum_{k=m}^{n} {n \choose m} \, {n-m \choose k-m} \, p^k \, q^m \, (1-p)^{n-k} \, (1-q)^{k-m} \; .
diff --git a/P/lbf-lme.md b/P/lbf-lme.md
@@ -43,19 +43,23 @@ $$
 
 and the [log Bayes factor](/D/lbf) is defined as the logarithm of the Bayes factor
 
-$$ \label{eq:LBF}
-\mathrm{LBF}_{12} = \log \mathrm{BF}_{12} = \log \frac{p(y|m_1)}{p(y|m_2)} \; .
+$$ \label{eq:LBF-s1}
+\mathrm{LBF}_{12} = \log \mathrm{BF}_{12} \; .
 $$
 
-With the definition of the [log model evidence](/D/lme)
+Thus, the log Bayes factor can be expressed as
+
+$$ \label{eq:LBF-s2}
+\mathrm{LBF}_{12} = \log \frac{p(y|m_1)}{p(y|m_2)} \; .
+$$
+
+and, with the definition of the [log model evidence](/D/lme)
 
 $$ \label{eq:LME}
 \mathrm{LME}(m) = \log p(y|m)
 $$
 
-the log Bayes factor can be expressed as:
-
-Resolving the logarithm and applying the definition of the [log model evidence](/D/lme), we finally have:
+and resolving the logarithm, we finally have:
 
 $$ \label{eq:LBF-LME-qed}
 \begin{split}
diff --git a/P/matn-marg.md b/P/matn-marg.md
@@ -163,7 +163,7 @@ x_{ij} &= AXB = e_i X e_j^\mathrm{T} \\
 \end{split}
 $$
 
-As $x_{ij}$ is a scalar, this is equivalent to [a univariate normal distribution](/D/norm) as [a special case of](/P/norm-mvn) of [the matrix-normal distribution](/P/mvn-matn):
+As $x_{ij}$ is a scalar, this is equivalent to [a univariate normal distribution](/D/norm) as [a special case](/P/norm-mvn) of [the matrix-normal distribution](/P/mvn-matn):
 
 $$ \label{eq:xij-marg-norm}
 x_{ij} \sim \mathcal{N}(m_{ij}, u_{ii} v_{jj}) \; .
diff --git a/P/norm-chi2.md b/P/norm-chi2.md
@@ -76,8 +76,8 @@ $$ \label{eq:sum-Ui2-s1}
 \begin{split}
 \sum_{i=1}^{n} U_i^2 &= \sum_{i=1}^{n} \left( \frac{X_i - \mu}{\sigma} \right)^2 \\
 &= \sum_{i=1}^{n} \left( \frac{(X_i - \bar{X}) + (\bar{X} - \mu)}{\sigma} \right)^2 \\
-&= \sum_{i=1}^{n} \frac{(X_i - \bar{X})^2}{\sigma^2} + \sum_{i=1}^{n} \frac{(\bar{X} - \mu)^2}{\sigma^2} + \sum_{i=1}^{n} \frac{(X_i - \bar{X})(\bar{X} - \mu)}{\sigma^2} \\
-&= \sum_{i=1}^{n} \left( \frac{X_i - \bar{X}}{\sigma^2} \right)^2 + \sum_{i=1}^{n} \left( \frac{\bar{X} - \mu}{\sigma^2} \right)^2 + \frac{(\bar{X} - \mu)}{\sigma^2} \sum_{i=1}^{n} (X_i - \bar{X}) \; .
+&= \sum_{i=1}^{n} \frac{(X_i - \bar{X})^2}{\sigma^2} + \sum_{i=1}^{n} \frac{(\bar{X} - \mu)^2}{\sigma^2} + 2 \sum_{i=1}^{n} \frac{(X_i - \bar{X})(\bar{X} - \mu)}{\sigma^2} \\
+&= \sum_{i=1}^{n} \left( \frac{X_i - \bar{X}}{\sigma} \right)^2 + \sum_{i=1}^{n} \left( \frac{\bar{X} - \mu}{\sigma} \right)^2 + \frac{(\bar{X} - \mu)}{\sigma^2} \sum_{i=1}^{n} (X_i - \bar{X}) \; .
 \end{split}
 $$
 
