0022 One-Way ANOVA Analysis of Variance.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'2.7.8 |Anaconda 2.1.0 (64-bit)| (default, Jul  2 2014, 15:12:11) [MSC v.1500 64 bit (AMD64)]'"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import sys\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "import statsmodels.api as sm\n",
    "from scipy import stats\n",
    "from __future__ import print_function\n",
    "\n",
    "# turn of data table rendering\n",
    "pd.set_option('display.notebook_repr_html', False)\n",
    "sns.set_palette(['#00A99D', '#F5CA0C', '#B6129F', '#76620C', '#095C57'])\n",
    "sys.version"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Testing Multiple Independent Samples\n",
    "If we want to test for significance in mean differences between more than two independent samples, we would have to perform a lot of t-tests. If, for example, we have 6 independent samples, we must conduct 15 t-tests to know what samples significantly differ from each other, as shown here in the binomial:\n",
    "\n",
    "$$\n",
    "\\binom{n}{2}=\\frac{n!}{2!(n-2)!}=\\frac{6\\cdot5\\cdot4\\cdot3\\cdot2\\cdot1}{(2\\cdot1)\\cdot(4\\cdot3\\cdot2\\cdot1)}=\\frac{6\\cdot5}{2}=\\frac{30}{2}=15\n",
    "$$\n",
    "\n",
    "\n",
    "That's a lot of, and even unnecessary, work. That's where ANOVA comes in. With Analysis of Variance we can test multiple independent sample sets to see if there is a significant difference between at least two samples. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Case\n",
    "A manufacturer of electric guitar strings 'Olixir', claims to have the longest sustaining tone in the market. They conducted a comparitive test with a group of three competing brands. The guitar used in the test was the same for all tests. All strings were plucked by the same person, with the same technique and performed a number of times. The tone sustain length is measured in milliseconds. The resulting data set looks like this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x18a80b00>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# The four sample sets with guitar string sustain lengths\n",
    "olixir = np.array([8021., 6947., 8411., 7593., 7781.])\n",
    "ernies = np.array([7013., 6222., 7202., 6437., 5946., 6917., 6152.])\n",
    "fendor = np.array([9896., 9497., 8986., 8496.])\n",
    "daddar = np.array([8253., 8467., 7898., 7699., 8738., 8638.])\n",
    "\n",
    "# Construct kernel density estimate plot\n",
    "sns.kdeplot(olixir, label='olixer', shade=True)\n",
    "sns.kdeplot(ernies, label='ernies', shade=True)\n",
    "sns.kdeplot(fendor, label='fendor', shade=True)\n",
    "sns.kdeplot(daddar, label='daddar', shade=True)\n",
    "\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Grand Mean\n",
    "We can compare three or more samples by finding the average squared deviation of each sample from the [grand mean](http://en.wikipedia.org/wiki/Grand_mean) of all sample groups $\\bar{\\bar{x}}$. The grand mean is calculated by taking all sample values and divide them by the number of samples (where $k$ is the number of groups and $n$ is the number of observations in a group and $x$ is single sample).\n",
    "\n",
    "$$\n",
    "\\bar{\\bar{x}}=\\frac{1}{kn}\\sum_{k}\\sum_{n}x\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Hypothesis\n",
    "While Fendor seems to have the highest mean average tone sustain, is the difference compared to the other samples statistical significant or due to random chance? We state our hypothesis: If there is no statistical significant difference between any of the tested guitar strings brands at an alpha level of .05, then we fail to reject the null hypothesis.\n",
    "\n",
    "|Hypothesis|$\\alpha = .05$||\n",
    "|---:|:---:|---|:---|\n",
    "|$H_0:$|$\\mu_o = \\mu_e = \\mu_f = \\mu_d$|There is no significant difference between any of the brands|\n",
    "|$H_a:$|$\\mu_o \\neq \\mu_e \\neq \\mu_f \\neq \\mu_d$|There is a significant difference between at least two brands|"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "7782.27272727\n"
     ]
    }
   ],
   "source": [
    "# Calculate the grand mean and sample group means\n",
    "N = np.concatenate([olixir, ernies, fendor, daddar]).size\n",
    "xbar_grand = np.concatenate([olixir, ernies, fendor, daddar]).mean()\n",
    "\n",
    "print(xbar_grand)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "7750.6\n",
      "6555.57142857\n",
      "9218.75\n",
      "8282.16666667\n"
     ]
    }
   ],
   "source": [
    "# Calculate the group means\n",
    "xbar_olixir = olixir.mean()\n",
    "xbar_ernies = ernies.mean()\n",
    "xbar_fendor = fendor.mean()\n",
    "xbar_daddar = daddar.mean()\n",
    "\n",
    "print(xbar_olixir, xbar_ernies, xbar_fendor, xbar_daddar, sep='\\n')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## ANOVA\n",
    "The [One-way Analysis of Variance](http://en.wikipedia.org/wiki/One-way_analysis_of_variance) results in a test statistic (F) which is a ratio of the 'between-group variability' and the 'within-group variability'. The between-group variability denotes the distance the group means have from each other, while the within-group variability is a measure of how wide or skinny the distribution of a group is. The larger the variance the wider the distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Between-group Variability\n",
    "To calculate the between-group variability we take mean difference of each sample (SSB) in relation to the grand mean. We multiple the number of groups times the sum of the differences between the group mean and grand mean, squared:\n",
    "\n",
    "$$\n",
    "\\sum n_k(\\bar{x}_k-\\bar{\\bar{x}})^2\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "20291819.866\n"
     ]
    }
   ],
   "source": [
    "# Calculate the sum of squares between groups (SSB)\n",
    "ssb_olixir = len(olixir) * np.square(xbar_olixir - xbar_grand)\n",
    "ssb_ernies = len(ernies) * np.square(xbar_ernies - xbar_grand)\n",
    "ssb_fendor = len(fendor) * np.square(xbar_fendor - xbar_grand)\n",
    "ssb_daddar = len(daddar) * np.square(xbar_daddar - xbar_grand)\n",
    "\n",
    "SSB = (ssb_olixir + ssb_ernies + ssb_fendor + ssb_daddar)\n",
    "print(SSB)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Within-group Variability\n",
    "To calculate the within-group variability we take the sum of squares of each sample point in relation to the group mean (SSW). We sum up these differences, squared:\n",
    "\n",
    "$$\n",
    "\\sum(x_i-\\bar{x}_k)^2\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4568010.49762\n"
     ]
    }
   ],
   "source": [
    "# Calculate the sum of squares within groups (SSW)\n",
    "ssw_olixir = np.square(olixir - xbar_olixir).sum()\n",
    "ssw_ernies = np.square(ernies - xbar_ernies).sum()\n",
    "ssw_fendor = np.square(fendor - xbar_fendor).sum()\n",
    "ssw_daddar = np.square(daddar - xbar_daddar).sum()\n",
    "\n",
    "SSW = ssw_olixir + ssw_ernies + ssw_fendor + ssw_daddar\n",
    "print(SSW)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## F-statistic\n",
    "Now we have the sum of squares for both within-group and between-group, we can calculate the F-statistic. For this we need the degrees of freedom for between-groups (k-1) and within-groups (N-k). We then calculate the mean square between groups (MSB) and the mean square within groups (MSW). The complete formula for calculating the F-statistic is shown here:\n",
    "\n",
    "$$\n",
    "F=\\frac{\\sum n_k(\\bar{x}_k-\\bar{\\bar{x}})^2/(k-1)}{\\sum(x_i-\\bar{x}_k)^2/(N-k)}=\\frac{SSB/dfB}{SSW/dfW}=\\frac{MSB}{MSW}\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3, 18\n"
     ]
    }
   ],
   "source": [
    "# Calculate the degrees of freedom\n",
    "k = 4        # Number of sample groups\n",
    "dfB = k - 1  # Degrees of freedom between groups\n",
    "dfW = N - k  # Degrees of freedom within groups\n",
    "print(dfB, dfW, sep=', ')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "6763939.95534\n",
      "253778.360979\n"
     ]
    }
   ],
   "source": [
    "# Calculate the Mean squared difference\n",
    "MSB = SSB / dfB \n",
    "MSW = SSW / dfW \n",
    "print(MSB, MSW, sep='\\n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "26.6529420761\n"
     ]
    }
   ],
   "source": [
    "# Calculate the F-statistic\n",
    "F = MSB / MSW\n",
    "print(F)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Hypothesis Testing\n",
    "We test the hypothesis by comparing the calculated f-statistic against the F-critical value for an alpha level and the degreed of freedom between and within groups. The F-test is always a one-sided test against an alpha level on the F-distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.1599075898\n"
     ]
    }
   ],
   "source": [
    "# Find the F-critical value for alpha level .05\n",
    "F_critical = stats.distributions.f.ppf(1 - .05, dfB, dfW)\n",
    "print(F_critical)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "AAAAAElFTkSuQmCC\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x18d736d8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot the F-Distribution for dfB and dfW\n",
    "x_plot = np.linspace(0, 5, 250)\n",
    "y_plot = stats.distributions.f.pdf(x_plot, dfB, dfW)\n",
    "plt.fill_between(x_plot, y_plot, alpha=.2)\n",
    "plt.plot(x_plot, y_plot)\n",
    "\n",
    "# Plot the F-critical value at alpha .05\n",
    "x_fcrit = np.linspace(F_critical, 5, 250)\n",
    "y_fcrit = stats.distributions.f.pdf(x_fcrit, dfB, dfW)\n",
    "plt.fill_between(x_fcrit, y_fcrit, alpha=.5)\n",
    "plt.text(F_critical, .07, '$F_{critical}=$'+str(F_critical))\n",
    "plt.xlabel('F-values')\n",
    "plt.ylabel('Probability')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Test the null hypothesis\n",
    "F < F_critical"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since the F-statistic is (much) larger than the F-critical value, we reject the null hypothesis. At least two of the sample groups tested are have a significant different mean."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## ANOVA with f_oneway\n",
    "All the calculations shown here can be efficiently done by calling [scipy stats f_oneway](http://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.f_oneway.html), given the sample groups. The function returns the F-statistic as well as the p-value."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "26.6529420761\n",
      "7.6879993126e-07\n"
     ]
    }
   ],
   "source": [
    "# One-way ANOVA the convenient way\n",
    "f_value, p_value = stats.f_oneway(olixir, ernies, fendor, daddar)\n",
    "print(f_value)\n",
    "print(p_value)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Null hypothesis test at alpha level .05\n",
    "p_value > .05"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## ANOVA Table\n",
    "We often use and encounter [analysis of variance tables](https://onlinecourses.science.psu.edu/stat414/node/218) like this one to report our findings.\n",
    "\n",
    "|Source|DF|SS|MS|F|P|\n",
    "|:---|---:|---:|---:|---:|---:|\n",
    "|Factor|3|20291819.866|6763939.955|26.653|7.688e-07|\n",
    "|Error|18|4568010.498|253778.361|||\n",
    "|Total|21|24859830.364||||"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Eta Squared\n",
    "The effect size used to interpret the outcome of the ANOVA is called [eta squared](http://en.wikiversity.org/wiki/Eta-squared) $\\eta^2$. This value is the proportion of the difference due to a treatment, in this case the guitar string brand. Like $r^2$, we use $\\eta^2$ to denote how much of the difference in means is explained by the input variable. Eta squared is calculated as follows:\n",
    "\n",
    "$$\n",
    "\\eta^2=\\frac{\\sum n_k(\\bar{x}_k-\\bar{\\bar{x}})^2}{\\sum n_k(\\bar{x}_k-\\bar{\\bar{x}})^2+\\sum(x_i-\\bar{x}_k)^2}=\\frac{SSB}{SSB+SSW}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.816249329509\n"
     ]
    }
   ],
   "source": [
    "# Calculate eta squared\n",
    "eta_squared = SSB / (SSB + SSW)\n",
    "print(eta_squared)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This value points out that approximately 82% of the guitar string tone sustain can be explained by the different guitar brands."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tukey's HSD\n",
    "We can use Tukey's Honnestly Significant Difference ([Tukey-kramer HSD](http://en.wikipedia.org/wiki/Tukey%27s_range_test)) to find out which of the sample groups have a significant difference in relation to the other sample groups. To do this, we need to construct a single data set with all the observations and group labels."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "   observation grouplabel\n",
       "0         8021     olixer\n",
       "1         6947     olixer\n",
       "2         8411     olixer\n",
       "3         7593     olixer\n",
       "4         7781     olixer"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Construct single dataset for Tukey-kramer HSD test\n",
    "x1 = pd.DataFrame(olixir, columns=['observation'])\n",
    "x1['grouplabel'] = 'olixer'\n",
    "x2 = pd.DataFrame(ernies, columns=['observation'])\n",
    "x2['grouplabel'] = 'ernies'\n",
    "x3 = pd.DataFrame(fendor, columns=['observation'])\n",
    "x3['grouplabel'] = 'fendor'\n",
    "x4 = pd.DataFrame(daddar, columns=['observation'])\n",
    "x4['grouplabel'] = 'daddar'\n",
    "\n",
    "data = x1.append(x2).append(x3).append(x4)\n",
    "data.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " Multiple Comparison of Means - Tukey HSD,FWER=0.05 \n",
      "====================================================\n",
      "group1 group2  meandiff    lower      upper   reject\n",
      "----------------------------------------------------\n",
      "daddar ernies -1726.5952 -2518.7099 -934.4805  True \n",
      "daddar fendor  936.5833   17.5412   1855.6255  True \n",
      "daddar olixer -531.5667  -1393.7046  330.5713 False \n",
      "ernies fendor 2663.1786   1770.781  3555.5761  True \n",
      "ernies olixer 1195.0286   361.3519  2028.7052  True \n",
      "fendor olixer  -1468.15  -2423.2466 -513.0534  True \n",
      "----------------------------------------------------\n"
     ]
    }
   ],
   "source": [
    "result = sm.stats.multicomp.pairwise_tukeyhsd(data.observation, data.grouplabel)\n",
    "print(result.summary())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['daddar' 'ernies' 'fendor' 'olixer']\n"
     ]
    }
   ],
   "source": [
    "# The labels for group 0 to 3 respectively\n",
    "print(result.groupsunique)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It seems that only in the case of 'Daddar' and 'Olixir' we can state that their sample means are *not* significantly different at this familywise error rate (FWER=.05). In this case we fail to reject the null hypothesis. In all other cases we reject $H_0$ in favor of the alternative hypothesis."
   ]
  }
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