SR04_Sample_Size_Calculations.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Statistics review 4: Sample Size calculations\n",
    "\n",
    "R code accompanying [paper](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC137461/pdf/cc1521.pdf)\n",
    "\n",
    "## Key learning points\n",
    "\n",
    "- Studies must be adequately powered to achieve their aims, and appropriate sample size calculations should be carried out at the design stage of any study.\n",
    "- Estimation of the expected size of effect can be difficult and should, wherever possible, be based on existing evidence and clinical expertise. It is important that any estimates be large enough to be clinically important while also remaining plausible.\n",
    "- Many apparently null studies may be under-powered rather than genuinely demonstrating no difference between groups; absence of evidence is not evidence of absence."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "suppressPackageStartupMessages(library(tidyverse))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "options(repr.plot.width=4, repr.plot.height=3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Power\n",
    "\n",
    "Power is is the probability of correctly identifying a difference between the two groups in the study sample when one genuinely exists in the populations from which the samples were drawn."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problem with small sample sizes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose that a drug can lower blood pressure by 10 mm Hg on average, and that the standard deviation of the drug effect is also 10 mm Hg. We will run some simulations with two populations each of size $n$ to see what sample size is needed to get a power of 0.9."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "n.expts <- 1000\n",
    "alpha <- 0.05\n",
    "\n",
    "ns <- seq(3, 25, by=1)\n",
    "\n",
    "data <- matrix(NA, nrow=length(ns), ncol=2, dimnames=list(NULL, c(\"n\", \"power\")))\n",
    "\n",
    "for (n in seq(3, 25, by=1)) {\n",
    "    success <- 0\n",
    "    for (expt in 1:n.expts) {\n",
    "        effect <- rnorm(n, mean=10, sd=10)\n",
    "        p <- t.test(effect)$p.value\n",
    "        if (p < alpha) {\n",
    "            success <- success + 1\n",
    "        }\n",
    "    }\n",
    "    data[n-2, 1] <- n\n",
    "    data[n-2, 2] <- success/n.expts\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>n</th><th scope=col>power</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td> 3   </td><td>0.172</td></tr>\n",
       "\t<tr><td> 4   </td><td>0.309</td></tr>\n",
       "\t<tr><td> 5   </td><td>0.397</td></tr>\n",
       "\t<tr><td> 6   </td><td>0.510</td></tr>\n",
       "\t<tr><td> 7   </td><td>0.585</td></tr>\n",
       "\t<tr><td> 8   </td><td>0.689</td></tr>\n",
       "\t<tr><td> 9   </td><td>0.769</td></tr>\n",
       "\t<tr><td>10   </td><td>0.788</td></tr>\n",
       "\t<tr><td>11   </td><td>0.842</td></tr>\n",
       "\t<tr><td>12   </td><td>0.883</td></tr>\n",
       "\t<tr><td>13   </td><td>0.911</td></tr>\n",
       "\t<tr><td>14   </td><td>0.940</td></tr>\n",
       "\t<tr><td>15   </td><td>0.948</td></tr>\n",
       "\t<tr><td>16   </td><td>0.968</td></tr>\n",
       "\t<tr><td>17   </td><td>0.972</td></tr>\n",
       "\t<tr><td>18   </td><td>0.974</td></tr>\n",
       "\t<tr><td>19   </td><td>0.978</td></tr>\n",
       "\t<tr><td>20   </td><td>0.989</td></tr>\n",
       "\t<tr><td>21   </td><td>0.993</td></tr>\n",
       "\t<tr><td>22   </td><td>0.996</td></tr>\n",
       "\t<tr><td>23   </td><td>0.996</td></tr>\n",
       "\t<tr><td>24   </td><td>0.995</td></tr>\n",
       "\t<tr><td>25   </td><td>0.998</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ll}\n",
       " n & power\\\\\n",
       "\\hline\n",
       "\t  3    & 0.172\\\\\n",
       "\t  4    & 0.309\\\\\n",
       "\t  5    & 0.397\\\\\n",
       "\t  6    & 0.510\\\\\n",
       "\t  7    & 0.585\\\\\n",
       "\t  8    & 0.689\\\\\n",
       "\t  9    & 0.769\\\\\n",
       "\t 10    & 0.788\\\\\n",
       "\t 11    & 0.842\\\\\n",
       "\t 12    & 0.883\\\\\n",
       "\t 13    & 0.911\\\\\n",
       "\t 14    & 0.940\\\\\n",
       "\t 15    & 0.948\\\\\n",
       "\t 16    & 0.968\\\\\n",
       "\t 17    & 0.972\\\\\n",
       "\t 18    & 0.974\\\\\n",
       "\t 19    & 0.978\\\\\n",
       "\t 20    & 0.989\\\\\n",
       "\t 21    & 0.993\\\\\n",
       "\t 22    & 0.996\\\\\n",
       "\t 23    & 0.996\\\\\n",
       "\t 24    & 0.995\\\\\n",
       "\t 25    & 0.998\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "n | power | \n",
       "|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|\n",
       "|  3    | 0.172 | \n",
       "|  4    | 0.309 | \n",
       "|  5    | 0.397 | \n",
       "|  6    | 0.510 | \n",
       "|  7    | 0.585 | \n",
       "|  8    | 0.689 | \n",
       "|  9    | 0.769 | \n",
       "| 10    | 0.788 | \n",
       "| 11    | 0.842 | \n",
       "| 12    | 0.883 | \n",
       "| 13    | 0.911 | \n",
       "| 14    | 0.940 | \n",
       "| 15    | 0.948 | \n",
       "| 16    | 0.968 | \n",
       "| 17    | 0.972 | \n",
       "| 18    | 0.974 | \n",
       "| 19    | 0.978 | \n",
       "| 20    | 0.989 | \n",
       "| 21    | 0.993 | \n",
       "| 22    | 0.996 | \n",
       "| 23    | 0.996 | \n",
       "| 24    | 0.995 | \n",
       "| 25    | 0.998 | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "   n  power\n",
       "1   3 0.172\n",
       "2   4 0.309\n",
       "3   5 0.397\n",
       "4   6 0.510\n",
       "5   7 0.585\n",
       "6   8 0.689\n",
       "7   9 0.769\n",
       "8  10 0.788\n",
       "9  11 0.842\n",
       "10 12 0.883\n",
       "11 13 0.911\n",
       "12 14 0.940\n",
       "13 15 0.948\n",
       "14 16 0.968\n",
       "15 17 0.972\n",
       "16 18 0.974\n",
       "17 19 0.978\n",
       "18 20 0.989\n",
       "19 21 0.993\n",
       "20 22 0.996\n",
       "21 23 0.996\n",
       "22 24 0.995\n",
       "23 25 0.998"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "as.data.frame(data)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Interpretation of simulation results\n",
    "\n",
    "It appears as if a sample size of 13 will give a power of 0.9 with the given effect size. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Factors that affect sample size calculations\n",
    "\n",
    "The following affect sample size calculations\n",
    "\n",
    "- p value threshold (a 0.01 threshold requires a larger sample size than a 0.05 threshold)\n",
    "- power (0.9 power requires a larger sample size than 0.8 power)\n",
    "- effect size (small effect sizes require a larger sample size than large effect sizes)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Using a library to calculate power\n",
    "\n",
    "One of the most convenient libraries in R for simple sample size calculations is `pwr`. We will use it to illustrate many of the calculations done in the paper."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "library(pwr)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Check simulation results"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "     One-sample t test power calculation \n",
       "\n",
       "              n = 12.58546\n",
       "              d = 1\n",
       "      sig.level = 0.05\n",
       "          power = 0.9\n",
       "    alternative = two.sided\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "d <- 10/10 # effect size is change over standard deviaiton\n",
    "pwr.t.test(d = d, sig.level = 0.05, power = 0.9, type = \"one.sample\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Small p-value thresholds need larger sample sizes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "     One-sample t test power calculation \n",
       "\n",
       "              n = 18.30346\n",
       "              d = 1\n",
       "      sig.level = 0.01\n",
       "          power = 0.9\n",
       "    alternative = two.sided\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pwr.t.test(d = 1, sig.level = 0.01, power = 0.9, type = \"one.sample\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "     One-sample t test power calculation \n",
       "\n",
       "              n = 10.08101\n",
       "              d = 1\n",
       "      sig.level = 0.1\n",
       "          power = 0.9\n",
       "    alternative = two.sided\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pwr.t.test(d = 1, sig.level = 0.1, power = 0.9, type = \"one.sample\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Large power needs larger sample sizes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "     One-sample t test power calculation \n",
       "\n",
       "              n = 9.93785\n",
       "              d = 1\n",
       "      sig.level = 0.05\n",
       "          power = 0.8\n",
       "    alternative = two.sided\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pwr.t.test(d = 1, sig.level = 0.05, power = 0.8, type = \"one.sample\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "     One-sample t test power calculation \n",
       "\n",
       "              n = 12.58546\n",
       "              d = 1\n",
       "      sig.level = 0.05\n",
       "          power = 0.9\n",
       "    alternative = two.sided\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pwr.t.test(d = 1, sig.level = 0.05, power = 0.9, type = \"one.sample\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Small effect size needs larger sample size"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "     One-sample t test power calculation \n",
       "\n",
       "              n = 1052.665\n",
       "              d = 0.1\n",
       "      sig.level = 0.05\n",
       "          power = 0.9\n",
       "    alternative = two.sided\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pwr.t.test(d = 0.1, sig.level = 0.05, power = 0.9, type = \"one.sample\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "     One-sample t test power calculation \n",
       "\n",
       "              n = 12.58546\n",
       "              d = 1\n",
       "      sig.level = 0.05\n",
       "          power = 0.9\n",
       "    alternative = two.sided\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pwr.t.test(d = 1.0, sig.level = 0.05, power = 0.9, type = \"one.sample\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that an effect size of 1.0 is considered to be large. The before/after changes look like this when d=1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x <- seq(-3, 4, length.out = 100)\n",
    "y1 <- dnorm(x, 0, 1)\n",
    "y2 <- dnorm(x, 1, 1)\n",
    "df <- data.frame(x=x, y1=y1, y2=y2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {},
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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heXLsIIQKkg9GTPnj21QMF27jxFr1pD6VeJ0JMiD64ZlJCQQG3btqU9e/bQb7/9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s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\ncMB22QNpIAABCEDAEQL/AGCodI922Gb7AAAAAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ggplot(df, aes(x=x, y=y1)) +\n",
    "geom_line() + \n",
    "geom_line(aes(y = y2), color=\"red\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Sample size calculation for a difference in means (equal sized groups)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "     Two-sample t test power calculation \n",
       "\n",
       "              n = 22.02109\n",
       "              d = 1\n",
       "      sig.level = 0.05\n",
       "          power = 0.9\n",
       "    alternative = two.sided\n",
       "\n",
       "NOTE: n is number in *each* group\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pwr.t.test(d = 1.0, sig.level = 0.05, power = 0.9, type = \"two.sample\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Sample size calculation for a difference in means (different sized groups)\n",
    "\n",
    "Suppose one of the treatment groups is fixed in size."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "     t test power calculation \n",
       "\n",
       "             n1 = 20\n",
       "             n2 = 24.47031\n",
       "              d = 1\n",
       "      sig.level = 0.05\n",
       "          power = 0.9\n",
       "    alternative = two.sided\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pwr.t2n.test(d = 1.0, n1 = 20, sig.level = 0.05, power = 0.9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Sample size calculation for a difference in proportions (equal sized groups)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "     Difference of proportion power calculation for binomial distribution (arcsine transformation) \n",
       "\n",
       "              h = 0.4082483\n",
       "              n = 126.089\n",
       "      sig.level = 0.05\n",
       "          power = 0.9\n",
       "    alternative = two.sided\n",
       "\n",
       "NOTE: same sample sizes\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p1 <- 0.4\n",
    "p2 <- 0.6\n",
    "p <- mean(p1, p2)\n",
    "h <- (p1 - p2)/sqrt(p*(1-p))\n",
    "\n",
    "pwr.2p.test(h = h, sig.level = 0.05, power = 0.9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Sample size calculation for a difference in proportions (different sized groups)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "     difference of proportion power calculation for binomial distribution (arcsine transformation) \n",
       "\n",
       "              h = 0.4082483\n",
       "             n1 = 100\n",
       "             n2 = 170.5958\n",
       "      sig.level = 0.05\n",
       "          power = 0.9\n",
       "    alternative = two.sided\n",
       "\n",
       "NOTE: different sample sizes\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p1 <- 0.4\n",
    "p2 <- 0.6\n",
    "p <- mean(p1, p2)\n",
    "h <- (p1 - p2)/sqrt(p*(1-p))\n",
    "\n",
    "pwr.2p2n.test(h = h, n1 = 100, sig.level = 0.05, power = 0.9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Loss, dropout etc\n",
    "\n",
    "Obvious adjustments have to be made if loss or dropout is expected. For example, if 10% of the patients in the study are expected to be lost to follow up, we need $n/0.9$ patients to be enrolled."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercises"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose we are interested in whether a particular diet results in change of weight. We enroll subjects into two group - group A follows the special diet, and group B follows their regular diet (control). Suppose also that the standard deviation in both groups is the same at 25 lbs. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**1**. If we want to detect an effect of the diet of 10 lbs or more with 0.9 power and 0.05 significance level, how many subjects do we need for each group (assume equal size groups)?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**2**. If 20% of the patients are expected to drop-out before the study is completed, how many subjects would need to be enrolled for the same study design as in Ex. 1?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**3** Make a graph where the x-axis shows the sample size from 10 to 100, the y-axis shows the power from 0 to 1, and plot separate curves for effect sizes of 10, 25 and 40 lbs."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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