Diet_Case_Study-checkpoint.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Importing the necessary libraries"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-05-06T07:28:53.286508Z",
     "start_time": "2020-05-06T07:28:24.935193Z"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import seaborn as sns\n",
    "import scipy.stats as stats\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Reading the data into a dataframe"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-05-06T07:28:53.356337Z",
     "start_time": "2020-05-06T07:28:53.304280Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
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       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
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       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>diet</th>\n",
       "      <th>preweight</th>\n",
       "      <th>weight6weeks</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>B</td>\n",
       "      <td>60</td>\n",
       "      <td>60.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>B</td>\n",
       "      <td>103</td>\n",
       "      <td>103.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>A</td>\n",
       "      <td>58</td>\n",
       "      <td>54.2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>A</td>\n",
       "      <td>60</td>\n",
       "      <td>54.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>A</td>\n",
       "      <td>64</td>\n",
       "      <td>63.3</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "  diet  preweight  weight6weeks\n",
       "0    B         60          60.0\n",
       "1    B        103         103.0\n",
       "2    A         58          54.2\n",
       "3    A         60          54.0\n",
       "4    A         64          63.3"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df=pd.read_csv('diet.csv')\n",
    "df.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(78, 3)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# shape of the dataset\n",
    "df.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* The dataset contains 78 rows and 3 columns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<class 'pandas.core.frame.DataFrame'>\n",
      "RangeIndex: 78 entries, 0 to 77\n",
      "Data columns (total 3 columns):\n",
      " #   Column        Non-Null Count  Dtype  \n",
      "---  ------        --------------  -----  \n",
      " 0   diet          78 non-null     object \n",
      " 1   preweight     78 non-null     int64  \n",
      " 2   weight6weeks  78 non-null     float64\n",
      "dtypes: float64(1), int64(1), object(1)\n",
      "memory usage: 2.0+ KB\n"
     ]
    }
   ],
   "source": [
    "# information about the columns in the dataset\n",
    "df.info()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* The `Diet` column is categorical. While, `preweight` and `weight6weeks` are numerical."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-05-06T07:29:25.010688Z",
     "start_time": "2020-05-06T07:29:24.973174Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "C    27\n",
       "B    27\n",
       "A    24\n",
       "Name: diet, dtype: int64"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# distinct values of Diet\n",
    "df.diet.value_counts()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* As mentioned in the problem statement, there are three types of diet plans: A, B, and C"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-05-06T07:29:17.231853Z",
     "start_time": "2020-05-06T07:29:16.338089Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
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       "    }\n",
       "\n",
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       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>preweight</th>\n",
       "      <th>weight6weeks</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>count</th>\n",
       "      <td>78.000000</td>\n",
       "      <td>78.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean</th>\n",
       "      <td>72.525641</td>\n",
       "      <td>68.680769</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>std</th>\n",
       "      <td>8.723344</td>\n",
       "      <td>8.924504</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>min</th>\n",
       "      <td>58.000000</td>\n",
       "      <td>53.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25%</th>\n",
       "      <td>66.000000</td>\n",
       "      <td>61.850000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50%</th>\n",
       "      <td>72.000000</td>\n",
       "      <td>68.950000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75%</th>\n",
       "      <td>78.000000</td>\n",
       "      <td>73.825000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>max</th>\n",
       "      <td>103.000000</td>\n",
       "      <td>103.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "        preweight  weight6weeks\n",
       "count   78.000000     78.000000\n",
       "mean    72.525641     68.680769\n",
       "std      8.723344      8.924504\n",
       "min     58.000000     53.000000\n",
       "25%     66.000000     61.850000\n",
       "50%     72.000000     68.950000\n",
       "75%     78.000000     73.825000\n",
       "max    103.000000    103.000000"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# summary of the dataset\n",
    "df.describe()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* The mean weight before any diet plan is 72.53 kg, while the mean weight after 6 weeks of a diet plan is 68.68 kg.\n",
    "* Overall it can be seen that weights have been reduced after taking a diet."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Creating a new column 'weightloss'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>diet</th>\n",
       "      <th>preweight</th>\n",
       "      <th>weight6weeks</th>\n",
       "      <th>weightloss</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>B</td>\n",
       "      <td>60</td>\n",
       "      <td>60.0</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>B</td>\n",
       "      <td>103</td>\n",
       "      <td>103.0</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>A</td>\n",
       "      <td>58</td>\n",
       "      <td>54.2</td>\n",
       "      <td>-3.8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>A</td>\n",
       "      <td>60</td>\n",
       "      <td>54.0</td>\n",
       "      <td>-6.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>A</td>\n",
       "      <td>64</td>\n",
       "      <td>63.3</td>\n",
       "      <td>-0.7</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "  diet  preweight  weight6weeks  weightloss\n",
       "0    B         60          60.0         0.0\n",
       "1    B        103         103.0         0.0\n",
       "2    A         58          54.2        -3.8\n",
       "3    A         60          54.0        -6.0\n",
       "4    A         64          63.3        -0.7"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# weightloss = (post-weight) - (pre-weight)\n",
    "df['weightloss'] = df['weight6weeks'] - df['preweight']\n",
    "df.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-05-06T07:29:19.193946Z",
     "start_time": "2020-05-06T07:29:19.179044Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "diet            0\n",
       "preweight       0\n",
       "weight6weeks    0\n",
       "weightloss      0\n",
       "dtype: int64"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# check for missing values\n",
    "df.isnull().sum()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* There are no missing values in the dataset"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# visual analysis of the weightloss for the three diet-plans\n",
    "sns.boxplot(x=\"diet\", y=\"weightloss\", data = df)\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* The distribution of weight loss differs among the three groups.\n",
    "* Diet plan C is impacting the highest weight loss."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Step 1: Define null and alternate hypotheses"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$H_0$ : The mean weight losses with respect to each diet category is equal.\n",
    "\n",
    "$H_a$ : At least one of the mean weight losses with respect to the three diet category is different."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Step 2: Select Appropriate test"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is a problem, concerning three population means. One-way ANOVA could be the appropriate test here provided normality and equality of variance assumptions are verified.\n",
    "\n",
    "* For testing of normality, Shapiro-Wilk’s test is applied to the response variable.\n",
    "\n",
    "* For equality of variance, Levene test is applied to the response variable."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Shapiro-Wilk’s test\n",
    "\n",
    "We will test the null hypothesis\n",
    "\n",
    ">$H_0:$ The weight losses follow a normal distribution\n",
    "\n",
    "against the alternative hypothesis\n",
    "\n",
    ">$H_a:$ The weight losses do not not follow a normal distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The p-value is 0.8019816875457764\n"
     ]
    }
   ],
   "source": [
    "# Assumption 1: Normality\n",
    "# Use the shapiro function for the scipy.stats library for this test\n",
    "\n",
    "# find the p-value\n",
    "w, p_value = stats.shapiro(df['weightloss']) \n",
    "print('The p-value is', p_value)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since p-value of the test is very large than the 5% significance level, we fail to reject the null hypothesis that the response follows the normal distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Levene’s test\n",
    "\n",
    "We will test the null hypothesis\n",
    "\n",
    ">$H_0$: All the population variances are equal\n",
    "\n",
    "against the alternative hypothesis\n",
    "\n",
    ">$H_a$: At least one variance is different from the rest"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The p-value is 0.5376731304274011\n"
     ]
    }
   ],
   "source": [
    "#Assumption 2: Homogeneity of Variance\n",
    "# use levene function from scipy.stats library for this test\n",
    "\n",
    "# find the p-value\n",
    "statistic, p_value = stats.levene(df[df['diet']=='A']['weightloss'],\n",
    "                            df[df['diet']=='B']['weightloss'],\n",
    "                            df[df['diet']=='C']['weightloss'])\n",
    "print('The p-value is', p_value)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since the p-value is large than the 5% significance level, we fail to reject the null hypothesis of homogeneity of variances."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Step 3: Decide the significance level"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As given in the problem statement, we select α = 0.05."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Step 4: Collect and prepare data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "# separate the weightlosses with respect to the three diet-plans\n",
    "weightloss_diet_A = df[df['diet']=='A']['weightloss']\n",
    "weightloss_diet_B = df[df['diet']=='B']['weightloss']\n",
    "weightloss_diet_C = df[df['diet']=='C']['weightloss']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Step 5: Calculate the p-value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-05-06T07:31:40.967079Z",
     "start_time": "2020-05-06T07:31:40.922403Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The p-value is  0.0032290142385893524\n"
     ]
    }
   ],
   "source": [
    "# perform one-way anova test using the f_oneway function from scipy.stats library\n",
    "test_stat, p_value = stats.f_oneway(weightloss_diet_A, weightloss_diet_B, weightloss_diet_C)\n",
    "print('The p-value is ', p_value)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Step 6: Compare the p-value with $\\alpha$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "As the p-value 0.0032290142385893524 is less than the level of significance, we reject the null hypothesis.\n"
     ]
    }
   ],
   "source": [
    "# print the conclusion based on p-value\n",
    "if p_value < 0.05:\n",
    "    print(f'As the p-value {p_value} is less than the level of significance, we reject the null hypothesis.')\n",
    "else:\n",
    "    print(f'As the p-value {p_value} is greater than the level of significance, we fail to reject the null hypothesis.')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Step 7:  Draw inference"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since the p-value is less than the 5% significance level, we reject the null hypothesis. Hence, we have enough statistical evidence to say that the mean weight losses with respect to the three diet plans are different."
   ]
  }
 ],
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