Regression Modelling 09. Analysis of interactions in multivariate analysis .ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Regression Modelling 9\n",
    "## Analysis of interactions in multivariate analysis "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "sales <- read.table(url(\"http://jupyter-regressions.s3.eu-west-1.amazonaws.com/sales1.csv\"), \n",
    "                           sep=\",\", header = T, stringsAsFactors = F)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>Sales</th><th scope=col>Advert</th><th scope=col>Interest</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>4.0</td><td>3.5</td><td>5  </td></tr>\n",
       "\t<tr><td>6.0</td><td>5.5</td><td>5  </td></tr>\n",
       "\t<tr><td>8.0</td><td>7.0</td><td>5  </td></tr>\n",
       "\t<tr><td>2.0</td><td>1.0</td><td>4  </td></tr>\n",
       "\t<tr><td>4.5</td><td>3.0</td><td>4  </td></tr>\n",
       "\t<tr><td>9.0</td><td>6.5</td><td>4  </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|lll}\n",
       " Sales & Advert & Interest\\\\\n",
       "\\hline\n",
       "\t 4.0 & 3.5 & 5  \\\\\n",
       "\t 6.0 & 5.5 & 5  \\\\\n",
       "\t 8.0 & 7.0 & 5  \\\\\n",
       "\t 2.0 & 1.0 & 4  \\\\\n",
       "\t 4.5 & 3.0 & 4  \\\\\n",
       "\t 9.0 & 6.5 & 4  \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| Sales | Advert | Interest |\n",
       "|---|---|---|\n",
       "| 4.0 | 3.5 | 5   |\n",
       "| 6.0 | 5.5 | 5   |\n",
       "| 8.0 | 7.0 | 5   |\n",
       "| 2.0 | 1.0 | 4   |\n",
       "| 4.5 | 3.0 | 4   |\n",
       "| 9.0 | 6.5 | 4   |\n",
       "\n"
      ],
      "text/plain": [
       "  Sales Advert Interest\n",
       "1 4.0   3.5    5       \n",
       "2 6.0   5.5    5       \n",
       "3 8.0   7.0    5       \n",
       "4 2.0   1.0    4       \n",
       "5 4.5   3.0    4       \n",
       "6 9.0   6.5    4       "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "head(sales)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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LvZxA5pNiERQ1ft/f27AeUREm/eHhJb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",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(sales)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Interactions\n",
    "But what if we suspected increase in Sales from Advertising efffects depends on the interest rate? \n",
    "\n",
    "Is it enough to add another term to the model? - No\n",
    "\n",
    "If the effect of Advertising depends on the current interest rate, they are interacting, and we should include an **interaction term**.\n",
    "\n",
    "If the interest rate effected Sales directly and not via an interaction we would just include that term.\n",
    "\n",
    "The term is added as a standalone before the interaction term though, to preserve hirearchy, this is the same justification when adding quadratic terms to a model. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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PFZGMUdzEgvOjzIH0Ry9xOSvxFSeJPMSA85O8TmxiKLLULyN0IKb5YZaZyzQ/QX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\nRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(Sales~Advert, sales)\n",
    "text(Sales~Advert, labels=Interest, sales, cex=2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = Sales ~ Advert + Interest + Advert * Interest, data = sales)\n",
       "\n",
       "Residuals:\n",
       "     Min       1Q   Median       3Q      Max \n",
       "-0.42247 -0.20707  0.03773  0.16418  0.39588 \n",
       "\n",
       "Coefficients:\n",
       "                Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept)      5.94148    0.66169   8.979 1.88e-05 ***\n",
       "Advert           1.83632    0.13492  13.611 8.17e-07 ***\n",
       "Interest        -1.31187    0.19670  -6.669 0.000158 ***\n",
       "Advert:Interest -0.12601    0.03864  -3.261 0.011507 *  \n",
       "---\n",
       "Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1\n",
       "\n",
       "Residual standard error: 0.3075 on 8 degrees of freedom\n",
       "Multiple R-squared:  0.9944,\tAdjusted R-squared:  0.9923 \n",
       "F-statistic: 472.1 on 3 and 8 DF,  p-value: 2.445e-09\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "lmSales <- lm(Sales~Advert+Interest+Advert*Interest, sales)\n",
    "summary(lmSales)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>Sales</th><th scope=col>Advert</th><th scope=col>Interest</th><th scope=col>fit</th><th scope=col>lwr</th><th scope=col>upr</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td> 4.0     </td><td>3.5      </td><td>5        </td><td> 3.604124</td><td> 3.187454</td><td> 4.020794</td></tr>\n",
       "\t<tr><td> 6.0     </td><td>5.5      </td><td>5        </td><td> 6.016682</td><td> 5.652537</td><td> 6.380827</td></tr>\n",
       "\t<tr><td> 8.0     </td><td>7.0      </td><td>5        </td><td> 7.826101</td><td> 7.304394</td><td> 8.347808</td></tr>\n",
       "\t<tr><td> 2.0     </td><td>1.0      </td><td>4        </td><td> 2.026300</td><td> 1.549333</td><td> 2.503266</td></tr>\n",
       "\t<tr><td> 4.5     </td><td>3.0      </td><td>4        </td><td> 4.690874</td><td> 4.396657</td><td> 4.985090</td></tr>\n",
       "\t<tr><td> 9.0     </td><td>6.5      </td><td>4        </td><td> 9.353879</td><td> 9.036187</td><td> 9.671570</td></tr>\n",
       "\t<tr><td> 4.5     </td><td>2.0      </td><td>3        </td><td> 4.922468</td><td> 4.618412</td><td> 5.226524</td></tr>\n",
       "\t<tr><td> 8.0     </td><td>4.0      </td><td>3        </td><td> 7.839058</td><td> 7.614363</td><td> 8.063753</td></tr>\n",
       "\t<tr><td>10.5     </td><td>6.0      </td><td>3        </td><td>10.755648</td><td>10.450814</td><td>11.060482</td></tr>\n",
       "\t<tr><td> 5.0     </td><td>1.0      </td><td>2        </td><td> 4.902047</td><td> 4.316695</td><td> 5.487398</td></tr>\n",
       "\t<tr><td>10.0     </td><td>4.0      </td><td>2        </td><td> 9.654955</td><td> 9.308292</td><td>10.001619</td></tr>\n",
       "\t<tr><td>14.5     </td><td>7.0      </td><td>2        </td><td>14.407864</td><td>13.842830</td><td>14.972898</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|llllll}\n",
       " Sales & Advert & Interest & fit & lwr & upr\\\\\n",
       "\\hline\n",
       "\t  4.0      & 3.5       & 5         &  3.604124 &  3.187454 &  4.020794\\\\\n",
       "\t  6.0      & 5.5       & 5         &  6.016682 &  5.652537 &  6.380827\\\\\n",
       "\t  8.0      & 7.0       & 5         &  7.826101 &  7.304394 &  8.347808\\\\\n",
       "\t  2.0      & 1.0       & 4         &  2.026300 &  1.549333 &  2.503266\\\\\n",
       "\t  4.5      & 3.0       & 4         &  4.690874 &  4.396657 &  4.985090\\\\\n",
       "\t  9.0      & 6.5       & 4         &  9.353879 &  9.036187 &  9.671570\\\\\n",
       "\t  4.5      & 2.0       & 3         &  4.922468 &  4.618412 &  5.226524\\\\\n",
       "\t  8.0      & 4.0       & 3         &  7.839058 &  7.614363 &  8.063753\\\\\n",
       "\t 10.5      & 6.0       & 3         & 10.755648 & 10.450814 & 11.060482\\\\\n",
       "\t  5.0      & 1.0       & 2         &  4.902047 &  4.316695 &  5.487398\\\\\n",
       "\t 10.0      & 4.0       & 2         &  9.654955 &  9.308292 & 10.001619\\\\\n",
       "\t 14.5      & 7.0       & 2         & 14.407864 & 13.842830 & 14.972898\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| Sales | Advert | Interest | fit | lwr | upr |\n",
       "|---|---|---|---|---|---|\n",
       "|  4.0      | 3.5       | 5         |  3.604124 |  3.187454 |  4.020794 |\n",
       "|  6.0      | 5.5       | 5         |  6.016682 |  5.652537 |  6.380827 |\n",
       "|  8.0      | 7.0       | 5         |  7.826101 |  7.304394 |  8.347808 |\n",
       "|  2.0      | 1.0       | 4         |  2.026300 |  1.549333 |  2.503266 |\n",
       "|  4.5      | 3.0       | 4         |  4.690874 |  4.396657 |  4.985090 |\n",
       "|  9.0      | 6.5       | 4         |  9.353879 |  9.036187 |  9.671570 |\n",
       "|  4.5      | 2.0       | 3         |  4.922468 |  4.618412 |  5.226524 |\n",
       "|  8.0      | 4.0       | 3         |  7.839058 |  7.614363 |  8.063753 |\n",
       "| 10.5      | 6.0       | 3         | 10.755648 | 10.450814 | 11.060482 |\n",
       "|  5.0      | 1.0       | 2         |  4.902047 |  4.316695 |  5.487398 |\n",
       "| 10.0      | 4.0       | 2         |  9.654955 |  9.308292 | 10.001619 |\n",
       "| 14.5      | 7.0       | 2         | 14.407864 | 13.842830 | 14.972898 |\n",
       "\n"
      ],
      "text/plain": [
       "   Sales Advert Interest fit       lwr       upr      \n",
       "1   4.0  3.5    5         3.604124  3.187454  4.020794\n",
       "2   6.0  5.5    5         6.016682  5.652537  6.380827\n",
       "3   8.0  7.0    5         7.826101  7.304394  8.347808\n",
       "4   2.0  1.0    4         2.026300  1.549333  2.503266\n",
       "5   4.5  3.0    4         4.690874  4.396657  4.985090\n",
       "6   9.0  6.5    4         9.353879  9.036187  9.671570\n",
       "7   4.5  2.0    3         4.922468  4.618412  5.226524\n",
       "8   8.0  4.0    3         7.839058  7.614363  8.063753\n",
       "9  10.5  6.0    3        10.755648 10.450814 11.060482\n",
       "10  5.0  1.0    2         4.902047  4.316695  5.487398\n",
       "11 10.0  4.0    2         9.654955  9.308292 10.001619\n",
       "12 14.5  7.0    2        14.407864 13.842830 14.972898"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "lmPred <- cbind(sales, predict(lmSales, interval=\"confidence\"))\n",
    "lmPred"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "1.45829498367233"
      ],
      "text/latex": [
       "1.45829498367233"
      ],
      "text/markdown": [
       "1.45829498367233"
      ],
      "text/plain": [
       "[1] 1.458295"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "coefficients <- coef(lmSales)\n",
    "names(coefficients) <- NULL\n",
    "\n",
    "intercept <- coefficients[1]\n",
    "advert <- coefficients[2]\n",
    "interest <- coefficients[3]\n",
    "adv_int <- coefficients[4]\n",
    "\n",
    "sales_3mAds <- intercept + advert*3 + interest*3 + adv_int*(3*3)\n",
    "sales_4mAds <- intercept + advert*4 + interest*3 + adv_int*(4*3)\n",
    "\n",
    "sales_4mAds - sales_3mAds"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In other words: When the prevailing interest rate is 3%, we expect to increase sales by 1.46mill for each additional lm we spend on advertising"
   ]
  }
 ],
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