Chapter4ANOVAII.ipynb

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   "source": [
    "# Chapter 4 Analysis of Variance\n",
    "     "
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "## 4.2 Two-way ANOVA"
   ]
  },
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   "source": [
    "### 4.2.1 Motivating example and model\n",
    "\n",
    "Consider the Hey Fever dataset. Nine compounds for Hay Fever Relief are made by varying levels of two basic ingredients. Ingredient 1 (denoted factor A) has $a = 3$ levels: low $(i = 1)$, medium $(i = 2)$ and high $(i = 3)$. And Ingredient 2 (factor B) has $b = 3$ levels: low $(j = 1)$,\n",
    "medium $(j = 2)$ and high $(j = 3)$. A total of 36 subjects suffering from hay fever are selected and each of the 9 compounds are given to randomly selected $n = 4$ individuals.\n",
    "\n",
    "\n",
    "The cell means model for a *balanced* two-way ANOVA takes the following form \n",
    "$$\n",
    "Y_{ijk} = \\mu_{ij} +\\epsilon_{ijk}, \\ k=1,\\ldots, n, j=1,\\ldots, b, i=1,\\ldots, a,\n",
    "$$\n",
    "where $\\{\\epsilon_{ijk}\\}$ are i.i.d. $N(0,\\sigma^2)$. There are $ab$ unknown means and one unknown $\\sigma$ in the cell means model. \n",
    "\n",
    "In practice, we often prefer the fhe factor effect form \n",
    "$$\n",
    "Y_{ijk} = \\mu_{\\cdot\\cdot} + \\alpha_i+\\beta_j + (\\alpha\\beta)_{ij}+\\epsilon_{ijk}, \\ k=1,\\ldots, n, j=1,\\ldots, b, i=1,\\ldots, a,\n",
    "$$\n",
    "\n",
    "其实这里$\\alpha\\beta$ 叫 $\\gamma$ 更好，因为这个值不是 $\\alpha$与$\\beta$的乘积，而是他们的相互作用interaction.\n",
    "\n",
    "如果只有$\\alpha$与$\\beta$,则模型少了好多变量（数量从$\\cdot\\times\\cdot$变成$\\cdot+\\cdot$，只有+是不够的）\n",
    "\n",
    "where $\\{\\epsilon_{ijk}\\}$ are i.i.d. $N(0,\\sigma^2)$. A quick count shows that there are $a$ $\\alpha$s, $b$ $\\beta$s, $ab$ interaction terms, one $\\mu$ and one $\\sigma$ in the factor effect form, which amount to $ab+a+b+2$ parameters. Indeed, the parameters in the factor effect form are no longer \"free\" as in the cell means model. A set of constraints are in placed is in place with the factor effect form, which will be discussed in Section 4.2.2. \n",
    "\n",
    "\n",
    "**Remark:** Note that $(\\alpha\\beta)_{ij}$ is a new parameter standing for the interaction effect, but **not** the product of $\\alpha_i$ and $\\beta_j$. We can replace it with $\\gamma_{ij}$ to avoid confusion. However, the advantage of using  $(\\alpha\\beta)_{ij}$ is self-explanatory; that is the expression  $(\\alpha\\beta)_{ij}$ shows that it is an effect jointly controlled by factor A and factor B. \n"
   ]
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   "execution_count": 1,
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   "outputs": [
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UpWLgDWtK1tUbJyAbCmbW2LkpULgDVt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\n7aazAAAAAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "Hay <- read.csv(file=\"../Data/HayFever.csv\", header=TRUE, sep=\",\")\n",
    "\n",
    "# Use a slightly different visualization:\n",
    "pairs(Hay,pch=16,col='red',cex=1.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "### 4.2.2 Constraints on parameters\n",
    "\n",
    "In the two-way model, the mapping between the cell means form and the factor effects form is slightly more complicated that that in the one-way case. We consider the balanced design here for simplicity. \n",
    "\n",
    "The cell means $\\{\\mu_{ij}: i =1,\\ldots, a, j=1,\\ldots, b\\}$ are naturally defined as the population mean (or expectation) within each cell determined by one unique combination of two factors. We proceed to define the factor effects using the cell means. \n",
    "\n",
    "We first look at the over mean $\\mu_{\\cdot \\cdot}$  \n",
    "$$\n",
    "\\mu_{\\cdot \\cdot} =\\sum_{i=1}^a \\sum_{j=1}^ b \\mu_{ij}/(ab), \\ \\mu_{i\\cdot} = \\sum_{j=1}^b \\mu_{ij} /b, \\ \\mu_{\\cdot j}=\\sum_{i=1}^a \\mu_{ij}/a.\n",
    "$$\n",
    "We can then define the factor effects as \n",
    "$$\n",
    "\\alpha_i=\\mu_{i\\cdot} - \\mu_{\\cdot \\cdot},\\ \\beta_j=\\mu_{\\cdot j}-\\mu_{\\cdot\\cdot},\\ (\\alpha\\beta)_{ij} =\\mu_{ij}-\\mu_{i\\cdot}-\\mu_{\\cdot j}+\\mu_{\\cdot\\cdot}. \n",
    "$$\n",
    "From these definitions, we can show that there are  natural constraints on these effects \n",
    "\\begin{align}\n",
    "\\sum \\alpha_i & = \\sum \\beta_j=0\\\\\n",
    "\\sum_{i=1}^a (\\alpha\\beta)_{ij} & =\\sum_{j=1}^b (\\alpha\\beta)_{ij} =0\n",
    "\\end{align}\n",
    "\n",
    "\n",
    "【这里才是最终形式！！！！factor-effect form two-way ANOVA model with constraints:】\n",
    "\n",
    "We can now write down the factor effects form with constraints \n",
    "$$\n",
    "Y_{ijk} = \\mu_{\\cdot\\cdot} + \\alpha_i+\\beta_j + (\\alpha\\beta)_{ij}+\\epsilon_{ijk}, \\ k=1,\\ldots, n, j=1,\\ldots, b, i=1,\\ldots, a,\n",
    "$$\n",
    "where $\\{\\epsilon_{ijk}\\}$ are i.i.d. $N(0,\\sigma^2)$ and \n",
    "\\begin{align}\n",
    "\\sum_i \\alpha_i & = \\sum_j \\beta_j=0\\\\\n",
    "\\sum_{i=1}^a (\\alpha\\beta)_{ij} & =\\sum_{j=1}^b (\\alpha\\beta)_{ij} =0\n",
    "\\end{align}\n",
    "\n",
    "Very often you may see an additive model $Y_{ijk} = \\mu_{\\cdot\\cdot} + \\alpha_i+\\beta_j + \\epsilon_{ijk}$, where the interactions $\\{(\\alpha\\beta)_{ij}\\}$ are dropped. Dropping the interaction terms reduces the number of unknown parameters to estimate, which improves efficiency if the reduced model does not fall too far away from the truth. However, a reduced model places additional assumptions which may not be true in the real world. In practice, we need to carefully decide the form of model based on, but not limited to, background, questions of interest, prior knowledge, exploratory analysis, etc. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "### 4.2.3 Estimation\n",
    "\n",
    "We can estimate the cell means as before \n",
    "$\\hat{\\mu}_{ij}=\\bar{Y}_{ij\\cdot}, \\hat{\\mu}_{\\cdot\\cdot}=\\bar{Y}_{\\cdot\\cdot\\cdot}, \\hat{\\mu}_{i\\cdot} =\\bar{Y}_{i\\cdot\\cdot}, \\hat{\\mu}_{\\cdot j}=\\bar{Y}_{\\cdot j \\cdot}$. \n",
    "Noting that the effects are linear combinations of cell means, we can estimate them using the equalities in Section 4.2.2\n",
    "$$\n",
    "\\hat{\\alpha}_i = \\bar{Y}_{i\\cdot \\cdot}-\\bar{Y}_{\\cdot \\cdot \\cdot}, \\hat{\\beta}_{j}=\\bar{Y}_{\\cdot j \\cdot} -\\bar{Y}_{\\cdot \\cdot \\cdot}, (\\hat{\\alpha\\beta})_{ij}=\\bar{Y}_{ij}-(\\bar{Y}_{\\cdot\\cdot\\cdot} +\\hat{\\alpha}_i+\\hat{\\beta}_j).\n",
    "$$\n",
    "As a result, the same set of constraints still hold on the estimators. \n",
    "\n",
    "\n",
    "For each $Y_{ijk}$, the fitted value is $\\bar{Y}_{ij\\cdot}$. The residual is thus $e_{ijk} \\equiv Y_{ijk}-\\bar{Y}_{ij\\cdot}$. We can show that the fitted value is independent with the residual. Just like in one-way ANOVA, this independence is the backbone for testing. \n",
    "\n",
    "The sum of squares are as follow.  \n",
    "\n",
    "**Residual sum of squares.**\n",
    "$$\n",
    "{\\rm SSE} = \\sum_i \\sum_j \\sum_k e_{ijk}^2 = \\sum_i \\sum_j \\sum_k \\big(Y_{ijk}-\\bar{Y}_{ij\\cdot}\\big)^2, df({\\rm SSE})=n_T -ab =(n-1)ab.\n",
    "$$\n",
    "As before, we can define the mean squared errors as ${\\rm MSE} ={ {\\rm SSE}}/{ df({\\rm SSE})}$, where we know that $\\mathbb{E}[{\\rm MSE}]=\\sigma^2$. \n",
    "\n",
    "**Total sum of squares.**\n",
    "$$\n",
    "{\\rm SSTO} =\\sum_i \\sum_j \\sum_k \\big(Y_{ijk}-\\bar{Y}_{\\cdot \\cdot \\cdot}\\big)^2, \\ df({\\rm SSTO})=n_T -1 = nab-1.\n",
    "$$\n",
    "\n",
    "**Sum of squares due to the main effect of factor A.(Balanced design)** \n",
    "$$\n",
    "{\\rm SSA} = \\sum_i \\sum_j \\sum_k\\hat{\\alpha}_i^2 = nb\\sum_{i=1}^a \\hat{\\alpha}_i^2, \\ df({\\rm SSA})=a-1.\n",
    "$$\n",
    "\n",
    "**Sum of squares due to the main effect of factor B.(Balanced design)** \n",
    "$$\n",
    "{\\rm SSB} =\\sum_i \\sum_j \\sum_k \\hat{\\beta}_j^2 = na\\sum_{j=1}^a \\hat{\\beta}_j^2, \\ df({\\rm SSB})=b-1.\n",
    "$$\n",
    "\n",
    "**Sum of squares due to the interaction effects.(Balanced design)**\n",
    "$$\n",
    "{\\rm SSAB} = \\sum_i \\sum_j \\sum_k \\hat{(\\alpha\\beta)}_{ij}^2 = n \\sum_i \\sum_j \\hat{(\\alpha\\beta)}_{ij}^2, \\ df({\\rm SSAB})=(a-1)(b-1).\n",
    "$$\n",
    "\n",
    "We have the following equalities\n",
    "$$\n",
    "{\\rm SSTO}={\\rm SSA}+{\\rm SSB} + {\\rm SSAB}+{\\rm SSE}$$\n",
    "and \n",
    "$$df({\\rm SSTO})=df({\\rm SSA})+df({\\rm SSB})+df({\\rm SSAB})+df({\\rm SSE}).\n",
    "$$\n",
    "\n",
    "For the mean squares, when $\\epsilon$s are i.i.d. $N(0,\\sigma^2)$, we have $\\hat{Y}_{ijk} \\perp e_{ijk}$, which yields ${\\rm MSE} \\perp {\\rm MSA}, {\\rm MSB}, {\\rm MSAB}$. \n",
    "\n",
    "\n",
    "**Properties**\n",
    "1. $\\mathbb{E}[{\\rm MSE}]=\\sigma^2$; \n",
    "2. $\\mathbb{E}[{\\rm MSA}]=\\sigma^2 + nb \\sum_i \\alpha_i^2 /(a-1)$; \n",
    "3. $\\mathbb{E}[{\\rm MSB}]=\\sigma^2 +na \\sum_j \\beta_j^2 /(b-1)$; \n",
    "4. $\\mathbb{E}[{\\rm MSAB}]=\\sigma^2 + n \\sum_i \\sum_j ({\\alpha\\beta})_{ij}^2/(a-1)(b-1)$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 4.2.4 Inference \n",
    "\n",
    "Tests for two-way ANOVA models are essentially identical to those for one-way ANOVA model. We discuss only the F-tests for the interactions and one main effect here. \n",
    "\n",
    "**F-test for interactions.** Consider the following null and alternative hypotheses. \n",
    "$$\n",
    "H_0: (\\alpha\\beta)_{ij}=0 \\  \\forall i, j  \\ \\ {\\rm v.s.} \\ \\ \\ H_1:\\  {\\rm not \\ all \\ } (\\alpha\\beta)_{ij} \\ {\\rm are \\ zero}.\n",
    "$$\n",
    "The F-statistics is $F^*={\\rm MSAB}/{\\rm MSE}$. Under the null, $F^*$ follows an $F$-distribution with $df=( (a-1)(b-1),(n-1)ab)$. \n",
    "\n",
    "**Test for the main effect of factor A.** Consider the following null and alternative hypotheses. \n",
    "$$\n",
    "H_0: \\alpha_{i}=0 \\ \\ \\forall i \\ \\ \\  {\\rm v.s.}\\ \\  \\ H_1:\\  {\\rm not \\ all \\ } \\alpha_i \\ {\\rm are \\ zero}.\n",
    "$$\n",
    "We have $F^*={\\rm MSA}/{\\rm MSE}$ that follows an F-distribution with $df=(a-1,(n-1)ab)$. \n",
    "\n",
    "\n",
    "Similar results hold for factor B. Theory and justifications for these tests are almost identical to those for one-way ANOVA. \n",
    "\n",
    "**MSE without interactions.** In the absence of interactions, the MSE defined in the previous section no long makes sense. Instead, we can define $\\hat{Y}_{ijk}=\\hat{\\mu}_{\\cdot\\cdot} +\\hat{\\alpha}_i + \\hat{\\beta}_j$. This  leads to $e_{ijk}'=Y_{ijk}-\\hat{Y}_{ijk}$.  Consequently, ${\\rm SSE}= \\sum \\sum \\sum \\big(Y_{ijk}-\\hat{Y}_{ijk}\\big)^2$ and $df({\\rm SSE})=nab-a-b+1$. Finally, we have that ${\\rm MSE} = {\\rm SSE} /(nab-a-b+1)$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Methods for testing linear combinations and simultaneous inference generalize to the case with two-way, and higher order, ANOVA model. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "### 4.2.5 Special case: one observation per case\n",
    "\n",
    "When there is only one observation per cell, the number of observations ($ab$) equals to the number of unknown means $(ab)$. It is not possible to estimate the full model with interactions with proper measures of uncertainty. However, the number of observed cases does not alter the truth underneath the data. We may still want to test whether interactions are present. \n",
    "\n",
    "**Tukey's test for additivity.**\n",
    "Consider the following model\n",
    "$$\n",
    "Y_{ij}=\\mu_{\\cdot\\cdot} +\\alpha_i+\\beta_j+D \\alpha_i\\beta_j+\\epsilon_{ij}, \n",
    "$$\n",
    "where $\\epsilon_{ij}$ are i.i.d. $N(0,\\sigma^2)$. In this model, the fourth term on the right-hand side attempts to capture the interaction effects with limited capacity. The specific form $D \\alpha_i\\beta_j$ does not capture all possible interactions, but it is a work-around with limited observations. In this model, we can test \n",
    "$$\n",
    "H_0: D=0 \\ \\ \\ {\\rm v.s.}\\ \\ \\ H_a: D \\neq 0.\n",
    "$$\n",
    "\n",
    "We can derive that \n",
    "$$\n",
    "\\hat{D} = \\frac{\\sum_i \\sum_j Y_{ij} \\hat{\\alpha}_i \\hat{\\beta}_j}{\\sum_i \\hat{\\alpha}_i^2 \\sum_j \\hat{\\beta}_j^2 }. \n",
    "$$\n",
    "We can construct the F-test for $H_0: D=0$ in a similar manner as in Section 4.2.4. You can read more about this test in, e.g., [here](https://en.wikipedia.org/wiki/Tukey's_test_of_additivity). \n",
    "\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 4.2.6 Imbalanced two-way ANOVA \n",
    "\n",
    "A two-way ANOVA when the numbers of observations vary across cells takes the following form \n",
    "$$\n",
    "Y_{ijk} =  \\mu_{ijk}+\\epsilon_{ijk}= \\mu_{\\cdot\\cdot} + \\alpha_i+\\beta_j + (\\alpha\\beta)_{ij}+\\epsilon_{ijk}, \\ k=1,\\ldots, n_{ij}, j=1, \\ldots, b, i=1,\\ldots, a,\n",
    "$$\n",
    "where $\\{\\epsilon_{ijk}\\}$ are i.i.d. $N(0,\\sigma^2)$. The same constraints hold on the parameters. \n",
    "\n",
    "The factor effects are still defined as follows \n",
    "$$\n",
    "\\mu_{\\cdot \\cdot} =\\sum_{i=1}^a \\sum_{j=1}^ b \\mu_{ij}/(ab), \\ \\mu_{i\\cdot} = \\sum_{j=1}^b \\mu_{ij} /b, \\ \\mu_{\\cdot j}=\\sum_{i=1}^a \\mu_{ij}/a.\n",
    "$$\n",
    "And furthermore,  we have \n",
    "$$\n",
    "\\alpha_i=\\mu_{i\\cdot} - \\mu_{\\cdot \\cdot},\\ \\beta_j=\\mu_{\\cdot j}-\\mu_{\\cdot\\cdot},\\ (\\alpha\\beta)_{ij} =\\mu_{ij}-\\mu_{i\\cdot}-\\mu_{\\cdot j}+\\mu_{\\cdot\\cdot}. \n",
    "$$\n",
    "Estimators follow the same form by plugging the sample means in the equations above.\n",
    "\n",
    "However, it is **no longer true** that \n",
    "$$\n",
    "\\hat{\\mu}_{\\cdot\\cdot} =\\bar{Y}_{\\cdot\\cdot\\cdot}, \\ \\hat{\\mu}_{i\\cdot} = \\bar{Y}_{i\\cdot\\cdot}, \\ \\hat{\\mu}_{\\cdot j}=\\bar{Y}_{\\cdot j \\cdot},  \\ {\\rm SSTO}={\\rm SSA}+{\\rm SSB} + {\\rm SSAB}+{\\rm SSE}.\n",
    "$$\n",
    "To see this, we just need to expand $\\bar{Y}_{\\cdot\\cdot\\cdot}$ as weighted average of the sample cell means. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Sum of squares.** In an imbalanced design, the sum of squares decomposition depends on the *order* of the decomposition, which gives us three types of sums of squares. Let us denote ${\\rm SSE}_{A}$ as the residual sum of squares from the model with only the main effect term of Factor A. Similarly, we can define ${\\rm SSE}_{B}$, ${\\rm SSE}_{A,B}$, ${\\rm SSE}_{A,AB}$,  ${\\rm SSE}_{A,B,AB}$, etc. \n",
    "\n",
    "1. Type I (the sequential sum of squares). Suppose that the order is A, B, and AB. We can define the sum of squares due to the main effect of factor A as ${\\rm SSTO}-{\\rm SSE}_{A}$, the sum of squares due to the main effect of factor B as ${\\rm SSE}_{A}-{\\rm SSE}_{A,B}$, and the sum of squares due to the interaction as ${\\rm SSE}_{A,B}-{\\rm SSE}_{A,B,AB}$. ($Y\\sim X_1+X_2$变量顺序很重要)\n",
    "\n",
    "2. Type II. We define the sum of squares due to the main effect of Factor A as ${\\rm SSE}_{B}-{\\rm SSE}_{A,B}$, and the main effect due to Factor B as ${\\rm SSE}_{A}-{\\rm SSE}_{A,B}$. Note that this type should only be employed if there is no interaction term, either from a previous test or prior knowledge. \n",
    "\n",
    "3. Type III. We define the sum of squares due to the main effect of Factor A as ${\\rm SSE}_{B,AB}-{\\rm SSE}_{A,B,AB}$, and the sum of squares due to the main effect of Factor B as ${\\rm SSE}_{A,AB}-{\\rm SSE}_{A,B,AB}$. \n",
    " \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "In a balanced design, all three types of decompositions are equivalent because the factors are *orthogonal*. One can think about the orthogonal design using the formula for LSE in a linear regression $\\hat{\\beta}=(X^TX)^{-1} X^T Y$ and that $\\mathrm{var}(\\hat{\\beta})=(X^TX)^{-1}$. When the factors are orthogonal, the variance-covariance matrix of $\\hat{\\beta}$ is diagonal."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "Suppose that we are interested in testing the presence of interactions. \n",
    "$$\n",
    "H_0: (\\alpha\\beta)_{ij}=0 \\ {\\rm v.s.} \\ H_1: {\\rm not \\ all \\ } (\\alpha\\beta)_{ij} \\ {\\rm are \\ zero}.\n",
    "$$\n",
    "We now need to use the following framework for testing. \n",
    "* Full model: $Y_{ijk} =\\mu_{\\cdot\\cdot} + \\alpha_i+\\beta_j + (\\alpha\\beta)_{ij}+\\epsilon_{ijk}$\n",
    "* Reduced model: $Y_{ijk} =\\mu_{\\cdot\\cdot} + \\alpha_i+\\beta_j +\\epsilon_{ijk}$\n",
    "\n",
    "The F-statistic is then \n",
    "$$\n",
    "F^*=\\frac{ [{\\rm SSE}_{\\rm red}-{\\rm SSE}_{\\rm full}  ] / [ df_{\\rm red}-df_{\\rm full}   ]  }{ {\\rm SSE}_{\\rm full}/df_{\\rm full}},\n",
    "$$\n",
    "where $F^* \\sim F( (a-1)(b-1), n_T-ab)$ under the null hypothesis. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note: you can use the `Anova()` function in the `car` package if you want to fit an imbalanced ANOVA model. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To use the Type I sum of squares, we can use the `aov` function and carefully specify the order of factors in the model formula. For type II and type III ANOVA, we can use the `Anova()` function in the `car` package.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "      wage         ethnicity          occupation       \n",
       " Min.   : 1.000   Length:534         Length:534        \n",
       " 1st Qu.: 5.250   Class :character   Class :character  \n",
       " Median : 7.780   Mode  :character   Mode  :character  \n",
       " Mean   : 9.024                                        \n",
       " 3rd Qu.:11.250                                        \n",
       " Max.   :44.500                                        "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "library(car)\n",
    "wage <- read.csv(file=\"../Data/Wage.csv\", header=TRUE, sep=\",\")\n",
    "summary(wage)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "t1.a=aov(wage~ethnicity+occupation,data=wage);\n",
    "t1.b=aov(wage~occupation+ethnicity,data=wage);\n",
    "\n",
    "t2=Anova(lm(wage~ethnicity+occupation, data=wage), type=2)\n",
    "\n",
    "t3=Anova(lm(wage~ethnicity*occupation, data=wage, \n",
    "            contrasts=list(ethnicity=contr.sum, occupation=contr.sum)), type=3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "             Df Sum Sq Mean Sq F value Pr(>F)    \n",
       "ethnicity     2    173    86.3   3.967 0.0195 *  \n",
       "occupation    5   2459   491.7  22.598 <2e-16 ***\n",
       "Residuals   526  11445    21.8                   \n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "             Df Sum Sq Mean Sq F value Pr(>F)    \n",
       "occupation    5   2538   507.5  23.325 <2e-16 ***\n",
       "ethnicity     2     94    46.8   2.149  0.118    \n",
       "Residuals   526  11445    21.8                   \n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "summary(t1.a)\n",
    "summary(t1.b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A anova: 3 × 4</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>Sum Sq</th><th scope=col>Df</th><th scope=col>F value</th><th scope=col>Pr(&gt;F)</th></tr>\n",
       "\t<tr><th></th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>ethnicity</th><td>   93.52643</td><td>  2</td><td> 2.149098</td><td>1.176119e-01</td></tr>\n",
       "\t<tr><th scope=row>occupation</th><td> 2458.56892</td><td>  5</td><td>22.597703</td><td>1.467948e-20</td></tr>\n",
       "\t<tr><th scope=row>Residuals</th><td>11445.47501</td><td>526</td><td>       NA</td><td>          NA</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A anova: 3 × 4\n",
       "\\begin{tabular}{r|llll}\n",
       "  & Sum Sq & Df & F value & Pr(>F)\\\\\n",
       "  & <dbl> & <dbl> & <dbl> & <dbl>\\\\\n",
       "\\hline\n",
       "\tethnicity &    93.52643 &   2 &  2.149098 & 1.176119e-01\\\\\n",
       "\toccupation &  2458.56892 &   5 & 22.597703 & 1.467948e-20\\\\\n",
       "\tResiduals & 11445.47501 & 526 &        NA &           NA\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A anova: 3 × 4\n",
       "\n",
       "| <!--/--> | Sum Sq &lt;dbl&gt; | Df &lt;dbl&gt; | F value &lt;dbl&gt; | Pr(&gt;F) &lt;dbl&gt; |\n",
       "|---|---|---|---|---|\n",
       "| ethnicity |    93.52643 |   2 |  2.149098 | 1.176119e-01 |\n",
       "| occupation |  2458.56892 |   5 | 22.597703 | 1.467948e-20 |\n",
       "| Residuals | 11445.47501 | 526 |        NA |           NA |\n",
       "\n"
      ],
      "text/plain": [
       "           Sum Sq      Df  F value   Pr(>F)      \n",
       "ethnicity     93.52643   2  2.149098 1.176119e-01\n",
       "occupation  2458.56892   5 22.597703 1.467948e-20\n",
       "Residuals  11445.47501 526        NA           NA"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A anova: 5 × 4</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>Sum Sq</th><th scope=col>Df</th><th scope=col>F value</th><th scope=col>Pr(&gt;F)</th></tr>\n",
       "\t<tr><th></th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>(Intercept)</th><td> 7403.5772</td><td>  1</td><td>341.846822</td><td>6.135442e-59</td></tr>\n",
       "\t<tr><th scope=row>ethnicity</th><td>  104.0006</td><td>  2</td><td>  2.401019</td><td>9.163740e-02</td></tr>\n",
       "\t<tr><th scope=row>occupation</th><td>  549.9500</td><td>  5</td><td>  5.078589</td><td>1.483054e-04</td></tr>\n",
       "\t<tr><th scope=row>ethnicity:occupation</th><td>  270.1603</td><td> 10</td><td>  1.247417</td><td>2.578712e-01</td></tr>\n",
       "\t<tr><th scope=row>Residuals</th><td>11175.3147</td><td>516</td><td>        NA</td><td>          NA</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A anova: 5 × 4\n",
       "\\begin{tabular}{r|llll}\n",
       "  & Sum Sq & Df & F value & Pr(>F)\\\\\n",
       "  & <dbl> & <dbl> & <dbl> & <dbl>\\\\\n",
       "\\hline\n",
       "\t(Intercept) &  7403.5772 &   1 & 341.846822 & 6.135442e-59\\\\\n",
       "\tethnicity &   104.0006 &   2 &   2.401019 & 9.163740e-02\\\\\n",
       "\toccupation &   549.9500 &   5 &   5.078589 & 1.483054e-04\\\\\n",
       "\tethnicity:occupation &   270.1603 &  10 &   1.247417 & 2.578712e-01\\\\\n",
       "\tResiduals & 11175.3147 & 516 &         NA &           NA\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A anova: 5 × 4\n",
       "\n",
       "| <!--/--> | Sum Sq &lt;dbl&gt; | Df &lt;dbl&gt; | F value &lt;dbl&gt; | Pr(&gt;F) &lt;dbl&gt; |\n",
       "|---|---|---|---|---|\n",
       "| (Intercept) |  7403.5772 |   1 | 341.846822 | 6.135442e-59 |\n",
       "| ethnicity |   104.0006 |   2 |   2.401019 | 9.163740e-02 |\n",
       "| occupation |   549.9500 |   5 |   5.078589 | 1.483054e-04 |\n",
       "| ethnicity:occupation |   270.1603 |  10 |   1.247417 | 2.578712e-01 |\n",
       "| Residuals | 11175.3147 | 516 |         NA |           NA |\n",
       "\n"
      ],
      "text/plain": [
       "                     Sum Sq     Df  F value    Pr(>F)      \n",
       "(Intercept)           7403.5772   1 341.846822 6.135442e-59\n",
       "ethnicity              104.0006   2   2.401019 9.163740e-02\n",
       "occupation             549.9500   5   5.078589 1.483054e-04\n",
       "ethnicity:occupation   270.1603  10   1.247417 2.578712e-01\n",
       "Residuals            11175.3147 516         NA           NA"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t2\n",
    "t3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 4.2.7 Example: Hay Fever data\n",
    "\n",
    "Let's return to the Hay Fever data. If an analysis plan has been determined, as is usually the case for randomized control trials, we need to strictly follow the analysis plan. The pre-determined analysis plan makes sure that there won't be data mining attempts that inflates the type I errors.  \n",
    "\n",
    "In this example, we will pretend that we have no such plans. Instead, our task is to explore and investigate the effectiveness of the two ingradients. We still, however, need to be mindful with type I errors from our inference. Think about the type I error as what determines the *weight* your results carry: the larger the type I error is, the more negligible your statistical inference is. \n",
    " "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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UpWLgDWtK1tUbJyAbCmbW2LkpULgDVt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\n7aazAAAAAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "Hay <- read.csv(file=\"../Data/HayFever.csv\", header=TRUE, sep=\",\")\n",
    "\n",
    "# Pairwise scatter plot\n",
    "pairs(Hay,pch=16,col='red',cex=1.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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VRaqbDpyYb5dnD/d2PMRlgkYwKwSMYE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\nxgRgkYwJwCIZE4BFMiaA/wFW1mzu9o0nzgAAAABJRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "Hay.jittered = Hay;\n",
    "Hay.jittered$Ingredient.2 = jitter(Hay.jittered$Ingredient.2)\n",
    "# Jittered one factor \n",
    "pairs(Hay.jittered,pch=16,col='red',cex=1.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "There are two factors with a total of 9 combinations in this data with 36 samples. It is natural to consider a two-way ANOVA model here. The next question is then whether the interaction effects are present, i.e., whether we want to use a two-way ANOVA model with or without interactions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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4QOdJgnzwcCzStzzqdAc0HcQMeq806g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\nSUVORK5CYII=",
      "text/plain": [
       "Plot with title \"Main  effect, Ingredient 2\""
      ]
     },
     "metadata": {
      "image/png": {
       "height": 720,
       "width": 720
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Exploratory analysis\n",
    "library(gplots)\n",
    "Hay$Ingredient.1=as.factor(Hay$Ingredient.1)\n",
    "Hay$Ingredient.2=as.factor(Hay$Ingredient.2)\n",
    "\n",
    "options(repr.plot.width=12, repr.plot.height=12)\n",
    "par(mfrow=c(2,2))\n",
    "# Main effect plot for ingredient 1\n",
    "plotmeans(Relief~Ingredient.1,data=Hay,xlab=\"Ingredient 1\",ylab=\"Relief\",\n",
    "          main=\"Main  effect, Ingredient 1\",cex.lab=1.5) \n",
    "# Main effect plot for ingredient 2\n",
    "plotmeans(Relief~Ingredient.2,data=Hay,xlab=\"Ingredient 2\",ylab=\"Relief\", \n",
    "          main=\"Main  effect, Ingredient 2\",cex.lab=1.5) \n",
    "#Interaction plot\n",
    "interaction.plot(Hay$Ingredient.1, Hay$Ingredient.2, Hay$Relief\n",
    "                ,cex.lab=1.5,ylab=\"Relief\",xlab='Ingredient 1')\n",
    "par(mfrow=c(1,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<caption>A anova: 2 × 6</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>Res.Df</th><th scope=col>RSS</th><th scope=col>Df</th><th scope=col>Sum of Sq</th><th scope=col>F</th><th scope=col>Pr(&gt;F)</th></tr>\n",
       "\t<tr><th></th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>1</th><td>31</td><td>31.050</td><td>NA</td><td>    NA</td><td>      NA</td><td>          NA</td></tr>\n",
       "\t<tr><th scope=row>2</th><td>27</td><td> 1.625</td><td> 4</td><td>29.425</td><td>122.2269</td><td>6.972083e-17</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A anova: 2 × 6\n",
       "\\begin{tabular}{r|llllll}\n",
       "  & Res.Df & RSS & Df & Sum of Sq & F & Pr(>F)\\\\\n",
       "  & <dbl> & <dbl> & <dbl> & <dbl> & <dbl> & <dbl>\\\\\n",
       "\\hline\n",
       "\t1 & 31 & 31.050 & NA &     NA &       NA &           NA\\\\\n",
       "\t2 & 27 &  1.625 &  4 & 29.425 & 122.2269 & 6.972083e-17\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A anova: 2 × 6\n",
       "\n",
       "| <!--/--> | Res.Df &lt;dbl&gt; | RSS &lt;dbl&gt; | Df &lt;dbl&gt; | Sum of Sq &lt;dbl&gt; | F &lt;dbl&gt; | Pr(&gt;F) &lt;dbl&gt; |\n",
       "|---|---|---|---|---|---|---|\n",
       "| 1 | 31 | 31.050 | NA |     NA |       NA |           NA |\n",
       "| 2 | 27 |  1.625 |  4 | 29.425 | 122.2269 | 6.972083e-17 |\n",
       "\n"
      ],
      "text/plain": [
       "  Res.Df RSS    Df Sum of Sq F        Pr(>F)      \n",
       "1 31     31.050 NA     NA          NA           NA\n",
       "2 27      1.625  4 29.425    122.2269 6.972083e-17"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Test for interactions \n",
    "full_model=lm(Relief~as.factor(Ingredient.1)+as.factor(Ingredient.2)+as.factor(Ingredient.1)*as.factor(Ingredient.2),data=Hay);\n",
    "reduced_model=lm(Relief~as.factor(Ingredient.1)+as.factor(Ingredient.2),data=Hay);\n",
    "anova(reduced_model,full_model)\n",
    "# Conclusion?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "                          Df Sum Sq Mean Sq F value Pr(>F)    \n",
       "Ingredient.1               2 220.02  110.01  1827.9 <2e-16 ***\n",
       "Ingredient.2               2 123.66   61.83  1027.3 <2e-16 ***\n",
       "Ingredient.1:Ingredient.2  4  29.43    7.36   122.2 <2e-16 ***\n",
       "Residuals                 27   1.62    0.06                   \n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "                          Df Sum Sq Mean Sq F value Pr(>F)    \n",
       "Ingredient.1               2 220.02  110.01  1827.9 <2e-16 ***\n",
       "Ingredient.2               2 123.66   61.83  1027.3 <2e-16 ***\n",
       "Ingredient.1:Ingredient.2  4  29.43    7.36   122.2 <2e-16 ***\n",
       "Residuals                 27   1.62    0.06                   \n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Fit the chosen model:\n",
    "library(stats)\n",
    "sig.level=0.05;\n",
    "anova.fit<-aov(Relief~Ingredient.1+Ingredient.2+Ingredient.1*Ingredient.2,data=Hay)\n",
    "summary(anova.fit)\n",
    "\n",
    "anova.fit<-aov(Relief~Ingredient.1*Ingredient.2+Ingredient.2+Ingredient.1,data=Hay)\n",
    "summary(anova.fit)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It seems that both integredients and their interaction may have (statistically) significant impact on the response variable. We may want to explore what the best combination of ingredients is, or whether such combination exists. The search of \"best combination\" can be translated as finding the cell with the highest cell mean compared to other cells, which leads us to the Tukey-Kramer method."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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EQRaJIINFEEmiQCTRSBJolAE0WgSSLQ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\nTPQAAAAASUVORK5CYII=",
      "text/plain": [
       "Plot with title \"95% family-wise confidence level\n",
       "\""
      ]
     },
     "metadata": {
      "image/png": {
       "height": 720,
       "width": 720
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Find the best combination\n",
    "\n",
    "T.ci=TukeyHSD(anova.fit,conf.level = 1-sig.level)\n",
    "par(mfrow=c(2,2))\n",
    "plot(T.ci, las=1 , col=\"brown\")\n",
    "par(mfrow=c(1,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<caption>A matrix: 3 × 3 of type dbl</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>1</th><th scope=col>2</th><th scope=col>3</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>1</th><td>2.475</td><td> 4.600</td><td> 4.575</td></tr>\n",
       "\t<tr><th scope=row>2</th><td>5.450</td><td> 8.925</td><td> 9.125</td></tr>\n",
       "\t<tr><th scope=row>3</th><td>5.975</td><td>10.275</td><td>13.250</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A matrix: 3 × 3 of type dbl\n",
       "\\begin{tabular}{r|lll}\n",
       "  & 1 & 2 & 3\\\\\n",
       "\\hline\n",
       "\t1 & 2.475 &  4.600 &  4.575\\\\\n",
       "\t2 & 5.450 &  8.925 &  9.125\\\\\n",
       "\t3 & 5.975 & 10.275 & 13.250\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A matrix: 3 × 3 of type dbl\n",
       "\n",
       "| <!--/--> | 1 | 2 | 3 |\n",
       "|---|---|---|---|\n",
       "| 1 | 2.475 |  4.600 |  4.575 |\n",
       "| 2 | 5.450 |  8.925 |  9.125 |\n",
       "| 3 | 5.975 | 10.275 | 13.250 |\n",
       "\n"
      ],
      "text/plain": [
       "  1     2      3     \n",
       "1 2.475  4.600  4.575\n",
       "2 5.450  8.925  9.125\n",
       "3 5.975 10.275 13.250"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# We only need to pay attention to the differences of the two largest means\n",
    "idx=list();\n",
    "idx[[1]]=Hay$Ingredient.1;idx[[2]]=Hay$Ingredient.2;\n",
    "(means.comb=tapply( Hay$Relief, INDEX=idx,mean))\n",
    "# From this table, the two cells are (3,3) and (3,2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<caption>A matrix: 36 × 4 of type dbl</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>diff</th><th scope=col>lwr</th><th scope=col>upr</th><th scope=col>p adj</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>2:1-1:1</th><td> 2.975</td><td> 2.3913187</td><td> 3.5586813</td><td>2.142730e-14</td></tr>\n",
       "\t<tr><th scope=row>3:1-1:1</th><td> 3.500</td><td> 2.9163187</td><td> 4.0836813</td><td>4.440892e-15</td></tr>\n",
       "\t<tr><th scope=row>1:2-1:1</th><td> 2.125</td><td> 1.5413187</td><td> 2.7086813</td><td>5.300671e-11</td></tr>\n",
       "\t<tr><th scope=row>2:2-1:1</th><td> 6.450</td><td> 5.8663187</td><td> 7.0336813</td><td>4.218847e-15</td></tr>\n",
       "\t<tr><th scope=row>3:2-1:1</th><td> 7.800</td><td> 7.2163187</td><td> 8.3836813</td><td>4.218847e-15</td></tr>\n",
       "\t<tr><th scope=row>1:3-1:1</th><td> 2.100</td><td> 1.5163187</td><td> 2.6836813</td><td>6.947942e-11</td></tr>\n",
       "\t<tr><th scope=row>2:3-1:1</th><td> 6.650</td><td> 6.0663187</td><td> 7.2336813</td><td>4.218847e-15</td></tr>\n",
       "\t<tr><th scope=row>3:3-1:1</th><td>10.775</td><td>10.1913187</td><td>11.3586813</td><td>4.218847e-15</td></tr>\n",
       "\t<tr><th scope=row>3:1-2:1</th><td> 0.525</td><td>-0.0586813</td><td> 1.1086813</td><td>1.033088e-01</td></tr>\n",
       "\t<tr><th scope=row>1:2-2:1</th><td>-0.850</td><td>-1.4336813</td><td>-0.2663187</td><td>1.142444e-03</td></tr>\n",
       "\t<tr><th scope=row>2:2-2:1</th><td> 3.475</td><td> 2.8913187</td><td> 4.0586813</td><td>4.551914e-15</td></tr>\n",
       "\t<tr><th scope=row>3:2-2:1</th><td> 4.825</td><td> 4.2413187</td><td> 5.4086813</td><td>4.218847e-15</td></tr>\n",
       "\t<tr><th scope=row>1:3-2:1</th><td>-0.875</td><td>-1.4586813</td><td>-0.2913187</td><td>7.862241e-04</td></tr>\n",
       "\t<tr><th scope=row>2:3-2:1</th><td> 3.675</td><td> 3.0913187</td><td> 4.2586813</td><td>4.218847e-15</td></tr>\n",
       "\t<tr><th scope=row>3:3-2:1</th><td> 7.800</td><td> 7.2163187</td><td> 8.3836813</td><td>4.218847e-15</td></tr>\n",
       "\t<tr><th scope=row>1:2-3:1</th><td>-1.375</td><td>-1.9586813</td><td>-0.7913187</td><td>5.283997e-07</td></tr>\n",
       "\t<tr><th scope=row>2:2-3:1</th><td> 2.950</td><td> 2.3663187</td><td> 3.5336813</td><td>2.520206e-14</td></tr>\n",
       "\t<tr><th scope=row>3:2-3:1</th><td> 4.300</td><td> 3.7163187</td><td> 4.8836813</td><td>4.218847e-15</td></tr>\n",
       "\t<tr><th scope=row>1:3-3:1</th><td>-1.400</td><td>-1.9836813</td><td>-0.8163187</td><td>3.744504e-07</td></tr>\n",
       "\t<tr><th scope=row>2:3-3:1</th><td> 3.150</td><td> 2.5663187</td><td> 3.7336813</td><td>8.659740e-15</td></tr>\n",
       "\t<tr><th scope=row>3:3-3:1</th><td> 7.275</td><td> 6.6913187</td><td> 7.8586813</td><td>4.218847e-15</td></tr>\n",
       "\t<tr><th scope=row>2:2-1:2</th><td> 4.325</td><td> 3.7413187</td><td> 4.9086813</td><td>4.218847e-15</td></tr>\n",
       "\t<tr><th scope=row>3:2-1:2</th><td> 5.675</td><td> 5.0913187</td><td> 6.2586813</td><td>4.218847e-15</td></tr>\n",
       "\t<tr><th scope=row>1:3-1:2</th><td>-0.025</td><td>-0.6086813</td><td> 0.5586813</td><td>1.000000e+00</td></tr>\n",
       "\t<tr><th scope=row>2:3-1:2</th><td> 4.525</td><td> 3.9413187</td><td> 5.1086813</td><td>4.218847e-15</td></tr>\n",
       "\t<tr><th scope=row>3:3-1:2</th><td> 8.650</td><td> 8.0663187</td><td> 9.2336813</td><td>4.218847e-15</td></tr>\n",
       "\t<tr><th scope=row>3:2-2:2</th><td> 1.350</td><td> 0.7663187</td><td> 1.9336813</td><td>7.475322e-07</td></tr>\n",
       "\t<tr><th scope=row>1:3-2:2</th><td>-4.350</td><td>-4.9336813</td><td>-3.7663187</td><td>4.218847e-15</td></tr>\n",
       "\t<tr><th scope=row>2:3-2:2</th><td> 0.200</td><td>-0.3836813</td><td> 0.7836813</td><td>9.596929e-01</td></tr>\n",
       "\t<tr><th scope=row>3:3-2:2</th><td> 4.325</td><td> 3.7413187</td><td> 4.9086813</td><td>4.218847e-15</td></tr>\n",
       "\t<tr><th scope=row>1:3-3:2</th><td>-5.700</td><td>-6.2836813</td><td>-5.1163187</td><td>4.218847e-15</td></tr>\n",
       "\t<tr><th scope=row>2:3-3:2</th><td>-1.150</td><td>-1.7336813</td><td>-0.5663187</td><td>1.305447e-05</td></tr>\n",
       "\t<tr><th scope=row>3:3-3:2</th><td> 2.975</td><td> 2.3913187</td><td> 3.5586813</td><td>2.142730e-14</td></tr>\n",
       "\t<tr><th scope=row>2:3-1:3</th><td> 4.550</td><td> 3.9663187</td><td> 5.1336813</td><td>4.218847e-15</td></tr>\n",
       "\t<tr><th scope=row>3:3-1:3</th><td> 8.675</td><td> 8.0913187</td><td> 9.2586813</td><td>4.218847e-15</td></tr>\n",
       "\t<tr><th scope=row>3:3-2:3</th><td> 4.125</td><td> 3.5413187</td><td> 4.7086813</td><td>4.218847e-15</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A matrix: 36 × 4 of type dbl\n",
       "\\begin{tabular}{r|llll}\n",
       "  & diff & lwr & upr & p adj\\\\\n",
       "\\hline\n",
       "\t2:1-1:1 &  2.975 &  2.3913187 &  3.5586813 & 2.142730e-14\\\\\n",
       "\t3:1-1:1 &  3.500 &  2.9163187 &  4.0836813 & 4.440892e-15\\\\\n",
       "\t1:2-1:1 &  2.125 &  1.5413187 &  2.7086813 & 5.300671e-11\\\\\n",
       "\t2:2-1:1 &  6.450 &  5.8663187 &  7.0336813 & 4.218847e-15\\\\\n",
       "\t3:2-1:1 &  7.800 &  7.2163187 &  8.3836813 & 4.218847e-15\\\\\n",
       "\t1:3-1:1 &  2.100 &  1.5163187 &  2.6836813 & 6.947942e-11\\\\\n",
       "\t2:3-1:1 &  6.650 &  6.0663187 &  7.2336813 & 4.218847e-15\\\\\n",
       "\t3:3-1:1 & 10.775 & 10.1913187 & 11.3586813 & 4.218847e-15\\\\\n",
       "\t3:1-2:1 &  0.525 & -0.0586813 &  1.1086813 & 1.033088e-01\\\\\n",
       "\t1:2-2:1 & -0.850 & -1.4336813 & -0.2663187 & 1.142444e-03\\\\\n",
       "\t2:2-2:1 &  3.475 &  2.8913187 &  4.0586813 & 4.551914e-15\\\\\n",
       "\t3:2-2:1 &  4.825 &  4.2413187 &  5.4086813 & 4.218847e-15\\\\\n",
       "\t1:3-2:1 & -0.875 & -1.4586813 & -0.2913187 & 7.862241e-04\\\\\n",
       "\t2:3-2:1 &  3.675 &  3.0913187 &  4.2586813 & 4.218847e-15\\\\\n",
       "\t3:3-2:1 &  7.800 &  7.2163187 &  8.3836813 & 4.218847e-15\\\\\n",
       "\t1:2-3:1 & -1.375 & -1.9586813 & -0.7913187 & 5.283997e-07\\\\\n",
       "\t2:2-3:1 &  2.950 &  2.3663187 &  3.5336813 & 2.520206e-14\\\\\n",
       "\t3:2-3:1 &  4.300 &  3.7163187 &  4.8836813 & 4.218847e-15\\\\\n",
       "\t1:3-3:1 & -1.400 & -1.9836813 & -0.8163187 & 3.744504e-07\\\\\n",
       "\t2:3-3:1 &  3.150 &  2.5663187 &  3.7336813 & 8.659740e-15\\\\\n",
       "\t3:3-3:1 &  7.275 &  6.6913187 &  7.8586813 & 4.218847e-15\\\\\n",
       "\t2:2-1:2 &  4.325 &  3.7413187 &  4.9086813 & 4.218847e-15\\\\\n",
       "\t3:2-1:2 &  5.675 &  5.0913187 &  6.2586813 & 4.218847e-15\\\\\n",
       "\t1:3-1:2 & -0.025 & -0.6086813 &  0.5586813 & 1.000000e+00\\\\\n",
       "\t2:3-1:2 &  4.525 &  3.9413187 &  5.1086813 & 4.218847e-15\\\\\n",
       "\t3:3-1:2 &  8.650 &  8.0663187 &  9.2336813 & 4.218847e-15\\\\\n",
       "\t3:2-2:2 &  1.350 &  0.7663187 &  1.9336813 & 7.475322e-07\\\\\n",
       "\t1:3-2:2 & -4.350 & -4.9336813 & -3.7663187 & 4.218847e-15\\\\\n",
       "\t2:3-2:2 &  0.200 & -0.3836813 &  0.7836813 & 9.596929e-01\\\\\n",
       "\t3:3-2:2 &  4.325 &  3.7413187 &  4.9086813 & 4.218847e-15\\\\\n",
       "\t1:3-3:2 & -5.700 & -6.2836813 & -5.1163187 & 4.218847e-15\\\\\n",
       "\t2:3-3:2 & -1.150 & -1.7336813 & -0.5663187 & 1.305447e-05\\\\\n",
       "\t3:3-3:2 &  2.975 &  2.3913187 &  3.5586813 & 2.142730e-14\\\\\n",
       "\t2:3-1:3 &  4.550 &  3.9663187 &  5.1336813 & 4.218847e-15\\\\\n",
       "\t3:3-1:3 &  8.675 &  8.0913187 &  9.2586813 & 4.218847e-15\\\\\n",
       "\t3:3-2:3 &  4.125 &  3.5413187 &  4.7086813 & 4.218847e-15\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A matrix: 36 × 4 of type dbl\n",
       "\n",
       "| <!--/--> | diff | lwr | upr | p adj |\n",
       "|---|---|---|---|---|\n",
       "| 2:1-1:1 |  2.975 |  2.3913187 |  3.5586813 | 2.142730e-14 |\n",
       "| 3:1-1:1 |  3.500 |  2.9163187 |  4.0836813 | 4.440892e-15 |\n",
       "| 1:2-1:1 |  2.125 |  1.5413187 |  2.7086813 | 5.300671e-11 |\n",
       "| 2:2-1:1 |  6.450 |  5.8663187 |  7.0336813 | 4.218847e-15 |\n",
       "| 3:2-1:1 |  7.800 |  7.2163187 |  8.3836813 | 4.218847e-15 |\n",
       "| 1:3-1:1 |  2.100 |  1.5163187 |  2.6836813 | 6.947942e-11 |\n",
       "| 2:3-1:1 |  6.650 |  6.0663187 |  7.2336813 | 4.218847e-15 |\n",
       "| 3:3-1:1 | 10.775 | 10.1913187 | 11.3586813 | 4.218847e-15 |\n",
       "| 3:1-2:1 |  0.525 | -0.0586813 |  1.1086813 | 1.033088e-01 |\n",
       "| 1:2-2:1 | -0.850 | -1.4336813 | -0.2663187 | 1.142444e-03 |\n",
       "| 2:2-2:1 |  3.475 |  2.8913187 |  4.0586813 | 4.551914e-15 |\n",
       "| 3:2-2:1 |  4.825 |  4.2413187 |  5.4086813 | 4.218847e-15 |\n",
       "| 1:3-2:1 | -0.875 | -1.4586813 | -0.2913187 | 7.862241e-04 |\n",
       "| 2:3-2:1 |  3.675 |  3.0913187 |  4.2586813 | 4.218847e-15 |\n",
       "| 3:3-2:1 |  7.800 |  7.2163187 |  8.3836813 | 4.218847e-15 |\n",
       "| 1:2-3:1 | -1.375 | -1.9586813 | -0.7913187 | 5.283997e-07 |\n",
       "| 2:2-3:1 |  2.950 |  2.3663187 |  3.5336813 | 2.520206e-14 |\n",
       "| 3:2-3:1 |  4.300 |  3.7163187 |  4.8836813 | 4.218847e-15 |\n",
       "| 1:3-3:1 | -1.400 | -1.9836813 | -0.8163187 | 3.744504e-07 |\n",
       "| 2:3-3:1 |  3.150 |  2.5663187 |  3.7336813 | 8.659740e-15 |\n",
       "| 3:3-3:1 |  7.275 |  6.6913187 |  7.8586813 | 4.218847e-15 |\n",
       "| 2:2-1:2 |  4.325 |  3.7413187 |  4.9086813 | 4.218847e-15 |\n",
       "| 3:2-1:2 |  5.675 |  5.0913187 |  6.2586813 | 4.218847e-15 |\n",
       "| 1:3-1:2 | -0.025 | -0.6086813 |  0.5586813 | 1.000000e+00 |\n",
       "| 2:3-1:2 |  4.525 |  3.9413187 |  5.1086813 | 4.218847e-15 |\n",
       "| 3:3-1:2 |  8.650 |  8.0663187 |  9.2336813 | 4.218847e-15 |\n",
       "| 3:2-2:2 |  1.350 |  0.7663187 |  1.9336813 | 7.475322e-07 |\n",
       "| 1:3-2:2 | -4.350 | -4.9336813 | -3.7663187 | 4.218847e-15 |\n",
       "| 2:3-2:2 |  0.200 | -0.3836813 |  0.7836813 | 9.596929e-01 |\n",
       "| 3:3-2:2 |  4.325 |  3.7413187 |  4.9086813 | 4.218847e-15 |\n",
       "| 1:3-3:2 | -5.700 | -6.2836813 | -5.1163187 | 4.218847e-15 |\n",
       "| 2:3-3:2 | -1.150 | -1.7336813 | -0.5663187 | 1.305447e-05 |\n",
       "| 3:3-3:2 |  2.975 |  2.3913187 |  3.5586813 | 2.142730e-14 |\n",
       "| 2:3-1:3 |  4.550 |  3.9663187 |  5.1336813 | 4.218847e-15 |\n",
       "| 3:3-1:3 |  8.675 |  8.0913187 |  9.2586813 | 4.218847e-15 |\n",
       "| 3:3-2:3 |  4.125 |  3.5413187 |  4.7086813 | 4.218847e-15 |\n",
       "\n"
      ],
      "text/plain": [
       "        diff   lwr        upr        p adj       \n",
       "2:1-1:1  2.975  2.3913187  3.5586813 2.142730e-14\n",
       "3:1-1:1  3.500  2.9163187  4.0836813 4.440892e-15\n",
       "1:2-1:1  2.125  1.5413187  2.7086813 5.300671e-11\n",
       "2:2-1:1  6.450  5.8663187  7.0336813 4.218847e-15\n",
       "3:2-1:1  7.800  7.2163187  8.3836813 4.218847e-15\n",
       "1:3-1:1  2.100  1.5163187  2.6836813 6.947942e-11\n",
       "2:3-1:1  6.650  6.0663187  7.2336813 4.218847e-15\n",
       "3:3-1:1 10.775 10.1913187 11.3586813 4.218847e-15\n",
       "3:1-2:1  0.525 -0.0586813  1.1086813 1.033088e-01\n",
       "1:2-2:1 -0.850 -1.4336813 -0.2663187 1.142444e-03\n",
       "2:2-2:1  3.475  2.8913187  4.0586813 4.551914e-15\n",
       "3:2-2:1  4.825  4.2413187  5.4086813 4.218847e-15\n",
       "1:3-2:1 -0.875 -1.4586813 -0.2913187 7.862241e-04\n",
       "2:3-2:1  3.675  3.0913187  4.2586813 4.218847e-15\n",
       "3:3-2:1  7.800  7.2163187  8.3836813 4.218847e-15\n",
       "1:2-3:1 -1.375 -1.9586813 -0.7913187 5.283997e-07\n",
       "2:2-3:1  2.950  2.3663187  3.5336813 2.520206e-14\n",
       "3:2-3:1  4.300  3.7163187  4.8836813 4.218847e-15\n",
       "1:3-3:1 -1.400 -1.9836813 -0.8163187 3.744504e-07\n",
       "2:3-3:1  3.150  2.5663187  3.7336813 8.659740e-15\n",
       "3:3-3:1  7.275  6.6913187  7.8586813 4.218847e-15\n",
       "2:2-1:2  4.325  3.7413187  4.9086813 4.218847e-15\n",
       "3:2-1:2  5.675  5.0913187  6.2586813 4.218847e-15\n",
       "1:3-1:2 -0.025 -0.6086813  0.5586813 1.000000e+00\n",
       "2:3-1:2  4.525  3.9413187  5.1086813 4.218847e-15\n",
       "3:3-1:2  8.650  8.0663187  9.2336813 4.218847e-15\n",
       "3:2-2:2  1.350  0.7663187  1.9336813 7.475322e-07\n",
       "1:3-2:2 -4.350 -4.9336813 -3.7663187 4.218847e-15\n",
       "2:3-2:2  0.200 -0.3836813  0.7836813 9.596929e-01\n",
       "3:3-2:2  4.325  3.7413187  4.9086813 4.218847e-15\n",
       "1:3-3:2 -5.700 -6.2836813 -5.1163187 4.218847e-15\n",
       "2:3-3:2 -1.150 -1.7336813 -0.5663187 1.305447e-05\n",
       "3:3-3:2  2.975  2.3913187  3.5586813 2.142730e-14\n",
       "2:3-1:3  4.550  3.9663187  5.1336813 4.218847e-15\n",
       "3:3-1:3  8.675  8.0913187  9.2586813 4.218847e-15\n",
       "3:3-2:3  4.125  3.5413187  4.7086813 4.218847e-15"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Find the confidence interval among the many comparisons...\n",
    "T.ci[['Ingredient.1:Ingredient.2']]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To wrap up this exploratory analysis, we want to investigate the plausibility of the assumptions we make. In `R`, the basic diagnostic plots can be conveniently called using the `plot()` function once we feed it with a fitted model. If you find any red flags, you may proceed to conduct formal testings or sensitivity analysis (i.e., fitting alternative models)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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EtHslGG3Qf0v4qA/sLmmsr7D9tj2ssM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\nacdBb5px0NlqqzoOus/kmYQW52lAcvYkG4UU6JxJOPEkFmkBl4K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      "text/plain": [
       "Plot with title \"\""
      ]
     },
     "metadata": {
      "image/png": {
       "height": 360,
       "width": 720
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Diagnostic plots\n",
    "options(repr.plot.width=12, repr.plot.height=6)\n",
    "par(mfrow=c(1,2))\n",
    "plot(anova.fit,cex.lab=1.2,which=1:2)\n",
    "par(mfrow=c(1,1))"
   ]
  }
 ],
 "metadata": {
  "author": "mes ",
  "kernelspec": {
   "display_name": "R",
   "language": "R",
   "name": "ir"
  },
  "language_info": {
   "codemirror_mode": "r",
   "file_extension": ".r",
   "mimetype": "text/x-r-source",
   "name": "R",
   "pygments_lexer": "r",
   "version": "4.1.3"
  },
  "latex_envs": {
   "LaTeX_envs_menu_present": true,
   "autoclose": false,
   "autocomplete": true,
   "bibliofile": "biblio.bib",
   "cite_by": "apalike",
   "current_citInitial": 1,
   "eqLabelWithNumbers": true,
   "eqNumInitial": 1,
   "hotkeys": {
    "equation": "Ctrl-E",
    "itemize": "Ctrl-I"
   },
   "labels_anchors": false,
   "latex_user_defs": false,
   "report_style_numbering": false,
   "user_envs_cfg": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}